problem solving real numbers

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).

Module 1: Algebra Essentials

Real numbers, learning outcomes.

• Classify a real number.
• Perform calculations using order of operations.
• Use the properties of real numbers.
• Evaluate and simplify algebraic expressions.

Because of the evolution of the number system, we can now perform complex calculations using several categories of real numbers. In this section we will explore sets of numbers, perform calculations with different kinds of numbers, and begin to learn about the use of numbers in algebraic expressions.

Classify 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 {1, 2, 3, …} 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: {0, 1, 2, 3,…}.

The set of integers adds the opposites of the natural numbers to the set of whole numbers: {…,-3, -2, -1, 0, 1, 2, 3,…}. 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.

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.

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

Example: Identifying Rational Numbers

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

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

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

• [latex]-\dfrac{5}{7}=-0.\overline{714285}[/latex], a repeating decimal
• [latex]\dfrac{15}{5}=3[/latex] (or 3.0), a terminating decimal
• [latex]\dfrac{13}{25}=0.52[/latex], a terminating decimal

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.

Example: 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]\dfrac{33}{9}[/latex]
• [latex]\sqrt{11}[/latex]
• [latex]\dfrac{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.
• [latex]\sqrt{11}:[/latex] This cannot be simplified any further. Therefore, [latex]\sqrt{11}[/latex] is an irrational number.
• 0.3033033303333… 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.

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 .

The real number line

Example: 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]-\dfrac{10}{3}[/latex]
• [latex]\sqrt{5}[/latex]
• [latex]-\sqrt{289}[/latex]
• [latex]-6\pi[/latex]
• [latex]0.616161\dots[/latex]
• [latex] 0.13 [/latex]
• [latex]-\dfrac{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]0.616161\dots [/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.
• [latex] 0.13 [/latex] is a finite decimal and may be written as 13/100.  So it is rational and positive.
• [latex]\sqrt{73}[/latex]
• [latex]-11.411411411\dots [/latex]
• [latex]\dfrac{47}{19}[/latex]
• [latex]-\dfrac{\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.

Sets of numbers.   N : the set of natural numbers   W : the set of whole numbers   I : the set of integers   Q : the set of rational numbers   Q´ : the set of irrational numbers

A General Note: Sets of Numbers

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

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

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

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

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

Example: Differentiating the Sets of Numbers

Classify each number as being a natural number, whole number, integer, rational number, and/or irrational number.

• [latex]\sqrt{36}[/latex]
• [latex]\dfrac{8}{3}[/latex]
• [latex]-6[/latex]
• [latex]3.2121121112\dots [/latex]

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

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

Properties of Real Numbers

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 \dfrac{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.

A General Note: 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)

How To: 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.

Example: 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]\dfrac{{5}^{2}-4}{7}-\sqrt{11 - 2}[/latex]
• [latex]6-|5 - 8|+3\left(4 - 1\right)[/latex]
• [latex]\dfrac{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{align}\left(3\cdot 2\right)^{2} & =\left(6\right)^{2}-4\left(8\right) && \text{Simplify parentheses} \\ & =36-4\left(8\right) && \text{Simplify exponent} \\ & =36-32 && \text{Simplify multiplication} \\ & =4 && \text{Simplify subtraction}\end{align}[/latex]

[latex]\begin{align}\frac{5^{2}-4}{7}-\sqrt{11-2} & =\frac{5^{2}-4}{7}-\sqrt{9} && \text{Simplify grouping systems (radical)} \\ & =\frac{5^{2}-4}{7}-3 && \text{Simplify radical} \\ & =\frac{25-4}{7}-3 && \text{Simplify exponent} \\ & =\frac{21}{7}-3 && \text{Simplify subtraction in numerator} \\ & =3-3 && \text{Simplify division} \\ & =0 && \text{Simplify subtraction}\end{align}[/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{align}6-|5-8|+3\left(4-1\right) & =6-|-3|+3\left(3\right) && \text{Simplify inside grouping system} \\ & =6-3+3\left(3\right) && \text{Simplify absolute value} \\ & =6-3+9 && \text{Simplify multiplication} \\ & =3+9 && \text{Simplify subtraction} \\ & =12 && \text{Simplify addition}\end{align}[/latex]

