homework 1 the real numbers

Chapter 1 Prerequisites

1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

• Classify a real number as a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate, items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting or enumerating items are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as[latex]\,\left\{1,2,3,...\right\}\,[/latex], where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero:[latex]\,\left\{0,1,2,3,...\right\}.[/latex]

The set of integers adds the opposites of the natural numbers to the set of whole numbers:[latex]\,\left\{...,-3,-2,-1,0,1,2,3,...\right\}.[/latex] It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

[latex]\begin{array}{lllll}\stackrel{\text{negative integers}}{\stackrel{}{\dots ,-3,-2,-1,}}\hfill & \hfill & \stackrel{\text{zero}}{\stackrel{}{0,}}\hfill & \hfill & \stackrel{\text{positive integers}}{\stackrel{}{1,2,3,\cdots }}\hfill \end{array}[/latex]

The set of rational numbers is written as[latex]\,\left\{\frac{m}{n}\,|m\text{ and }n\text{ are integers and }n\ne 0\right\}.\,[/latex]Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

• a terminating decimal:[latex]\,\frac{15}{8}=1.875,[/latex] or
• a repeating decimal:[latex]\,\frac{4}{11}=0.36363636\dots =0.\overline{36}[/latex]

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

• [latex]7=\frac{7}{1}[/latex]
• [latex]0=\frac{0}{1}[/latex]
• [latex]-8=-\frac{8}{1}[/latex]
• [latex]\frac{11}{1}[/latex]
• [latex]\frac{3}{1}[/latex]
• [latex]-\frac{4}{1}[/latex]

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

• [latex]-\frac{5}{7}[/latex]
• [latex]\frac{15}{5}[/latex]
• [latex]\frac{13}{25}[/latex]

Write each fraction as a decimal by dividing the numerator by the denominator.

• [latex]-\frac{5}{7}=-0.\stackrel{\text{———}}{714285},[/latex] a repeating decimal
• [latex]\frac{15}{5}=3\,[/latex](or 3.0), a terminating decimal
• [latex]\frac{13}{25}=0.52,[/latex] a terminating decimal
• [latex]\frac{68}{17}[/latex]
• [latex]\frac{8}{13}[/latex]
• [latex]-\frac{17}{20}[/latex]
• 4 (or 4.0), terminating;
• [latex]0.\overline{615384},[/latex]repeating;
• –0.85, terminating

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even[latex]\,\frac{3}{2},[/latex]but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

• [latex]\sqrt{25}[/latex]
• [latex]\frac{33}{9}[/latex]
• [latex]\sqrt{11}[/latex]
• [latex]\frac{17}{34}[/latex]
• [latex]0.3033033303333\dots[/latex]
• [latex]\sqrt{25}:\,[/latex]This can be simplified as[latex]\,\sqrt{25}=5.\,[/latex]Therefore,[latex]\sqrt{25}\,[/latex]is rational.

So,[latex]\,\frac{33}{9}\,[/latex]is rational and a repeating decimal.

• [latex]\sqrt{11}:\,[/latex]This cannot be simplified any further. Therefore,[latex]\,\sqrt{11}\,[/latex]is an irrational number.

So,[latex]\,\frac{17}{34}\,[/latex]is rational and a terminating decimal.

• [latex]0.3033033303333\dots \,[/latex]is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
• [latex]\frac{7}{77}[/latex]
• [latex]\sqrt{81}[/latex]
• [latex]4.27027002700027\dots[/latex]
• [latex]\frac{91}{13}[/latex]
• [latex]\sqrt{39}[/latex]
• rational and repeating;
• rational and terminating;
• irrational;

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line. The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line, as shown in (Figure 1) .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

• [latex]-\frac{10}{3}[/latex]
• [latex]\sqrt{5}[/latex]
• [latex]-\sqrt{289}[/latex]
• [latex]-6\pi[/latex]
• [latex]0.615384615384\dots[/latex]
• [latex]-\frac{10}{3}\,[/latex]is negative and rational. It lies to the left of 0 on the number line.
• [latex]\sqrt{5}\,[/latex]is positive and irrational. It lies to the right of 0.
• [latex]-\sqrt{289}=-\sqrt{{17}^{2}}=-17\,[/latex]is negative and rational. It lies to the left of 0.
• [latex]-6\pi \,[/latex]is negative and irrational. It lies to the left of 0.
• [latex]0.615384615384\dots \,[/latex]is a repeating decimal so it is rational and positive. It lies to the right of 0.
• [latex]\sqrt{73}[/latex]
• [latex]-11.411411411\dots[/latex]
• [latex]\frac{47}{19}[/latex]
• [latex]-\frac{\sqrt{5}}{2}[/latex]
• [latex]6.210735[/latex]
• positive, irrational; right
• negative, rational; left
• positive, rational; right
• negative, irrational; left

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2.

Sets of Numbers

The set of natural numbers includes the numbers used for counting:[latex]\,\left\{1,2,3,...\right\}.[/latex]

The set of whole numbers is the set of natural numbers plus zero:[latex]\,\left\{0,1,2,3,...\right\}.[/latex]

The set of integers adds the negative natural numbers to the set of whole numbers:[latex]\,\left\{...,-3,-2,-1,0,1,2,3,...\right\}.[/latex]

The set of rational numbers includes fractions written as[latex]\,\left\{\frac{m}{n}\,|m\text{ and }n\text{ are integers and }n\ne 0\right\}.[/latex]

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating:[latex]\,\left\{h|h\text{ is not a rational number}\right\}.[/latex]

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

• [latex]\sqrt{36}[/latex]
• [latex]\frac{8}{3}[/latex]

[latex]-6[/latex]

• [latex]3.2121121112\dots[/latex]
• [latex]-\frac{35}{7}[/latex]
• [latex]0[/latex]
• [latex]\sqrt{169}[/latex]
• [latex]\sqrt{24}[/latex]
• [latex]4.763763763\dots[/latex]

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example,[latex]\,{4}^{2}=4\cdot 4=16.\,[/latex]We can raise any number to any power. In general, the exponential notation[latex]\,{a}^{n}\,[/latex]means that the number or variable[latex]\,a\,[/latex]is used as a factor[latex]\,n\,[/latex]times.

In this notation,[latex]\,{a}^{n}\,[/latex]is read as the n th power of[latex]\,a,\,[/latex]where[latex]\,a\,[/latex]is called the base and[latex]\,n\,[/latex]is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example,[latex]\,24+6\cdot \frac{2}{3}-{4}^{2}\,[/latex]is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify[latex]\,{4}^{2}\,[/latex]as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore,[latex]\,24+6\cdot \frac{2}{3}-{4}^{2}=12.[/latex]

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses)

E (xponents)

M (ultiplication) and D (ivision)

A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

• Simplify any expressions within grouping symbols.
• Simplify any expressions containing exponents or radicals.
• Perform any multiplication and division in order, from left to right.
• Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

• [latex]{\left(3\cdot 2\right)}^{2}-4\left(6+2\right)[/latex]
• [latex]\frac{{5}^{2}-4}{7}-\sqrt{11-2}[/latex]
• [latex]6-|5-8|+3\left(4-1\right)[/latex]
• [latex]\frac{14-3\cdot 2}{2\cdot 5-{3}^{2}}[/latex]
• [latex]7\left(5\cdot 3\right)-2\left[\left(6-3\right)-{4}^{2}\right]+1[/latex]
• [latex]\begin{array}{cccc}\hfill{\left(3\cdot 2\right)}^{2}-4\left(6+2\right) =& {\left(6\right)}^{2}-4\left(8\right)\phantom{\rule{1em}{0ex}} \text{Simplify parentheses}\end{array}[/latex] [latex]\begin{array}\\ =& 36-4\left(8\right) & \phantom{\rule{2em}{0ex}}\text{Simplify exponent}\end{array}[/latex] [latex]\begin{array}\\ =& 36-32 & \phantom{\rule{2em}{0ex}}\text{Simplify multiplication}\end{array}[/latex] [latex]\begin{array}\\ =& 4 & \phantom{\rule{2em}{0ex}}\text{Simplify subtraction}\end{array}[/latex]

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol, so the numerator is considered to be grouped.

• [latex]\begin{array}{cccc}{ 6-|5-8|+3\left(4-1\right)} \end{array}[/latex] [latex]\begin{array}= 6-|-3|+3\left(3\right) \phantom{\rule{2em}{0ex}}\text{Simplify inside grouping symbols} \end{array}[/latex] [latex]\begin{array}\\&= 6-3+3\left(3\right)\phantom{\rule{2em}{0ex}}\text{Simplify absolute value} \end{array}[/latex] [latex]\begin{array}\\&= 6-3+9\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify multiplication}\end{array}[/latex] [latex]\begin{array}\\&= 3+9\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify subtraction}\end{array}[/latex] [latex]\begin{array}\\&= 12\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify addition} \end{array}[/latex]

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

• [latex]\sqrt{{5}^{2}-{4}^{2}}+7{\left(5-4\right)}^{2}[/latex]
• [latex]1+\frac{7\cdot 5-8\cdot 4}{9-6}[/latex]
• [latex]|1.8-4.3|+0.4\sqrt{15+10}[/latex]
• [latex]\frac{1}{2}\left[5\cdot {3}^{2}-{7}^{2}\right]+\frac{1}{3}\cdot {9}^{2}[/latex]
• [latex]\left[{\left(3-8\right)}^{2}-4\right]-\left(3-8\right)[/latex]

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example,[latex]\,17-5\,[/latex]is not the same as[latex]\,5-17.\,[/latex]Similarly,[latex]\,20÷5\ne 5÷20.[/latex]

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference[latex]\,12-\left(5+3\right).\,[/latex]We can rewrite the difference of the two terms 12 and[latex]\,\left(5+3\right)\,[/latex]by turning the subtraction expression into addition of the opposite. So instead of subtracting[latex]\,\left(5+3\right),[/latex]we add the opposite.

Now, distribute[latex]\,-1\,[/latex]and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have[latex]\,\left(-6\right)+0=-6\,[/latex]and[latex]\,23\cdot 1=23.\,[/latex]There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that for every real number a , there is a unique number, called the additive inverse (or opposite), denoted− a , that, when added to the original number, results in the additive identity, 0.

