hypothesis in introduction

Writing an Introduction for a Scientific Paper

Dr. michelle harris, dr. janet batzli, biocore.

This section provides guidelines on how to construct a solid introduction to a scientific paper including background information, study question , biological rationale, hypothesis , and general approach . If the Introduction is done well, there should be no question in the reader’s mind why and on what basis you have posed a specific hypothesis.

Broad Question : based on an initial observation (e.g., “I see a lot of guppies close to the shore. Do guppies like living in shallow water?”). This observation of the natural world may inspire you to investigate background literature or your observation could be based on previous research by others or your own pilot study. Broad questions are not always included in your written text, but are essential for establishing the direction of your research.

Background Information : key issues, concepts, terminology, and definitions needed to understand the biological rationale for the experiment. It often includes a summary of findings from previous, relevant studies. Remember to cite references, be concise, and only include relevant information given your audience and your experimental design. Concisely summarized background information leads to the identification of specific scientific knowledge gaps that still exist. (e.g., “No studies to date have examined whether guppies do indeed spend more time in shallow water.”)

Testable Question : these questions are much more focused than the initial broad question, are specific to the knowledge gap identified, and can be addressed with data. (e.g., “Do guppies spend different amounts of time in water <1 meter deep as compared to their time in water that is >1 meter deep?”)

Biological Rationale : describes the purpose of your experiment distilling what is known and what is not known that defines the knowledge gap that you are addressing. The “BR” provides the logic for your hypothesis and experimental approach, describing the biological mechanism and assumptions that explain why your hypothesis should be true.

The biological rationale is based on your interpretation of the scientific literature, your personal observations, and the underlying assumptions you are making about how you think the system works. If you have written your biological rationale, your reader should see your hypothesis in your introduction section and say to themselves, “Of course, this hypothesis seems very logical based on the rationale presented.”

• A thorough rationale defines your assumptions about the system that have not been revealed in scientific literature or from previous systematic observation. These assumptions drive the direction of your specific hypothesis or general predictions.
• Defining the rationale is probably the most critical task for a writer, as it tells your reader why your research is biologically meaningful. It may help to think about the rationale as an answer to the questions— how is this investigation related to what we know, what assumptions am I making about what we don’t yet know, AND how will this experiment add to our knowledge? *There may or may not be broader implications for your study; be careful not to overstate these (see note on social justifications below).
• Expect to spend time and mental effort on this. You may have to do considerable digging into the scientific literature to define how your experiment fits into what is already known and why it is relevant to pursue.
• Be open to the possibility that as you work with and think about your data, you may develop a deeper, more accurate understanding of the experimental system. You may find the original rationale needs to be revised to reflect your new, more sophisticated understanding.
• As you progress through Biocore and upper level biology courses, your rationale should become more focused and matched with the level of study e ., cellular, biochemical, or physiological mechanisms that underlie the rationale. Achieving this type of understanding takes effort, but it will lead to better communication of your science.

***Special note on avoiding social justifications: You should not overemphasize the relevance of your experiment and the possible connections to large-scale processes. Be realistic and logical —do not overgeneralize or state grand implications that are not sensible given the structure of your experimental system. Not all science is easily applied to improving the human condition. Performing an investigation just for the sake of adding to our scientific knowledge (“pure or basic science”) is just as important as applied science. In fact, basic science often provides the foundation for applied studies.

Hypothesis / Predictions : specific prediction(s) that you will test during your experiment. For manipulative experiments, the hypothesis should include the independent variable (what you manipulate), the dependent variable(s) (what you measure), the organism or system , the direction of your results, and comparison to be made.

If you are doing a systematic observation , your hypothesis presents a variable or set of variables that you predict are important for helping you characterize the system as a whole, or predict differences between components/areas of the system that help you explain how the system functions or changes over time.

Experimental Approach : Briefly gives the reader a general sense of the experiment, the type of data it will yield, and the kind of conclusions you expect to obtain from the data. Do not confuse the experimental approach with the experimental protocol . The experimental protocol consists of the detailed step-by-step procedures and techniques used during the experiment that are to be reported in the Methods and Materials section.

Some Final Tips on Writing an Introduction

• As you progress through the Biocore sequence, for instance, from organismal level of Biocore 301/302 to the cellular level in Biocore 303/304, we expect the contents of your “Introduction” paragraphs to reflect the level of your coursework and previous writing experience. For example, in Biocore 304 (Cell Biology Lab) biological rationale should draw upon assumptions we are making about cellular and biochemical processes.
• Be Concise yet Specific: Remember to be concise and only include relevant information given your audience and your experimental design. As you write, keep asking, “Is this necessary information or is this irrelevant detail?” For example, if you are writing a paper claiming that a certain compound is a competitive inhibitor to the enzyme alkaline phosphatase and acts by binding to the active site, you need to explain (briefly) Michaelis-Menton kinetics and the meaning and significance of Km and Vmax. This explanation is not necessary if you are reporting the dependence of enzyme activity on pH because you do not need to measure Km and Vmax to get an estimate of enzyme activity.
• Another example: if you are writing a paper reporting an increase in Daphnia magna heart rate upon exposure to caffeine you need not describe the reproductive cycle of magna unless it is germane to your results and discussion. Be specific and concrete, especially when making introductory or summary statements.

Where Do You Discuss Pilot Studies? Many times it is important to do pilot studies to help you get familiar with your experimental system or to improve your experimental design. If your pilot study influences your biological rationale or hypothesis, you need to describe it in your Introduction. If your pilot study simply informs the logistics or techniques, but does not influence your rationale, then the description of your pilot study belongs in the Materials and Methods section.  

How will introductions be evaluated? The following is part of the rubric we will be using to evaluate your papers.

What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By:  Derek Jansen (MBA)  | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing. 

Research Hypothesis 101

• What is a hypothesis ?
• What is a research hypothesis (scientific hypothesis)?
• Requirements for a research hypothesis
• Definition of a research hypothesis
• The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

Need a helping hand?

Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.  

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference. 

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell. 

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

Psst... there’s more!

This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...

You Might Also Like:

16 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

this is very important note help me much more

Trackbacks/Pingbacks

• What Is Research Methodology? Simple Definition (With Examples) - Grad Coach - […] Contrasted to this, a quantitative methodology is typically used when the research aims and objectives are confirmatory in nature. For example,…

Submit a Comment Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

• Print Friendly

• USC Libraries
• Research Guides

Organizing Your Social Sciences Research Paper

• 4. The Introduction
• Purpose of Guide
• Design Flaws to Avoid
• Independent and Dependent Variables
• Glossary of Research Terms
• Reading Research Effectively
• Narrowing a Topic Idea
• Broadening a Topic Idea
• Extending the Timeliness of a Topic Idea
• Academic Writing Style
• Applying Critical Thinking
• Choosing a Title
• Making an Outline
• Paragraph Development
• Research Process Video Series
• Executive Summary
• The C.A.R.S. Model
• Background Information
• The Research Problem/Question
• Theoretical Framework
• Citation Tracking
• Content Alert Services
• Evaluating Sources
• Primary Sources
• Secondary Sources
• Tiertiary Sources
• Scholarly vs. Popular Publications
• Qualitative Methods
• Quantitative Methods
• Insiderness
• Using Non-Textual Elements
• Limitations of the Study
• Common Grammar Mistakes
• Writing Concisely
• Avoiding Plagiarism
• Footnotes or Endnotes?
• Further Readings
• Generative AI and Writing
• USC Libraries Tutorials and Other Guides
• Bibliography

The introduction leads the reader from a general subject area to a particular topic of inquiry. It establishes the scope, context, and significance of the research being conducted by summarizing current understanding and background information about the topic, stating the purpose of the work in the form of the research problem supported by a hypothesis or a set of questions, explaining briefly the methodological approach used to examine the research problem, highlighting the potential outcomes your study can reveal, and outlining the remaining structure and organization of the paper.

Key Elements of the Research Proposal. Prepared under the direction of the Superintendent and by the 2010 Curriculum Design and Writing Team. Baltimore County Public Schools.

Importance of a Good Introduction

Think of the introduction as a mental road map that must answer for the reader these four questions:

• What was I studying?
• Why was this topic important to investigate?
• What did we know about this topic before I did this study?
• How will this study advance new knowledge or new ways of understanding?

According to Reyes, there are three overarching goals of a good introduction: 1) ensure that you summarize prior studies about the topic in a manner that lays a foundation for understanding the research problem; 2) explain how your study specifically addresses gaps in the literature, insufficient consideration of the topic, or other deficiency in the literature; and, 3) note the broader theoretical, empirical, and/or policy contributions and implications of your research.

A well-written introduction is important because, quite simply, you never get a second chance to make a good first impression. The opening paragraphs of your paper will provide your readers with their initial impressions about the logic of your argument, your writing style, the overall quality of your research, and, ultimately, the validity of your findings and conclusions. A vague, disorganized, or error-filled introduction will create a negative impression, whereas, a concise, engaging, and well-written introduction will lead your readers to think highly of your analytical skills, your writing style, and your research approach. All introductions should conclude with a brief paragraph that describes the organization of the rest of the paper.

Hirano, Eliana. “Research Article Introductions in English for Specific Purposes: A Comparison between Brazilian, Portuguese, and English.” English for Specific Purposes 28 (October 2009): 240-250; Samraj, B. “Introductions in Research Articles: Variations Across Disciplines.” English for Specific Purposes 21 (2002): 1–17; Introductions. The Writing Center. University of North Carolina; “Writing Introductions.” In Good Essay Writing: A Social Sciences Guide. Peter Redman. 4th edition. (London: Sage, 2011), pp. 63-70; Reyes, Victoria. Demystifying the Journal Article. Inside Higher Education.

Structure and Writing Style

I.  Structure and Approach

The introduction is the broad beginning of the paper that answers three important questions for the reader:

• What is this?
• Why should I read it?
• What do you want me to think about / consider doing / react to?

Think of the structure of the introduction as an inverted triangle of information that lays a foundation for understanding the research problem. Organize the information so as to present the more general aspects of the topic early in the introduction, then narrow your analysis to more specific topical information that provides context, finally arriving at your research problem and the rationale for studying it [often written as a series of key questions to be addressed or framed as a hypothesis or set of assumptions to be tested] and, whenever possible, a description of the potential outcomes your study can reveal.

These are general phases associated with writing an introduction: 1.  Establish an area to research by:

• Highlighting the importance of the topic, and/or
• Making general statements about the topic, and/or
• Presenting an overview on current research on the subject.

2.  Identify a research niche by:

• Opposing an existing assumption, and/or
• Revealing a gap in existing research, and/or
• Formulating a research question or problem, and/or
• Continuing a disciplinary tradition.

3.  Place your research within the research niche by:

• Stating the intent of your study,
• Outlining the key characteristics of your study,
• Describing important results, and
• Giving a brief overview of the structure of the paper.

NOTE:   It is often useful to review the introduction late in the writing process. This is appropriate because outcomes are unknown until you've completed the study. After you complete writing the body of the paper, go back and review introductory descriptions of the structure of the paper, the method of data gathering, the reporting and analysis of results, and the conclusion. Reviewing and, if necessary, rewriting the introduction ensures that it correctly matches the overall structure of your final paper.

II.  Delimitations of the Study

Delimitations refer to those characteristics that limit the scope and define the conceptual boundaries of your research . This is determined by the conscious exclusionary and inclusionary decisions you make about how to investigate the research problem. In other words, not only should you tell the reader what it is you are studying and why, but you must also acknowledge why you rejected alternative approaches that could have been used to examine the topic.

Obviously, the first limiting step was the choice of research problem itself. However, implicit are other, related problems that could have been chosen but were rejected. These should be noted in the conclusion of your introduction. For example, a delimitating statement could read, "Although many factors can be understood to impact the likelihood young people will vote, this study will focus on socioeconomic factors related to the need to work full-time while in school." The point is not to document every possible delimiting factor, but to highlight why previously researched issues related to the topic were not addressed.

Examples of delimitating choices would be:

• The key aims and objectives of your study,
• The research questions that you address,
• The variables of interest [i.e., the various factors and features of the phenomenon being studied],
• The method(s) of investigation,
• The time period your study covers, and
• Any relevant alternative theoretical frameworks that could have been adopted.

Review each of these decisions. Not only do you clearly establish what you intend to accomplish in your research, but you should also include a declaration of what the study does not intend to cover. In the latter case, your exclusionary decisions should be based upon criteria understood as, "not interesting"; "not directly relevant"; “too problematic because..."; "not feasible," and the like. Make this reasoning explicit!

NOTE:   Delimitations refer to the initial choices made about the broader, overall design of your study and should not be confused with documenting the limitations of your study discovered after the research has been completed.

ANOTHER NOTE : Do not view delimitating statements as admitting to an inherent failing or shortcoming in your research. They are an accepted element of academic writing intended to keep the reader focused on the research problem by explicitly defining the conceptual boundaries and scope of your study. It addresses any critical questions in the reader's mind of, "Why the hell didn't the author examine this?"

III.  The Narrative Flow

Issues to keep in mind that will help the narrative flow in your introduction :

• Your introduction should clearly identify the subject area of interest . A simple strategy to follow is to use key words from your title in the first few sentences of the introduction. This will help focus the introduction on the topic at the appropriate level and ensures that you get to the subject matter quickly without losing focus, or discussing information that is too general.
• Establish context by providing a brief and balanced review of the pertinent published literature that is available on the subject. The key is to summarize for the reader what is known about the specific research problem before you did your analysis. This part of your introduction should not represent a comprehensive literature review--that comes next. It consists of a general review of the important, foundational research literature [with citations] that establishes a foundation for understanding key elements of the research problem. See the drop-down menu under this tab for " Background Information " regarding types of contexts.
• Clearly state the hypothesis that you investigated . When you are first learning to write in this format it is okay, and actually preferable, to use a past statement like, "The purpose of this study was to...." or "We investigated three possible mechanisms to explain the...."
• Why did you choose this kind of research study or design? Provide a clear statement of the rationale for your approach to the problem studied. This will usually follow your statement of purpose in the last paragraph of the introduction.

IV.  Engaging the Reader

A research problem in the social sciences can come across as dry and uninteresting to anyone unfamiliar with the topic . Therefore, one of the goals of your introduction is to make readers want to read your paper. Here are several strategies you can use to grab the reader's attention:

• Open with a compelling story . Almost all research problems in the social sciences, no matter how obscure or esoteric , are really about the lives of people. Telling a story that humanizes an issue can help illuminate the significance of the problem and help the reader empathize with those affected by the condition being studied.
• Include a strong quotation or a vivid, perhaps unexpected, anecdote . During your review of the literature, make note of any quotes or anecdotes that grab your attention because they can used in your introduction to highlight the research problem in a captivating way.
• Pose a provocative or thought-provoking question . Your research problem should be framed by a set of questions to be addressed or hypotheses to be tested. However, a provocative question can be presented in the beginning of your introduction that challenges an existing assumption or compels the reader to consider an alternative viewpoint that helps establish the significance of your study. 
• Describe a puzzling scenario or incongruity . This involves highlighting an interesting quandary concerning the research problem or describing contradictory findings from prior studies about a topic. Posing what is essentially an unresolved intellectual riddle about the problem can engage the reader's interest in the study.
• Cite a stirring example or case study that illustrates why the research problem is important . Draw upon the findings of others to demonstrate the significance of the problem and to describe how your study builds upon or offers alternatives ways of investigating this prior research.

NOTE:   It is important that you choose only one of the suggested strategies for engaging your readers. This avoids giving an impression that your paper is more flash than substance and does not distract from the substance of your study.

Freedman, Leora  and Jerry Plotnick. Introductions and Conclusions. University College Writing Centre. University of Toronto; Introduction. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Introductions. The Writing Center. University of North Carolina; Introductions. The Writer’s Handbook. Writing Center. University of Wisconsin, Madison; Introductions, Body Paragraphs, and Conclusions for an Argument Paper. The Writing Lab and The OWL. Purdue University; “Writing Introductions.” In Good Essay Writing: A Social Sciences Guide . Peter Redman. 4th edition. (London: Sage, 2011), pp. 63-70; Resources for Writers: Introduction Strategies. Program in Writing and Humanistic Studies. Massachusetts Institute of Technology; Sharpling, Gerald. Writing an Introduction. Centre for Applied Linguistics, University of Warwick; Samraj, B. “Introductions in Research Articles: Variations Across Disciplines.” English for Specific Purposes 21 (2002): 1–17; Swales, John and Christine B. Feak. Academic Writing for Graduate Students: Essential Skills and Tasks . 2nd edition. Ann Arbor, MI: University of Michigan Press, 2004 ; Writing Your Introduction. Department of English Writing Guide. George Mason University.

Writing Tip

Avoid the "Dictionary" Introduction

Giving the dictionary definition of words related to the research problem may appear appropriate because it is important to define specific terminology that readers may be unfamiliar with. However, anyone can look a word up in the dictionary and a general dictionary is not a particularly authoritative source because it doesn't take into account the context of your topic and doesn't offer particularly detailed information. Also, placed in the context of a particular discipline, a term or concept may have a different meaning than what is found in a general dictionary. If you feel that you must seek out an authoritative definition, use a subject specific dictionary or encyclopedia [e.g., if you are a sociology student, search for dictionaries of sociology]. A good database for obtaining definitive definitions of concepts or terms is Credo Reference .

Saba, Robert. The College Research Paper. Florida International University; Introductions. The Writing Center. University of North Carolina.

Another Writing Tip

When Do I Begin?

A common question asked at the start of any paper is, "Where should I begin?" An equally important question to ask yourself is, "When do I begin?" Research problems in the social sciences rarely rest in isolation from history. Therefore, it is important to lay a foundation for understanding the historical context underpinning the research problem. However, this information should be brief and succinct and begin at a point in time that illustrates the study's overall importance. For example, a study that investigates coffee cultivation and export in West Africa as a key stimulus for local economic growth needs to describe the beginning of exporting coffee in the region and establishing why economic growth is important. You do not need to give a long historical explanation about coffee exports in Africa. If a research problem requires a substantial exploration of the historical context, do this in the literature review section. In your introduction, make note of this as part of the "roadmap" [see below] that you use to describe the organization of your paper.

Introductions. The Writing Center. University of North Carolina; “Writing Introductions.” In Good Essay Writing: A Social Sciences Guide . Peter Redman. 4th edition. (London: Sage, 2011), pp. 63-70.

Yet Another Writing Tip

Always End with a Roadmap

The final paragraph or sentences of your introduction should forecast your main arguments and conclusions and provide a brief description of the rest of the paper [the "roadmap"] that let's the reader know where you are going and what to expect. A roadmap is important because it helps the reader place the research problem within the context of their own perspectives about the topic. In addition, concluding your introduction with an explicit roadmap tells the reader that you have a clear understanding of the structural purpose of your paper. In this way, the roadmap acts as a type of promise to yourself and to your readers that you will follow a consistent and coherent approach to addressing the topic of inquiry. Refer to it often to help keep your writing focused and organized.

Cassuto, Leonard. “On the Dissertation: How to Write the Introduction.” The Chronicle of Higher Education , May 28, 2018; Radich, Michael. A Student's Guide to Writing in East Asian Studies . (Cambridge, MA: Harvard University Writing n. d.), pp. 35-37.

