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Intellectual Mathematics

How to write a history of mathematics essay

This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.

The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.

So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.

Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.

The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.

Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.

For many examples of how one might write a critique, see my book reviews categorised as “critical.”

Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.

Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?

Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.

When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?

Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.

The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.

Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.

Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn ? Or does mathematics work according to the model of Kitcher ? Or that of Lakatos or Crowe ? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska ? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?

Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.

Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.

In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book . It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper ) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis , 1621, or many other works listed in a bibliography by Schüling .

Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.

Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.

Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:

The table shows the state of the literature before and after your research project has been carried out.

A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.

You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.

An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.

Many areas of the history of mathematics have been thoroughly Described but never Explained.

What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.

Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.

See also History of mathematics literature guide .

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History of mathematics.

Through in-depth exploration, the titles in this series offer compelling historical perspectives on the individuals and communities that have profoundly influenced the development of mathematics Whether you are a historian, mathematician, or simply an avid reader, each book in this series will be a valuable addition to your book collection.

Readers with institutional access to History of Mathematics titles can find individual volumes listed in the above menu.

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ISSN 2472-4785 (online); ISSN 0899-2428 (print)

MA 410: History of Mathematics: Sample Paper Topics

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Suggested Topics

Here are some topics suggested by your teacher:

• History of perfect numbers from Euclid to the day before yesterday • The influence of social needs on the uses of mathematics • History of the binomial theorem • The quadratic equation • An insight into Islamic mathematics • Goldbach and his famous conjecture • The history of logarithms and logarithm tables • Geometrical mathematical recreations • Prime numbers • The history of trigonometry • Charles Babbage and his engines • Some revolutionary curves • Angle trisection • Relationship of mathematics to music • Comparison of algebra texts from 1900 to today • American mathematics prior to 1875 • How our view of the early history of mathematics has changed in the last 100 years • The dark ages were dark for mathematics, too • Is mathematics discovered or invented? • How new was the new math? • Female mathematicians in history • The books that most changed our present geometry texts • The remarkable Bernoulli family • The rise of periodical literature and how it affected mathematics • The changing nature of function • The calculus priority dispute • The history of linear programming • Quality control, with emphasis on control charts • The role of cryptology in World War II • Navigation and mathematics • The Platonic solids • Mathematics and the early Rabbis • History of Hindu mathematics • The history of pi

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The history of mathematics: a very short introduction.

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Jacqueline Stedall

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

• Table of Contents

The Oxford University Press series of Very Short Introductions has existed since 1995; there seem to be more than 350 of them. The idea is to ask an expert to introduce readers to the essential ideas of a subject. The books are small (roughly 4 × 7 inches) and, of course, short. For the most part, the series seems to focus on the humanities; the web site currently lists as forthcoming, among others, volumes on Comedy , Rastafari , American Politics , The Avant Garde , Thought , The Napoleonic Wars , and Buddhism . But, of course, the sciences do get a look in; among the recent volumes we see Networks , Magnetism , Stars , The Antarctic , and Robotics .

The first mathematical book in the series (ten years ago) was a very short introduction to Mathematics , by Tim Gowers. (One book for all of mathematics!) I am aware of only a few others: an introduction to Probability was reviewed here recently, and there are volumes on Statistics and Numbers . Presumably some portions of the volumes on Isaac Newton, Cryptography, Logic, Bertrand Russell , and Galileo are mathematical, but in general we are under-represented. (The volume on The Elements is, alas, not about Euclid.) So it is great news that now The History of Mathematics has its own VSI.

An introduction to the history of mathematics might be two different things. One possibility is to provide a kind of summary of what is known about the history of mathematics. The other is to write an introduction to the discipline called “history of mathematics,” focusing on what mathematical historians do and how they do it. The first kind of book would really be an introductory textbook; there are many of those and it’s hard to imagine how it might be done in a “very short” book. What we have here, appropriately for this series, is the latter.

Jackie Stedall is a prolific author and a respected historian whose work deals mostly with the history of algebra. She has written a terrific little book that will be useful both to students and to the “educated general reader” who wants to know more about what historians of mathematics do. She starts gently, with a chapter that focuses on Fermat’s Last Theorem and visits several moments in its history. This chapter is really making an argument: to investigate the history of mathematics, she claims, one should go beyond the story of individuals and their “contributions” to the development of mathematical theories.