For example, if[latex]\,a=-8,[/latex]the additive inverse is 8, since[latex]\,\left(-8\right)+8=0.[/latex]

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted[latex]\,\frac{1}{a},[/latex] that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if[latex]\,a=-\frac{2}{3},[/latex] the reciprocal, denoted[latex]\,\frac{1}{a},[/latex] is[latex]\,-\frac{3}{2}\,[/latex] because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

• [latex]\,3\cdot 6+3\cdot 4[/latex]
• [latex]\,\left(5+8\right)+\left(-8\right)[/latex]
• [latex]\,6-\left(15+9\right)[/latex]
• [latex]\,\frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)[/latex]
• [latex]\,100\cdot \left[0.75+\left(-2.38\right)\right][/latex]
• [latex]\,\left(-\frac{23}{5}\right)\cdot \left[11\cdot \left(-\frac{5}{23}\right)\right][/latex]
• [latex]\,5\cdot \left(6.2+0.4\right)[/latex]
• [latex]\,18-\left(7-15\right)[/latex]
• [latex]\,\frac{17}{18}+\left[\frac{4}{9}+\left(-\frac{17}{18}\right)\right][/latex]
• [latex]\,6\cdot \left(-3\right)+6\cdot 3[/latex]
• 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
• 33, distributive property;
• 26, distributive property;
• [latex]\,\frac{4}{9},[/latex] commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
• 0, distributive property, inverse property of addition, identity property of addition

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as[latex]\,x+5,\frac{4}{3}\pi {r}^{3},[/latex] or[latex]\,\sqrt{2{m}^{3}{n}^{2}}.\,[/latex]In the expression[latex]\,x+5,[/latex] 5 is called a constant because it does not vary, and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

• [latex]\frac{4}{3}\pi {r}^{3}[/latex]
• [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]
• [latex]2\pi r\left(r+h\right)[/latex]
• [latex]4{y}^{3}+y[/latex]

Evaluating an Algebraic Expression at Different Values

Evaluate the expression[latex]\,2x-7\,[/latex]for each value for x.

• [latex]\,x=0[/latex]
• [latex]\,x=1[/latex]
• [latex]\,x=\frac{1}{2}[/latex]
• [latex]\,x=-4[/latex]
• Substitute 0 for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(0\right)-7\\ & =& 0-7\hfill \\ & =& -7\hfill \end{array}[/latex]
• Substitute 1 for[latex]\,x.[/latex] [latex]\begin{array}{ccc}2x-7& =& 2\left(1\right)-7\hfill \\ & =& 2-7\hfill \\ & =& -5\hfill \end{array}[/latex]
• Substitute[latex]\,\frac{1}{2}\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(\frac{1}{2}\right)-7\hfill \\ & =& 1-7\hfill \\ & =& -6\hfill \end{array}[/latex]
• Substitute[latex]\,-4\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(-4\right)-7\\ & =& -8-7\hfill \\ & =& -15\hfill \end{array}[/latex]

Evaluate the expression[latex]\,11-3y\,[/latex]for each value for y.

• [latex]\,y=2[/latex]
• [latex]\,y=0[/latex]
• [latex]\,y=\frac{2}{3}[/latex]
• [latex]\,y=-5[/latex]

Evaluate each expression for the given values.

• [latex]\,x+5\,[/latex]for[latex]\,x=-5[/latex]
• [latex]\,\frac{t}{2t-1}\,[/latex]for[latex]\,t=10[/latex]
• [latex]\,\frac{4}{3}\pi {r}^{3}\,[/latex]for[latex]\,r=5[/latex]
• [latex]\,a+ab+b\,[/latex]for[latex]a=11,b=-8[/latex]
• [latex]\,\sqrt{2{m}^{3}{n}^{2}}\,[/latex]for[latex]\,m=2,n=3[/latex]
• Substitute[latex]\,-5\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill x+5& =& \left(-5\right)+5\hfill \\ & =& 0\hfill \end{array}[/latex]
• Substitute 10 for[latex]\,t.[/latex] [latex]\begin{array}{ccc}\hfill \frac{t}{2t-1}& =& \frac{\left(10\right)}{2\left(10\right)-1}\hfill \\ & =& \frac{10}{20-1}\hfill \\ & =& \frac{10}{19}\hfill \end{array}[/latex]
• Substitute 5 for[latex]\,r.[/latex] [latex]\begin{array}{ccc}\hfill \frac{4}{3}\pi {r}^{3}& =& \frac{4}{3}\pi {\left(5\right)}^{3}\\ & =& \frac{4}{3}\pi \left(125\right)\hfill \\ & =& \frac{500}{3}\pi \hfill \end{array}[/latex]
• Substitute 11 for[latex]\,a\,[/latex]and –8 for[latex]\,b.[/latex] [latex]\begin{array}{ccc}\hfill a+ab+b& =& \left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right)\\ & =& 11-88-8\hfill \\ & =& -85\hfill \end{array}[/latex]
• Substitute 2 for[latex]\,m\,[/latex]and 3 for[latex]\,n.[/latex] [latex]\begin{array}{ccc}\hfill \sqrt{2{m}^{3}{n}^{2}}& =& \sqrt{2{\left(2\right)}^{3}{\left(3\right)}^{2}}\hfill \\ & =& \sqrt{2\left(8\right)\left(9\right)}\hfill \\ & =& \sqrt{144}\hfill \\ & =& 12\hfill \end{array}[/latex]
• [latex]\,\frac{y+3}{y-3}\,[/latex]for[latex]\,y=5[/latex]
• [latex]\,7-2t\,[/latex]for[latex]\,t=-2[/latex]
• [latex]\,\frac{1}{3}\pi {r}^{2}\,[/latex]for[latex]\,r=11[/latex]
• [latex]\,{\left({p}^{2}q\right)}^{3}\,[/latex]for[latex]\,p=-2,q=3[/latex]
• [latex]\,4\left(m-n\right)-5\left(n-m\right)\,[/latex]for[latex]\,m=\frac{2}{3},n=\frac{1}{3}[/latex]
• [latex]\,\frac{121}{3}\pi[/latex];

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation[latex]\,2x+1=7\,[/latex]has the unique solution of 3[latex][/latex] because when we substitute 3 for[latex]\,x\,[/latex]in the equation, we obtain the true statement[latex]\2\left(3\right)+1=7.[/latex]

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area[latex]\,A\,[/latex]of a circle in terms of the radius[latex]\,r\,[/latex]of the circle:[latex]\,A=\pi {r}^{2}.\,[/latex]For any value of[latex]\,r,[/latex] the area[latex]\,A\,[/latex]can be found by evaluating the expression[latex]\,\pi {r}^{2}.[/latex]

Using a Formula

A right circular cylinder with radius[latex]\,r\,[/latex]and height[latex]\,h\,[/latex]has the surface area[latex]\,S\,[/latex](in square units) given by the formula[latex]\,S=2\pi r\left(r+h\right).\,[/latex]See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of[latex]\,\pi .[/latex]

Evaluate the expression[latex]\,2\pi r\left(r+h\right)\,[/latex]for[latex]\,r=6\,[/latex]and[latex]\,h=9.[/latex]

The surface area is[latex]\,180\pi \,[/latex]square inches.

A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm 2 ) is found to be[latex]\,A=\left(L+16\right)\left(W+16\right)-L\cdot W.\,[/latex]See Figure 4. Find the area of a matte for a photograph with length 32 cm and width 24 cm.

Given that [latex]L=32[/latex] and [latex]W=24[/latex], plug the numbers into the formula [latex]\,A=\left(L+16\right)\left(W+16\right)-L\cdot W.\,[/latex]

[latex]\,A=\left(32+16\right)\left(24+16\right)-32\cdot 16.\,[/latex]

[latex]\,A=\left(48\right)\left(40\right)-512.\,[/latex]

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

• [latex]3x-2y+x-3y-7[/latex]
• [latex]2r-5\left(3-r\right)+4[/latex]
• [latex]\left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)[/latex]
• [latex]2mn-5m+3mn+n[/latex]
• [latex]\begin{array}{cccc}\hfill 3x-2y+x-3y-7& =& 3x+x-2y-3y-7\hfill & \phantom{\rule{6.5em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& 4x-5y-7\hfill & \phantom{\rule{6.5em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill 2r-5\left(3-r\right)+4& =& 2r-15+5r+4\hfill & \phantom{\rule{10em}{0ex}}\text{Distributive property}\hfill \\ & =& 2r+5r-15+4\hfill & \phantom{\rule{10em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& 7r-11\hfill & \phantom{\rule{10em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)& =& 4t-\frac{5}{4}s-\frac{2}{3}t-2s\hfill & \phantom{\rule{4em}{0ex}}\text{Distributive property}\hfill \\ & =& 4t-\frac{2}{3}t-\frac{5}{4}s-2s\hfill & \phantom{\rule{4em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& \frac{10}{3}t-\frac{13}{4}s\hfill & \phantom{\rule{4em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}2mn-5m+3mn+n& =& 2mn+3mn-5m+n& \phantom{\rule{5em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& \text{ }5mn-5m+n\hfill & \phantom{\rule{5em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\frac{2}{3}y-2\left(\frac{4}{3}y+z\right)[/latex]
• [latex]\frac{5}{t}-2-\frac{3}{t}+1[/latex]
• [latex]4p\left(q-1\right)+q\left(1-p\right)[/latex]
• [latex]9r-\left(s+2r\right)+\left(6-s\right)[/latex]
• [latex]\,-2y-2z\text{ or }-2\left(y+z\right);[/latex]
• [latex]\,\frac{2}{t}-1;[/latex]
• [latex]\,3pq-4p+q;[/latex]
• [latex]\,7r-2s+6[/latex]

Simplifying a Formula

A rectangle with length[latex]\,L\,[/latex]and width[latex]\,W\,[/latex]has a perimeter[latex]\,P\,[/latex]given by[latex]\,P=L+W+L+W.\,[/latex]Simplify this expression.

If the amount[latex]\,P\,[/latex]is deposited into an account paying simple interest[latex]\,r\,[/latex]for time[latex]\,t,[/latex] the total value of the deposit[latex]\,A\,[/latex]is given by[latex]\,A=P+Prt.\,[/latex]Simplify the expression. (This formula will be explored in more detail later in the course.)