• << Previous: Executive Summary
• Next: The C.A.R.S. Model >>
• Last Updated: May 15, 2024 9:53 AM
• URL: https://libguides.usc.edu/writingguide

• Manuscript Preparation

What is and How to Write a Good Hypothesis in Research?

• 4 minute read
• 313.6K views

Table of Contents

One of the most important aspects of conducting research is constructing a strong hypothesis. But what makes a hypothesis in research effective? In this article, we’ll look at the difference between a hypothesis and a research question, as well as the elements of a good hypothesis in research. We’ll also include some examples of effective hypotheses, and what pitfalls to avoid.

What is a Hypothesis in Research?

Simply put, a hypothesis is a research question that also includes the predicted or expected result of the research. Without a hypothesis, there can be no basis for a scientific or research experiment. As such, it is critical that you carefully construct your hypothesis by being deliberate and thorough, even before you set pen to paper. Unless your hypothesis is clearly and carefully constructed, any flaw can have an adverse, and even grave, effect on the quality of your experiment and its subsequent results.

Research Question vs Hypothesis

It’s easy to confuse research questions with hypotheses, and vice versa. While they’re both critical to the Scientific Method, they have very specific differences. Primarily, a research question, just like a hypothesis, is focused and concise. But a hypothesis includes a prediction based on the proposed research, and is designed to forecast the relationship of and between two (or more) variables. Research questions are open-ended, and invite debate and discussion, while hypotheses are closed, e.g. “The relationship between A and B will be C.”

A hypothesis is generally used if your research topic is fairly well established, and you are relatively certain about the relationship between the variables that will be presented in your research. Since a hypothesis is ideally suited for experimental studies, it will, by its very existence, affect the design of your experiment. The research question is typically used for new topics that have not yet been researched extensively. Here, the relationship between different variables is less known. There is no prediction made, but there may be variables explored. The research question can be casual in nature, simply trying to understand if a relationship even exists, descriptive or comparative.

How to Write Hypothesis in Research

Writing an effective hypothesis starts before you even begin to type. Like any task, preparation is key, so you start first by conducting research yourself, and reading all you can about the topic that you plan to research. From there, you’ll gain the knowledge you need to understand where your focus within the topic will lie.

Remember that a hypothesis is a prediction of the relationship that exists between two or more variables. Your job is to write a hypothesis, and design the research, to “prove” whether or not your prediction is correct. A common pitfall is to use judgments that are subjective and inappropriate for the construction of a hypothesis. It’s important to keep the focus and language of your hypothesis objective.

An effective hypothesis in research is clearly and concisely written, and any terms or definitions clarified and defined. Specific language must also be used to avoid any generalities or assumptions.

Use the following points as a checklist to evaluate the effectiveness of your research hypothesis:

• Predicts the relationship and outcome
• Simple and concise – avoid wordiness
• Clear with no ambiguity or assumptions about the readers’ knowledge
• Observable and testable results
• Relevant and specific to the research question or problem

Research Hypothesis Example

Perhaps the best way to evaluate whether or not your hypothesis is effective is to compare it to those of your colleagues in the field. There is no need to reinvent the wheel when it comes to writing a powerful research hypothesis. As you’re reading and preparing your hypothesis, you’ll also read other hypotheses. These can help guide you on what works, and what doesn’t, when it comes to writing a strong research hypothesis.

Here are a few generic examples to get you started.

Eating an apple each day, after the age of 60, will result in a reduction of frequency of physician visits.

Budget airlines are more likely to receive more customer complaints. A budget airline is defined as an airline that offers lower fares and fewer amenities than a traditional full-service airline. (Note that the term “budget airline” is included in the hypothesis.

Workplaces that offer flexible working hours report higher levels of employee job satisfaction than workplaces with fixed hours.

Each of the above examples are specific, observable and measurable, and the statement of prediction can be verified or shown to be false by utilizing standard experimental practices. It should be noted, however, that often your hypothesis will change as your research progresses.

Language Editing Plus

Elsevier’s Language Editing Plus service can help ensure that your research hypothesis is well-designed, and articulates your research and conclusions. Our most comprehensive editing package, you can count on a thorough language review by native-English speakers who are PhDs or PhD candidates. We’ll check for effective logic and flow of your manuscript, as well as document formatting for your chosen journal, reference checks, and much more.

• Research Process

Systematic Literature Review or Literature Review?

What is a Problem Statement? [with examples]

You may also like.

Make Hook, Line, and Sinker: The Art of Crafting Engaging Introductions

Can Describing Study Limitations Improve the Quality of Your Paper?

A Guide to Crafting Shorter, Impactful Sentences in Academic Writing

6 Steps to Write an Excellent Discussion in Your Manuscript

How to Write Clear and Crisp Civil Engineering Papers? Here are 5 Key Tips to Consider

The Clear Path to An Impactful Paper: ②

The Essentials of Writing to Communicate Research in Medicine

Changing Lines: Sentence Patterns in Academic Writing

Input your search keywords and press Enter.

• Bipolar Disorder
• Therapy Center
• When To See a Therapist
• Types of Therapy
• Best Online Therapy
• Best Couples Therapy
• Best Family Therapy
• Managing Stress
• Sleep and Dreaming
• Understanding Emotions
• Self-Improvement
• Healthy Relationships
• Student Resources
• Personality Types
• Guided Meditations
• Verywell Mind Insights
• 2024 Verywell Mind 25
• Mental Health in the Classroom
• Editorial Process
• Meet Our Review Board
• Crisis Support

How to Write an Introduction for a Psychology Paper

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Emily is a board-certified science editor who has worked with top digital publishing brands like Voices for Biodiversity, Study.com, GoodTherapy, Vox, and Verywell.

• Writing Tips

If you are writing a psychology paper, it is essential to kick things off with a strong introduction. The introduction to a psychology research paper helps your readers understand why the topic is important and what they need to know before they delve deeper.

Your goal in this section is to introduce the topic to the reader, provide an overview of previous research on the topic, and identify your own hypothesis .

At a Glance

Writing a great introduction can be a great foundation for the rest of your psychology paper. To create a strong intro:

• Research your topic
• Outline your paper
• Introduce your topic
• Summarize the previous research
• Present your hypothesis or main argument

Before You Write an Introduction

There are some important steps you need to take before you even begin writing your introduction. To know what to write, you need to collect important background information and create a detailed plan.

Research Your Topic

Search a journal database, PsychInfo or ERIC, to find articles on your subject. Once you have located an article, look at the reference section to locate other studies cited in the article. As you take notes from these articles, be sure to write down where you found the information.

A simple note detailing the author's name, journal, and date of publication can help you keep track of sources and avoid plagiarism.

Create a Detailed Outline

This is often one of the most boring and onerous steps, so students tend to skip outlining and go straight to writing. Creating an outline might seem tedious, but it can be an enormous time-saver down the road and will make the writing process much easier.

Start by looking over the notes you made during the research process and consider how you want to present all of your ideas and research.

Introduce the Topic

Once you are ready to write your introduction, your first task is to provide a brief description of the research question. What is the experiment or study attempting to demonstrate? What phenomena are you studying? Provide a brief history of your topic and explain how it relates to your current research.

As you are introducing your topic, consider what makes it important. Why should it matter to your reader? The goal of your introduction is not only to let your reader know what your paper is about, but also to justify why it is important for them to learn more.

If your paper tackles a controversial subject and is focused on resolving the issue, it is important to summarize both sides of the controversy in a fair and impartial way. Consider how your paper fits in with the relevant research on the topic.

The introduction of a research paper is designed to grab interest. It should present a compelling look at the research that already exists and explain to readers what questions your own paper will address.

Summarize Previous Research

The second task of your introduction is to provide a well-rounded summary of previous research that is relevant to your topic. So, before you begin to write this summary, it is important to research your topic thoroughly.

Finding appropriate sources amid thousands of journal articles can be a daunting task, but there are several steps you can take to simplify your research. If you have completed the initial steps of researching and keeping detailed notes, writing your introduction will be much easier.

It is essential to give the reader a good overview of the historical context of the issue you are writing about, but do not feel like you must provide an exhaustive review of the subject. Focus on hitting the main points, and try to include the most relevant studies.

You might describe previous research findings and then explain how the current study differs or expands upon earlier research.

Provide Your Hypothesis

Once you have summarized the previous research, explain areas where the research is lacking or potentially flawed. What is missing from previous studies on your topic? What research questions have yet to be answered? Your hypothesis should lead to these questions.

At the end of your introduction, offer your hypothesis and describe what you expected to find in your experiment or study.

The introduction should be relatively brief. You want to give your readers an overview of a topic, explain why you are addressing it, and provide your arguments.

Tips for Writing Your Psychology Paper Intro

• Use 3x5 inch note cards to write down notes and sources.
• Look in professional psychology journals for examples of introductions.
• Remember to cite your sources.
• Maintain a working bibliography with all of the sources you might use in your final paper. This will make it much easier to prepare your reference section later on.
• Use a copy of the APA style manual to ensure that your introduction and references are in proper APA format .

What This Means For You

Before you delve into the main body of your paper, you need to give your readers some background and present your main argument in the introduction of you paper. You can do this by first explaining what your topic is about, summarizing past research, and then providing your thesis.

Armağan A. How to write an introduction section of a scientific article ?  Turk J Urol . 2013;39(Suppl 1):8-9. doi:10.5152/tud.2013.046

Fried T, Foltz C, Lendner M, Vaccaro AR. How to write an effective introduction .  Clin Spine Surg . 2019;32(3):111-112. doi:10.1097/BSD.0000000000000714

Jawaid SA, Jawaid M. How to write introduction and discussion .  Saudi J Anaesth . 2019;13(Suppl 1):S18-S19. doi:10.4103/sja.SJA_584_18

American Psychological Association. Information Recommended for Inclusion in Manuscripts That Report New Data Collections Regardless of Research Design . Published 2020.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

• Affiliate Program

• UNITED STATES
• 台灣 (TAIWAN)
• TÜRKIYE (TURKEY)
• Academic Editing Services
• - Research Paper
• - Journal Manuscript
• - Dissertation
• - College & University Assignments
• Admissions Editing Services
• - Application Essay
• - Personal Statement
• - Recommendation Letter
• - Cover Letter
• - CV/Resume
• Business Editing Services
• - Business Documents
• - Report & Brochure
• - Website & Blog
• Writer Editing Services
• - Script & Screenplay
• Our Editors
• Client Reviews
• Editing & Proofreading Prices
• Wordvice Points
• Partner Discount
• Plagiarism Checker
• APA Citation Generator
• MLA Citation Generator
• Chicago Citation Generator
• Vancouver Citation Generator
• - APA Style
• - MLA Style
• - Chicago Style
• - Vancouver Style
• Writing & Editing Guide
• Academic Resources
• Admissions Resources

How to Write a Research Hypothesis: Good & Bad Examples

What is a research hypothesis?

A research hypothesis is an attempt at explaining a phenomenon or the relationships between phenomena/variables in the real world. Hypotheses are sometimes called “educated guesses”, but they are in fact (or let’s say they should be) based on previous observations, existing theories, scientific evidence, and logic. A research hypothesis is also not a prediction—rather, predictions are ( should be) based on clearly formulated hypotheses. For example, “We tested the hypothesis that KLF2 knockout mice would show deficiencies in heart development” is an assumption or prediction, not a hypothesis. 

The research hypothesis at the basis of this prediction is “the product of the KLF2 gene is involved in the development of the cardiovascular system in mice”—and this hypothesis is probably (hopefully) based on a clear observation, such as that mice with low levels of Kruppel-like factor 2 (which KLF2 codes for) seem to have heart problems. From this hypothesis, you can derive the idea that a mouse in which this particular gene does not function cannot develop a normal cardiovascular system, and then make the prediction that we started with. 

What is the difference between a hypothesis and a prediction?

You might think that these are very subtle differences, and you will certainly come across many publications that do not contain an actual hypothesis or do not make these distinctions correctly. But considering that the formulation and testing of hypotheses is an integral part of the scientific method, it is good to be aware of the concepts underlying this approach. The two hallmarks of a scientific hypothesis are falsifiability (an evaluation standard that was introduced by the philosopher of science Karl Popper in 1934) and testability —if you cannot use experiments or data to decide whether an idea is true or false, then it is not a hypothesis (or at least a very bad one).

So, in a nutshell, you (1) look at existing evidence/theories, (2) come up with a hypothesis, (3) make a prediction that allows you to (4) design an experiment or data analysis to test it, and (5) come to a conclusion. Of course, not all studies have hypotheses (there is also exploratory or hypothesis-generating research), and you do not necessarily have to state your hypothesis as such in your paper. 

But for the sake of understanding the principles of the scientific method, let’s first take a closer look at the different types of hypotheses that research articles refer to and then give you a step-by-step guide for how to formulate a strong hypothesis for your own paper.

Types of Research Hypotheses

Hypotheses can be simple , which means they describe the relationship between one single independent variable (the one you observe variations in or plan to manipulate) and one single dependent variable (the one you expect to be affected by the variations/manipulation). If there are more variables on either side, you are dealing with a complex hypothesis. You can also distinguish hypotheses according to the kind of relationship between the variables you are interested in (e.g., causal or associative ). But apart from these variations, we are usually interested in what is called the “alternative hypothesis” and, in contrast to that, the “null hypothesis”. If you think these two should be listed the other way round, then you are right, logically speaking—the alternative should surely come second. However, since this is the hypothesis we (as researchers) are usually interested in, let’s start from there.

Alternative Hypothesis

If you predict a relationship between two variables in your study, then the research hypothesis that you formulate to describe that relationship is your alternative hypothesis (usually H1 in statistical terms). The goal of your hypothesis testing is thus to demonstrate that there is sufficient evidence that supports the alternative hypothesis, rather than evidence for the possibility that there is no such relationship. The alternative hypothesis is usually the research hypothesis of a study and is based on the literature, previous observations, and widely known theories. 

Null Hypothesis

The hypothesis that describes the other possible outcome, that is, that your variables are not related, is the null hypothesis ( H0 ). Based on your findings, you choose between the two hypotheses—usually that means that if your prediction was correct, you reject the null hypothesis and accept the alternative. Make sure, however, that you are not getting lost at this step of the thinking process: If your prediction is that there will be no difference or change, then you are trying to find support for the null hypothesis and reject H1. 

Directional Hypothesis

While the null hypothesis is obviously “static”, the alternative hypothesis can specify a direction for the observed relationship between variables—for example, that mice with higher expression levels of a certain protein are more active than those with lower levels. This is then called a one-tailed hypothesis. 

Another example for a directional one-tailed alternative hypothesis would be that 

H1: Attending private classes before important exams has a positive effect on performance. 

Your null hypothesis would then be that

H0: Attending private classes before important exams has no/a negative effect on performance.

Nondirectional Hypothesis

A nondirectional hypothesis does not specify the direction of the potentially observed effect, only that there is a relationship between the studied variables—this is called a two-tailed hypothesis. For instance, if you are studying a new drug that has shown some effects on pathways involved in a certain condition (e.g., anxiety) in vitro in the lab, but you can’t say for sure whether it will have the same effects in an animal model or maybe induce other/side effects that you can’t predict and potentially increase anxiety levels instead, you could state the two hypotheses like this:

H1: The only lab-tested drug (somehow) affects anxiety levels in an anxiety mouse model.

You then test this nondirectional alternative hypothesis against the null hypothesis:

H0: The only lab-tested drug has no effect on anxiety levels in an anxiety mouse model.

How to Write a Hypothesis for a Research Paper

Now that we understand the important distinctions between different kinds of research hypotheses, let’s look at a simple process of how to write a hypothesis.

Writing a Hypothesis Step:1

Ask a question, based on earlier research. Research always starts with a question, but one that takes into account what is already known about a topic or phenomenon. For example, if you are interested in whether people who have pets are happier than those who don’t, do a literature search and find out what has already been demonstrated. You will probably realize that yes, there is quite a bit of research that shows a relationship between happiness and owning a pet—and even studies that show that owning a dog is more beneficial than owning a cat ! Let’s say you are so intrigued by this finding that you wonder: 

What is it that makes dog owners even happier than cat owners? 

Let’s move on to Step 2 and find an answer to that question.

Writing a Hypothesis Step 2:

Formulate a strong hypothesis by answering your own question. Again, you don’t want to make things up, take unicorns into account, or repeat/ignore what has already been done. Looking at the dog-vs-cat papers your literature search returned, you see that most studies are based on self-report questionnaires on personality traits, mental health, and life satisfaction. What you don’t find is any data on actual (mental or physical) health measures, and no experiments. You therefore decide to make a bold claim come up with the carefully thought-through hypothesis that it’s maybe the lifestyle of the dog owners, which includes walking their dog several times per day, engaging in fun and healthy activities such as agility competitions, and taking them on trips, that gives them that extra boost in happiness. You could therefore answer your question in the following way:

Dog owners are happier than cat owners because of the dog-related activities they engage in.

Now you have to verify that your hypothesis fulfills the two requirements we introduced at the beginning of this resource article: falsifiability and testability . If it can’t be wrong and can’t be tested, it’s not a hypothesis. We are lucky, however, because yes, we can test whether owning a dog but not engaging in any of those activities leads to lower levels of happiness or well-being than owning a dog and playing and running around with them or taking them on trips.  

Writing a Hypothesis Step 3:

Make your predictions and define your variables. We have verified that we can test our hypothesis, but now we have to define all the relevant variables, design our experiment or data analysis, and make precise predictions. You could, for example, decide to study dog owners (not surprising at this point), let them fill in questionnaires about their lifestyle as well as their life satisfaction (as other studies did), and then compare two groups of active and inactive dog owners. Alternatively, if you want to go beyond the data that earlier studies produced and analyzed and directly manipulate the activity level of your dog owners to study the effect of that manipulation, you could invite them to your lab, select groups of participants with similar lifestyles, make them change their lifestyle (e.g., couch potato dog owners start agility classes, very active ones have to refrain from any fun activities for a certain period of time) and assess their happiness levels before and after the intervention. In both cases, your independent variable would be “ level of engagement in fun activities with dog” and your dependent variable would be happiness or well-being . 

Examples of a Good and Bad Hypothesis

Let’s look at a few examples of good and bad hypotheses to get you started.

Good Hypothesis Examples

Bad hypothesis examples, tips for writing a research hypothesis.

If you understood the distinction between a hypothesis and a prediction we made at the beginning of this article, then you will have no problem formulating your hypotheses and predictions correctly. To refresh your memory: We have to (1) look at existing evidence, (2) come up with a hypothesis, (3) make a prediction, and (4) design an experiment. For example, you could summarize your dog/happiness study like this:

(1) While research suggests that dog owners are happier than cat owners, there are no reports on what factors drive this difference. (2) We hypothesized that it is the fun activities that many dog owners (but very few cat owners) engage in with their pets that increases their happiness levels. (3) We thus predicted that preventing very active dog owners from engaging in such activities for some time and making very inactive dog owners take up such activities would lead to an increase and decrease in their overall self-ratings of happiness, respectively. (4) To test this, we invited dog owners into our lab, assessed their mental and emotional well-being through questionnaires, and then assigned them to an “active” and an “inactive” group, depending on… 

Note that you use “we hypothesize” only for your hypothesis, not for your experimental prediction, and “would” or “if – then” only for your prediction, not your hypothesis. A hypothesis that states that something “would” affect something else sounds as if you don’t have enough confidence to make a clear statement—in which case you can’t expect your readers to believe in your research either. Write in the present tense, don’t use modal verbs that express varying degrees of certainty (such as may, might, or could ), and remember that you are not drawing a conclusion while trying not to exaggerate but making a clear statement that you then, in a way, try to disprove . And if that happens, that is not something to fear but an important part of the scientific process.