In the rest of the book, Stedall considers questions such as

What is mathematics and who counts as a mathematician? How do mathematical ideas get disseminated? How was mathematics learned and taught? How did mathematicians earn their living?

Considering this kind of question gives Stedall the chance to give many examples that demonstrate the richness of this “thick” approach to the history of mathematics. For the most part, these are both interesting and accessible to readers who may not themselves be mathematicians. Of course, the “history of ideas” aspect also comes in, but at the end, where it can enter into useful conversation with the more external and social issues. A final short chapter addresses the history of the history of mathematics, focusing especially on the changes that happened over the previous fifty or so years.

I once read of a historian of mathematics who was asked what she did by a cab driver. When she said she studied the history of ancient mathematics, they driver pulled out a notebook and wrote it down. She asked him why, and he said he collected exotic occupations, and that this was one of the strangest ever. So there’s a need for a little book like this. Perhaps it will enlarge the circle at least a little.

I will be teaching Colby’s History of Mathematics course this spring, and this little book will be the first piece of required reading. I am hoping that it will introduce my students to a subject that is broader than they think it is, and perhaps get them excited for what comes after.

Mathematicians should read it too.

Fernando Q. Gouvêa is a mathematician who has fallen in love with the history of mathematics. With William P. Berlinghoff, he is the author of Math through the Ages: A Gentle History for Teachers and Others .

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History of Mathematics

A virtual interactive exhibit being developed for the National Museum of Mathematics in New York City

Interactive Exhibits

The modern use and development of numbers and counting began with the rise of cities as a result of the need to organize people and allocate goods and resources.

Addition, subtraction, multiplication and division of numbers are important to trade and have been employed by civilizations for thousands of years.

Algebra deals with solving problems that involve mathematical symbols. The simplest of such problems were studied as long ago as 1900 BCE.

Pythagorean Theorem

The Pythagorean theorem relates the side lengths of a right triangle and was known to the ancient scholars and builders of Babylonia, Egypt, Greece, China and India.

Geometry focuses on the properties of space and the size and shape of objects. Its investigation dates back to the earliest recorded civilizations.

Primes are numbers having exactly one divisor other than 1. They are the building blocks of all counting numbers and were studied as early as 250 BCE.

The computation of π, the ratio of a circle's circumference to its diameter, has been of practical and mathematical interest in both the ancient and modern world.

Polyhedra are solids consisting of polygons joined at their edges. They were known to the ancients from nature and used in art, architecture and games of chance.

Mathematics Education

While the Greeks studied mathematics for its own sake, it has also been seen as needed only for certain trades. It is now a central part of school curricula.

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Mathematical Beans and Knotted Strings

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History Of Mathematics Essay

Mathematics is a certain way of thinking and doing that has been around since the dawn of humanity. Mathematics as a whole can be seen through history as a steady evolution, starting from simple hand calculations to modern-day computing machinery.

Mathematics allows people to understand not only what’s happening in the world but also why these things are happening. Mathematics has helped humanity to understand the laws of nature, predict where certain places are on Earth, travel into space and new worlds, estimate population size, track economies, etc. Mathematics is an integral part of human life – without it society would be all but lost.

Math was first used by humans as a system for counting things they wanted or needed. They started with simple things, like how many eggs are in a nest or how many fish are in a pond. Mathematics developed quickly to give us knowledge of geometry and measure the world around us. Mathematics was used to make advancements in astronomy; using math, we were able to calculate Earth’s circumference (Eratosthenes), create star catalogues (Hipparchus and Ptolemy), predict eclipses (Eudoxus and Calliphenes). Mathematics also helped us create calendars, which we use to this day.

The next step for Mathematics was during the Renaissance period with people like Fibonacci and Descartes. This is where the base of Mathematics as we know it today was laid. Mathematics became for the first time a discipline independent of astronomy and physics, having its own rules and proofs. Mathematics continued to be developed with mathematicians like Newton and Euler and then Mathematics finally reached what we know as ‘modern Mathematics’.