[latex]A=P\left(1+rt\right)[/latex]

Key Concepts

• Rational numbers may be written as fractions or terminating or repeating decimals.
• Determine whether a number is rational or irrational by writing it as a decimal.
• The rational numbers and irrational numbers make up the set of real numbers. A number can be classified as natural, whole, integer, rational, or irrational.
• The order of operations is used to evaluate expressions.
• The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.
• Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.
• Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression.
• Is[latex]\,\sqrt{2}\,[/latex]an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

• What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?
• What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

For the following exercises, simplify the given expression.

• [latex]10+2\,×\,\left(5-3\right)[/latex]
• [latex]6÷2-\left(81÷{3}^{2}\right)[/latex]
• [latex]18+{\left(6-8\right)}^{3}[/latex]
• [latex]-2\,×\,{\left[16÷{\left(8-4\right)}^{2}\right]}^{2}[/latex]

[latex]-2[/latex]

• [latex]4-6+2\,×\,7[/latex]
• [latex]3\left(5-8\right)[/latex]

[latex]-9[/latex]

• [latex]4+6-10÷2[/latex]
• [latex]12÷\left(36÷9\right)+6[/latex]
• [latex]{\left(4+5\right)}^{2}÷3[/latex]
• [latex]3-12\,×\,2+19[/latex]
• [latex]2+8\,×\,7÷4[/latex]
• [latex]5+\left(6+4\right)-11[/latex]
• [latex]9-18÷{3}^{2}[/latex]
• [latex]14\,×\,3÷7-6[/latex]
• [latex]9-\left(3+11\right)\,×\,2[/latex]
• [latex]6+2\,×\,2-1[/latex]
• [latex]64÷\left(8+4\,×\,2\right)[/latex]
• [latex]9+4\left({2}^{2}\right)[/latex]
• [latex]{\left(12÷3\,×\,3\right)}^{2}[/latex]
• [latex]25÷{5}^{2}-7[/latex]
• [latex]\left(15-7\right)\,×\,\left(3-7\right)[/latex]
• [latex]2\,×\,4-9\left(-1\right)[/latex]
• [latex]{4}^{2}-25\,×\,\frac{1}{5}[/latex]
• [latex]12\left(3-1\right)÷6[/latex]

For the following exercises, solve for the variable.

• [latex]8\left(x+3\right)=64[/latex]
• [latex]4y+8=2y[/latex]

[latex]-4[/latex]

• [latex]\left(11a+3\right)-18a=-4[/latex]
• [latex]4z-2z\left(1+4\right)=36[/latex]
• [latex]4y{\left(7-2\right)}^{2}=-200[/latex]
• [latex]-{\left(2x\right)}^{2}+1=-3[/latex]

[latex]±1[/latex]

• [latex]8\left(2+4\right)-15b=b[/latex]
• [latex]2\left(11c-4\right)=36[/latex]
• [latex]4\left(3-1\right)x=4[/latex]
• [latex]\frac{1}{4}\left(8w-{4}^{2}\right)=0[/latex]

For the following exercises, simplify the expression.

• [latex]4x+x\left(13-7\right)[/latex]
• [latex]2y-{\left(4\right)}^{2}y-11[/latex]

[latex]-14y-11[/latex]

• [latex]\frac{a}{{2}^{3}}\left(64\right)-12a÷6[/latex]
• [latex]8b-4b\left(3\right)+1[/latex]

[latex]-4b+1[/latex]

• [latex]5l÷3l\,×\,\left(9-6\right)[/latex]
• [latex]7z-3+z\,×\,{6}^{2}[/latex]

[latex]43z-3[/latex]

• [latex]4\,×\,3+18x÷9-12[/latex]
• [latex]9\left(y+8\right)-27[/latex]

[latex]9y+45[/latex]

• [latex]\left(\frac{9}{6}t-4\right)2[/latex]
• [latex]6+12b-3\,×\,6b[/latex]

[latex]-6b+6[/latex]

• [latex]18y-2\left(1+7y\right)[/latex]
• [latex]{\left(\frac{4}{9}\right)}^{2}\,×\,27x[/latex]

[latex]\frac{16x}{3}[/latex]

• [latex]8\left(3-m\right)+1\left(-8\right)[/latex]
• [latex]9x+4x\left(2+3\right)-4\left(2x+3x\right)[/latex]

[latex]9x[/latex]

• [latex]{5}^{2}-4\left(3x\right)[/latex]

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends $10 on mp3s, puts half of what is left in a savings account, and gets another $5 for washing his neighbor’s car.

• Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

[latex]\frac{1}{2}\left(40-10\right)+5[/latex]

• How much money does Fred keep?

For the following exercises, solve the given problem.

• According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by[latex]\,\pi .\,[/latex]Is the circumference of a quarter a whole number, a rational number, or an irrational number?

irrational number

• Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of[latex]\,g\,[/latex]pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

• Write the equation that describes the situation.

[latex]g+400-2\left(600\right)=1200[/latex]

• Solve for g .

For the following exercise, solve the given problem.

• Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. He must spend the budget such that[latex]\,2,500,000-x=0.\,[/latex]What property of addition tells us what the value of x must be?

inverse property of addition

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

• [latex]0.5{\left(12.3\right)}^{2}-48x=\frac{3}{5}[/latex]
• [latex]{\left(0.25-0.75\right)}^{2}x-7.2=9.9[/latex]
• If a whole number is not a natural number, what must the number be?
• Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.
• Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.
• Determine whether the simplified expression is rational or irrational:[latex]\,\sqrt{-18-4\left(5\right)\left(-1\right)}.[/latex]
• Determine whether the simplified expression is rational or irrational:[latex]\,\sqrt{-16+4\left(5\right)+5}.[/latex]
• The division of two whole numbers will always result in what type of number?
• What property of real numbers would simplify the following expression:[latex]\,4+7\left(x-1\right)?[/latex]

Media Attributions

• 1.1 Figure 1 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.1 Figure 2 © OpenStax Algebra and Trignometry is licensed under a CC BY (Attribution) license
• 1.1 Distributive Property Graphic © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.1 Figure 3 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.1 Figure 4 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license

College Algebra Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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1.8 The Real Numbers

Learning objectives.

By the end of this section, you will be able to:

• Simplify expressions with square roots
• Identify integers, rational numbers, irrational numbers, and real numbers
• Locate fractions on the number line
• Locate decimals on the number line

Be Prepared 1.8

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapters, Decimals and Properties of Real Numbers .

Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write n 2 n 2 and read it “n squared.” The result is called the square of n . For example,

Similarly, 121 is the square of 11, because 11 2 11 2 is 121.

Square of a Number

If n 2 = m , n 2 = m , then m is the square of n .

Manipulative Mathematics

Complete the following table to show the squares of the counting numbers 1 through 15.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 10 2 = 100 , 10 2 = 100 , we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m m is called a square root of m .

Square Root of a Number

If n 2 = m , n 2 = m , then n is a square root of m .

Notice ( −10 ) 2 = 100 ( −10 ) 2 = 100 also, so −10 −10 is also a square root of 100. Therefore, both 10 and −10 −10 are square roots of 100.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign , m , m , denotes the positive square root. The positive square root is called the principal square root . When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because 0 2 = 0 , 0 2 = 0 , 0 = 0 . 0 = 0 . Notice that zero has only one square root.

Square Root Notation

m m is read “the square root of m ”

If m = n 2 , m = n 2 , then m = n , m = n , for n ≥ 0 . n ≥ 0 .

The square root of m , m , m , is the positive number whose square is m .

Since 10 is the principal square root of 100, we write 100 = 10 . 100 = 10 . You may want to complete the following table to help you recognize square roots.

Example 1.108

Simplify: ⓐ 25 25 ⓑ 121 . 121 .

Try It 1.215

Simplify: ⓐ 36 36 ⓑ 169 . 169 .

Try It 1.216

Simplify: ⓐ 16 16 ⓑ 196 . 196 .

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 100 = 10 . 100 = 10 . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, − 100 = −10 . − 100 = −10 . We read − 100 − 100 as “the opposite of the square root of 10.”

Example 1.109

Simplify: ⓐ − 9 − 9 ⓑ − 144 . − 144 .

Try It 1.217

Simplify: ⓐ − 4 − 4 ⓑ − 225 . − 225 .

Try It 1.218

Simplify: ⓐ − 81 − 81 ⓑ − 100 . − 100 .

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting number s , whole number s , and integers . What is the difference between these types of numbers?

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

Rational Number

A rational number is a number of the form p q , p q , where p and q are integers and q ≠ 0 . q ≠ 0 .

A rational number can be written as the ratio of two integers.

All signed fractions, such as 4 5 , − 7 8 , 13 4 , − 20 3 4 5 , − 7 8 , 13 4 , − 20 3 are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to 3 1 , 6 2 , 9 3 , 12 4 , 15 5 … 3 1 , 6 2 , 9 3 , 12 4 , 15 5 …

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers ! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer −8 −8 could be written as the decimal −8.0 . −8.0 . So, clearly, some decimals are rational.

Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means 7 3 10 , 7 3 10 , we can write it as an improper fraction, 73 10 . 73 10 . So 7.3 is the ratio of the integers 73 and 10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 or −1.2684 ) −1.2684 ) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.

Example 1.110

Write as the ratio of two integers: ⓐ −27 −27 ⓑ 7.31.

So we see that −27 −27 and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

Try It 1.219

Write as the ratio of two integers: ⓐ −24 −24 ⓑ 3.57.

Try It 1.220

Write as the ratio of two integers: ⓐ −19 −19 ⓑ 8.41.

Let’s look at the decimal form of the numbers we know are rational.

We have seen that every integer is a rational number , since a = a 1 a = a 1 for any integer, a . We can also change any integer to a decimal by adding a decimal point and a zero.

We have also seen that every fraction is a rational number . Look at the decimal form of the fractions we considered above.

What do these examples tell us?

Every rational number can be written both as a ratio of integers , ( p q , ( p q , where p and q are integers and q ≠ 0 ) , q ≠ 0 ) , and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes!

The number π π (the Greek letter pi , pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

We can even create a decimal pattern that does not stop or repeat, such as

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

Rational or Irrational?