Similarly, don’t use “we hypothesize” when you explain the implications of your research or make predictions in the conclusion section of your manuscript, since these are clearly not hypotheses in the true sense of the word. As we said earlier, you will find that many authors of academic articles do not seem to care too much about these rather subtle distinctions, but thinking very clearly about your own research will not only help you write better but also ensure that even that infamous Reviewer 2 will find fewer reasons to nitpick about your manuscript. 

Perfect Your Manuscript With Professional Editing

Now that you know how to write a strong research hypothesis for your research paper, you might be interested in our free AI proofreader , Wordvice AI, which finds and fixes errors in grammar, punctuation, and word choice in academic texts. Or if you are interested in human proofreading , check out our English editing services , including research paper editing and manuscript editing .

On the Wordvice academic resources website , you can also find many more articles and other resources that can help you with writing the other parts of your research paper , with making a research paper outline before you put everything together, or with writing an effective cover letter once you are ready to submit.

HOW TO: Use Articles for Research: Introduction: Hypothesis/Thesis

• What's a Scholarly Journal?
• Reading the Citation
• Authors' Credentials
• Introduction: Hypothesis/Thesis
• Literature Review
• Research Method
• Results/Data
• Discussion/Conclusions

Hypothesis or Thesis

The first few paragraphs of a journal article serve to introduce the topic, to provide the author's hypothesis or thesis, and to indicate why the research was done.  A thesis or hypothesis is not always clearly labled; you may need to read through the introductory paragraphs to determine what the authors are proposing.

• << Previous: Abstract
• Next: Literature Review >>
• Last Updated: Jan 29, 2024 3:35 PM
• URL: https://libguides.cayuga-cc.edu/1ST-PRIORITY/articles

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, automatically generate references for free.

• Knowledge Base
• Dissertation

How to Write a Thesis or Dissertation Introduction

Published on 9 September 2022 by Tegan George and Shona McCombes.

The introduction is the first section of your thesis or dissertation , appearing right after the table of contents . Your introduction draws your reader in, setting the stage for your research with a clear focus, purpose, and direction.

Your introduction should include:

• Your topic, in context: what does your reader need to know to understand your thesis dissertation?
• Your focus and scope: what specific aspect of the topic will you address?
• The relevance of your research: how does your work fit into existing studies on your topic?
• Your questions and objectives: what does your research aim to find out, and how?
• An overview of your structure: what does each section contribute to the overall aim?

Instantly correct all language mistakes in your text

Be assured that you'll submit flawless writing. Upload your document to correct all your mistakes.

Table of contents

How to start your introduction, topic and context, focus and scope, relevance and importance, questions and objectives, overview of the structure, thesis introduction example, introduction checklist, frequently asked questions about introductions.

Although your introduction kicks off your dissertation, it doesn’t have to be the first thing you write – in fact, it’s often one of the very last parts to be completed (just before your abstract ).

It’s a good idea to write a rough draft of your introduction as you begin your research, to help guide you. If you wrote a research proposal , consider using this as a template, as it contains many of the same elements. However, be sure to revise your introduction throughout the writing process, making sure it matches the content of your ensuing sections.

The only proofreading tool specialized in correcting academic writing

The academic proofreading tool has been trained on 1000s of academic texts and by native English editors. Making it the most accurate and reliable proofreading tool for students.

Correct my document today

Begin by introducing your research topic and giving any necessary background information. It’s important to contextualise your research and generate interest. Aim to show why your topic is timely or important. You may want to mention a relevant news item, academic debate, or practical problem.

After a brief introduction to your general area of interest, narrow your focus and define the scope of your research.

You can narrow this down in many ways, such as by:

• Geographical area
• Time period
• Demographics or communities
• Themes or aspects of the topic

It’s essential to share your motivation for doing this research, as well as how it relates to existing work on your topic. Further, you should also mention what new insights you expect it will contribute.

Start by giving a brief overview of the current state of research. You should definitely cite the most relevant literature, but remember that you will conduct a more in-depth survey of relevant sources in the literature review section, so there’s no need to go too in-depth in the introduction.

Depending on your field, the importance of your research might focus on its practical application (e.g., in policy or management) or on advancing scholarly understanding of the topic (e.g., by developing theories or adding new empirical data). In many cases, it will do both.

Ultimately, your introduction should explain how your thesis or dissertation:

• Helps solve a practical or theoretical problem
• Addresses a gap in the literature
• Builds on existing research
• Proposes a new understanding of your topic

Prevent plagiarism, run a free check.

Perhaps the most important part of your introduction is your questions and objectives, as it sets up the expectations for the rest of your thesis or dissertation. How you formulate your research questions and research objectives will depend on your discipline, topic, and focus, but you should always clearly state the central aim of your research.

If your research aims to test hypotheses , you can formulate them here. Your introduction is also a good place for a conceptual framework that suggests relationships between variables .

• Conduct surveys to collect data on students’ levels of knowledge, understanding, and positive/negative perceptions of government policy.
• Determine whether attitudes to climate policy are associated with variables such as age, gender, region, and social class.
• Conduct interviews to gain qualitative insights into students’ perspectives and actions in relation to climate policy.

To help guide your reader, end your introduction with an outline  of the structure of the thesis or dissertation to follow. Share a brief summary of each chapter, clearly showing how each contributes to your central aims. However, be careful to keep this overview concise: 1-2 sentences should be enough.

I. Introduction

Human language consists of a set of vowels and consonants which are combined to form words. During the speech production process, thoughts are converted into spoken utterances to convey a message. The appropriate words and their meanings are selected in the mental lexicon (Dell & Burger, 1997). This pre-verbal message is then grammatically coded, during which a syntactic representation of the utterance is built.

Speech, language, and voice disorders affect the vocal cords, nerves, muscles, and brain structures, which result in a distorted language reception or speech production (Sataloff & Hawkshaw, 2014). The symptoms vary from adding superfluous words and taking pauses to hoarseness of the voice, depending on the type of disorder (Dodd, 2005). However, distortions of the speech may also occur as a result of a disease that seems unrelated to speech, such as multiple sclerosis or chronic obstructive pulmonary disease.

This study aims to determine which acoustic parameters are suitable for the automatic detection of exacerbations in patients suffering from chronic obstructive pulmonary disease (COPD) by investigating which aspects of speech differ between COPD patients and healthy speakers and which aspects differ between COPD patients in exacerbation and stable COPD patients.

Checklist: Introduction

I have introduced my research topic in an engaging way.

I have provided necessary context to help the reader understand my topic.

I have clearly specified the focus of my research.

I have shown the relevance and importance of the dissertation topic .

I have clearly stated the problem or question that my research addresses.

I have outlined the specific objectives of the research .

I have provided an overview of the dissertation’s structure .

You've written a strong introduction for your thesis or dissertation. Use the other checklists to continue improving your dissertation.

The introduction of a research paper includes several key elements:

• A hook to catch the reader’s interest
• Relevant background on the topic
• Details of your research problem
• A thesis statement or research question
• Sometimes an outline of the paper

Don’t feel that you have to write the introduction first. The introduction is often one of the last parts of the research paper you’ll write, along with the conclusion.

This is because it can be easier to introduce your paper once you’ve already written the body ; you may not have the clearest idea of your arguments until you’ve written them, and things can change during the writing process .

Research objectives describe what you intend your research project to accomplish.

They summarise the approach and purpose of the project and help to focus your research.

Your objectives should appear in the introduction of your research paper , at the end of your problem statement .

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the ‘Cite this Scribbr article’ button to automatically add the citation to our free Reference Generator.

George, T. & McCombes, S. (2022, September 09). How to Write a Thesis or Dissertation Introduction. Scribbr. Retrieved 14 May 2024, from https://www.scribbr.co.uk/thesis-dissertation/introduction/

Is this article helpful?

Tegan George

Other students also liked, what is a dissertation | 5 essential questions to get started, how to write an abstract | steps & examples, how to write a thesis or dissertation conclusion.

How to Develop a Good Research Hypothesis

The story of a research study begins by asking a question. Researchers all around the globe are asking curious questions and formulating research hypothesis. However, whether the research study provides an effective conclusion depends on how well one develops a good research hypothesis. Research hypothesis examples could help researchers get an idea as to how to write a good research hypothesis.

This blog will help you understand what is a research hypothesis, its characteristics and, how to formulate a research hypothesis

Table of Contents

What is Hypothesis?

Hypothesis is an assumption or an idea proposed for the sake of argument so that it can be tested. It is a precise, testable statement of what the researchers predict will be outcome of the study.  Hypothesis usually involves proposing a relationship between two variables: the independent variable (what the researchers change) and the dependent variable (what the research measures).

What is a Research Hypothesis?

Research hypothesis is a statement that introduces a research question and proposes an expected result. It is an integral part of the scientific method that forms the basis of scientific experiments. Therefore, you need to be careful and thorough when building your research hypothesis. A minor flaw in the construction of your hypothesis could have an adverse effect on your experiment. In research, there is a convention that the hypothesis is written in two forms, the null hypothesis, and the alternative hypothesis (called the experimental hypothesis when the method of investigation is an experiment).

Characteristics of a Good Research Hypothesis

As the hypothesis is specific, there is a testable prediction about what you expect to happen in a study. You may consider drawing hypothesis from previously published research based on the theory.

A good research hypothesis involves more effort than just a guess. In particular, your hypothesis may begin with a question that could be further explored through background research.

To help you formulate a promising research hypothesis, you should ask yourself the following questions:

• Is the language clear and focused?
• What is the relationship between your hypothesis and your research topic?
• Is your hypothesis testable? If yes, then how?
• What are the possible explanations that you might want to explore?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate your variables without hampering the ethical standards?
• Does your research predict the relationship and outcome?
• Is your research simple and concise (avoids wordiness)?
• Is it clear with no ambiguity or assumptions about the readers’ knowledge
• Is your research observable and testable results?
• Is it relevant and specific to the research question or problem?

The questions listed above can be used as a checklist to make sure your hypothesis is based on a solid foundation. Furthermore, it can help you identify weaknesses in your hypothesis and revise it if necessary.

Source: Educational Hub

How to formulate a research hypothesis.

A testable hypothesis is not a simple statement. It is rather an intricate statement that needs to offer a clear introduction to a scientific experiment, its intentions, and the possible outcomes. However, there are some important things to consider when building a compelling hypothesis.

1. State the problem that you are trying to solve.

Make sure that the hypothesis clearly defines the topic and the focus of the experiment.

2. Try to write the hypothesis as an if-then statement.

Follow this template: If a specific action is taken, then a certain outcome is expected.

3. Define the variables

Independent variables are the ones that are manipulated, controlled, or changed. Independent variables are isolated from other factors of the study.

Dependent variables , as the name suggests are dependent on other factors of the study. They are influenced by the change in independent variable.

4. Scrutinize the hypothesis

Evaluate assumptions, predictions, and evidence rigorously to refine your understanding.

Types of Research Hypothesis

The types of research hypothesis are stated below:

1. Simple Hypothesis

It predicts the relationship between a single dependent variable and a single independent variable.

2. Complex Hypothesis

It predicts the relationship between two or more independent and dependent variables.

3. Directional Hypothesis

It specifies the expected direction to be followed to determine the relationship between variables and is derived from theory. Furthermore, it implies the researcher’s intellectual commitment to a particular outcome.

4. Non-directional Hypothesis

It does not predict the exact direction or nature of the relationship between the two variables. The non-directional hypothesis is used when there is no theory involved or when findings contradict previous research.

5. Associative and Causal Hypothesis

The associative hypothesis defines interdependency between variables. A change in one variable results in the change of the other variable. On the other hand, the causal hypothesis proposes an effect on the dependent due to manipulation of the independent variable.

6. Null Hypothesis

Null hypothesis states a negative statement to support the researcher’s findings that there is no relationship between two variables. There will be no changes in the dependent variable due the manipulation of the independent variable. Furthermore, it states results are due to chance and are not significant in terms of supporting the idea being investigated.

7. Alternative Hypothesis

It states that there is a relationship between the two variables of the study and that the results are significant to the research topic. An experimental hypothesis predicts what changes will take place in the dependent variable when the independent variable is manipulated. Also, it states that the results are not due to chance and that they are significant in terms of supporting the theory being investigated.

Research Hypothesis Examples of Independent and Dependent Variables

Research Hypothesis Example 1 The greater number of coal plants in a region (independent variable) increases water pollution (dependent variable). If you change the independent variable (building more coal factories), it will change the dependent variable (amount of water pollution).

Research Hypothesis Example 2 What is the effect of diet or regular soda (independent variable) on blood sugar levels (dependent variable)? If you change the independent variable (the type of soda you consume), it will change the dependent variable (blood sugar levels)

You should not ignore the importance of the above steps. The validity of your experiment and its results rely on a robust testable hypothesis. Developing a strong testable hypothesis has few advantages, it compels us to think intensely and specifically about the outcomes of a study. Consequently, it enables us to understand the implication of the question and the different variables involved in the study. Furthermore, it helps us to make precise predictions based on prior research. Hence, forming a hypothesis would be of great value to the research. Here are some good examples of testable hypotheses.

More importantly, you need to build a robust testable research hypothesis for your scientific experiments. A testable hypothesis is a hypothesis that can be proved or disproved as a result of experimentation.

Importance of a Testable Hypothesis

To devise and perform an experiment using scientific method, you need to make sure that your hypothesis is testable. To be considered testable, some essential criteria must be met:

• There must be a possibility to prove that the hypothesis is true.
• There must be a possibility to prove that the hypothesis is false.
• The results of the hypothesis must be reproducible.

Without these criteria, the hypothesis and the results will be vague. As a result, the experiment will not prove or disprove anything significant.

What are your experiences with building hypotheses for scientific experiments? What challenges did you face? How did you overcome these challenges? Please share your thoughts with us in the comments section.

Frequently Asked Questions

The steps to write a research hypothesis are: 1. Stating the problem: Ensure that the hypothesis defines the research problem 2. Writing a hypothesis as an 'if-then' statement: Include the action and the expected outcome of your study by following a ‘if-then’ structure. 3. Defining the variables: Define the variables as Dependent or Independent based on their dependency to other factors. 4. Scrutinizing the hypothesis: Identify the type of your hypothesis

Hypothesis testing is a statistical tool which is used to make inferences about a population data to draw conclusions for a particular hypothesis.

Hypothesis in statistics is a formal statement about the nature of a population within a structured framework of a statistical model. It is used to test an existing hypothesis by studying a population.

Research hypothesis is a statement that introduces a research question and proposes an expected result. It forms the basis of scientific experiments.

The different types of hypothesis in research are: • Null hypothesis: Null hypothesis is a negative statement to support the researcher’s findings that there is no relationship between two variables. • Alternate hypothesis: Alternate hypothesis predicts the relationship between the two variables of the study. • Directional hypothesis: Directional hypothesis specifies the expected direction to be followed to determine the relationship between variables. • Non-directional hypothesis: Non-directional hypothesis does not predict the exact direction or nature of the relationship between the two variables. • Simple hypothesis: Simple hypothesis predicts the relationship between a single dependent variable and a single independent variable. • Complex hypothesis: Complex hypothesis predicts the relationship between two or more independent and dependent variables. • Associative and casual hypothesis: Associative and casual hypothesis predicts the relationship between two or more independent and dependent variables. • Empirical hypothesis: Empirical hypothesis can be tested via experiments and observation. • Statistical hypothesis: A statistical hypothesis utilizes statistical models to draw conclusions about broader populations.

Wow! You really simplified your explanation that even dummies would find it easy to comprehend. Thank you so much.

Thanks a lot for your valuable guidance.

I enjoy reading the post. Hypotheses are actually an intrinsic part in a study. It bridges the research question and the methodology of the study.

Useful piece!

This is awesome.Wow.

It very interesting to read the topic, can you guide me any specific example of hypothesis process establish throw the Demand and supply of the specific product in market

Nicely explained

It is really a useful for me Kindly give some examples of hypothesis

It was a well explained content ,can you please give me an example with the null and alternative hypothesis illustrated

clear and concise. thanks.

So Good so Amazing

Good to learn

Thanks a lot for explaining to my level of understanding

Explained well and in simple terms. Quick read! Thank you

It awesome. It has really positioned me in my research project

Rate this article Cancel Reply

Your email address will not be published.

Enago Academy's Most Popular Articles

• Reporting Research

Choosing the Right Analytical Approach: Thematic analysis vs. content analysis for data interpretation

In research, choosing the right approach to understand data is crucial for deriving meaningful insights.…

Comparing Cross Sectional and Longitudinal Studies: 5 steps for choosing the right approach

The process of choosing the right research design can put ourselves at the crossroads of…

• Industry News

COPE Forum Discussion Highlights Challenges and Urges Clarity in Institutional Authorship Standards

The COPE forum discussion held in December 2023 initiated with a fundamental question — is…

• Career Corner

Unlocking the Power of Networking in Academic Conferences

Embarking on your first academic conference experience? Fear not, we got you covered! Academic conferences…

Research Recommendations – Guiding policy-makers for evidence-based decision making

Research recommendations play a crucial role in guiding scholars and researchers toward fruitful avenues of…

Choosing the Right Analytical Approach: Thematic analysis vs. content analysis for…

Comparing Cross Sectional and Longitudinal Studies: 5 steps for choosing the right…

How to Design Effective Research Questionnaires for Robust Findings

Sign-up to read more

Subscribe for free to get unrestricted access to all our resources on research writing and academic publishing including:

• 2000+ blog articles
• 50+ Webinars
• 10+ Expert podcasts
• 50+ Infographics
• 10+ Checklists
• Research Guides

We hate spam too. We promise to protect your privacy and never spam you.

I am looking for Editing/ Proofreading services for my manuscript Tentative date of next journal submission:

As a researcher, what do you consider most when choosing an image manipulation detector?

Holman Library

Research Guide: Scholarly Journals

• Introduction: Hypothesis/Thesis
• Why Use Scholarly Journals?
• What does "Peer-Reviewed" mean?
• What is *NOT* a Scholarly Journal Article?
• Interlibrary Loan for Journal Articles
• Reading the Citation
• Authors' Credentials
• Literature Review
• Methodology
• Results/Data
• Discussion/Conclusions
• APA Citations for Scholarly Journal Articles
• MLA Citations for Scholarly Journal Articles

Hypothesis or Thesis

Looking for the author's thesis or hypothesis.

The image below shows the part of the scholarly article that shows where the authors are making their argument. 

(click on image to enlarge)

• The first few paragraphs of a journal article serve to introduce the topic, to provide the author's hypothesis or thesis, and to indicate why the research was done.  
• A thesis or hypothesis is not always clearly labeled; you may need to read through the introductory paragraphs to determine what the authors are proposing.
• << Previous: How to Read a Scholarly Article
• Next: Reading the Citation >>
• Last Updated: May 4, 2024 2:04 PM
• URL: https://libguides.greenriver.edu/scholarlyjournals

Get science-backed answers as you write with Paperpal's Research feature

How to Write a Research Paper Introduction (with Examples)

The research paper introduction section, along with the Title and Abstract, can be considered the face of any research paper. The following article is intended to guide you in organizing and writing the research paper introduction for a quality academic article or dissertation.