The history of Mathematics is an interesting topic that can be looked at from many different points of view. Mathematics has been used for thousands of years and, when Mathematics is so integral in all our lives, Mathematics’ history is important in understanding how we got to where we are today.

Mathematics, therefore, has become the basic tool of physical science and must be included in the social sciences. Mathematics is also indispensable in many forms of human endeavor (e.g., music, philosophy) that would not ordinarily be classed as scientific.

Mathematics has its roots in counting, calculation, measurement, and the formulation of quantitative laws. Mathematics is used in the study of all branches of the physical sciences, biological sciences, earth sciences, and social sciences; to solve problems in pure and applied arts; to formulate the rules governing games, sports, and gambling; to devise coding systems for transmitting messages or storing information; to carry out actuarial computations for insurance companies; and so on. Mathematics has made possible human progress by furnishing means for dealing with natural phenomena in ways that are precise rather than intuitively apparent.

The word mathematics comes from mathesis , a form of address derived from muses (Gr., “the patron goddesses of creative arts”), thus meaning literally “that which is learned.” The Greeks called mathematics, or sometimes “philosophy,” “the knowledge of things that are,” and the division of the quadrivium (arithmetic, geometry, astronomy, and music) recognized by ancient scholars may be interpreted as a classification of all branches of knowledge.

Mathematics is distinguished from other sciences in several ways: mathematicians seek to know pure truth without considering its application; mathematicians seek necessary truths whereas other scientists seek empirical laws; mathematics studies abstract patterns whereas science concerns itself with concrete objects.

The history of Mathematics can be seen as an ever-increasing series of abstractions. The earliest methods by which man obtained a measure for a quantity were based only on the properties of concrete objects such as a string or a stick. A length was determined by using the human body as the standard unit of measure. Only by degrees did man progress to the invention of simple tools such as the divided segment, marked stick, and marked pebble.

In Mathematics, history is important. Mathematics as a whole would not exist without history. Mathematics is the study of numbers and figures as far as we know it today.

Rigorous mathematics as it exists now was started in India by Aryabhata I, who lived from 476-550 CE . He introduced zero to mathematics and he attempted to solve quadratic equations. From India, mathematics went to China where it flourished until around 1200 A.D., when a general disinterest in Chinese Mathematics caused it to decline until its re-discovery during the Renaissance Period.

The first Mathematics book was written was by Euclid of Alexandria around 300 B.C.. In this book The Elements, Euclid set out to prove Pythagoras’ Theorem of right triangles by using a process known as deductive reasoning, or proof.

Only two other Mathematics books were written after this for about 1000 years – one by Al-Khowarizmi and one by Plato of Alexandria.

In 1400 A.D., Mathematics was brought to Europe from Africa by the Moors when they invaded Spain. It remained in Spain until 1492, then it spread throughout Western Europe. In 1545 A.D., Francois Viete wrote on imaginary numbers, which were a major focus of Mathematics at the time, Blaise Pascal had his first thoughts on what is now known as infinitesimal calculus in 1644 A.D.. Three years later, John Wallis published works on calculus, and this is the first known works on calculus. In 1665 Isaac Newton published his work on infinitesimal calculus, which was a major advancement from Pascal’s work.

In 1748 A.D., Mathematics took a big step forward when Leonard Euler’s Seven Bridges of Königsberg Problem was solved. This problem involving walking over bridges to cross rivers with different numbers of arches had been around since the mid-1700s. Leonhard Euler set out to solve it through real analysis by looking at what shapes were possible for traversing each bridge only once. He found that one shape worked for all seven bridges and proposed a solution in 1736 A.D.. His solution used something now called graph theory, which is the study of points that are joined by lines. Graph theory may sound familiar because it plays an important role in Mathematics today, but this was just the beginning.