If the decimal form of a number

• repeats or stops , the number is rational .
• does not repeat and does not stop , the number is irrational .

Example 1.111

Given the numbers 0.58 3 – , 0.47 , 3.605551275 . . . 0.58 3 – , 0.47 , 3.605551275 . . . list the ⓐ rational numbers ⓑ irrational numbers.

Try It 1.221

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: 0.29 , 0.81 6 – , 2.515115111 … . 0.29 , 0.81 6 – , 2.515115111 … .

Try It 1.222

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: 2.6 3 – , 0.125 , 0.418302 … 2.6 3 – , 0.125 , 0.418302 …

Example 1.112

For each number given, identify whether it is rational or irrational: ⓐ 36 36 ⓑ 44 . 44 .

• ⓐ Recognize that 36 is a perfect square, since 6 2 = 36 . 6 2 = 36 . So 36 = 6 , 36 = 6 , therefore 36 36 is rational.
• ⓑ Remember that 6 2 = 36 6 2 = 36 and 7 2 = 49 , 7 2 = 49 , so 44 is not a perfect square. Therefore, the decimal form of 44 44 will never repeat and never stop, so 44 44 is irrational.

Try It 1.223

For each number given, identify whether it is rational or irrational: ⓐ 81 81 ⓑ 17 . 17 .

Try It 1.224

For each number given, identify whether it is rational or irrational: ⓐ 116 116 ⓑ 121 . 121 .

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real number s .

Real Number

A real number is a number that is either rational or irrational.

All the numbers we use in elementary algebra are real numbers. Figure 1.15 illustrates how the number sets we’ve discussed in this section fit together.

Can we simplify −25 ? −25 ? Is there a number whose square is −25 ? −25 ?

None of the numbers that we have dealt with so far has a square that is −25 . −25 . Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to −25 . −25 .

The square root of a negative number is not a real number.

Example 1.113

For each number given, identify whether it is a real number or not a real number: ⓐ −169 −169 ⓑ − 64 . − 64 .

• ⓐ There is no real number whose square is −169 . −169 . Therefore, −169 −169 is not a real number.
• ⓑ Since the negative is in front of the radical, − 64 − 64 is −8 , −8 , Since −8 −8 is a real number, − 64 − 64 is a real number.

Try It 1.225

For each number given, identify whether it is a real number or not a real number: ⓐ −196 −196 ⓑ − 81 . − 81 .

Try It 1.226

For each number given, identify whether it is a real number or not a real number: ⓐ − 49 − 49 ⓑ −121 . −121 .

Example 1.114

Given the numbers −7 , 14 5 , 8 , 5 , 5.9 , − 64 , −7 , 14 5 , 8 , 5 , 5.9 , − 64 , list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers.

• ⓐ Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
• ⓑ The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and −7 −7 is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so − 64 = −8 . − 64 = −8 . So the integers are −7 , 8 , − 64 . −7 , 8 , − 64 .
• ⓒ Since all integers are rational, then −7 , 8 , − 64 −7 , 8 , − 64 are rational. Rational numbers also include fractions and decimals that repeat or stop, so 14 5 and 5.9 14 5 and 5.9 are rational. So the list of rational numbers is −7 , 14 5 , 8 , 5.9 , − 64 . −7 , 14 5 , 8 , 5.9 , − 64 .
• ⓓ Remember that 5 is not a perfect square, so 5 5 is irrational.
• ⓔ All the numbers listed are real numbers.

Try It 1.227

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: −3 , − 2 , 0. 3 – , 9 5 , 4 , 49 . −3 , − 2 , 0. 3 – , 9 5 , 4 , 49 .

Try It 1.228

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: − 25 , − 3 8 , −1 , 6 , 121 , 2.041975 … − 25 , − 3 8 , −1 , 6 , 121 , 2.041975 …

Locate Fractions on the Number Line

The last time we looked at the number line , it only had positive and negative integers on it. We now want to include fraction s and decimals on it.

Let’s start with fractions and locate 1 5 , − 4 5 , 3 , 7 4 , − 9 2 , −5 , and 8 3 1 5 , − 4 5 , 3 , 7 4 , − 9 2 , −5 , and 8 3 on the number line.

We’ll start with the whole numbers 3 3 and −5 . −5 . because they are the easiest to plot. See Figure 1.16 .

The proper fractions listed are 1 5 and − 4 5 . 1 5 and − 4 5 . We know the proper fraction 1 5 1 5 has value less than one and so would be located between 0 and 1. 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts 1 5 , 2 5 , 3 5 , 4 5 . 1 5 , 2 5 , 3 5 , 4 5 . We plot 1 5 . 1 5 . See Figure 1.16 .

Similarly, − 4 5 − 4 5 is between 0 and −1 . −1 . After dividing the unit into 5 equal parts we plot − 4 5 . − 4 5 . See Figure 1.16 .

Finally, look at the improper fractions 7 4 , − 9 2 , 8 3 . 7 4 , − 9 2 , 8 3 . These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See Figure 1.16 .

Figure 1.16 shows the number line with all the points plotted.

Example 1.115

Locate and label the following on a number line: 4 , 3 4 , − 1 4 , −3 , 6 5 , − 5 2 , and 7 3 . 4 , 3 4 , − 1 4 , −3 , 6 5 , − 5 2 , and 7 3 .

Locate and plot the integers, 4 , −3 . 4 , −3 .

Locate the proper fraction 3 4 3 4 first. The fraction 3 4 3 4 is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot 3 4 . 3 4 . Similarly plot − 1 4 . − 1 4 .

Now locate the improper fractions 6 5 , − 5 2 , 7 3 . 6 5 , − 5 2 , 7 3 . It is easier to plot them if we convert them to mixed numbers and then plot them as described above: 6 5 = 1 1 5 , − 5 2 = −2 1 2 , 7 3 = 2 1 3 . 6 5 = 1 1 5 , − 5 2 = −2 1 2 , 7 3 = 2 1 3 .

Try It 1.229

Locate and label the following on a number line: −1 , 1 3 , 6 5 , − 7 4 , 9 2 , 5 , − 8 3 . −1 , 1 3 , 6 5 , − 7 4 , 9 2 , 5 , − 8 3 .

Try It 1.230

Locate and label the following on a number line: −2 , 2 3 , 7 5 , − 7 4 , 7 2 , 3 , − 7 3 . −2 , 2 3 , 7 5 , − 7 4 , 7 2 , 3 , − 7 3 .

In Example 1.116 , we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• a < b “ a is less than b ” when a is to the left of b on the number line
• a > b “ a is greater than b ” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Example 1.116

Order each of the following pairs of numbers, using < or >. It may be helpful to refer Figure 1.17 .

ⓐ − 2 3 ___ −1 − 2 3 ___ −1 ⓑ −3 1 2 ___ −3 −3 1 2 ___ −3 ⓒ − 3 4 ___ − 1 4 − 3 4 ___ − 1 4 ⓓ −2 ___ − 8 3 −2 ___ − 8 3

Try It 1.231

Order each of the following pairs of numbers, using < or >:

ⓐ − 1 3 ___ −1 − 1 3 ___ −1 ⓑ −1 1 2 ___ −2 −1 1 2 ___ −2 ⓒ − 2 3 ___ − 1 3 − 2 3 ___ − 1 3 ⓓ −3 ___ − 7 3 . −3 ___ − 7 3 .

Try It 1.232

ⓐ −1 ___ − 2 3 −1 ___ − 2 3 ⓑ −2 1 4 ___ −2 −2 1 4 ___ −2 ⓒ − 3 5 ___ − 4 5 − 3 5 ___ − 4 5 ⓓ −4 ___ − 10 3 . −4 ___ − 10 3 .

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Example 1.117

Locate 0.4 on the number line.

A proper fraction has value less than one. The decimal number 0.4 is equivalent to 4 10 , 4 10 , a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See Figure 1.18 .

Try It 1.233

Locate on the number line: 0.6.

Try It 1.234

Locate on the number line: 0.9.

Example 1.118

Locate −0.74 −0.74 on the number line.

The decimal −0.74 −0.74 is equivalent to − 74 100 , − 74 100 , so it is located between 0 and −1 . −1 . On a number line, mark off and label the hundredths in the interval between 0 and −1 . −1 . See Figure 1.19 .

Try It 1.235

Locate on the number line: −0.6 . −0.6 .

Try It 1.236

Locate on the number line: −0.7 . −0.7 .

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,

0.40 > 0.04 0.40 > 0.04

Again, we can use the number line to order numbers.

Where are 0.04 and 0.40 located on the number line? See Figure 1.20 .

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04.

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

Because 310 > 308, we know that 310 1000 > 308 1000 . 310 1000 > 308 1000 . Therefore, 0.31 > 0.308.

Notice what we did in converting 0.31 to a fraction—we started with the fraction 31 100 31 100 and ended with the equivalent fraction 310 1000 . 310 1000 . Converting 310 1000 310 1000 back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

We say 0.31 and 0.310 are equivalent decimals .

Equivalent Decimals

Two decimals are equivalent if they convert to equivalent fractions.

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

Order Decimals.

• Step 1. Write the numbers one under the other, lining up the decimal points.
• Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
• Step 3. Compare the numbers as if they were whole numbers.
• Step 4. Order the numbers using the appropriate inequality sign.

Example 1.119

Order 0.64 ___ 0.6 0.64 ___ 0.6 using < < or > . > .

Try It 1.237

Order each of the following pairs of numbers, using < or > : 0.42 ___ 0.4 . < or > : 0.42 ___ 0.4 .

Try It 1.238

Order each of the following pairs of numbers, using < or > : 0.18 ___ 0.1 . < or > : 0.18 ___ 0.1 .

Example 1.120

Order 0.83 ___ 0.803 0.83 ___ 0.803 using < < or > . > .

Try It 1.239

Order the following pair of numbers, using < or > : 0.76 ___ 0.706 . < or > : 0.76 ___ 0.706 .

Try It 1.240

Order the following pair of numbers, using < or > : 0.305 ___ 0.35 . < or > : 0.305 ___ 0.35 .