The research paper introduction aims to present the topic to the reader. A study will only be accepted for publishing if you can ascertain that the available literature cannot answer your research question. So it is important to ensure that you have read important studies on that particular topic, especially those within the last five to ten years, and that they are properly referenced in this section. 1 What should be included in the research paper introduction is decided by what you want to tell readers about the reason behind the research and how you plan to fill the knowledge gap. The best research paper introduction provides a systemic review of existing work and demonstrates additional work that needs to be done. It needs to be brief, captivating, and well-referenced; a well-drafted research paper introduction will help the researcher win half the battle.

The introduction for a research paper is where you set up your topic and approach for the reader. It has several key goals:

• Present your research topic
• Capture reader interest
• Summarize existing research
• Position your own approach
• Define your specific research problem and problem statement
• Highlight the novelty and contributions of the study
• Give an overview of the paper’s structure

The research paper introduction can vary in size and structure depending on whether your paper presents the results of original empirical research or is a review paper. Some research paper introduction examples are only half a page while others are a few pages long. In many cases, the introduction will be shorter than all of the other sections of your paper; its length depends on the size of your paper as a whole.

• Break through writer’s block. Write your research paper introduction with Paperpal Copilot

Table of Contents

What is the introduction for a research paper, why is the introduction important in a research paper, craft a compelling introduction section with paperpal. try now, 1. introduce the research topic:, 2. determine a research niche:, 3. place your research within the research niche:, craft accurate research paper introductions with paperpal. start writing now, frequently asked questions on research paper introduction, key points to remember.

The introduction in a research paper is placed at the beginning to guide the reader from a broad subject area to the specific topic that your research addresses. They present the following information to the reader

• Scope: The topic covered in the research paper
• Context: Background of your topic
• Importance: Why your research matters in that particular area of research and the industry problem that can be targeted

The research paper introduction conveys a lot of information and can be considered an essential roadmap for the rest of your paper. A good introduction for a research paper is important for the following reasons:

• It stimulates your reader’s interest: A good introduction section can make your readers want to read your paper by capturing their interest. It informs the reader what they are going to learn and helps determine if the topic is of interest to them.
• It helps the reader understand the research background: Without a clear introduction, your readers may feel confused and even struggle when reading your paper. A good research paper introduction will prepare them for the in-depth research to come. It provides you the opportunity to engage with the readers and demonstrate your knowledge and authority on the specific topic.
• It explains why your research paper is worth reading: Your introduction can convey a lot of information to your readers. It introduces the topic, why the topic is important, and how you plan to proceed with your research.
• It helps guide the reader through the rest of the paper: The research paper introduction gives the reader a sense of the nature of the information that will support your arguments and the general organization of the paragraphs that will follow. It offers an overview of what to expect when reading the main body of your paper.

What are the parts of introduction in the research?

A good research paper introduction section should comprise three main elements: 2

• What is known: This sets the stage for your research. It informs the readers of what is known on the subject.
• What is lacking: This is aimed at justifying the reason for carrying out your research. This could involve investigating a new concept or method or building upon previous research.
• What you aim to do: This part briefly states the objectives of your research and its major contributions. Your detailed hypothesis will also form a part of this section.

How to write a research paper introduction?

The first step in writing the research paper introduction is to inform the reader what your topic is and why it’s interesting or important. This is generally accomplished with a strong opening statement. The second step involves establishing the kinds of research that have been done and ending with limitations or gaps in the research that you intend to address. Finally, the research paper introduction clarifies how your own research fits in and what problem it addresses. If your research involved testing hypotheses, these should be stated along with your research question. The hypothesis should be presented in the past tense since it will have been tested by the time you are writing the research paper introduction.

The following key points, with examples, can guide you when writing the research paper introduction section:

• Highlight the importance of the research field or topic
• Describe the background of the topic
• Present an overview of current research on the topic

Example: The inclusion of experiential and competency-based learning has benefitted electronics engineering education. Industry partnerships provide an excellent alternative for students wanting to engage in solving real-world challenges. Industry-academia participation has grown in recent years due to the need for skilled engineers with practical training and specialized expertise. However, from the educational perspective, many activities are needed to incorporate sustainable development goals into the university curricula and consolidate learning innovation in universities.

• Reveal a gap in existing research or oppose an existing assumption
• Formulate the research question

Example: There have been plausible efforts to integrate educational activities in higher education electronics engineering programs. However, very few studies have considered using educational research methods for performance evaluation of competency-based higher engineering education, with a focus on technical and or transversal skills. To remedy the current need for evaluating competencies in STEM fields and providing sustainable development goals in engineering education, in this study, a comparison was drawn between study groups without and with industry partners.

• State the purpose of your study
• Highlight the key characteristics of your study
• Describe important results
• Highlight the novelty of the study.
• Offer a brief overview of the structure of the paper.

Example: The study evaluates the main competency needed in the applied electronics course, which is a fundamental core subject for many electronics engineering undergraduate programs. We compared two groups, without and with an industrial partner, that offered real-world projects to solve during the semester. This comparison can help determine significant differences in both groups in terms of developing subject competency and achieving sustainable development goals.

Write a Research Paper Introduction in Minutes with Paperpal

Paperpal Copilot is a generative AI-powered academic writing assistant. It’s trained on millions of published scholarly articles and over 20 years of STM experience. Paperpal Copilot helps authors write better and faster with:

• Real-time writing suggestions
• In-depth checks for language and grammar correction
• Paraphrasing to add variety, ensure academic tone, and trim text to meet journal limits

With Paperpal Copilot, create a research paper introduction effortlessly. In this step-by-step guide, we’ll walk you through how Paperpal transforms your initial ideas into a polished and publication-ready introduction.

How to use Paperpal to write the Introduction section

Step 1: Sign up on Paperpal and click on the Copilot feature, under this choose Outlines > Research Article > Introduction

Step 2: Add your unstructured notes or initial draft, whether in English or another language, to Paperpal, which is to be used as the base for your content.

Step 3: Fill in the specifics, such as your field of study, brief description or details you want to include, which will help the AI generate the outline for your Introduction.

Step 4: Use this outline and sentence suggestions to develop your content, adding citations where needed and modifying it to align with your specific research focus.

Step 5: Turn to Paperpal’s granular language checks to refine your content, tailor it to reflect your personal writing style, and ensure it effectively conveys your message.

You can use the same process to develop each section of your article, and finally your research paper in half the time and without any of the stress.

The purpose of the research paper introduction is to introduce the reader to the problem definition, justify the need for the study, and describe the main theme of the study. The aim is to gain the reader’s attention by providing them with necessary background information and establishing the main purpose and direction of the research.

The length of the research paper introduction can vary across journals and disciplines. While there are no strict word limits for writing the research paper introduction, an ideal length would be one page, with a maximum of 400 words over 1-4 paragraphs. Generally, it is one of the shorter sections of the paper as the reader is assumed to have at least a reasonable knowledge about the topic. 2 For example, for a study evaluating the role of building design in ensuring fire safety, there is no need to discuss definitions and nature of fire in the introduction; you could start by commenting upon the existing practices for fire safety and how your study will add to the existing knowledge and practice.

When deciding what to include in the research paper introduction, the rest of the paper should also be considered. The aim is to introduce the reader smoothly to the topic and facilitate an easy read without much dependency on external sources. 3 Below is a list of elements you can include to prepare a research paper introduction outline and follow it when you are writing the research paper introduction. Topic introduction: This can include key definitions and a brief history of the topic. Research context and background: Offer the readers some general information and then narrow it down to specific aspects. Details of the research you conducted: A brief literature review can be included to support your arguments or line of thought. Rationale for the study: This establishes the relevance of your study and establishes its importance. Importance of your research: The main contributions are highlighted to help establish the novelty of your study Research hypothesis: Introduce your research question and propose an expected outcome. Organization of the paper: Include a short paragraph of 3-4 sentences that highlights your plan for the entire paper

Cite only works that are most relevant to your topic; as a general rule, you can include one to three. Note that readers want to see evidence of original thinking. So it is better to avoid using too many references as it does not leave much room for your personal standpoint to shine through. Citations in your research paper introduction support the key points, and the number of citations depend on the subject matter and the point discussed. If the research paper introduction is too long or overflowing with citations, it is better to cite a few review articles rather than the individual articles summarized in the review. A good point to remember when citing research papers in the introduction section is to include at least one-third of the references in the introduction.

The literature review plays a significant role in the research paper introduction section. A good literature review accomplishes the following: Introduces the topic – Establishes the study’s significance – Provides an overview of the relevant literature – Provides context for the study using literature – Identifies knowledge gaps However, remember to avoid making the following mistakes when writing a research paper introduction: Do not use studies from the literature review to aggressively support your research Avoid direct quoting Do not allow literature review to be the focus of this section. Instead, the literature review should only aid in setting a foundation for the manuscript.

Remember the following key points for writing a good research paper introduction: 4

• Avoid stuffing too much general information: Avoid including what an average reader would know and include only that information related to the problem being addressed in the research paper introduction. For example, when describing a comparative study of non-traditional methods for mechanical design optimization, information related to the traditional methods and differences between traditional and non-traditional methods would not be relevant. In this case, the introduction for the research paper should begin with the state-of-the-art non-traditional methods and methods to evaluate the efficiency of newly developed algorithms.
• Avoid packing too many references: Cite only the required works in your research paper introduction. The other works can be included in the discussion section to strengthen your findings.
• Avoid extensive criticism of previous studies: Avoid being overly critical of earlier studies while setting the rationale for your study. A better place for this would be the Discussion section, where you can highlight the advantages of your method.
• Avoid describing conclusions of the study: When writing a research paper introduction remember not to include the findings of your study. The aim is to let the readers know what question is being answered. The actual answer should only be given in the Results and Discussion section.

To summarize, the research paper introduction section should be brief yet informative. It should convince the reader the need to conduct the study and motivate him to read further. If you’re feeling stuck or unsure, choose trusted AI academic writing assistants like Paperpal to effortlessly craft your research paper introduction and other sections of your research article.

1. Jawaid, S. A., & Jawaid, M. (2019). How to write introduction and discussion. Saudi Journal of Anaesthesia, 13(Suppl 1), S18.

2. Dewan, P., & Gupta, P. (2016). Writing the title, abstract and introduction: Looks matter!. Indian pediatrics, 53, 235-241.

3. Cetin, S., & Hackam, D. J. (2005). An approach to the writing of a scientific Manuscript1. Journal of Surgical Research, 128(2), 165-167.

4. Bavdekar, S. B. (2015). Writing introduction: Laying the foundations of a research paper. Journal of the Association of Physicians of India, 63(7), 44-6.

Paperpal is a comprehensive AI writing toolkit that helps students and researchers achieve 2x the writing in half the time. It leverages 21+ years of STM experience and insights from millions of research articles to provide in-depth academic writing, language editing, and submission readiness support to help you write better, faster.  

Get accurate academic translations, rewriting support, grammar checks, vocabulary suggestions, and generative AI assistance that delivers human precision at machine speed. Try for free or upgrade to Paperpal Prime starting at US$19 a month to access premium features, including consistency, plagiarism, and 30+ submission readiness checks to help you succeed.  

Experience the future of academic writing – Sign up to Paperpal and start writing for free!  

Related Reads:

• Scientific Writing Style Guides Explained
• 5 Reasons for Rejection After Peer Review
• Ethical Research Practices For Research with Human Subjects
• 8 Most Effective Ways to Increase Motivation for Thesis Writing 

Practice vs. Practise: Learn the Difference

Academic paraphrasing: why paperpal’s rewrite should be your first choice , you may also like, how to write a high-quality conference paper, academic editing: how to self-edit academic text with..., measuring academic success: definition & strategies for excellence, phd qualifying exam: tips for success , ai in education: it’s time to change the..., is it ethical to use ai-generated abstracts without..., what are journal guidelines on using generative ai..., quillbot review: features, pricing, and free alternatives, what is an academic paper types and elements , should you use ai tools like chatgpt for....

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

• Advanced Search
• Journal List
• Turk J Urol
• v.39(Suppl 1); 2013 Sep

How to write an introduction section of a scientific article?

An article primarily includes the following sections: introduction, materials and methods, results, discussion, and conclusion. Before writing the introduction, the main steps, the heading and the familiarity level of the readers should be considered. Writing should begin when the experimental system and the equipment are available. The introduction section comprises the first portion of the manuscript, and it should be written using the simple present tense. Additionally, abbreviations and explanations are included in this section. The main goal of the introduction is to convey basic information to the readers without obligating them to investigate previous publications and to provide clues as to the results of the present study. To do this, the subject of the article should be thoroughly reviewed, and the aim of the study should be clearly stated immediately after discussing the basic references. In this review, we aim to convey the principles of writing the introduction section of a manuscript to residents and young investigators who have just begun to write a manuscript.

Introduction

When entering a gate of a magnificent city we can make a prediction about the splendor, pomposity, history, and civilization we will encounter in the city. Occasionally, gates do not give even a glimpse of the city, and it can mislead the visitors about inner sections of the city. Introduction sections of the articles are like gates of a city. It is a presentation aiming at introducing itself to the readers, and attracting their attention. Attractiveness, clarity, piquancy, and analytical capacity of the presentation will urge the reader to read the subsequent sections of the article. On the other hand as is understood from the motto of antique Greek poet Euripides “a bad beginning makes a bad ending”, ‘Introduction’ section of a scientific article is important in that it can reveal the conclusion of the article. [ 1 ]

It is useful to analyze the issues to be considered in the ‘Introduction’ section under 3 headings. Firstly, information should be provided about the general topic of the article in the light of the current literature which paves the way for the disclosure of the objective of the manuscript. Then the specific subject matter, and the issue to be focused on should be dealt with, the problem should be brought forth, and fundamental references related to the topic should be discussed. Finally, our recommendations for solution should be described, in other words our aim should be communicated. When these steps are followed in that order, the reader can track the problem, and its solution from his/her own perspective under the light of current literature. Otherwise, even a perfect study presented in a non-systematized, confused design will lose the chance of reading. Indeed inadequate information, inability to clarify the problem, and sometimes concealing the solution will keep the reader who has a desire to attain new information away from reading the manuscript. [ 1 – 3 ]

First of all, explanation of the topic in the light of the current literature should be made in clear, and precise terms as if the reader is completely ignorant of the subject. In this section, establishment of a warm rapport between the reader, and the manuscript is aimed. Since frantic plunging into the problem or the solution will push the reader into the dilemma of either screening the literature about the subject matter or refraining from reading the article. Updated, and robust information should be presented in the ‘Introduction’ section.

Then main topic of our manuscript, and the encountered problem should be analyzed in the light of the current literature following a short instance of brain exercise. At this point the problems should be reduced to one issue as far as possible. Of course, there might be more than one problem, however this new issue, and its solution should be the subject matter of another article. Problems should be expressed clearly. If targets are more numerous, and complex, solutions will be more than one, and confusing.

Finally, the last paragraphs of the ‘Introduction’ section should include the solution in which we will describe the information we generated, and related data. Our sentences which arouse curiosity in the readers should not be left unanswered. The reader who thinks to obtain the most effective information in no time while reading a scientific article should not be smothered with mysterious sentences, and word plays, and the readers should not be left alone to arrive at a conclusion by themselves. If we have contrary expectations, then we might write an article which won’t have any reader. A clearly expressed or recommended solutions to an explicitly revealed problem is also very important for the integrity of the ‘Introduction’ section. [ 1 – 5 ]

We can summarize our arguments with the following example ( Figure 1 ). The introduction section of the exemplary article is written in simple present tense which includes abbreviations, acronyms, and their explanations. Based on our statements above we can divide the introduction section into 3 parts. In the first paragraph, miniaturization, and evolvement of pediatric endourological instruments, and competitions among PNL, ESWL, and URS in the treatment of urinary system stone disease are described, in other words the background is prepared. In the second paragraph, a newly defined system which facilitates intrarenal access in PNL procedure has been described. Besides basic references related to the subject matter have been given, and their outcomes have been indicated. In other words, fundamental references concerning main subject have been discussed. In the last paragraph the aim of the researchers to investigate the outcomes, and safety of the application of this new method in the light of current information has been indicated.

An exemplary introduction section of an article

Apart from the abovementioned information about the introduction section of a scientific article we will summarize a few major issues in brief headings

Important points which one should take heed of:

• Abbreviations should be given following their explanations in the ‘Introduction’ section (their explanations in the summary does not count)
• Simple present tense should be used.
• References should be selected from updated publication with a higher impact factor, and prestigous source books.
• Avoid mysterious, and confounding expressions, construct clear sentences aiming at problematic issues, and their solutions.
• The sentences should be attractive, tempting, and comjprehensible.
• Firstly general, then subject-specific information should be given. Finally our aim should be clearly explained.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

9.1: Introduction to Hypothesis Testing

• Last updated
• Save as PDF
• Page ID 10211

• Kyle Siegrist
• University of Alabama in Huntsville via Random Services

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

Basic Theory

Preliminaries.

As usual, our starting point is a random experiment with an underlying sample space and a probability measure \(\P\). In the basic statistical model, we have an observable random variable \(\bs{X}\) taking values in a set \(S\). In general, \(\bs{X}\) can have quite a complicated structure. For example, if the experiment is to sample \(n\) objects from a population and record various measurements of interest, then \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] where \(X_i\) is the vector of measurements for the \(i\)th object. The most important special case occurs when \((X_1, X_2, \ldots, X_n)\) are independent and identically distributed. In this case, we have a random sample of size \(n\) from the common distribution.

The purpose of this section is to define and discuss the basic concepts of statistical hypothesis testing . Collectively, these concepts are sometimes referred to as the Neyman-Pearson framework, in honor of Jerzy Neyman and Egon Pearson, who first formalized them.

A statistical hypothesis is a statement about the distribution of \(\bs{X}\). Equivalently, a statistical hypothesis specifies a set of possible distributions of \(\bs{X}\): the set of distributions for which the statement is true. A hypothesis that specifies a single distribution for \(\bs{X}\) is called simple ; a hypothesis that specifies more than one distribution for \(\bs{X}\) is called composite .

In hypothesis testing , the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis . The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\).

An hypothesis test is a statistical decision ; the conclusion will either be to reject the null hypothesis in favor of the alternative, or to fail to reject the null hypothesis. The decision that we make must, of course, be based on the observed value \(\bs{x}\) of the data vector \(\bs{X}\). Thus, we will find an appropriate subset \(R\) of the sample space \(S\) and reject \(H_0\) if and only if \(\bs{x} \in R\). The set \(R\) is known as the rejection region or the critical region . Note the asymmetry between the null and alternative hypotheses. This asymmetry is due to the fact that we assume the null hypothesis, in a sense, and then see if there is sufficient evidence in \(\bs{x}\) to overturn this assumption in favor of the alternative.

An hypothesis test is a statistical analogy to proof by contradiction, in a sense. Suppose for a moment that \(H_1\) is a statement in a mathematical theory and that \(H_0\) is its negation. One way that we can prove \(H_1\) is to assume \(H_0\) and work our way logically to a contradiction. In an hypothesis test, we don't prove anything of course, but there are similarities. We assume \(H_0\) and then see if the data \(\bs{x}\) are sufficiently at odds with that assumption that we feel justified in rejecting \(H_0\) in favor of \(H_1\).