Besides Mathematics becoming more general in its study, to include all possible Mathematics, Mathematics also became much more abstracted away from real-world problems and examples. This abstraction began in 1854 A.D., when George Boole published his work on symbolic logic. His work introduced numbers called 0 and 1 along with logical operators for not ( ), and ( & ) along with parentheses, making expressions like ((A & B) | ~C), which would be read as “A and B or C”. This system turned out to be very useful for Mathematics later.

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Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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Deepansh Gautam

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The History of Mathematics: A Very Short Introduction

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The History of Mathematics: A Very Short Introduction is arranged thematically to exemplify the varied contexts in which people have learned, used, and handed on mathematics. Using illustrative case studies drawn from a range of times and places, including early imperial China, the medieval Islamic world, and nineteenth-century Britain, the rich historical and cultural diversity of mathematical endeavour from the distant past to the present day is explored. Mathematics is a fundamental human activity that can be practised and understood in a multitude of ways; indeed, mathematical ideas themselves are far from being fixed; they have been adapted and changed by their passage across periods and cultures.

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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‘It Feels Like I Am Screaming Into the Void With Each Application’

By Peter Coy

Opinion Writer

When I asked new college graduates last month to tell me about their job searches, I got back a ton of heartache. Unanswered applications. Lowered expectations. For some, a sense that college was a waste of time and money.

John York wrote that he was about to earn a master’s degree in mathematics from New York University. “I have submitted close to 400 applications. I have heard back from less than 40, all rejections,” he wrote. “I essentially cannot get any job, because there are no entry-level positions anywhere at all.” He has a patent, he passed the first-level exam for Chartered Financial Analysts and he’s getting his Series 3 license, another financial credential. Nevertheless, he wrote, “It is just so silent, it feels like I am screaming into the void with each application I am filling out.”

Mauricio Naranjo, who is seeking work as a graphic designer, wrote, “Over the past year, I have submitted more than 400 applications and consistently receive a response that appears to be A.I.-generated, stating that unfortunately, they have moved forward with another candidate who better fits their expectations. This is the exact phrasing every time. Very few respond, as most do not reply at all.”

“Exhausting. Utterly demoralizing,” wrote Beth Donnelly, who is graduating this month with a major in linguistics and minors in German and teaching English as a second language. “I’ve been searching since early August for full-time, part-time or internship positions after I graduate. I’ve started putting my ‘desired salary’ at $35,000 in hope that just one person will think, ‘Oh, I won’t have to pay this person a large wage, so they get a leg up in the hiring process.’”

I got some positive responses, too. Lucinda Warnke, who landed a job in journalism as a general assignment reporter, wrote: “I am optimistic and excited! I feel confident in my career trajectory and my ability to build a stable, satisfying career. The job I got out of school comes with a livable wage and benefits, so I can build savings in the event that I am laid off or have some other financially demanding emergency. I feel like I made a good investment in my education because I went to a school that was affordable and studied subjects that balanced my interests with my professional needs.”

A majority of responses were grim, though. That’s not too surprising, given that half of college graduates are underemployed a year after graduation, meaning that they are working in jobs that don’t require the degrees they earned, as I wrote in my April 29 newsletter.

There’s clearly something wrong when young graduates can’t find jobs at the same time that employers complain of not being able to find qualified workers. As of March, there were still fewer unemployed people than job openings, according to the Bureau of Labor Statistics. In April the unemployment rate remained below average at 3.9 percent.

The responses I got aren’t a representative sample of all college graduates. It’s possible that unhappy people were more likely to write in. (I had to leave out some of the angriest and most dejected people because they didn’t want their names to appear.) Separately, my informal impression is that the people who wrote — happy or sad — were more likely to have attended a highly ranked school and to have graduated without student loans than the general student population.

Many students wrote that the jobs they were seeking or secured didn’t draw on what they learned in the classroom. “I will be using the skills I picked up in my data science minor, but nothing from my major (international relations),” Rain Orsi, a 2024 graduate, wrote. “A lot of the educational stuff could’ve been condensed to a 20-page PDF and I probably would be at the same knowledge level,” another student wrote. Jackeline Arcara wrote that if she had it to do over again, “I wouldn’t go to a four-year, fancy-pants school. I would take classes at a local college part-time and see where that takes me.”