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because −2 −2 lies to the right of −3 −3 on the number line, we know that −2 > −3 . −2 > −3 . Similarly, smaller numbers lie to the left on the number line. For example, because −9 −9 lies to the left of −6 −6 on the number line, we know that −9 < −6 . −9 < −6 . See Figure 1.21 .

If we zoomed in on the interval between 0 and −1 , −1 , as shown in Example 1.121 , we would see in the same way that −0.2 > −0.3 and − 0.9 < −0.6 . −0.2 > −0.3 and − 0.9 < −0.6 .

Example 1.121

Use < < or > > to order −0.1 ___ −0.8 . −0.1 ___ −0.8 .

Try It 1.241

Order the following pair of numbers, using < or >: −0.3 ___ −0.5 . −0.3 ___ −0.5 .

Try It 1.242

Order the following pair of numbers, using < or >: −0.6 ___ −0.7 . −0.6 ___ −0.7 .

Section 1.8 Exercises

Practice makes perfect.

In the following exercises, simplify.

− 100 − 100

− 121 − 121

In the following exercises, write as the ratio of two integers.

ⓐ − 12 − 12 ⓑ 9.279

ⓐ − 16 − 16 ⓑ 4.399

In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers

0.75 , 0.22 3 – , 1.39174 … 0.75 , 0.22 3 – , 1.39174 …

0.36 , 0.94729 … , 2.52 8 – 0.36 , 0.94729 … , 2.52 8 –

0.4 5 – , 1.919293 … , 3.59 0.4 5 – , 1.919293 … , 3.59

0.1 3 – , 0.42982 … , 1.875 0.1 3 – , 0.42982 … , 1.875

In the following exercises, identify whether each number is rational or irrational.

ⓐ 25 25 ⓑ 30 30

ⓐ 44 44 ⓑ 49 49

ⓐ 164 164 ⓑ 169 169

ⓐ 225 225 ⓑ 216 216

In the following exercises, identify whether each number is a real number or not a real number.

ⓐ − 81 − 81 ⓑ −121 −121

ⓐ − 64 − 64 ⓑ −9 −9

ⓐ −36 −36 ⓑ − 144 − 144

ⓐ −49 −49 ⓑ − 144 − 144

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

−8 , 0 , 1.95286 … , 12 5 , 36 , 9 −8 , 0 , 1.95286 … , 12 5 , 36 , 9

−9 , −3 4 9 , − 9 , 0.40 9 – , 11 6 , 7 −9 , −3 4 9 , − 9 , 0.40 9 – , 11 6 , 7

− 100 , −7 , − 8 3 , −1 , 0.77 , 3 1 4 − 100 , −7 , − 8 3 , −1 , 0.77 , 3 1 4

−6 , − 5 2 , 0 , 0. 714285 ——— , 2 1 5 , 14 −6 , − 5 2 , 0 , 0. 714285 ——— , 2 1 5 , 14

In the following exercises, locate the numbers on a number line.

3 4 , 8 5 , 10 3 3 4 , 8 5 , 10 3

1 4 , 9 5 , 11 3 1 4 , 9 5 , 11 3

3 10 , 7 2 , 11 6 , 4 3 10 , 7 2 , 11 6 , 4

7 10 , 5 2 , 13 8 , 3 7 10 , 5 2 , 13 8 , 3

2 5 , − 2 5 2 5 , − 2 5

3 4 , − 3 4 3 4 , − 3 4

3 4 , − 3 4 , 1 2 3 , −1 2 3 , 5 2 , − 5 2 3 4 , − 3 4 , 1 2 3 , −1 2 3 , 5 2 , − 5 2

2 5 , − 2 5 , 1 3 4 , −1 3 4 , 8 3 , − 8 3 2 5 , − 2 5 , 1 3 4 , −1 3 4 , 8 3 , − 8 3

In the following exercises, order each of the pairs of numbers, using < or >.

−1 ___ − 1 4 −1 ___ − 1 4

−1 ___ − 1 3 −1 ___ − 1 3

−2 1 2 ___ −3 −2 1 2 ___ −3

−1 3 4 ___ −2 −1 3 4 ___ −2

− 5 12 ___ − 7 12 − 5 12 ___ − 7 12

− 9 10 ___ − 3 10 − 9 10 ___ − 3 10

−3 ___ − 13 5 −3 ___ − 13 5

−4 ___ − 23 6 −4 ___ − 23 6

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

In the following exercises, order each pair of numbers, using < or >.

0.37 ___ 0.63 0.37 ___ 0.63

0.86 ___ 0.69 0.86 ___ 0.69

0.91 ___ 0.901 0.91 ___ 0.901

0.415 ___ 0.41 0.415 ___ 0.41

−0.5 ___ −0.3 −0.5 ___ −0.3

−0.1 ___ −0.4 −0.1 ___ −0.4

−0.62 ___ −0.619 −0.62 ___ −0.619

−7.31 ___ −7.3 −7.31 ___ −7.3

Everyday Math

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

ⓐ How many busses will be needed? ⓑ Why must the answer be a whole number? ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.

ⓐ How many teachers will be needed? ⓑ Why must the answer be a whole number? ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Writing Exercises

In your own words, explain the difference between a rational number and an irrational number.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.

ⓑ On a scale of 1 − 10 , 1 − 10 , how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Real Numbers

Real numbers are just numbers like:.

Nearly any number you can think of is a Real Number

Real Numbers include:

Real Numbers can also be positive , negative or zero .

So ... what is NOT a Real Number?

Mathematicians also play with some special numbers that aren't Real Numbers.

The Real Number Line

The Real Number Line is like a geometric line .

A point is chosen on the line to be the "origin" . Points to the right are positive, and points to the left are negative.

A distance is chosen to be "1", then whole numbers are marked off: {1,2,3,...}, and also in the negative direction: {...,−3,−2,−1}

Any point on the line is a Real Number:

• The numbers could be whole (like 7)
• or rational (like 20/9)
• or irrational (like π )

But we won't find Infinity, or an Imaginary Number.

Any Number of Digits

A Real Number can have any number of digits either side of the decimal point

• 0.000 000 0001

There can be an infinite number of digits, such as 1 3 = 0.333...

Why are they called "Real" Numbers?

Because they are not Imaginary Numbers

The Real Numbers had no name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!

Real does not mean they are in the real world

They are not called "Real" because they show the value of something real .

In mathematics we like our numbers pure, when we write 0.5 we mean exactly half.

But in the real world half may not be exact (try cutting an apple exactly in half).

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Unit 1 Algebra Basics Homework 1 The Real Numbers

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Learning Objectives

By the end of this section, you will be able to:

• Simplify expressions with square roots
• Identify integers, rational numbers, irrational numbers, and real numbers
• Locate fractions on the number line
• Locate decimals on the number line

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapters, Decimals and Properties of Real Numbers .

Simplify Expressions with Square Roots

Remember that when a number \(n\) is multiplied by itself, we write \(n^{2}\) and read it “\(n\) squared.” The result is called the square of \(n\). For example,

\[\begin{array} { l l } { 8 ^ { 2 } } & { \text { read '8 squared' } } \\ { 64 } & { 64 \text { is called the square of } 8 \text { . } } \end{array}\]

Similarly, 121 is the square of 11, because \(11^{2}\) is 121.

SQUARE OF A NUMBER

If \(n^{2}=m\), then \(m\) is the square of \(n\).

Doing the Manipulative Mathematics activity “Square Numbers” will help you develop a better understanding of perfect square numbers.

Complete the following table to show the squares of the counting numbers 1 through 15.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

\[( - 3 ) ^ { 2 } = 9 \quad ( - 8 ) ^ { 2 } = 64 \quad ( - 11 ) ^ { 2 } = 121 \quad ( - 15 ) ^ { 2 } = 225\]

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because \(10^{2}=100\), we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is mm is called a square root of \(m\).

SQUARE ROOT OF A NUMBER

If \(n^{2}=m\), then \(n\) is a square root of \(m\).

Notice \((−10)^{2}=100\) also, so \(−10\) is also a square root of \(100\). Therefore, both \(10\) and \(−10\) are square roots of \(100\).

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign , \(\sqrt{m}\), denotes the positive square root. The positive square root is called the principal square root . When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because \(0^{2}=0, \sqrt{0}=0\). Notice that zero has only one square root.

SQUARE ROOT NOTATION

\(\sqrt{m}\) is read “the square root of \(m\)”

If \(m = n^{2}\), then \(\sqrt{m} = n\), for \(n\geq 0\).

The square root of \(m\), \(\sqrt{m}\), is the positive number whose square is \(m\).

Since 10 is the principal square root of 100, we write \(\sqrt{100}=10\). You may want to complete the following table to help you recognize square roots.

Exercise \(\PageIndex{1}\)

• \(\sqrt{25}\)
• \(\sqrt{121}\)
• \[\begin{array} {ll} {} &{\sqrt{25}} \\ {\text {Since }5^{2} = 25} &{5} \end{array}\]
• \[\begin{array} {ll} {} &{\sqrt{121}} \\ {\text {Since }11^{2} = 121} &{11} \end{array}\]

Exercise \(\PageIndex{2}\)

• \(\sqrt{36}\)
• \(\sqrt{169}\)

Exercise \(\PageIndex{3}\)

• \(\sqrt{16}\)
• \(\sqrt{196}\)

We know that every positive number has two square roots and the radical sign indicates the positive one. We write \(\sqrt{100)=10\). If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \(-\sqrt{100)=-10\). We read \(-\sqrt{100)\) as “the opposite of the square root of 10.”

Exercise \(\PageIndex{4}\)

• \(-\sqrt{9}\)
• \(-\sqrt{144}\)
• \[\begin{array} {ll} {} &{-\sqrt{9}} \\ {\text {The negative is in front of the radical sign.}} &{-3} \end{array}\]
• \[\begin{array} {ll} {} &{-\sqrt{144}} \\ {\text {The negative is in front of the radical sign.}} &{-12} \end{array}\]

Exercise \(\PageIndex{5}\)

Exercise \(\pageindex{6}\), identify integers, rational numbers, irrational numbers, and real numbers.