Often, the critical region is defined in terms of a statistic \(w(\bs{X})\), known as a test statistic , where \(w\) is a function from \(S\) into another set \(T\). We find an appropriate rejection region \(R_T \subseteq T\) and reject \(H_0\) when the observed value \(w(\bs{x}) \in R_T\). Thus, the rejection region in \(S\) is then \(R = w^{-1}(R_T) = \left\{\bs{x} \in S: w(\bs{x}) \in R_T\right\}\). As usual, the use of a statistic often allows significant data reduction when the dimension of the test statistic is much smaller than the dimension of the data vector.

The ultimate decision may be correct or may be in error. There are two types of errors, depending on which of the hypotheses is actually true.

Types of errors:

• A type 1 error is rejecting the null hypothesis \(H_0\) when \(H_0\) is true.
• A type 2 error is failing to reject the null hypothesis \(H_0\) when the alternative hypothesis \(H_1\) is true.

Similarly, there are two ways to make a correct decision: we could reject \(H_0\) when \(H_1\) is true or we could fail to reject \(H_0\) when \(H_0\) is true. The possibilities are summarized in the following table:

Of course, when we observe \(\bs{X} = \bs{x}\) and make our decision, either we will have made the correct decision or we will have committed an error, and usually we will never know which of these events has occurred. Prior to gathering the data, however, we can consider the probabilities of the various errors.

If \(H_0\) is true (that is, the distribution of \(\bs{X}\) is specified by \(H_0\)), then \(\P(\bs{X} \in R)\) is the probability of a type 1 error for this distribution. If \(H_0\) is composite, then \(H_0\) specifies a variety of different distributions for \(\bs{X}\) and thus there is a set of type 1 error probabilities.

The maximum probability of a type 1 error, over the set of distributions specified by \( H_0 \), is the significance level of the test or the size of the critical region.

The significance level is often denoted by \(\alpha\). Usually, the rejection region is constructed so that the significance level is a prescribed, small value (typically 0.1, 0.05, 0.01).

If \(H_1\) is true (that is, the distribution of \(\bs{X}\) is specified by \(H_1\)), then \(\P(\bs{X} \notin R)\) is the probability of a type 2 error for this distribution. Again, if \(H_1\) is composite then \(H_1\) specifies a variety of different distributions for \(\bs{X}\), and thus there will be a set of type 2 error probabilities. Generally, there is a tradeoff between the type 1 and type 2 error probabilities. If we reduce the probability of a type 1 error, by making the rejection region \(R\) smaller, we necessarily increase the probability of a type 2 error because the complementary region \(S \setminus R\) is larger.

The extreme cases can give us some insight. First consider the decision rule in which we never reject \(H_0\), regardless of the evidence \(\bs{x}\). This corresponds to the rejection region \(R = \emptyset\). A type 1 error is impossible, so the significance level is 0. On the other hand, the probability of a type 2 error is 1 for any distribution defined by \(H_1\). At the other extreme, consider the decision rule in which we always rejects \(H_0\) regardless of the evidence \(\bs{x}\). This corresponds to the rejection region \(R = S\). A type 2 error is impossible, but now the probability of a type 1 error is 1 for any distribution defined by \(H_0\). In between these two worthless tests are meaningful tests that take the evidence \(\bs{x}\) into account.

If \(H_1\) is true, so that the distribution of \(\bs{X}\) is specified by \(H_1\), then \(\P(\bs{X} \in R)\), the probability of rejecting \(H_0\) is the power of the test for that distribution.

Thus the power of the test for a distribution specified by \( H_1 \) is the probability of making the correct decision.

Suppose that we have two tests, corresponding to rejection regions \(R_1\) and \(R_2\), respectively, each having significance level \(\alpha\). The test with region \(R_1\) is uniformly more powerful than the test with region \(R_2\) if \[ \P(\bs{X} \in R_1) \ge \P(\bs{X} \in R_2) \text{ for every distribution of } \bs{X} \text{ specified by } H_1 \]

Naturally, in this case, we would prefer the first test. Often, however, two tests will not be uniformly ordered; one test will be more powerful for some distributions specified by \(H_1\) while the other test will be more powerful for other distributions specified by \(H_1\).

If a test has significance level \(\alpha\) and is uniformly more powerful than any other test with significance level \(\alpha\), then the test is said to be a uniformly most powerful test at level \(\alpha\).

Clearly a uniformly most powerful test is the best we can do.

\(P\)-value

In most cases, we have a general procedure that allows us to construct a test (that is, a rejection region \(R_\alpha\)) for any given significance level \(\alpha \in (0, 1)\). Typically, \(R_\alpha\) decreases (in the subset sense) as \(\alpha\) decreases.

The \(P\)-value of the observed value \(\bs{x}\) of \(\bs{X}\), denoted \(P(\bs{x})\), is defined to be the smallest \(\alpha\) for which \(\bs{x} \in R_\alpha\); that is, the smallest significance level for which \(H_0\) is rejected, given \(\bs{X} = \bs{x}\).

Knowing \(P(\bs{x})\) allows us to test \(H_0\) at any significance level for the given data \(\bs{x}\): If \(P(\bs{x}) \le \alpha\) then we would reject \(H_0\) at significance level \(\alpha\); if \(P(\bs{x}) \gt \alpha\) then we fail to reject \(H_0\) at significance level \(\alpha\). Note that \(P(\bs{X})\) is a statistic . Informally, \(P(\bs{x})\) can often be thought of as the probability of an outcome as or more extreme than the observed value \(\bs{x}\), where extreme is interpreted relative to the null hypothesis \(H_0\).

Analogy with Justice Systems

There is a helpful analogy between statistical hypothesis testing and the criminal justice system in the US and various other countries. Consider a person charged with a crime. The presumed null hypothesis is that the person is innocent of the crime; the conjectured alternative hypothesis is that the person is guilty of the crime. The test of the hypotheses is a trial with evidence presented by both sides playing the role of the data. After considering the evidence, the jury delivers the decision as either not guilty or guilty . Note that innocent is not a possible verdict of the jury, because it is not the point of the trial to prove the person innocent. Rather, the point of the trial is to see whether there is sufficient evidence to overturn the null hypothesis that the person is innocent in favor of the alternative hypothesis of that the person is guilty. A type 1 error is convicting a person who is innocent; a type 2 error is acquitting a person who is guilty. Generally, a type 1 error is considered the more serious of the two possible errors, so in an attempt to hold the chance of a type 1 error to a very low level, the standard for conviction in serious criminal cases is beyond a reasonable doubt .

Tests of an Unknown Parameter

Hypothesis testing is a very general concept, but an important special class occurs when the distribution of the data variable \(\bs{X}\) depends on a parameter \(\theta\) taking values in a parameter space \(\Theta\). The parameter may be vector-valued, so that \(\bs{\theta} = (\theta_1, \theta_2, \ldots, \theta_n)\) and \(\Theta \subseteq \R^k\) for some \(k \in \N_+\). The hypotheses generally take the form \[ H_0: \theta \in \Theta_0 \text{ versus } H_1: \theta \notin \Theta_0 \] where \(\Theta_0\) is a prescribed subset of the parameter space \(\Theta\). In this setting, the probabilities of making an error or a correct decision depend on the true value of \(\theta\). If \(R\) is the rejection region, then the power function \( Q \) is given by \[ Q(\theta) = \P_\theta(\bs{X} \in R), \quad \theta \in \Theta \] The power function gives a lot of information about the test.

The power function satisfies the following properties:

• \(Q(\theta)\) is the probability of a type 1 error when \(\theta \in \Theta_0\).
• \(\max\left\{Q(\theta): \theta \in \Theta_0\right\}\) is the significance level of the test.
• \(1 - Q(\theta)\) is the probability of a type 2 error when \(\theta \notin \Theta_0\).
• \(Q(\theta)\) is the power of the test when \(\theta \notin \Theta_0\).

If we have two tests, we can compare them by means of their power functions.

Suppose that we have two tests, corresponding to rejection regions \(R_1\) and \(R_2\), respectively, each having significance level \(\alpha\). The test with rejection region \(R_1\) is uniformly more powerful than the test with rejection region \(R_2\) if \( Q_1(\theta) \ge Q_2(\theta)\) for all \( \theta \notin \Theta_0 \).

Most hypothesis tests of an unknown real parameter \(\theta\) fall into three special cases:

Suppose that \( \theta \) is a real parameter and \( \theta_0 \in \Theta \) a specified value. The tests below are respectively the two-sided test , the left-tailed test , and the right-tailed test .

• \(H_0: \theta = \theta_0\) versus \(H_1: \theta \ne \theta_0\)
• \(H_0: \theta \ge \theta_0\) versus \(H_1: \theta \lt \theta_0\)
• \(H_0: \theta \le \theta_0\) versus \(H_1: \theta \gt \theta_0\)

Thus the tests are named after the conjectured alternative. Of course, there may be other unknown parameters besides \(\theta\) (known as nuisance parameters ).

Equivalence Between Hypothesis Test and Confidence Sets

There is an equivalence between hypothesis tests and confidence sets for a parameter \(\theta\).

Suppose that \(C(\bs{x})\) is a \(1 - \alpha\) level confidence set for \(\theta\). The following test has significance level \(\alpha\) for the hypothesis \( H_0: \theta = \theta_0 \) versus \( H_1: \theta \ne \theta_0 \): Reject \(H_0\) if and only if \(\theta_0 \notin C(\bs{x})\)

By definition, \(\P[\theta \in C(\bs{X})] = 1 - \alpha\). Hence if \(H_0\) is true so that \(\theta = \theta_0\), then the probability of a type 1 error is \(P[\theta \notin C(\bs{X})] = \alpha\).

Equivalently, we fail to reject \(H_0\) at significance level \(\alpha\) if and only if \(\theta_0\) is in the corresponding \(1 - \alpha\) level confidence set. In particular, this equivalence applies to interval estimates of a real parameter \(\theta\) and the common tests for \(\theta\) given above .

In each case below, the confidence interval has confidence level \(1 - \alpha\) and the test has significance level \(\alpha\).

• Suppose that \(\left[L(\bs{X}, U(\bs{X})\right]\) is a two-sided confidence interval for \(\theta\). Reject \(H_0: \theta = \theta_0\) versus \(H_1: \theta \ne \theta_0\) if and only if \(\theta_0 \lt L(\bs{X})\) or \(\theta_0 \gt U(\bs{X})\).
• Suppose that \(L(\bs{X})\) is a confidence lower bound for \(\theta\). Reject \(H_0: \theta \le \theta_0\) versus \(H_1: \theta \gt \theta_0\) if and only if \(\theta_0 \lt L(\bs{X})\).
• Suppose that \(U(\bs{X})\) is a confidence upper bound for \(\theta\). Reject \(H_0: \theta \ge \theta_0\) versus \(H_1: \theta \lt \theta_0\) if and only if \(\theta_0 \gt U(\bs{X})\).

Pivot Variables and Test Statistics

Recall that confidence sets of an unknown parameter \(\theta\) are often constructed through a pivot variable , that is, a random variable \(W(\bs{X}, \theta)\) that depends on the data vector \(\bs{X}\) and the parameter \(\theta\), but whose distribution does not depend on \(\theta\) and is known. In this case, a natural test statistic for the basic tests given above is \(W(\bs{X}, \theta_0)\).

Purdue Online Writing Lab Purdue OWL® College of Liberal Arts

Writing the Experimental Report: Overview, Introductions, and Literature Reviews

Welcome to the Purdue OWL

This page is brought to you by the OWL at Purdue University. When printing this page, you must include the entire legal notice.

Copyright ©1995-2018 by The Writing Lab & The OWL at Purdue and Purdue University. All rights reserved. This material may not be published, reproduced, broadcast, rewritten, or redistributed without permission. Use of this site constitutes acceptance of our terms and conditions of fair use.

Experimental reports (also known as "lab reports") are reports of empirical research conducted by their authors. You should think of an experimental report as a "story" of your research in which you lead your readers through your experiment. As you are telling this story, you are crafting an argument about both the validity and reliability of your research, what your results mean, and how they fit into other previous work.

These next two sections provide an overview of the experimental report in APA format. Always check with your instructor, advisor, or journal editor for specific formatting guidelines.

General-specific-general format

Experimental reports follow a general to specific to general pattern. Your report will start off broadly in your introduction and discussion of the literature; the report narrows as it leads up to your specific hypotheses, methods, and results. Your discussion transitions from talking about your specific results to more general ramifications, future work, and trends relating to your research.

Experimental reports in APA format have a title page. Title page formatting is as follows:

• A running head and page number in the upper right corner (right aligned)
• A definition of running head in IN ALL CAPS below the running head (left aligned)
• Vertically and horizontally centered paper title, followed by author and affiliation

Please see our sample APA title page .

Crafting your story

Before you begin to write, carefully consider your purpose in writing: what is it that you discovered, would like to share, or would like to argue? You can see report writing as crafting a story about your research and your findings. Consider the following.

• What is the story you would like to tell?
• What literature best speaks to that story?
• How do your results tell the story?
• How can you discuss the story in broad terms?

During each section of your paper, you should be focusing on your story. Consider how each sentence, each paragraph, and each section contributes to your overall purpose in writing. Here is a description of one student's process.

Briel is writing an experimental report on her results from her experimental psychology lab class. She was interested in looking at the role gender plays in persuading individuals to take financial risks. After her data analysis, she finds that men are more easily persuaded by women to take financial risks and that men are generally willing to take more financial risks.

When Briel begins to write, she focuses her introduction on financial risk taking and gender, focusing on male behaviors. She then presents relevant literature on financial risk taking and gender that help illuminate her own study, but also help demonstrate the need for her own work. Her introduction ends with a study overview that directly leads from the literature review. Because she has already broadly introduced her study through her introduction and literature review, her readers can anticipate where she is going when she gets to her study overview. Her methods and results continue that story. Finally, her discussion concludes that story, discussing her findings, implications of her work, and the need for more research in the area of gender and financial risk taking.

The abstract gives a concise summary of the contents of the report.

• Abstracts should be brief (about 100 words)
• Abstracts should be self-contained and provide a complete picture of what the study is about
• Abstracts should be organized just like your experimental report—introduction, literature review, methods, results and discussion
• Abstracts should be written last during your drafting stage

Introduction

The introduction in an experimental article should follow a general to specific pattern, where you first introduce the problem generally and then provide a short overview of your own study. The introduction includes three parts: opening statements, literature review, and study overview.

Opening statements: Define the problem broadly in plain English and then lead into the literature review (this is the "general" part of the introduction). Your opening statements should already be setting the stage for the story you are going to tell.

Literature review: Discusses literature (previous studies) relevant to your current study in a concise manner. Keep your story in mind as you organize your lit review and as you choose what literature to include. The following are tips when writing your literature review.

• You should discuss studies that are directly related to your problem at hand and that logically lead to your own hypotheses.
• You do not need to provide a complete historical overview nor provide literature that is peripheral to your own study.
• Studies should be presented based on themes or concepts relevant to your research, not in a chronological format.
• You should also consider what gap in the literature your own research fills. What hasn't been examined? What does your work do that others have not?

Study overview: The literature review should lead directly into the last section of the introduction—your study overview. Your short overview should provide your hypotheses and briefly describe your method. The study overview functions as a transition to your methods section.

You should always give good, descriptive names to your hypotheses that you use consistently throughout your study. When you number hypotheses, readers must go back to your introduction to find them, which makes your piece more difficult to read. Using descriptive names reminds readers what your hypotheses were and allows for better overall flow.

In our example above, Briel had three different hypotheses based on previous literature. Her first hypothesis, the "masculine risk-taking hypothesis" was that men would be more willing to take financial risks overall. She clearly named her hypothesis in the study overview, and then referred back to it in her results and discussion sections.

Thais and Sanford (2000) recommend the following organization for introductions.

• Provide an introduction to your topic
• Provide a very concise overview of the literature
• State your hypotheses and how they connect to the literature
• Provide an overview of the methods for investigation used in your research

Bem (2006) provides the following rules of thumb for writing introductions.

• Write in plain English
• Take the time and space to introduce readers to your problem step-by-step; do not plunge them into the middle of the problem without an introduction
• Use examples to illustrate difficult or unfamiliar theories or concepts. The more complicated the concept or theory, the more important it is to have clear examples
• Open with a discussion about people and their behavior, not about psychologists and their research

Phosphorus Environmental Risk Assessment in Wetland Soil

• Original research article
• Published: 17 May 2024
• Volume 44 , article number  58 , ( 2024 )

Cite this article

• Ana Paula Marés Mikosik   ORCID: orcid.org/0000-0002-5486-8185 1 ,
• Nerilde Favaretto 1 ,
• Antonio Carlos Vargas Motta 1 ,
• Vander de Freitas Melo 1 ,
• Fabiane Machado Vezzani 1 ,
• Jairo Calderari de Oliveira Júnior 1 &
• Verediana Fernanda Cherobim 1  

At the interface between agricultural fields and water bodies there are wetlands constituted by hydromorphic soils. Our hypothesis is that hydromorphic soil acts a P sink and the P buffer capacity increases over time. To test our hypothesis, we apply tools to evaluate the P environmental risk via i) maximum phosphorus adsorption capacity (PMAC); ii) environmental soil phosphorus thresholds (P-threshold); iii) soil phosphorus storage capacity (SPSC) in hydromorphic soil (Histosol) and a non-hydromorphic soil (Cambisol) under application of mineral P. The PMAC was estimated by the Langmuir model in soil samples collected at four depths (0–10, 10–20, 20–40 and 40–60 cm). Soil samples were incubated for 30, 60 and 120 days with mineral P equivalent to 0, 25, 50, 75 and 100% of the PMAC. The P-threshold was determined from the degree of phosphorus saturation (DPS), estimated using PMAC and Fe + Al extracted with Mehlich-1. The SPSC was obtained from the 0–60 cm depth using the DPS% M1(CMAP) . The PMAC values ranged from 2321 to 3539 mg kg −1 with higher values in the Histosol compared to the Cambisol. The Histosol presented a P-threshold of 19% DPS (609 mg kg −1 ), while in the Cambisol it was 24% DPS (582 mg kg −1 of P Mehlich-1). According to the SPSC tool, the soil acted as a source of P when P Mehlich-1 exceeded 887 mg kg −1 in Histosol, while in Cambisol it was 773 mg kg −1 . Overall, the P buffering capacity was higher in the Histosol, indicating the importance of preserving wetlands for water quality.