Some students said that classroom learning was only part of what made college worthwhile to them. “College gives you four years to grow up — I have the maturity now to handle a full-time job. Before college, not so much,” wrote Caroline Lidz, who got a job in public relations after graduating in December with a degree in media studies and communications and a minor in art history.

Several said internships matter, a lot. “I wish I interned for a company outside of the school instead of being a research/lab assistant,” wrote Roger Vitek, who is graduating in June with a degree in product design and is still job hunting.

Economists have found that what you study in college is at least as important as where you study. As I wrote in my April 29 piece, there’s relatively strong demand for computer science, engineering, mathematics and math-intensive business fields such as finance and accounting.

But as I found out from the people who wrote in, that’s not always the case. Robert Vermeulen, a computer science major, wrote, “Out of the ~155 applications I haven’t had a reference on, I have gotten zero interviews.” Morgan Steckler wrote that he is looking for a software engineering or I.T. administration role paying at least $70,000 a year, but has had no luck so far. He said he’s thinking of bartending while continuing to send out applications. On the positive side, there are people like Warnke, who got a job as a reporter — not exactly a fast-growing profession.

As I read students’ responses, I had to remind myself that this is actually a relatively good year for finding a job. To a lot of members of the class of ’24, it doesn’t feel that way. Julia Brukx, who is graduating with a degree in history and art history, wrote, “I think I hit a new low just this morning when asked to write a cover letter for a retail position.”

Donnelly, the woman who described her job search as demoralizing, wrote: “I was told that if I was involved, active, kind, ready to learn, driven and intelligent, I would end up with a job out of college. This is evidently not true, and few older people seem to understand this.” She added, “I don’t have a backup plan besides working in the service industry.”

Elsewhere: Caps, Not Bans, for Short-Term Rentals

New York City’s Local Law 18, which was passed with the support of the hotel industry, tightens the rules on renting out rooms for less than 30 days. Supporters say renting rooms to tourists raises rents for New Yorkers. But an article published in Harvard Business Review by three scholars — one of whom used to work for Airbnb — calculates that Airbnb caused only about 1 percent of the aggregate increase in rents over the past decade or so. Hosts, guests and the businesses that serve them benefit. To keep certain neighborhoods from being overwhelmed by tourists, the authors recommend caps on how many nights per year a place may be rented out.

Quote of the Day

“The hedonistic conception of man is that of a lightning calculator of pleasures and pains who oscillates like a homogeneous globule of desire of happiness under the impulse of stimuli that shift him about the area, but leave him intact. He has neither antecedent nor consequent.”

— Thorstein Veblen, “Why Is Economics Not an Evolutionary Science?” (1898)

Peter Coy is a writer for the Opinion section of The Times, covering economics and business. Email him at [email protected] . @ petercoy

Opinion: My family’s generations of mothering

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Once, when I was in my preteens, my best friend and I wandered off at the local ice skating rink to browse the gift shop. We were gone for so long that my mother panicked and had our names called over the rink’s loudspeaker. When she eventually found us, I think she was just as angry as she was relieved, and I’ve never forgotten the look on her face.

Now, more than 20 years later, I watch my mother care for her own mother in stores, restaurants and the house with the same love and trepidation. My grandmother has dementia that has reached the stage where she doesn’t know who most of her relatives or friends are anymore. She needs to search for most words and isn’t always able to find them. She doesn’t know that her younger sister died last year. She disappears into certain rooms of my parents’ house for so long that the other day I asked my dad if we should check on her. He said not to worry, that she does this multiple times per day. Sure enough, she reappeared and sat back down in her recliner and then, about 15 minutes later, she wandered to another room again.

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For the past several years I’ve been watching my parents raise my grandmother. I always knew they were good parents — I had a wonderful childhood, something I really appreciated only in retrospect. But watching them now as they parent my grandmother — and I don’t know what other word to use to describe the way they feed her, clothe her, make sure she’s taking her medications, set up all her appointments — it hits me in a different way. I’m turning 36 this year and have no children. I’m almost certain I don’t want any, but it hasn’t escaped me that I’m reaching the point where I need to decide for sure.