We have already described numbers as counting number s , whole number s , and integers . What is the difference between these types of numbers?

\[\begin{array} { l l } { \text { Counting numbers } } & { 1,2,3,4 , \ldots } \\ { \text { Whole numbers } } & { 0,1,2,3,4 , \ldots } \\ { \text { Integers } } & { \dots - 3 , - 2 , - 1,0,1,2,3 , \ldots } \end{array}\]

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

RATIONAL NUMBER

A rational number is a number of the form \(\dfrac{p}{q}\), where p and q are integers and \(q \neq 0\)

A rational number can be written as the ratio of two integers.

All signed fractions, such as \(\dfrac{4}{5}\), \(-\dfrac{7}{8}\), \(\dfrac{13}{4}\), \(-\dfrac{20}{3}\) are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to \(\dfrac{3}{1}\), \(-\dfrac{6}{2}\), \(\dfrac{9}{3}\), \(\dfrac{12}{4}\),\(-\dfrac{15}{5} \ldots\)

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

\[3 = \frac { 3 } { 1 } \quad - 8 = - \frac { 8 } { 1 } \quad 0 = \frac { 0 } { 1 }\]

Since any integer can be written as the ratio of two integers, all integers are rational numbers ! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer \(−8\) could be written as the decimal \(−8.0\). So, clearly, some decimals are rational.

Think about the decimal \(7.3\). Can we write it as a ratio of two integers? Because \(7.3\) means \(7\dfrac{3}{10}\), we can write it as an improper fraction, \(\dfrac{73}{10}\). So \(7.3\) is the ratio of the integers \(73\) and \(10\). It is a rational number.

In general, any decimal that ends after a number of digits (such as \(7.3\) or \(−1.2684\)) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

Exercise \(\PageIndex{7}\)

Write as the ratio of two integers:

• \[\begin{array} {ll} {} &{-27} \\ {\text {Write it as a fraction with denominator 1.}} &{\dfrac{-27}{1}} \end{array}\]
• \[\begin{array} {ll} {} &{7.31} \\ {\text {Write is as a mixed number. Remember.}} &{} \\ {\text {7 is the whole number and the decimal}} &{7\dfrac{31}{100}} \\ {\text {part, 0.31, indicates hundredths.}} &{} \\ {\text{Convert to an improper fraction.}} &{\dfrac{731}{100}} \end{array}\]

So we see that −27 and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

Exercise \(\PageIndex{8}\)

• \(\dfrac{-24}{1}\)
• \(\dfrac{357}{100}\)

Exercise \(\PageIndex{9}\)

• \(\dfrac{-19}{1}\)
• \(\dfrac{841}{100}\)

We have seen that every integer is a rational number , since \(a = \dfrac{a}{1}\) for any integer, \( a\) . We can also change any integer to a decimal by adding a decimal point and a zero.

\[\begin{array} { l l l l l l l } { \text { Integer } } & { - 2 } & { - 1 } & { 0 } & { 1 } & { 2 } & { 3 } \\ { \text { Decimal form } } & { - 2.0 } & { - 1.0 } & { 0.0 } & { 1.0 } & { 2.0 } & { 3.0 } \\ { } & { \text { These decimal numbers stop. } } \end{array}\]

We have also seen that every fraction is a rational number . Look at the decimal form of the fractions we considered above.

\[\begin{array} { l l l l } { \text { Ratio of integers } } & { \frac { 4 } { 5 } } & { - \frac { 7 } { 8 } } & { \frac { 13 } { 4 } } & { - \frac { 20 } { 3 } } \\ { \text { The decimal form } } & { 0.8 } & { - 0.875 } & { 3.25 } & { - 6.666 \dots } \\ { } & { } & { } & { - 6.\overline{6} } \\ { } & { \text { These decimal either stop or repeat. } } \end{array}\]

What do these examples tell us?

Every rational number can be written both as a ratio of integers , (\(\dfrac{p}{q}\), where p and q are integers and \(q\neq 0\)),, and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

A rational number is a number of the form \(\frac{p}{q}\), where p and q are integers and \(q\neq 0\)

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes!

The number \(\pi\) (the Greek letter pi , pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

\[\pi =3.141592654\ldots\]

We can even create a decimal pattern that does not stop or repeat, such as

\[2.01001000100001\ldots\]

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

IRRATIONAL NUMBER

An irrational number is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

RATIONAL OR IRRATIONAL?

If the decimal form of a number

• repeats or stops , the number is rational .
• does not repeat and does not stop , the number is irrational .

Exercise \(\PageIndex{10}\)

Given the numbers \(0.58\overline{3}, 0.47, 3.605551275\ldots\) list the

• rational numbers
• irrational numbers.
• \[\begin{array} {ll} {\text{Look for decimals that repeat or stop}} &{\text{The 3 repeats in }0.58\overline{3}.} \\ {} &{\text {The decimal 0.47 stops after the 7.}}\\ {} &{\text {So } 0.58\overline{3} \text{ and } 0.47 \text{are rational}} \end{array}\]
• \[\begin{array} {ll} {\text{Look for decimals that repeat or stop}} &{3.605551275\ldots\text{has no repeating block of}} \\ {} &{\text {digits and it does not stop.}}\\ {} &{\text {So } 3.605551275\ldots \text{ is irrational.}} \end{array}\]

Exercise \(\PageIndex{11}\)

For the given numbers list the

• irrational numbers: \(0.29, 0.81\overline{6}, 2.515115111….\)
• \(0.29, 0.81\overline{6}\)
• \(2.515115111….\)

Exercise \(\PageIndex{12}\)

• irrational numbers: \(2.6\overline{3}, 0.125, 0.418302…\)
• \(2.6\overline{3}, 0.125\)
• \(0.418302…\)

Exercise \(\PageIndex{13}\)

For each number given, identify whether it is rational or irrational:

• \(\sqrt{44}\)
• Recognize that 36 is a perfect square, since \(6^{2} = 36\). So \(\sqrt{36} = 6\), therefore \(\sqrt{36}\) is rational.
• Remember that \(6^{2} = 36\) and \(7^{2} = 49\), so \(44\) is not a perfect square. Therefore, the decimal form of \(\sqrt{44}\) will never repeat and never stop, so \(\sqrt{44}\) is irrational.

Exercise \(\PageIndex{14}\)

• \(\sqrt{81}\)
• \(\sqrt{17}\)

Exercise \(\PageIndex{15}\)

• \(\sqrt{116}\)

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real number s .

REAL NUMBER

A real number is a number that is either rational or irrational.

All the numbers we use in elementary algebra are real numbers. Figure \(\PageIndex{3}\) illustrates how the number sets we’ve discussed in this section fit together.

Can we simplify \(\sqrt{-25}\)? Is there a number whose square is \(−25\)?

\[(\quad)^{2}=−25?\]

None of the numbers that we have dealt with so far has a square that is \(−25\). Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to \(\sqrt{-25}\).

The square root of a negative number is not a real number.

Exercise \(\PageIndex{16}\)

For each number given, identify whether it is a real number or not a real number:

• \(\sqrt{-169}\)
• \(-\sqrt{64}\)
• There is no real number whose square is \(−169\). Therefore, \(\sqrt{-169}\) is not a real number.
• Since the negative is in front of the radical, \(-\sqrt{64}\) is \(−8\), Since \(−8\) is a real number, \(-\sqrt{64}\) is a real number.

Exercise \(\PageIndex{17}\)

• \(\sqrt{-196}\)
• \(-\sqrt{81}\)
• not a real number
• real number

Exercise \(\PageIndex{18}\)

• \(-\sqrt{49}\)
• \(\sqrt{-121}\)

Exercise \(\PageIndex{19}\)

Given the numbers \(−7, \frac{14}{5}, 8, \sqrt{5}, 5.9, \sqrt{64}\), list the

• whole numbers
• irrational numbers
• real numbers
• Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
• The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and −7 is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so \(-\sqrt{64} = -8\). So the integers are \(−7, 8, \sqrt{64}\).
• Since all integers are rational, then \(-7, 8, -\sqrt{64}\) are rational. Rational numbers also include fractions and decimals that repeat or stop, so \(\frac{14}{5}\) and \(5.9\) are rational. So the list of rational numbers is \(−7, \frac{14}{5}, 8, 5.9, \sqrt{64}\)
• Remember that 5 is not a perfect square, so \(\sqrt{5}\) is irrational.
• All the numbers listed are real numbers.

Exercise \(\PageIndex{20}\)

For the given numbers, list the

• real numbers: \(−3, -\sqrt{2}, 0.\overline{3}, \frac{9}{5}, 4, \sqrt{49}\)
• \(4, \sqrt{49}\).
• \(−3, 4, \sqrt{49}\)
• \(−3, 0.\overline{3}, \frac{9}{5}, 4, \sqrt{49}\)
• \( -\sqrt{2}\)
• \(−3, \sqrt{2}, 0.\overline{3}, \frac{9}{5}, 4, \sqrt{49}\)

Exercise \(\PageIndex{21}\)

• real numbers: \(−\sqrt{25},−\frac{3}{8}, −1, 6, \sqrt{121}, 2.041975…\)
• \(6, \sqrt{121}\).
• \(−\sqrt{25}, −1, 6, \sqrt{121}\)
• \(−\sqrt{25},−\frac{3}{8}, −1, 6, \sqrt{121}\)
• \(2.041975…\)
• \(−\sqrt{25},−\frac{3}{8}, −1, 6, \sqrt{121}, 2.041975…\)

Locate Fractions on the Number Line

The last time we looked at the number line , it only had positive and negative integers on it. We now want to include fraction s and decimals on it.

Doing the Manipulative Mathematics activity “Number Line Part 3” will help you develop a better understanding of the location of fractions on the number line.

Let’s start with fractions and locate \(\frac{1}{5}, -\frac{4}{5}, 3, \frac{7}{4}, -\frac{9}{2}, -5\) and \(\frac{8}{3}\) on the number line.

We’ll start with the whole numbers 3 and −5. because they are the easiest to plot. See Figure \(\PageIndex{4}\).

The proper fractions listed are \(\frac{1}{5}\text{ and } -\frac{4}{5}\). We know the proper fraction \(\frac{1}{5}\) has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts \(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}\). We plot \(\frac{1}{5}\). See Figure \(\PageIndex{4}\).

Similarly, \(-\frac{4}{5}\) is between 0 and −1. After dividing the unit into 5 equal parts we plot \(-\frac{4}{5}\). See Figure \(\PageIndex{4}\).