Graphical Abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Abboud FY, Favaretto N, Motta ACV, Barth G, Goularte GD (2018) Phosphorus mobility and degree of saturation in oxisol under no-tillage after long-term dairy liquid manure application. Soil Tillage Res 177:45–53

Article   Google Scholar  

Albuquerque CG, Gavelaki F, Melo VF, Motta ACV, Zarbin AJG, Ferreira CM (2022) Model of inner-sphere adsorption of oxyanions in goethite – why is phosphate adsorption more significant than that of sulfate? Rev Bras Cienc Solo 46:e0210146

Alva AK, Larsen S, Bille SW (1980) The influence of rhizosphere in rice crop on resin-extractable phosphate in flooded soils at various levels of phosphate aplications. Plant Soil 56:17–25

Article   CAS   Google Scholar  

Alvares CA, Stape JL, Sentelhas PC, Moraes Gonçalves JL, Sparovek G (2013) Köppen’s climate classification map for Brazil. Meteorol Z 22:711–728

Alvarez VVH, Fonseca DM (1990) Definição de doses de fósforo para determinação de capacidade máxima de adsorção de fosfatos e para ensaios de casa de vegetação. Revista Brasileira de Ciência do Solo, Campinas 14:49–55

Google Scholar  

Alvarez VVH, Novais RF, Dias LE, Oliveira JA (2000) Determinação e uso do fósforo remanescente. Boletim Informativo Sociedade Brasileira De Ciência Do Solo 25:27–32

Akpan US, Nkanga NA (2017) Phosphate Sorption Capacity, Bonding Energy and Buffering Mechanisms of Wetland Soils in Akwa Ibom State Nigeria. Greener Journal of Agricultural Sciences 7:065–073

Bajwa MI (1982) Soil Clay Mineralogies in Relation to Fertility Management: Effect of Clay Mineral Types on Ammonium Fixation under Conditions of Wetland Rice Culture. Agronomy Journal 74:143–144

Barbier EB, Acreman M, Knowler D (1997) Valoración económica de los humedales. Guía para decisores y planificadores. Oficina de la Convención de Ramsar, Gland, Suiza, p 150

Barbosa J, Poggere G, Mancini M, Silva SHG, Motta ACV, Marques J, Sáe Melo JJG, Curi N (2022) National-scale spatial variations of soil phosphorus retention capacity in Brazil. Physics and Chemistry of the Earth 128:103271

Bedrock CN, Cheshire MV, Shand CA (1997) The involvement of iron and aluminum in the bonding of phosphorus to soil humic acid. Communications in Soil Science and Plant Analysis 28(11–12):961–971

Bloom PR (1981) Phosphorus Adsorption by an Aluminum-Peat Complex. Soil Science Society of America Journal 45:267–272

Bortolon L, Ernani PR, Bortolon ESO, Gianello C, Almeida RGO, Welter S, Rogeri DA (2016) Degree of phosphorus saturation threshold in soils from Southern Brazil. Pesquisa Agropecuária Brasileira [s. l.] 51(9):1088–1098

BRASIL. Decreto nº 86.146, de 23 de Junho de 1981. Dispõe sobre a criação do Programa Nacional para Aproveitamento de Várzeas Irrigáveis - PROVÁRZEAS NACIONAL. Brasília, 23 de junho de 1981

Breeuwsma A, Silva S (1992) Phosphorus fertilization and environmental effects in The Netherlands and the Po Region (Italy). In: Report 57, agricultural research department, the winand staring centre for integrated land, soil, and water research, Wageningen, p 39

Diegues ACS (2002) Povos e águas: inventário de áreas úmidas brasileiras. São Paulo, NUPAUB 1:597

Dunne EJ, Reddy R, Clark MW (2006) Biogeochemical indices of phosphorus retention and release by wetland soils and adjacent stream sediments. Wetlands 26(4):1026–1041

EMBRAPA (2011) Centro Nacional de Pesquisa de Solos. Manual de métodos de análise de solo. 2 rd edn. Rio de Janeiro: Embrapa Solos, p. 230

Eriksson AK, Gustafsson JP, Hesterberg D (2015) Phosphorus speciation of clay fractions from long-term fertility experiments in Sweden. Geoderma 241–242:68–74

Ferreira DF (2019) SISVAR: A Computer Analysis system to fixed effects split plot type designs. Revista Brasileira De Biometria 37(4):529

Gatiboni LC, Nicoloso RS, Mumbach GL, Souza AA, Dall’Orsoletta DJ, Schmitt DE, Smyth TJ (2020) Establishing environmental soil phosphorus thresholds to decrease the risk of losses to water in soils from Rio Grande do Sul, Brazil. Revista Brasileira de Ciência do Solo 44:1–14

Gatiboni LC, Smyth TJ, Schmitt DE, Cassol PC, Oliveira CMB (2015) Soil phosphorus thresholds in evaluating risk of environmental transfer to surface waters in Santa Catarina Brazil. Revista Brasileira de Ciência do Solo 39(4):1225–1234

Gerke J (2010) Humic (organic matter)-Al (Fe)-phosphate complexes: an underestimated phosphate form in soils and source of plant-available phosphate. Soil Science 175(9):417–425

Guilherme LRG, Curi N, Silva MLN, Renó NB, Machado RAF (2000) Adsorção de fósforo em solos de várzea do estado de Minas Gerais. Revista Brasileira de Ciência do Solo Viçosa 24:27–34

Guppy CN, Menzies NW, Moody PW, Blamey FPC (2005) Competitive sorption reactions between phosphorus and organic matter in soil: a review. Australian Journal of Soil Research 43:189–202

Hargrove WL, Thomas GW (1982) Conditional formation constants for aluminumorganic matter complexes. Canadian Journal of Soil Science 62:571–575

Hernández J, Meurer EJ (1998) Adsorção de fósforo e sua relação com formas de ferro em dez solos do Uruguai. Revista Brasileira de Ciência do Solo 22:223–230

Hue NV, Craddock CR, Adams F (1986) Effect of organic acids on aluminium toxicity in subsoils. Soil Science Society of America Journal 50:28–34

Janardhanan L, Daroub S (2010) Phosphorus sorption in organic soils in South Florida. Soil Sci Soc Am J 74:1597–1606

Juo ASR, Fox RL (1977) Phosphate sorption capacity of some benchmark soils in West Africa. Soil Science 124:370–376

Kay P, Edwards AC, Foulger M (2009) A review of the efficacy of contemporary agricultural stewardship measures for ameliorating water pollution problems of key concern to the UK water industry. Agricultural Systems 99:67–75

Larsen JE, Warren GF, Langston R (1959) Effect of iron, aluminum and humic acid on phosphorus fixation by organic soils. Soil Science Society of America Proceedings 23:438–440

Litaor M, Reichmann O, Belzer M, Auerswald K, Nishri A, Shenker M (2003) Spatial Analysis of Phosphorus Sorption Capacity in a Semiarid Altered Wetland. Journal of Environmental Quality 32:335–343

Article   CAS   PubMed   Google Scholar  

Maack R (2012) Geografia Física do estado do Paraná. 4 ed. Curitiba: Imprensa Oficial, p 526

Mcdowell RW, Sharpley AN, Brookes PC, Poulton PR (2001) Relationship between soil test phosphorus and phosphorus release to solution. Soil Science 166:137–149

Mcdowell RW, Sharpley AN, Withers P (2002) Indicator to Predict the Movement of Phosphorus from Soil to Subsurface Flow. Environmental Science & Technology 36:1505–1509

Mckeague JA (1978) Manual on Soil Sampling and Methods of Analysis, 2nd edn. Canadian Society of Soil Science, Ottawa, p 212

Mehra OP, Jackson ML (1960) Iron oxide removal from soils and clay by a dithionite-citrate system buffered with sodium bicarbonate. Clays Clay Mine 7:317–327

Mesquita Filho MV, Torrent J (1993) Phosphate sorption as related to mineralogy of a hydrosequence of soils from the Cerrado region (Brazil). Geoderma 58:107–123

Mitsch WJ, Gosselink JG (1993) Wetlands. 2nd ed. John Wiley and Sons, New York, p 722

Mng’ong’o M, Munishi L, Blake W, Ndakidemi P, Comber S, Hutchinson T (2021) Characterization of soil phosphate status, sorption and saturation in paddy wetlands in usangu basin-Tanzania. Chemosphere 278:130466

Article   PubMed   Google Scholar  

Mukherjee A, Nair V, Clark M (2009) Reddy K (2009) Development of Indices to Predict Phosphorus Release from Wetland Soils. Journal of Environmental Quality 38:878–886

Nair V (2014) Soil phosphorus saturation ratio for risk assessment in land use systems. Front Environ Sci 2:1–4

Nair VD, Clark M, Reddy KR (2015) Evaluation of Legacy Phosphorus Storage and Release from Wetland Soils. J Environmental Quality 44(6):1956–1964

Nair VD, Harris EG (2004) A capacity factor as an alternative to soil test phosphorus in phosphorus risk assessment. New Zealand Journal of Agricultural Research 47:491–497

Nair VD, Portier KM, Graetz DA, Walker ML (2004) An Environmental Threshold for Degree of Phosphorus Saturation in Sandy Soils. Journal of Environmental Quality 33:107–113

Nair VD, Reddy KR (2013) Phosphorus sorption and desorption in wetland soils. In: Delaune R, Reddy KR, Richardson CJ, Megonigal P. Methods in Biogeochemistry of Wetlands. Madison: Soil Science Society of America Publication, 667–678

Novak JM, Watts DW (2006) Phosphorus sorption by sediments in a southeastern Coastal Plain in-stream wetland. Journal of Environmental Quality 35:1975–1982

Ohno T, Hoskins B, Erich M (2007) Soil organic matter effects on plant available and water-soluble phosphorus. Biology and Fertility of Soils 43:683–690

Olsen SR, Watanabe FS (1957) A method to determine a phosphorus adsorption maximum of soil as measured by the Langmuir isotherm. Soil Science Society of American Proceedings, Madison 21:144–149

PARANÁ. Lei Nº 18295 de 10/11/2014. Súmula Instituição, nos termos do art. 24 da Constituição Federal, do Programa de Regularização Ambiental das propriedades e imóveis rurais, criado pela Lei Federal nº 12.651, de 25 de maio de 2012. Palácio do Governo, em 10 de novembro de 2014

PARANÁ (2008) Resolução Conjunta IBAMA/SEMA/IAP N° 005, de 28 de março de 2008. Casa Civil Governadoria do Poder Executivo do Estado do Paraná. Curitiba, PR

Pautler MC, Sims JT (2000) Relationships Between Soil Test Phosphorus, Soluble Phosphorus, and Phosphorus Saturation in Delaware Soils. Soil Science Society of America Journal [s. l.] 64(2):765

Pavan MA, Bloch, MF, Zempulski HC, Miyazawa M, Zocoler DC (1992) Manual de análise química de solo e controle de qualidade. Londrina. IAPAR, circular 76:38

Penn CJ, Mullins GL, Zelazny LW (2005) Mineralogy in relation to phosphorus sorption and dissolved phosphorus losses in runoff. Soil Science Society of America Journal 69:1532–1540

Raij BV, Andrade JC, Cantarella H, Quaggio JA (2001) Análise química para avaliação da fertilidade de solos tropicais. Instituto Agronômico de Campinas (IAC), Campinas

Raij BV, Quaggio JA, Silva NM (1986) Extraction of phosphorus, potassium, calcium, and magnesium from soils by an ion-exchange resin procedure. Communications in Soil Science and Plant Analysis 17(5):547–566

R CORE TEAM (2020) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/ . Accessed 16 May 2024

Ranno SK, Silva LS, Gatiboni LC, Rhoden AC (2007) Capacidade de adsorção de fósforo em solos de várzea do Estado do Rio Grande do Sul. Revista Brasileira de Ciência do Solo 31(1):21–28

Reddy KR, Flaig E, Graetz DA (1996) Phosphorus storage capacity of uplands, wetlands and streams of the Lake Okeechobee Watershed Florida. Agriculture, Ecosystems & Environment 59(3):203–216

Reddy KR, Kadlec RH, Flaig E, Gale PM (1999) Phosphorus retention in streams and wetlands: a review. Critical Reviews in Environmental Science and Technology 29:83–146

Reddy KR, Orlando AD, Leonard JS, Moshe A (1995) Phosphorus dynamics in selected wetlands and streams of the lake Okeechobee Basin. Ecological Engineering 5:183–207

Rhoades JD (1996) Salinity: Electrical conductivity and total dissolved solids. In: Sparks DL, Page AL, Helmeke PA, Leoppert RH, Soltanpour PN, Tabatabai MA, Johnston CT, Sumner ME (Eds.). Methods of soil analysis. Part 3. SSSA Book Ser. 5. SSSA, Madison, WI, p.417–435

Richardson CJ (1985) Mechanisms controlling phosphorus retention capacity in freshwater wetlands. Science 228:1424–1427

Richardson CJ, Marshall PE (1986) Processes controlling movement, storage, and export of phosphorus in a fen peatland. Ecological Monographs 56:279–302

Roderjan CV, Galvão F, Kuniyoshi YS, Hatschbach GG (2002) As unidades fitogeográficas do estado do Paraná Brasil. Ciência e Ambiente Santa Maria 24:75–92

Sanyal SK, De Datta SK (1991) Chemistry of Phosphorus Transformations in Soil. Advances in Soil Science 6:1–120

Sekhon BS, Bhumbla DK, Sencindiver J, Mcdonald LM (2014) Using soil survey data for series-level environmental phosphorus risk assessment. Environmental Earth Sciences 72:2345–2356

Sharpley A (1995) Dependence of runoff phosphorus on extractable soil phosphorus. Journal of Environmental Quality 24:920–926

Sharpley A, Wang X (2014) Managing agricultural phosphorus for water quality: Lessons from the USA and China. Journal of Environmental Sciences 16:1770–1782

Simonete MA, Ernani PR, Teixeira-Gandra CFA, Moro L (2018) Phosphorus adsorption in lowlands of Santa Catarina cultivated with rice and its relation with soil properties. Revista Ciência Agronômica 49:566–573

Sims JT (1993) Environmental Soil Testing for Phosphorus. Journal of Production Agriculture 6:501–507

Smith VH, Tilman GD, Nekola JC (1999) Eutrophication: impacts of excess nutrient inputs on freshwater, marine, and terrestrial ecosystems. Environmental Pollution 100:179–196

Sposito G (2008) The surface chemistry of soils, 2nd edn. Oxford. University Press, New York

Stevenson FJ (1986) Cycles of Soil: C, N, P, S, Micronutrients. John Wiley and Sons, New York

Valadares SV, Alvarez VH, Santos WO, Paes JLAG, Lins CB, Novais RF (2017) Sensitivity of soil P availability tests to ca-P in oxisols. Communications in Soil Science and Plant Analysis 48(15):1834–1842

Van Der Zee SEATM, Fokkink LGJ, Van Riemsdijk WH (1987) A new technique for assessment of reversibly adsorbed phosphate. Soil Science Society of America Journal [s. l] 51(3):599–604

Verhoeven JTA, Arheimer B, Yin C, Hefting MM (2006) Regional and global concerns over wetlands and water quality. Trends in Ecology & Evolution 21(2):96–103

Vepraskas MJ, Caldwell PV (2008) Interpreting morphological features in wetland soils with a hydrologic model. Catena 73:153–165

Villapando RR, Graetz DA (2001) Phosphorus Sorption and Desorption Properties of the Spodic Horizon from Selected Florida Spodosols. Soil Science Society of America Journal 65:331–339

Wang X, Hu Y, Tang Y, Yang P, Feng X, Xu W, Zhu M (2017) Phosphate and phytate adsorption and precipitation on ferrihydrite surfaces. Environmental Science: Nano Journal 4:2193–2204

CAS   Google Scholar  

Wang L, Liang T (2014) Effects of exogenous rare earth elements on phosphorus adsorption and desorption in different types of soils. Chemosphere 103:148–155

Wang X, Phillips BL, Boily J-F, Hu Y, Hu Z, Yang P, Feng X, Xu W, Zhu M (2019) Phosphate Sorption Speciation and Precipitation Mechanisms on Amorphous Aluminum Hydroxide. Soil Systems 3:20

Xu G, Ren Y, Yue M, Lv Y, Chen X, Hui H (2022) Phosphorus sorption capacity in soils from freshwater restored coastal wetlands increased with restoration age. Geoderma 422:115926

Xu T, Weng B, Yan D, Wang K, Li X, Bi W, Li M, Cheng X, Liu Y (2019) Wetlands of international importance: Status, threats, and future protection. International Journal of Environmental Research and Public Health, 16(10):1818

Yang X, Chen X, Yang X (2019) Effect of organic matter on phosphorus adsorption and desorption in a black soil from Northeast China. Soil and Tillage Research 187:85–91

Download references

Acknowledgements

The authors are grateful to the National Council for Scientific and Technological Development (CNPq/Brazil) and Coordination for the Improvement of Higher Education Personnel (CAPES/Brazil) for the financial support (grants and scholarships).

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and affiliations.

DSEA-UFPR - Department of Soil Science and Agricultural Engineering, Federal University of Paraná, Curitiba, Paraná, Brazil

Ana Paula Marés Mikosik, Nerilde Favaretto, Antonio Carlos Vargas Motta, Vander de Freitas Melo, Fabiane Machado Vezzani, Jairo Calderari de Oliveira Júnior & Verediana Fernanda Cherobim

You can also search for this author in PubMed   Google Scholar

Contributions

All authors contributed to the study conception and design. Field sampling by Ana Paula Marés Mikosik, Nerilde Favaretto, Verediana Fernanda Cherobim, Antonio Carlos Vargas Motta e Jairo Calderari de Oliveira Júnior. Material preparation and laboratories analyses by Ana Paula Marés Mikosik and Verediana Fernanda Cherobim. Date analyses by Ana Paula Marés Mikosik, Nerilde Favaretto, Antonio Carlos Vargas Motta, Vander de Freitas Melo, Fabiane Machado Vezzani. The first draft of the manuscript was written by Ana Paula Marés Mikosik and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Ana Paula Marés Mikosik .

Ethics declarations

Competing interests.

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Mikosik, A.P.M., Favaretto, N., Motta, A.C.V. et al. Phosphorus Environmental Risk Assessment in Wetland Soil. Wetlands 44 , 58 (2024). https://doi.org/10.1007/s13157-024-01812-9

Download citation

Received : 16 November 2023

Accepted : 26 April 2024

Published : 17 May 2024

DOI : https://doi.org/10.1007/s13157-024-01812-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Hydromorphic Soils
• Maximum Phosphorus Adsorption Capacity
• Degree of Phosphorus Saturation
• Environmental Soil Phosphorus Threshold
• Soil Phosphorus Storage Capacity
• Find a journal
• Publish with us
• Track your research

The Riemann hypothesis and dynamics of Backtracking New Q-Newton’s method

A new variant of Newton’s method - named Backtracking New Q-Newton’s method (BNQN) - was recently introduced by the second author. This method has good convergence guarantees, specially concerning finding roots of meromorphic functions. This paper explores using BNQN for the Riemann xi function. We show in particular that the Riemann hypothesis is equivalent to that all attractors of BNQN lie on the critical line. We also explain how an apparent relation between the basins of attraction of BNQN and Voronoi’s diagram can be helpful for verifying the Riemann hypothesis or finding a counterexample to it. Some illustrating experimental results are included, which convey some interesting phenomena. The experiments show that BNQN works very stably with highly transcendental functions like the Riemann xi function and its derivatives. Based on insights from the experiments, we discuss some concrete steps on using BNQN towards the Riemann hypothesis. Ideas and results from this paper can be extended to other zeta functions.