I am also starting to realize that even if I don’t birth my own children, I will probably still end up parenting some day. Dementia is often hereditary. My grandmother’s father died from it, and we have no idea if my mother will inherit it. But if it’s not that, there will most likely be something else that brings about the role-switch for us that she is currently experiencing. I also can’t help but wonder, if I don’t have any children, who will be there to parent me at the end? Yet I’m not sure I can commit to the idea of having a child just so that I’ll have someone to take care of me when my own mother is no longer here.

Opinion: Single mothers like me are easy scapegoats. But the case for marriage is a myth

Almost half of Americans think single mothers are bad for society. We’re being blamed for problems we didn’t create.

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My mom and I are different in a lot of ways. She’s very outgoing and naturally friendly, which I am not; she remembers every birthday and every important date, whereas I forget something immediately if I don’t write it down. She grew up a tomboy and has played sports her entire life; I have been awarded more than one participation medal for my feeble athletic efforts. Sometimes we’ve struggled to find common ground. But now that I am well into adulthood, we have a relationship that, for the first time, lets us see each other on the same level. Parenting is at the center of both our lives right now, as I decide whether or not I want to become one, and she is thrust back into a role that I’m not sure she ever imagined returning to after my sister and I grew up. If anything, she was supposed to get a promotion from mother to grandmother.

“Mother” is certainly not her only identity, but I know that sometimes she must feel like it is. And I know that, despite how much she loves her mom, she will be ready for the next phase of her life once my grandmother passes. She is ready to fully inhabit the other identities that I saw once I was old enough to realize that she had a right to exist outside of me: traveler, hiker, friend, reader, wife, gardener, sister, aunt.

Taking care of an elderly parent comes with a lot of complicated feelings, and my mom has begun to talk about things that show me she’s thinking about how she’ll spend the rest of her own life. There are countries she wants to explore, but also everyday activities that are easy to take for granted when you aren’t responsible for someone else, such as wanting to join me on some of my regular morning walks on the beach.

For now, I watch my mother hold my grandma’s hand as she walks her down the street. She tucks my grandmother’s scarf into her jacket to make sure she’s warm enough, and I can see that my grandma feels safe, content and protected, even if everything else is confusing for her. I can only hope that, if and when our own roles reverse, I will be as good of a parent to my mom.

Jackie DesForges is a writer and artist in Los Angeles. @jackie__writes

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Mathematics > Number Theory

Title: the fyodorov-hiary-keating conjecture on mesoscopic intervals.

Abstract: We derive precise upper bounds for the maximum of the Riemann zeta function on short intervals on the critical line, showing for any $\theta\in(-1,0]$, the set of $t\in [T,2T]$ for which $$\max_{|h|\leq \log^\theta T}|\zeta(\tfrac{1}{2}+it+ih)|>\exp\bigg({y+\sqrt{(\log\log T)|\theta|/2}\cdot {S}}\bigg)\frac{(\log T)^{(1+\theta)}}{(\log\log T)^{3/4}}$$ is bounded above by $Cy\exp({-2y-y^2/((1+\theta)\log\log T)})$ (where $S$ is a random variable that is approximately a standard Gaussian as $T$ tends to infinity). This settles a strong form of a conjecture of Fyodorov--Hiary--Keating in mesoscopic intervals which was only known in the leading order. Using similar techniques, we also derive upper bounds for the second moment of the zeta function on such intervals. Conditioning on the value of $S$, we show that for all $t\in[T,2T]$ outside a set of order $o(T)$, $$\frac{1}{\log^\theta T}\int_{|h|\in \log^\theta T} |\zeta(\tfrac{1}{2}+it+ih)|^2\mathrm{d}h \ll e^{2S}\cdot \left(\frac{(\log T)^{(1+\theta)}}{\sqrt{\log\log T}}\right).$$ This proves a weak form of another conjecture of Fyodorov-Keating and generalizes a result of Harper, which is recovered at $\theta = 0$ (in which case $S$ is defined to be zero). Our main tool is an adaptation of the recursive scheme introduced by one of the authors, Bourgade and Radziwiłł to mesoscopic intervals.

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