Finally, look at the improper fractions \(\frac{7}{4}, -\frac{9}{2}, \frac{8}{3}\). These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See Figure \(\PageIndex{4}\).

Figure \(\PageIndex{4}\) shows the number line with all the points plotted.

Exercise \(\PageIndex{22}\)

Locate and label the following on a number line: \(4, \frac{3}{4}, -\frac{1}{4}, -3, \frac{6}{5}, -\frac{5}{2}\) and \(\frac{7}{3}\).

Locate and plot the integers, 4,−3.

Locate the proper fraction \(\frac{3}{4}\) first. The fraction \(\frac{3}{4}\) is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot \(\frac{3}{4}\). Similarly plot \(-\frac{1}{4}\).

Now locate the improper fractions \(\frac{6}{5}\), \(-\frac{5}{2}\), \(\frac{7}{3}\). It is easier to plot them if we convert them to mixed numbers and then plot them as described above: \(\frac{6}{5} = 1\frac{1}{5}\), \(-\frac{5}{2} = -2\frac{1}{2}\), \(\frac{7}{3} = 2\frac{1}{3}\).

Exercise \(\PageIndex{23}\)

Locate and label the following on a number line: \(-1, \frac{1}{3}, \frac{6}{5}, -\frac{7}{4}, \frac{9}{2}, 5\) and \(-\frac{8}{3}\).

Exercise \(\PageIndex{24}\)

Locate and label the following on a number line: \(\frac{1}{5}, -\frac{4}{5}, 3, \frac{7}{4}, -\frac{9}{2}, -5\) and \(\frac{8}{3}\).

In Exercise \(\PageIndex{25}\), we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• \(a < b\) “ a is less than b ” when a is to the left of b on the number line
• \(a > b\) “ a is greater than b ” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Exercise \(\PageIndex{25}\)

Order each of the following pairs of numbers, using \(<\) or \(>\). It may be helpful to refer Figure \(\PageIndex{5}\).

• \(−\frac{2}{3}\text{___}-1\)
• \(−3\frac{1}{2}\text{___}-3\)
• \(−\frac{3}{4}\text{___}-\frac{1}{4}\)
• \(−2\text{___}-\frac{8}{3}\)

Be careful when ordering negative numbers.

• \(\begin{array} { r r } { } & { - \frac { 2 } { 3 } \text{ ___ } -1 } \\ { - \frac { 2 } { 3 } \text { is to the right of } - 1 \text { on the number line. } } & { - \frac { 2 } { 3 } > - 1 } \end{array}\)
• \(\begin{array} { r r } { } & { - 3\frac { 1 } { 2 } \text{ ___ } -3 } \\ { - 3\frac { 1 } { 2 } \text { is to the right of } - 3 \text { on the number line. } } & { - \frac { 2 } { 3 } > - 1 } \end{array}\)
• \(\begin{array} { r r } { } & { - \frac { 3 } { 4 } \text{ ___ } -\frac{1}{4} } \\ { - \frac { 3 } { 4 } \text { is to the right of } - \frac{1}{4} \text { on the number line. } } & { - \frac{3}{4} < - \frac{1}{4} } \end{array}\)
• \(\begin{array} { r r } { } & { - \-2 \text{ ___ } -\frac{8}{3} } \\ { -2 \text { is to the right of } - \frac{8}{3} \text { on the number line. } } & { -2 > -\frac{8}{3} } \end{array}\)

Exercise \(\PageIndex{26}\)

Order each of the following pairs of numbers, using \(<\) or \(>\).

• \(−\frac{1}{3}\text{___}-1\)
• \(−1\frac{1}{2}\text{___}-2\)
• \(−\frac{2}{3}\text{___}-\frac{1}{3}\)
• \(−3\text{___}-\frac{7}{3}\)

Exercise \(\PageIndex{27}\)

• \(−1\text{___}-\frac{2}{3}\)
• \(−2\frac{1}{4}\text{___}-2\)
• \(−\frac{3}{5}\text{___}-\frac{4}{5}\)
• \(−4\text{___}-\frac{10}{3}\)

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Exercise \(\PageIndex{28}\)

Locate 0.4 on the number line.

A proper fraction has value less than one. The decimal number \(0.4\) is equivalent to \(\frac{4}{10}\), a proper fraction, so \(0.4\) is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts \(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0\). We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark \(0.4\) on the number line. See Figure \(\PageIndex{6}\).

Exercise \(\PageIndex{29}\)

Locate on the number line: 0.6.

Exercise \(\PageIndex{30}\)

Locate on the number line: 0.9.

Exercise \(\PageIndex{31}\)

Locate \(−0.74\) on the number line.

The decimal (−0.74\) is equivalent to \(-\frac{74}{100}\), so it is located between 0 and −1. On a number line, mark off and label the hundredths in the interval between 0 and −1. See Figure \(\PageIndex{7}\).

Exercise \(\PageIndex{32}\)

Locate on the number line: −0.6.

Exercise \(\PageIndex{33}\)

Locate on the number line: −0.7.

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So, \(0.40 > 0.04\)

Again, we can use the number line to order numbers.

Where are 0.04 and 0.40 located on the number line? See Figure \(\PageIndex{8}\).

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that \(0.40 > 0.04\).

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

Because \(310 > 308\), we know that \(\frac{310}{1000} > \frac{308}{1000}\). Therefore, \(0.31 > 0.308\).

Notice what we did in converting \(0.31\) to a fraction—we started with the fraction \(\frac{31}{100}\) and ended with the equivalent fraction \(\frac{310}{1000}\).Converting \(\frac{310}{1000}\) back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

\[\frac { 31 } { 100 } = \frac { 310 } { 1000 } \quad \text { and } \quad 0.31 = 0.310\]

We say 0.31 and 0.310 are equivalent decimals .

EQUIVALENT DECIMALS

Two decimals are equivalent if they convert to equivalent fractions.

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

ORDER DECIMALS.

• Write the numbers one under the other, lining up the decimal points.
• Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
• Compare the numbers as if they were whole numbers.
• Order the numbers using the appropriate inequality sign.

Exercise \(\PageIndex{34}\)

Order \(0.64 \text{ ___ } 0.6\) using \(<\) or \(>\).

\(\begin{array} { ll } { \text {Write the numbers one under the other, } } &{0.64} \\ { \text {lining up the decimal points. } } &{0.6} \\ \\ { \text {Add a zero to 0.6 to make it a decimal } } &{0.64} \\ {\text{with 2 decimal places.}} &{0.60} \\ {\text{Now they are both hundredths.}} &{} \\ \\ {\text{64 is greater than 60.}} &{64 > 60} \\ \\ {\text{64 hundredths is greater than 60 hundredths.}} &{0.64 > 0.60} \\ \\ {} &{0.64 > 0.6}\end{array}\)

Exercise \(\PageIndex{35}\)

Order each of the following pairs of numbers, using \(<\) or \(>\): \(0.42 \text{ ___ } 0.4\).

Exercise \(\PageIndex{36}\)

Order each of the following pairs of numbers, using \(<\) or \(>\): \(0.18 \text{ ___ } 0.1\).

Exercise \(\PageIndex{37}\)

Order \(0.83 \text{ ___ } 0.803\) using \(<\) or \(>\).

\(\begin{array} { ll } {} &{0.83\text{ ___ }0.803} \\ \\{ \text {Write the numbers one under the other, } } &{0.83} \\ { \text {lining up the decimal points. } } &{0.803} \\ \\ { \text {They do not have the same number of} } &{0.830} \\ {\text{digits.}} &{0.803} \\ {\text{Write one zero at the end of 0.83.}} &{} \\ \\ {\text{Since 830 > 803, 830 hundredths is}} &{0.830 > 0.803} \\ {\text{greater than 803 thousandths.}} &{}\\ \\ {} &{0.83 > 0.803}\end{array}\)

Exercise \(\PageIndex{38}\)

Order each of the following pairs of numbers, using \(<\) or \(>\): \(0.76 \text{ ___ } 0.706\).

Exercise \(\PageIndex{39}\)

Order each of the following pairs of numbers, using \(<\) or \(>\): \(0.305 \text{ ___ } 0.35\).

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because −2 lies to the right of -3 on the number line, we know that \(−2>−3\). Similarly, smaller numbers lie to the left on the number line. For example, because −9 lies to the left of −6 on the number line, we know that \(−9<−6\). See Figure \(\PageIndex{9}\).

If we zoomed in on the interval between 0 and −1, as shown in Exercise \(\PageIndex{40}\), we would see in the same way that \(−0.2>−0.3\) and \(−0.9<−0.6\).

Exercise \(\PageIndex{40}\)

Use \(<\) or \(>\) to order \(−0.1\text{ ___ }−0.8\).

\(\begin{array} { ll } {} &{-0.1 \text{ ___ } -0.8} \\ \\ { \text { Write the numbers one under the other, lining up the } } &{-0.1} \\ { \text { decimal points. } } &{-0.8} \\ { \text { They have the same number of digits. } } &{} \\ \\ { \text { since } - 1 > - 8 , - 1 \text { tenth is greater than } - 8 \text { tenths. } } &{-0.1 > -0.8} \end{array}\)

Exercise \(\PageIndex{41}\)

Order the following pair of numbers, using \(<\) or \(>\): \(−0.3\text{ ___ }−0.5\).

Exercise \(\PageIndex{42}\)

Order the following pair of numbers, using \(<\) or \(>\): \(−0.6\text{ ___ }−0.7\).

Key Concepts

• Square Root Notation \(\sqrt{m}\) is read ‘the square root of \(m\).’ If \(m = n^{2}\), then \(\sqrt{m} = n\), for \(n \geq 0\).

Practice Makes Perfect

In the following exercises, simplify.

• For Small Business

The Ultimate Guide to Value-Based Selling: Driving Customer Success in 2024

• May 14, 2024
• by steven-austin

In today‘s highly competitive business landscape, customers are increasingly savvy and discerning. They don‘t just want to hear about your product‘s features – they want to know how it will drive real value and help them achieve their goals. That‘s where value-based selling comes in.