1. Introduction

The Riemann hypothesis is a famous long standing open problem in mathematics. It concerns the zeros of the Riemann zeta function ζ ⁢ ( s ) 𝜁 𝑠 \zeta(s) italic_ζ ( italic_s ) , which is defined in the domain ℛ ⁢ ( s ) > 1 ℛ 𝑠 1 \mathcal{R}(s)>1 caligraphic_R ( italic_s ) > 1 (where ℛ ( . ) \mathcal{R}(.) caligraphic_R ( . ) is the real part of a complex number) by an absolutely convergent power series:

where n s = e s ⁢ log ⁡ ( n ) superscript 𝑛 𝑠 superscript 𝑒 𝑠 𝑛 n^{s}=e^{s\log(n)} italic_n start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT italic_s roman_log ( italic_n ) end_POSTSUPERSCRIPT . The Riemann zeta function has an analytic continuation for ℛ ⁢ ( s ) > 0 ℛ 𝑠 0 \mathcal{R}(s)>0 caligraphic_R ( italic_s ) > 0 by presenting it as a Riemann–Stietjes integral [ 19 ] :

where { u } 𝑢 \{u\} { italic_u } is the fractional part of u 𝑢 u italic_u (for example, if u = 2.52 𝑢 2.52 u=2.52 italic_u = 2.52 , then { 2.52 } = 0.52 2.52 0.52 \{2.52\}=0.52 { 2.52 } = 0.52 ).

The integral formula helps to establish that the Riemann zeta function is a meromorphic function with only one simple pole at s = 1 𝑠 1 s=1 italic_s = 1 with residue 1 1 1 1 . Moreover, one has the following functional equation:

Some zeros of the Riemann hypothesis are easy to determine, and hence named trivial zeros. They are precisely those of the form: − 2 ⁢ n 2 𝑛 -2n - 2 italic_n , n ∈ 𝐍 > 0 𝑛 subscript 𝐍 absent 0 n\in\mathbf{N}_{>0} italic_n ∈ bold_N start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT , which are related to the Gamma function. The other zeros of the Riemann zeta function are called non-trivial zeros.

The Riemann hypothesis : Non-trivial zeros of the Riemann zeta function all lie on the critical line ℛ ⁢ ( s ) = 0.5 ℛ 𝑠 0.5 \mathcal{R}(s)=0.5 caligraphic_R ( italic_s ) = 0.5 .

For the purpose of this paper, it is more convenient to work with the Riemann xi function:

The Riemann xi function is an entire function of order 1, and satisfies a symmetric relation (functional equation): ξ ⁢ ( s ) = ξ ⁢ ( 1 − s ) 𝜉 𝑠 𝜉 1 𝑠 \xi(s)=\xi(1-s) italic_ξ ( italic_s ) = italic_ξ ( 1 - italic_s ) . This relation helps to show that ξ 𝜉 \xi italic_ξ has real values on the critical line.

The roots of the Riemann xi function are precisely the non-trivial roots of the Riemann zeta function. Hence, the Riemann hypothesis is then the statement that all zeros of the Riemann xi function belong to the critical line.

0.5 italic-ϵ \mathcal{R}(s)>0.5+\epsilon caligraphic_R ( italic_s ) > 0.5 + italic_ϵ and | ℐ ⁢ ( s ) | < T ℐ 𝑠 𝑇 |\mathcal{I}(s)|<T | caligraphic_I ( italic_s ) | < italic_T (negligible to the number of the zeros of the Riemann zeta function in the domain 0 < ℛ ⁢ ( s ) < 1 0 ℛ 𝑠 1 0<\mathcal{R}(s)<1 0 < caligraphic_R ( italic_s ) < 1 and | ℐ ⁢ ( s ) | < T ℐ 𝑠 𝑇 |\mathcal{I}(s)|<T | caligraphic_I ( italic_s ) | < italic_T ), see [ 14 ] for improvements on this. Additionally, it is known that a positive portion of the non-trivial roots of the Riemann zeta function are simple (i.e. of multiplicity 1), the current record is 41 % percent 41 41\% 41 % [ 8 ] . Moreover, there are works (in particular by Levison and Conrey [ 17 ] [ 9 ] ) which show that the density of zeros of the m-th derivative of the Riemann ξ 𝜉 \xi italic_ξ function on the critical line tends to 1 1 1 1 when m → ∞ → 𝑚 m\rightarrow\infty italic_m → ∞ . Also, it is verified by numerical methods that the Riemann hypothesis is true for zeros s 𝑠 s italic_s with large imaginary parts (starting with works by Riemann himself, Turing’s method for finding roots on the critical line, and specially works with very large heights by Odlyzko), the current record being | s | ≤ 3 × 10 12 𝑠 3 superscript 10 12 |s|\leq 3\times 10^{12} | italic_s | ≤ 3 × 10 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT [ 25 ] (where a good reference list on this topic can be found).

The Riemann hypothesis has wide connections to many fields in mathematics and physics. As such, there are many approaches and equivalences to it (see e.g. [ 5 ] , [ 6 ] , [ 7 ] ). This paper concerns the connections between the Riemann hypothesis and root finding algorithms, specifically Newton-type methods. We note that this direction is different from the works mentioned above on verifying the Riemann hypothesis up to a given height: those works are based on Turing’s method, and concentrate on the critical line ℛ ⁢ ( s ) = 0.5 ℛ 𝑠 0.5 \mathcal{R}(s)=0.5 caligraphic_R ( italic_s ) = 0.5 only. The Newton-type’s methods, on the other hand, work on the whole critical strip 0 ≤ ℛ ⁢ ( s ) ≤ 1 0 ℛ 𝑠 1 0\leq\mathcal{R}(s)\leq 1 0 ≤ caligraphic_R ( italic_s ) ≤ 1 , and can potentially produce explicit counterexamples in case the Riemann hypothesis is not true. According to Wikipedia, while the numerical evidence for Riemann hypothesis is abundant, analytic number theory has many conjectures (e.g. Skewes number) which are true up to a very large number, but are actually false.

We recall that Newton’s method, for finding roots of a meromorphic function g : 𝐂 → 𝐂 ∪ { ∞ } : 𝑔 → 𝐂 𝐂 g:\mathbf{C}\rightarrow\mathbf{C}\cup\{\infty\} italic_g : bold_C → bold_C ∪ { ∞ } , is the following iterative method:

Associated to it is Newton’s flow:

A direct variant of Newton’s method is Relaxed Newton’s method

and its randomized version:

Prior to this paper, there have been experimental works applying Newton’s method and Newton’s flow to finding roots of the Riemann zeta function, see for example [ 22 ] [ 23 ] . There have been also experiments applying Newton’s method to the Riemann xi function [ 16 ] . There, it is seen that in large scale Newton’s method and Relaxed Newton’s method for the Riemann xi function looks quite simple, however it is very fractal in the critical strip. Some experiments in this paper confirm this fractal feature of Newton’s method. We note that there are many other variants of Newton’s method, in both equation solving and optimization, such as Levenberg-Marquardt method and Regularized Newton’s method, but the existing literature concerns mainly local convergence properties, while for global convergence guarantees they require very strong conditions (e.g. convextity or Lipschitz continuity of the gradient or Hessian matrix) which are not satisfied by the Riemann zeta or xi function. We are not aware of any work applying these methods to the Riemann hypothesis.

6 𝑖 𝑐 𝑛 𝑙 𝑜 𝑔 𝑛 6+icn/logn 6 + italic_i italic_c italic_n / italic_l italic_o italic_g italic_n , where c 𝑐 c italic_c is a constant and n ∈ 𝐙 \ { 0 } 𝑛 \ 𝐙 0 n\in\mathbf{Z}\backslash\{0\} italic_n ∈ bold_Z \ { 0 } can find all roots of the ξ 𝜉 \xi italic_ξ function. However, global convergence is unknown, for example it is unclear whether Newton’s method applied to the Riemann xi function could have some strange attractors (e.g. attracting periodic cycles, as seen in the case of polynomials).

For a meromorphic function g ⁢ ( z ) 𝑔 𝑧 g(z) italic_g ( italic_z ) , [ 16 ] defines ν g ⁢ ( z ) = z − g ⁢ ( z ) z ⁢ g ′ ⁢ ( z ) subscript 𝜈 𝑔 𝑧 𝑧 𝑔 𝑧 𝑧 superscript 𝑔 ′ 𝑧 \nu_{g}(z)=z-\frac{g(z)}{zg^{\prime}(z)} italic_ν start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ( italic_z ) = italic_z - divide start_ARG italic_g ( italic_z ) end_ARG start_ARG italic_z italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) end_ARG . Concerning theoretical connections between the Riemann hypothesis and dynamical systems, there is the following interesting result.

Theorem 1.1 ( [ 16 ] ) .

A) The Riemann hypothesis is equivalent to that there is no topological disk D 𝐷 D italic_D contained in the critical strip such that N ζ ⁢ ( D ) ¯ ⊂ D ¯ subscript 𝑁 𝜁 𝐷 𝐷 \overline{N_{\zeta}(D)}\subset D over¯ start_ARG italic_N start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_D ) end_ARG ⊂ italic_D , where N g ⁢ ( z ) = z − g ⁢ ( z ) / g ′ ⁢ ( z ) subscript 𝑁 𝑔 𝑧 𝑧 𝑔 𝑧 superscript 𝑔 ′ 𝑧 N_{g}(z)=z-g(z)/g^{\prime}(z) italic_N start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ( italic_z ) = italic_z - italic_g ( italic_z ) / italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) is Newton’s method for g 𝑔 g italic_g .

B) The following statements are equivalent (and similarly in the case we replace ζ 𝜁 \zeta italic_ζ by ξ 𝜉 \xi italic_ξ ):

Statement 1: The Riemann hypothesis is true and the non-trivial zeros are simple.

Statement 2: The meromorphic function ν ζ ⁢ ( z ) subscript 𝜈 𝜁 𝑧 \nu_{\zeta}(z) italic_ν start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_z ) has no attracting fixed point.

By Statement 2 in part B of the above theorem, one sees that ν g subscript 𝜈 𝑔 \nu_{g} italic_ν start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT is not a good root finding method, specially for ζ 𝜁 \zeta italic_ζ or ξ 𝜉 \xi italic_ξ .

Another result which uses the Riemann xi function, indirectly related to Newton’s flow, is:

Theorem 1.2 ( [ 32 ] ) .

The following statements are equivalent.

Statement 1: The Riemann hypothesis holds. Moreover, all zeros of the Riemann xi function and of its first derivative are simple.

Statement 2: All lines of constant phase of the Riemann xi function corresponding to ± π plus-or-minus 𝜋 \pm\pi ± italic_π , ± 2 ⁢ π plus-or-minus 2 𝜋 \pm 2\pi ± 2 italic_π , ± 3 ⁢ π plus-or-minus 3 𝜋 \pm 3\pi ± 3 italic_π , … … \ldots … merge with the critical line.

𝑥 𝑖 𝑦 2 2 F(x,y)=|g(x+iy)|^{2}/2 italic_F ( italic_x , italic_y ) = | italic_g ( italic_x + italic_i italic_y ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 from 𝐑 2 superscript 𝐑 2 \mathbf{R}^{2} bold_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT to 𝐑 ∪ { ∞ } 𝐑 \mathbf{R}\cup\{\infty\} bold_R ∪ { ∞ } . It has very strong convergence guarantee for finding roots of meromorphic functions, and hence very appropriate to use to find roots of the Riemann zeta/xi functions or other zeta functions. Please see Section 2 for the precise algorithm and for some main properties of this method.

Let ϕ : 𝐂 → 𝐂 : italic-ϕ → 𝐂 𝐂 \phi:\mathbf{C}\rightarrow\mathbf{C} italic_ϕ : bold_C → bold_C be a function (not necessarily continuous). We use the following simple notion of attractors:

The basin of attraction for an attractor S 𝑆 S italic_S consists of initial points z 0 ∈ 𝐂 subscript 𝑧 0 𝐂 z_{0}\in\mathbf{C} italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ bold_C whose orbit under the dynamics of ϕ italic-ϕ \phi italic_ϕ converges to S 𝑆 S italic_S .

Our main theoretical results are the following two theorems.

Theorem 1.3 .

The following 3 statements are equivalent:

Statement 1: The Riemann hypothesis is true.

Statement 2: All attractors of the dynamics of BNQN applied to the Riemann xi function are contained in the critical line.

Statement 3: For any initial point z 0 subscript 𝑧 0 z_{0} italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , the constructed sequence { z n } subscript 𝑧 𝑛 \{z_{n}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } will either converge to a point on the critical line or to the point ∞ \infty ∞ .

Under stronger assumptions, we have the following result, which shows that BNQN is a good root finding method for ξ 𝜉 \xi italic_ξ .

Theorem 1.4 .

Assume that the Riemann hypothesis holds, and moreover all roots of the Riemann xi function and its derivative are simple. If the parameters of BNQN are randomly chosen, then there is an exceptional set ℰ ⊂ 𝐂 ℰ 𝐂 \mathcal{E}\subset\mathbf{C} caligraphic_E ⊂ bold_C of Lebesgue measure 0 so that the following is true. If z 0 ∈ 𝐂 \ ℰ subscript 𝑧 0 \ 𝐂 ℰ z_{0}\in\mathbf{C}\backslash\mathcal{E} italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ bold_C \ caligraphic_E is an initial point, then for BNQN applied to the Riemann xi function, the constructed sequence { z n } subscript 𝑧 𝑛 \{z_{n}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } satisfies the following two alternatives:

Note that in [ 11 ] , it was observed that the basins of attraction produced by BNQN look similar to the Voronoi’s diagram of the roots. Here we recall that the Voronoi diagram [ 38 ] [ 39 ] of a discrete set of points { z n ∗ } superscript subscript 𝑧 𝑛 \{z_{n}^{*}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT } in 𝐂 𝐂 \mathbf{C} bold_C is the union of Voronoi’s cells, where each Voronoi cell V ⁢ ( z n ∗ ) 𝑉 superscript subscript 𝑧 𝑛 V(z_{n}^{*}) italic_V ( italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) consists of points z ∈ 𝐂 𝑧 𝐂 z\in\mathbf{C} italic_z ∈ bold_C for which | z − z n ∗ | < min j ≠ n ⁡ | z − z j ∗ | 𝑧 superscript subscript 𝑧 𝑛 subscript 𝑗 𝑛 𝑧 superscript subscript 𝑧 𝑗 |z-z_{n}^{*}|<\min_{j\not=n}|z-z_{j}^{*}| | italic_z - italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | < roman_min start_POSTSUBSCRIPT italic_j ≠ italic_n end_POSTSUBSCRIPT | italic_z - italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | . The boundaries of Voronoi’s cells are line segments. Note that if the whole sequence { z n ∗ } superscript subscript 𝑧 𝑛 \{z_{n}^{*}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT } belongs to a line L 𝐿 L italic_L , then the boundaries of the Voronoi cells are lines orthogonal to L 𝐿 L italic_L and go through the middle points of the intervals connecting consecutive points in the sequence { z n ∗ } superscript subscript 𝑧 𝑛 \{z_{n}^{*}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT } . Figure 1 presents the Voronoi’s diagram for the first 8 roots of the Riemann xi function in comparison to the basins of attraction found by Newton’s method, Random Relaxed Newton’s method and BNQN for the polynomial of degree 8 8 8 8 whose roots are exactly the first 8 roots of the Riemann xi function.

In general, this similarity between Voronoi’s diagrams and basins of attraction for BNQN should be interpreted as follows. The boundary curves of the basins of attraction should go through the critical points of the meromorphic function g ⁢ ( z ) 𝑔 𝑧 g(z) italic_g ( italic_z ) , and they do not need to be literally line intervals but are curved, depending on the flows of the gradient of F ⁢ ( x , y ) 𝐹 𝑥 𝑦 F(x,y) italic_F ( italic_x , italic_y ) and related vector fields. For nice functions like g ⁢ ( z ) = sin ⁡ ( z ) 𝑔 𝑧 𝑧 g(z)=\sin(z) italic_g ( italic_z ) = roman_sin ( italic_z ) , where all roots are on the real axis and all critical points are precisely middle points of consecutive roots, the picture of basins of attraction is precisely that of Voronoi’s diagram. On the other hand, for other functions like the one in Figure 1 or a very complicated one like the Riemann xi function (see Section 3), the similarity must be viewed in a more broad sense.

In any case, this suggests the following idea to finding all roots of the Riemann xi function which are closest to the line x = 0 𝑥 0 x=0 italic_x = 0 :

Idea. We consider initial points ( 0 , y j ) 0 subscript 𝑦 𝑗 (0,y_{j}) ( 0 , italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) , where y j subscript 𝑦 𝑗 y_{j} italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is randomly chosen, and run BNQN. We should be able to find all roots of the Riemann zeta function which are closest to the line x = 0 𝑥 0 x=0 italic_x = 0 (in particular a counter example in case the Riemann hypothesis fails).

The organisation of this paper is as follows. In Section 2 we recall some necessary tools, and then prove Theorems 1.3 and 1.4 . In Section 3 we present some experiments concerning applied BNQN to finding roots of the Riemann xi function. Various interesting phenomena from experiments are given in Section 4, where we elaborate on the above idea to discuss some concrete steps on using BNQN towards the Riemann hypothesis

Acknowledgments : The authors thank Juan Arias de Reyna and Tomoki Kawahira for helping with some references, Viktor Balch Barth for translation of some articles from German, and Terje Kvernes for helping with experiments. This paper is part of the first author’s Master’s thesis. The second author is partially supported by Young Research Talents grant 300814 from Research Council of Norway.

2. Proofs of theoretical results

First we recall the definition of BNQN, and some of its main properties which are useful for the purpose of this paper. We also recall about the Laguerre-Pólya class. Then, we provide the proofs of the main theorems.

2.1. Backtracking New Q-Newton’s method

𝑥 𝑖 𝑦 2 F(x,y)=|g(x+iy)|^{2} italic_F ( italic_x , italic_y ) = | italic_g ( italic_x + italic_i italic_y ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . We note that F ≥ 0 𝐹 0 F\geq 0 italic_F ≥ 0 , hence if z ∗ = ( x ∗ , y ∗ ) superscript 𝑧 superscript 𝑥 superscript 𝑦 z^{*}=(x^{*},y^{*}) italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ( italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) is a root of g ⁢ ( z ) 𝑔 𝑧 g(z) italic_g ( italic_z ) , then ( x ∗ , y ∗ ) superscript 𝑥 superscript 𝑦 (x^{*},y^{*}) ( italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) is a global minimum of F 𝐹 F italic_F . Thus, one can apply an optimization method to find roots of g ⁢ ( z ) 𝑔 𝑧 g(z) italic_g ( italic_z ) .

The version of Newton’s method for optimization (for a function F 𝐹 F italic_F as in the previous paragraph) is as follows:

starting from an initial point z 0 = ( x 0 , y 0 ) subscript 𝑧 0 subscript 𝑥 0 subscript 𝑦 0 z_{0}=(x_{0},y_{0}) italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , provided the matrix ∇ 2 F ⁢ ( z n ) superscript ∇ 2 𝐹 subscript 𝑧 𝑛 \nabla^{2}F(z_{n}) ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F ( italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is invertible. Here, ∇ F ∇ 𝐹 \nabla F ∇ italic_F is the gradient, and ∇ 2 F superscript ∇ 2 𝐹 \nabla^{2}F ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F is the Hessian matrix. We hope that the sequence z n subscript 𝑧 𝑛 z_{n} italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT will converge to a root of g ⁢ ( z ) 𝑔 𝑧 g(z) italic_g ( italic_z ) .