Value-based selling is a customer-centric approach that focuses on understanding and aligning with the customer‘s needs, challenges, and objectives. Rather than simply pushing a product, value-based selling positions the seller as a trusted advisor who collaborates with the buyer to craft the best solution.

In this comprehensive guide, we‘ll dive deep into the principles, strategies, and emerging trends shaping value-based selling in 2024 and beyond. Whether you‘re a startup founder, sales leader, or aspiring entrepreneur, these insights will help you build lasting customer relationships and drive better business results.

Why Value-Based Selling Matters More Than Ever

The impact of value-based selling is clear and compelling. According to recent studies:

• Organizations using value-based selling outperform peers by 30% in revenue growth and 25% in profit margins (Forrester)
• 87% of buyers say sales reps can earning their trust by demonstrating deep knowledge of their business (LinkedIn)
• Companies focused on value delivery have net retention rates over 120% (ProfitWell)

In an age of empowered buyers and intense competition, value-based selling has become essential for several reasons:

Differentiation – When competitors are focused on product specs and pricing, emphasizing unique value sets you apart.

Trusted advisor status – By understanding the customer‘s world and providing expert guidance, you become a trusted partner.

Long-term relationships – Delivering ongoing value builds customer loyalty, retention, and advocacy.

Higher win rates – Value-based selling has been shown to increase deal size by 30-40% and win rates 3-4x (RAIN Group).

Margin protection – By quantifying your solution‘s business impact, you‘re less likely to get pushed into discounting.

The bottom line is – if you want to win in 2024 and beyond, value-based selling needs to be at the core of your sales approach.

The 7 Key Principles of Highly Effective Value-Based Selling

Excelling at value-based selling requires mastering a set of fundamental principles. These seven tenets form the foundation of a customer-centric, results-focused sales approach:

1. Lead with empathy and curiosity

The best value-based sellers don‘t just focus on closing the deal – they seek to deeply understand the customer‘s world. They put themselves in the buyer‘s shoes, asking thoughtful questions and actively listening to uncover unstated needs and aspirations.

Consider these findings:

• 69% of buyers say the best way to create a positive sales experience is to listen to their needs (Hubspot)
• Top-performing sales reps ask on average 60% more questions than their peers (Gong)
• 82% of buyers view sales reps as trusted advisors when they demonstrate empathy and problem-solving (LinkedIn)

To build empathy and curiosity in your sales conversations:

• Practice active listening, giving the buyer your full attention and reflecting back what you hear
• Ask open-ended questions that probe for challenges, objectives, and areas of opportunity
• Follow the "70/30 rule" – let the customer do 70% of the talking while you do 30%
• Seek to understand the buyer‘s personal wins and motivations, not just business goals

Expert Insight : "The best sales conversations feel like a collaborative exploration, not an interrogation. When you‘re genuinely curious about the other person‘s perspective, it creates space for them to share openly and discover value together." – John Barrows, CEO of JBarrows Sales Training

2. Do your homework

Value-based selling requires a deep understanding of the customer‘s business context. Top sellers thoroughly research their accounts, uncovering key stakeholders, industry trends, competitive dynamics, and potential pain points.

The impact of pre-call preparation is significant:

• Forrester found 62% of sales meetings fail due to lack of preparation
• 78% of buyers say sales calls feel more personalized when reps have done their research (LinkedIn)
• Gong‘s analysis of 519,291 discovery calls found a direct correlation between thorough research and win rates

To conduct effective pre-call research:

• Review the company‘s website, press releases, earnings reports, and strategic initiatives
• Leverage social selling tools to gain insights on key decision-makers
• Study industry analyst reports and competitor moves to identify market shifts
• Mine your CRM for past interactions and notes from colleagues
• Formulate a pre-call plan with key questions, hypotheses, and value points

Data Highlight : Sales reps using research and trigger event data have 2.1x higher quota attainment than non-users (SiriusDecisions)

3. Quantify value with real numbers

To make value tangible for buyers, top sellers quantify the business impact of their solutions. They work with customers to define success metrics, gather relevant data, and build credible models showing expected ROI.

The power of value quantification is proven:

• 74% of buyers choose the sales rep who first provides value quantification (Forrester)
• Deals where sellers quantified value saw 11% higher win rates and 35% higher average deal size (CEB)
• Value quantification tools can boost revenue per rep by 17% and deal size by 25% (Mediafly)

Keys to effective value quantification include:

• Aligning calculations to metrics the customer cares about most
• Using the customer‘s own data, benchmarks, and assumptions where possible
• Providing realistic low, medium, and high case scenarios vs. just best case
• Building in risk factors and intangible benefits for a balanced analysis
• Evolving and refining models throughout the sales process based on new information

Value quantification doesn‘t have to be overly complex to be impactful. Even a simple analysis showing how a solution could save time, reduce costs, or accelerate revenue can meaningfully shape buyer perceptions.

Customer Story : Blackbaud, a software provider for nonprofits, uses value-based selling to make a compelling case for its solutions. Sales reps work with prospects to understand their specific fundraising goals and challenges. They then create tailored "value roadmaps" showing how Blackbaud‘s platform can help increase donor engagement, retention, and gift size.

By translating product capabilities into measurable outcomes, Blackbaud‘s value-based approach has driven:

• 22% higher average win rate
• 12% increase in average sale price
• 30% reduction in sales cycle length

4. Engage buyers with insights and stories

In a world of information overload, buyers crave fresh insights that challenge their assumptions and expand their thinking. And they remember emotionally resonant stories far longer than dry facts and figures.

The most effective value-based sellers use a combination of insights and stories to create memorable, impactful sales conversations:

• Insights help reframe the customer‘s perspective by surfacing underappreciated problems, costs, or opportunities
• Stories make key points concrete and relatable by showing how similar customers have achieved value with your solution

When it comes to delivering insights:

• Focus on the "why" behind the "what" – don‘t just share data, but interpret what it means for the buyer
• Connect insights to the customer‘s specific situation vs. generic observations
• Use "you phrasing" to make insights personally relevant – "What this means for your business is…"
• Balance challenge and optimism – reveal costs of inaction while showing a path to positive outcomes

And for leveraging the power of storytelling:

• Choose stories that mirror the buyer‘s situation and demonstrate results they care about
• Use specific details and data points to make stories credible and concrete
• Emphasize the pivotal role your solution played in driving the customer‘s success
• Encourage the buyer to imagine themselves in the story – "Just like Company X, you could…"

Research Spotlight : Studies by Stanford professor Chip Heath found 63% could recall stories in a presentation vs. only 5% recalling statistics alone. Stories are up to 22x more memorable than facts.

5. Co-create solutions with customers

One of the most powerful principles of value-based selling is collaborating with customers to design and align solutions to their unique needs. Rather than pushing pre-baked offerings, top sellers act as "doctors of value" – diagnosing problems and prescribing tailored solutions.

The impact of this approach is compelling:

• 86% of buyers are more likely to purchase when sales reps collaborate with them (CSO Insights)
• Customers are 62% more likely to buy again from a vendor that engages them in co-creation (Accenture)

Tactics for co-creating value with buyers include:

• Uncovering and validating customer needs and use cases through discovery
• Sharing recommendations on how to best configure and deploy your solution
• Collaborating on success plans, timelines and KPIs for the engagement
• Facilitating ideation sessions to brainstorm applications and integrations
• Prototyping potential solutions and gathering feedback to refine the approach

Analogy : Think of value-based selling like building a custom home. You wouldn‘t construct the house and then try to sell it to the owner. You‘d work with them upfront to understand their vision, co-design blueprints, and adjust plans as you go to ensure the end result meets their needs.

6. Make selling a cross-functional sport

Delivering maximum customer value often requires marshalling expertise and resources across your organization. Value-based selling breaks down silos between sales, marketing, product, delivery and service to create a coordinated customer experience.

Consider these proof points:

• Companies with dynamic collaboration across the customer journey have 10% higher win rates (Altify)
• Firms with tightly aligned sales and marketing functions enjoy 38% higher sales win rates (MarketingProfs)

Some ways to enable cross-functional value selling:

• Conduct regular "win/loss reviews" with sales, product, and marketing to extract learnings
• Create councils with leaders from each function to share customer insights and align priorities
• Develop shared goals and metrics focused on customer value realization
• Leverage "voice of the customer" data across functions to identify improvement opportunities

Recommendation : Establish a recurring "Customer Value Council" with representatives from sales, success, services, marketing, and product. Have each function share insights on top value drivers, friction points, and opportunities to enhance the customer experience. Use this group to align priorities, message and enable the organization around customer value.

7. Align incentives to customer outcomes

What gets rewarded gets done. If you want your sales team to embrace value-based selling, align their incentives and enablement to customer success vs. just booking revenue.

Examples of this principle in practice:

• Tying a portion of variable comp to usage, retention, and expansion vs. just new bookings
• Basing sales quotas on long-term customer lifetime value vs just first year sales
• Providing SPIFs and recognition for delivering exceptional customer value and outcomes
• Measuring and rewarding key value-based selling behaviors vs. only results
• Investing in tools and training that support customer value discovery, analysis, and realization

Forward View : As recurring revenue models rise, expect to see sales compensation plans increasingly blend new sales with "consumption" metrics like adoption, utilization, and growth. Value to customers will be the ultimate measure of sales success.

Harnessing The Power of Value-Based Selling

Amidst rising buyer expectations and economic uncertainty, value-based selling has become an organizational imperative. By deeply understanding customers, quantifying impact, and aligning solutions to outcomes, sellers can elevate themselves from hype to substance, and transactions to partnerships.

But adopting this approach requires more than a quick fix sales methodology. It demands a fundamental shift in mindset, behaviors, and enablement. Sales leaders must infuse value-based principles into every aspect of their go-to-market – from messaging and content to sales plays and compensation.

The rewards of this shift are substantial – higher win rates, deal values, and customer loyalty. In a world where value is the ultimate differentiator, organizations that master value-based selling will enjoy a powerful edge over the competition.

As you seek to hone your own value-based selling skills, remember that success starts with genuine curiosity and customer-centricity. By putting your buyer‘s needs at the center of every interaction and decision, you lay the foundation for relationships built on trust, insight, and mutual value. Your customers won‘t see you as just another sales rep, but as an indispensable partner in their success. And that is the ultimate goal of mastering value-based selling.