Like (direct) Newton’s method, this optimization version also has issues with convergence guarantee. For example, it cannot avoid saddle points (this can be observed with such simple functions as g ⁢ ( z ) = z 2 − 1 𝑔 𝑧 superscript 𝑧 2 1 g(z)=z^{2}-1 italic_g ( italic_z ) = italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 , where 0 0 is the saddle point of the function F = | g | 2 / 2 𝐹 superscript 𝑔 2 2 F=|g|^{2}/2 italic_F = | italic_g | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 ).

In [ 36 ] , a new variant of Newton’s method for optimization, New Q-Newton’s method (NQN), is proposed. A crucial idea is that we should change the sign of negative eigenvalues of the Hessian matrix to positive, in order to avoid saddle points. It can avoid saddle points and still has fast rate of convergence near non-degenerate local minima, but it still does not have global convergence guarantee. An improvement of NQN is proposed in [ 35 ] , which incorporates also Armijo’s Backtracking line search [ 2 ] (a well known technique to boost convergence for algorithms), and is named Backtracking New Q-Newton’s method (BNQN).

The following is a modification from [ 12 ] :

Both NQN and BNQN can be applied to any dimensions and any C 2 superscript 𝐶 2 C^{2} italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT function. Concerning finding roots of meromorphic functions, we have the following result [ 35 ] [ 36 ] :

Theorem 2.1 .

𝑥 𝑖 𝑦 2 2 F(x,y)=|g(x+iy)|^{2}/2 italic_F ( italic_x , italic_y ) = | italic_g ( italic_x + italic_i italic_y ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 .

Given an initial point z 0 ∈ 𝐂 subscript 𝑧 0 𝐂 z_{0}\in\mathbf{C} italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ bold_C , which is not a pole of g 𝑔 g italic_g , we let { z n } subscript 𝑧 𝑛 \{z_{n}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } be the sequence constructed by BNQN applied to the function F 𝐹 F italic_F with initial point z 0 subscript 𝑧 0 z_{0} italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . If the objective function has compact sublevels (i.e. for all C ∈ 𝐑 𝐶 𝐑 C\in\mathbf{R} italic_C ∈ bold_R the set { ( x , y ) ∈ 𝐑 2 : F ⁢ ( x , y ) ≤ C } conditional-set 𝑥 𝑦 superscript 𝐑 2 𝐹 𝑥 𝑦 𝐶 \{(x,y)\in\mathbf{R}^{2}:~{}F(x,y)\leq C\} { ( italic_x , italic_y ) ∈ bold_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT : italic_F ( italic_x , italic_y ) ≤ italic_C } is compact), we choose θ ≥ 0 𝜃 0 \theta\geq 0 italic_θ ≥ 0 , while in general we choose θ > 0 𝜃 0 \theta>0 italic_θ > 0 .

1) Any critical point of F 𝐹 F italic_F is a root of g ⁢ ( z ) ⁢ g ′ ⁢ ( z ) = 0 𝑔 𝑧 superscript 𝑔 ′ 𝑧 0 g(z)g^{\prime}(z)=0 italic_g ( italic_z ) italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) = 0 .

3) If F 𝐹 F italic_F has compact sublevels, then { z n } subscript 𝑧 𝑛 \{z_{n}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } converges.

Option 1: { z n } subscript 𝑧 𝑛 \{z_{n}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } converges to a root z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT of g ⁢ ( z ) 𝑔 𝑧 g(z) italic_g ( italic_z ) , and if γ 0 = 1 subscript 𝛾 0 1 \gamma_{0}=1 italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 in the algorithm then the rate of convergence is quadratic.

\lim_{n\rightarrow\infty}|z_{n}|=+\infty roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT | italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = + ∞ .

In terms of computational resources, ensuring Armijo’s condition (the second While loop in the algorithm) is the most expensive aspect of BNQN. For example, if γ 𝛾 \gamma italic_γ (the learning rate) = 0.0625 = ( 1 / 2 ) 4 absent 0.0625 superscript 1 2 4 =0.0625=(1/2)^{4} = 0.0625 = ( 1 / 2 ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT after the While loop, it means that we had to check Armijo’s condition 4 4 4 4 times. Note that by part 4 of Theorem 2.1 , when we are close enough to a non-degenerate root, then γ = 1 𝛾 1 \gamma=1 italic_γ = 1 after the While loop, that is we only had to check Armijo’s condition once.

2.2. Laguerre-Pólya class

[ 4 ] [ 21 ] defines a family of functions { H t } t ∈ 𝐑 subscript subscript 𝐻 𝑡 𝑡 𝐑 \{H_{t}\}_{t\in\mathbf{R}} { italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_t ∈ bold_R end_POSTSUBSCRIPT

There is a constant − ∞ < Λ < 1 / 2 Λ 1 2 -\infty<\Lambda<1/2 - ∞ < roman_Λ < 1 / 2 (de Bruijn–Newman constant) so that the function H t subscript 𝐻 𝑡 H_{t} italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT has only real zeros if and only if t ≥ Λ 𝑡 Λ t\geq\Lambda italic_t ≥ roman_Λ . Otherwise, the function H t subscript 𝐻 𝑡 H_{t} italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT has some non-real zeros for t < Λ 𝑡 Λ t<\Lambda italic_t < roman_Λ . We have the following relation between the Riemann xi function and the function H 0 subscript 𝐻 0 H_{0} italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .

By the relation, RH is true if and only if H 0 subscript 𝐻 0 H_{0} italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT has only real zeros. Moreover, it is known that the Riemann hypothesis holds if and only if Λ = 0 Λ 0 \Lambda=0 roman_Λ = 0 [ 31 ] [ 28 ] . One can also improve the upper bound Λ Λ \Lambda roman_Λ in terms of a height H 𝐻 H italic_H for which the Riemann hypothesis is valid in the domain { z ∈ 𝐂 : 0 ≤ ℛ ⁢ ( z ) ≤ 1 , | ℐ ⁢ ( z ) | ≤ H } conditional-set 𝑧 𝐂 formulae-sequence 0 ℛ 𝑧 1 ℐ 𝑧 𝐻 \{z\in\mathbf{C}:~{}0\leq\mathcal{R}(z)\leq 1,~{}|\mathcal{I}(z)|\leq H\} { italic_z ∈ bold_C : 0 ≤ caligraphic_R ( italic_z ) ≤ 1 , | caligraphic_I ( italic_z ) | ≤ italic_H } [ 28 , Table 1] . Therefore, the Riemann hypothesis is intimately related to the following class of entire functions.

Definition 2.2 .

Let f : ℂ → ℂ : 𝑓 → ℂ ℂ f\colon\mathbb{C}\to\mathbb{C} italic_f : blackboard_C → blackboard_C be an entire function. Then f 𝑓 f italic_f belongs to the Laguerre–Pólya class, denoted by ℒ ⁢ 𝒫 ℒ 𝒫 \mathcal{LP} caligraphic_L caligraphic_P , if f 𝑓 f italic_f can be expressed as the Hadamard factorization

where a ≥ 0 𝑎 0 a\geq 0 italic_a ≥ 0 , m ∈ ℕ 𝑚 ℕ m\in\mathbb{N} italic_m ∈ blackboard_N and z k ∈ ℝ ∖ { 0 } subscript 𝑧 𝑘 ℝ 0 z_{k}\in\mathbb{R}\setminus\{0\} italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ blackboard_R ∖ { 0 } are zeros of f 𝑓 f italic_f such that

One has the following useful characterisation for that f ∈ ℒ ⁢ 𝒫 𝑓 ℒ 𝒫 f\in\mathcal{LP} italic_f ∈ caligraphic_L caligraphic_P .

Theorem 2.3 ( [ 27 , Laguerre–Pólya Theorem] ) .

Let f : ℂ → ℂ : 𝑓 → ℂ ℂ f\colon\mathbb{C}\to\mathbb{C} italic_f : blackboard_C → blackboard_C be a non-constant entire function. Then f ∈ ℒ ⁢ 𝒫 𝑓 ℒ 𝒫 f\in\mathcal{LP} italic_f ∈ caligraphic_L caligraphic_P if and only if there exists a sequence of (complex) polynomials { P n } n ∈ ℕ subscript subscript 𝑃 𝑛 𝑛 ℕ \{P_{n}\}_{n\in\mathbb{N}} { italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT with only real zeros such that it converges uniformly to f 𝑓 f italic_f in | z | ≤ R 𝑧 𝑅 |z|\leq R | italic_z | ≤ italic_R , for every R > 0 𝑅 0 R>0 italic_R > 0 .

𝑎 superscript 𝑧 2 𝑏 𝑧 superscript subscript product 𝑘 1 1 𝑧 subscript 𝑧 𝑘 superscript 𝑒 𝑧 subscript 𝑧 𝑘 f(z)=Az^{m}e^{-az^{2}+bz}\prod_{k=1}^{\infty}\left(1-\frac{z}{z_{k}}\right)e^{% \frac{z}{z_{k}}} italic_f ( italic_z ) = italic_A italic_z start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_a italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b italic_z end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( 1 - divide start_ARG italic_z end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ) italic_e start_POSTSUPERSCRIPT divide start_ARG italic_z end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT , then we can approximate it by

𝑏 superscript subscript 𝑘 1 𝑛 1 subscript 𝑧 𝑘 b_{n}=b+\sum_{k=1}^{n}\frac{1}{z_{k}} italic_b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_b + ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG . Then we use

𝑎 superscript 𝑧 2 𝑏 𝑧 e^{-az^{2}+bz} italic_e start_POSTSUPERSCRIPT - italic_a italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b italic_z end_POSTSUPERSCRIPT by polynomials of the form (where k n subscript 𝑘 𝑛 k_{n} italic_k start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is relatively large in comparison to n 𝑛 n italic_n and b n subscript 𝑏 𝑛 b_{n} italic_b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )

Since a ≥ 0 𝑎 0 a\geq 0 italic_a ≥ 0 and b ∈ 𝐑 𝑏 𝐑 b\in\mathbf{R} italic_b ∈ bold_R (and hence also b n ∈ 𝐑 subscript 𝑏 𝑛 𝐑 b_{n}\in\mathbf{R} italic_b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ bold_R ), the above polynomials have only real roots and have real values on the real axis.

2.3. Proofs of main theoretical results

Now we are ready to prove Theorems 1.3 and 1.4 .

Proof of Theorem 1.3 .

We will present the equivalence between Statement 1 and Statement 2. The other equivalences are similar.

( ⇒ ⇒ \Rightarrow ⇒ ) We first prove that Statement 1 implies Statement 2. Indeed, Statement 1 implies that H 0 ⁢ ( s ) subscript 𝐻 0 𝑠 H_{0}(s) italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_s ) is of Laguerre-Pólya class. Therefore, there is a sequence of polynomials which have only real zeros and which converge locally uniformly to H 0 ⁢ ( s ) subscript 𝐻 0 𝑠 H_{0}(s) italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_s ) . From the relations between ξ 𝜉 \xi italic_ξ and H 0 subscript 𝐻 0 H_{0} italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , this implies that there is a sequence of polynomials P n ⁢ ( s ) subscript 𝑃 𝑛 𝑠 P_{n}(s) italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_s ) which have only zeros on the critical line and which converges locally uniformly to ξ 𝜉 \xi italic_ξ . Then the derivatives P n ′ ⁢ ( s ) superscript subscript 𝑃 𝑛 ′ 𝑠 P_{n}^{\prime}(s) italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) converges locally uniformly to ξ ′ ⁢ ( s ) superscript 𝜉 ′ 𝑠 \xi^{\prime}(s) italic_ξ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) . By Gauss-Lucas’ theorem, roots of P n ′ ⁢ ( s ) superscript subscript 𝑃 𝑛 ′ 𝑠 P_{n}^{\prime}(s) italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) lie in the convex hull of roots of P n ⁢ ( s ) subscript 𝑃 𝑛 𝑠 P_{n}(s) italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_s ) , and hence are also on the critical line. Therefore, all roots of ξ ′ ⁢ ( s ) superscript 𝜉 ′ 𝑠 \xi^{\prime}(s) italic_ξ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) are also on the critical line. (The above argument is basically in [ 26 ] , which is in German, which we reproduce here for the convenience of readers.)

By Theorem 2.1 , the attractors of BNQN applied to F ⁢ ( s ) = | ξ ⁢ ( s ) | 2 / 2 𝐹 𝑠 superscript 𝜉 𝑠 2 2 F(s)=|\xi(s)|^{2}/2 italic_F ( italic_s ) = | italic_ξ ( italic_s ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 are either zeros of ξ ⁢ ( s ) 𝜉 𝑠 \xi(s) italic_ξ ( italic_s ) or zeros of ξ ′ ⁢ ( s ) superscript 𝜉 ′ 𝑠 \xi^{\prime}(s) italic_ξ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) , hence belong to the critical line.

( ⇐ ⇐ \Leftarrow ⇐ ) First, we recall that by Theorem 2.1 , if S 𝑆 S italic_S is an attractor for the dynamics of BNQN applied to F ⁢ ( s ) 𝐹 𝑠 F(s) italic_F ( italic_s ) , then S 𝑆 S italic_S must be a point and moreover, must be a root of ξ ⁢ ( s ) ⁢ ξ ′ ⁢ ( s ) 𝜉 𝑠 superscript 𝜉 ′ 𝑠 \xi(s)\xi^{\prime}(s) italic_ξ ( italic_s ) italic_ξ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) . In addition, we have:

Claim: If z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is a root of ξ ⁢ ( s ) 𝜉 𝑠 \xi(s) italic_ξ ( italic_s ) , then z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is an attractor for the dynamics of BNQN.

Proof of Claim: Indeed, Lemma A.2 in [ 36 ] shows that near z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , inequality (3.1) in the paper [ 1 ] is satisfied. Therefore, since z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is an isolated (global) minimum of the function F 𝐹 F italic_F , the last part of the proof of Theorem 3.2 in [ 1 ] shows that there is ϵ > 0 italic-ϵ 0 \epsilon>0 italic_ϵ > 0 (which can be chosen so small that z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is the only root of ξ ⁢ ( s ) ⁢ ξ ′ ⁢ ( s ) 𝜉 𝑠 superscript 𝜉 ′ 𝑠 \xi(s)\xi^{\prime}(s) italic_ξ ( italic_s ) italic_ξ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) inside the set { z : | z − z ∗ | ≤ 2 ⁢ ϵ } conditional-set 𝑧 𝑧 superscript 𝑧 2 italic-ϵ \{z:~{}|z-z^{*}|\leq 2\epsilon\} { italic_z : | italic_z - italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | ≤ 2 italic_ϵ } ) so that if z 0 ∈ { z : | z − z ∗ | < ϵ } subscript 𝑧 0 conditional-set 𝑧 𝑧 superscript 𝑧 italic-ϵ z_{0}\in\{z:~{}|z-z^{*}|<\epsilon\} italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ { italic_z : | italic_z - italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | < italic_ϵ } , then z n ∈ { z : | z − z ∗ | < 2 ⁢ ϵ } subscript 𝑧 𝑛 conditional-set 𝑧 𝑧 superscript 𝑧 2 italic-ϵ z_{n}\in\{z:~{}|z-z^{*}|<2\epsilon\} italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ { italic_z : | italic_z - italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | < 2 italic_ϵ } for all n 𝑛 n italic_n . Therefore, by Theorem 2.1 , the sequence { z n } subscript 𝑧 𝑛 \{z_{n}\} { italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } must converge to z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT . Hence, z ∗ superscript 𝑧 z^{*} italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is an attractor for the dynamics of BNQN applied to F ⁢ ( s ) 𝐹 𝑠 F(s) italic_F ( italic_s ) .

From the Claim, if all of attractors of BNQN belong to the critical line, then specially all roots of the Riemann xi function belongs to the critical line. Therefore, the Riemann hypothesis follows.

Proof of Theorem 1.4 .

𝑥 𝑖 𝑦 2 2 F(x,y)=|\xi(x+iy)|^{2}/2 italic_F ( italic_x , italic_y ) = | italic_ξ ( italic_x + italic_i italic_y ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 . Therefore, we can apply Theorem 2.1 to obtain Theorem 1.4 . ∎

3. Experimental results

𝑥 𝑖 𝑦 2 2 F(x,y)=|\xi(x+iy)|^{2}/2 italic_F ( italic_x , italic_y ) = | italic_ξ ( italic_x + italic_i italic_y ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 .

We first mention two difficulties concerning computations with transcendental functions like the Riemann xi function. First, transcendental functions or numbers cannot be accurately calculated on computers. Therefore, one has to use approximations. Specially, for the Riemann zeta/xi function, an effective tool is provided in [ 30 ] , which is implemented in the python library mpmath [ 20 ] . Second, there are many points in the critical strip which are not close to the critical line but at which the value of the Riemann xi function is extremely small.

The above two difficulties require one to be very carefully when numerically finding roots of the Riemann xi function. In particular, one has to use a machine precision much higher than usually needed (in the experiments here we need to choose the values 100, 300 or 1000 for the mpmath parameter mp.dps), and the experiments will take a long time to run. We also have to restrict the number of iterations to about tens to one hundred. An implementation in Python of BNQN accompanies the paper [ 36 ] , and this paper incorporates mpmath to deal with the transcendental function ξ 𝜉 \xi italic_ξ and the high machine precision.

Experiment 1: We compare the basin of attractions for various methods in finding the first 8 roots of the Riemann xi function. Here the roots are:

0.5 14.13472514173 𝑖 0.5+14.13472514173i 0.5 + 14.13472514173 italic_i , with points in the basin of attraction having green colour.

Root 2 ∼ similar-to \sim ∼ 0.5 − 14.13472514173 ⁢ i 0.5 14.13472514173 𝑖 0.5-14.13472514173i 0.5 - 14.13472514173 italic_i , with points in the basin of attraction having yellow colour.

0.5 21.02203963877 𝑖 0.5+21.02203963877i 0.5 + 21.02203963877 italic_i , with points in the basin of attraction having blue colour.

Root 4 ∼ similar-to \sim ∼ 0.5 − 21.02203963877 ⁢ i 0.5 21.02203963877 𝑖 0.5-21.02203963877i 0.5 - 21.02203963877 italic_i , with points in the basin of attraction having red colour.

0.5 25.01085758014 𝑖 0.5+25.01085758014i 0.5 + 25.01085758014 italic_i , with points in the basin of attraction having pink colour.

Root 6 ∼ similar-to \sim ∼ 0.5 − 25.01085758014 ⁢ i 0.5 25.01085758014 𝑖 0.5-25.01085758014i 0.5 - 25.01085758014 italic_i , with points in the basin of attraction having cyan colour.

0.5 30.42487612585 𝑖 0.5+30.42487612585i 0.5 + 30.42487612585 italic_i , with points in the basin of attraction having orange colour.

Root 8 ∼ similar-to \sim ∼ 0.5 − 30.42487612585 ⁢ i 0.5 30.42487612585 𝑖 0.5-30.42487612585i 0.5 - 30.42487612585 italic_i , with points in the basin of attraction having purple colour.

All other points in the grid have black colour. Among other options, they may represent initial points where the sequence converge to another root or to infinity or to strange attractors. By Theorem 2.1 , BNQN has no strange attractors, but it is unknown whether Newton’s method or Random Relaxed Newton’s method can have.