[latex]\begin{align}\frac{14-3\cdot2}{2\cdot5-3^{2}} & =\frac{14-3\cdot2}{2\cdot5-9} && \text{Simplify exponent} \\ & =\frac{14-6}{10-9} && \text{Simplify products} \\ & =\frac{8}{1} && \text{Simplify quotient} \\ & =8 && \text{Simplify quotient}\end{align}[/latex] In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

[latex]\begin{align}7\left(5\cdot3\right)-2[\left(6-3\right)-4^{2}]+1 & =7\left(15\right)-2[\left(3\right)-4^{2}]+1 && \text{Simplify inside parentheses} \\ & 7\left(15\right)-2\left(3-16\right)+1 && \text{Simplify exponent} \\ & =7\left(15\right)-2\left(-13\right)+1 && \text{Subtract} \\ & =105+26+1 && \text{Multiply} \\ & =132 && \text{Add}\end{align}[/latex]

Watch the following video for more examples of using the order of operations to simplify an expression.

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\div 5\ne 5\div 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.

Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.

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

A General Note: Properties of Real Numbers

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

Example: Using Properties of Real Numbers

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]\dfrac{4}{7}\cdot \left(\frac{2}{3}\cdot \dfrac{7}{4}\right)[/latex]
• [latex]100\cdot \left[0.75+\left(-2.38\right)\right][/latex]

[latex]\begin{align}3\cdot6+3\cdot4 &=3\cdot\left(6+4\right) && \text{Distributive property} \\ &=3\cdot10 && \text{Simplify} \\ & =30 && \text{Simplify}\end{align}[/latex]

[latex]\begin{align}\left(5+8\right)+\left(-8\right) &=5+\left[8+\left(-8\right)\right] &&\text{Associative property of addition} \\ &=5+0 && \text{Inverse property of addition} \\ &=5 &&\text{Identity property of addition}\end{align}[/latex]

[latex]\begin{align}6-\left(15+9\right) & =6+(-15-9) && \text{Distributive property} \\ & =6+\left(-24\right) && \text{Simplify} \\ & =-18 && \text{Simplify}\end{align}[/latex]

[latex]\begin{align}\frac{4}{7}\cdot\left(\frac{2}{3}\cdot\frac{7}{4}\right) & =\frac{4}{7} \cdot\left(\frac{7}{4}\cdot\frac{2}{3}\right) && \text{Commutative property of multiplication} \\ & =\left(\frac{4}{7}\cdot\frac{7}{4}\right)\cdot\frac{2}{3} && \text{Associative property of multiplication} \\ & =1\cdot\frac{2}{3} && \text{Inverse property of multiplication} \\ & =\frac{2}{3} && \text{Identity property of multiplication}\end{align}[/latex]

[latex]\begin{align}100\cdot[0.75+\left(-2.38\right)] & =100\cdot0.75+100\cdot\left(-2.38\right) && \text{Distributive property} \\ & =75+\left(-238\right) && \text{Simplify} \\ & =-163 && \text{Simplify}\end{align}[/latex]

• [latex]\left(-\dfrac{23}{5}\right)\cdot \left[11\cdot \left(-\dfrac{5}{23}\right)\right][/latex]
• [latex]5\cdot \left(6.2+0.4\right)[/latex]
• [latex]18-\left(7 - 15\right)[/latex]
• [latex]\dfrac{17}{18}+\cdot \left[\dfrac{4}{9}+\left(-\dfrac{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]\dfrac{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

Evaluate and Simplify 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, 5[/latex] 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.

Example: 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]

Example: 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=\dfrac{1}{2}[/latex]
• [latex]x=-4[/latex]
• Substitute 0 for [latex]x[/latex]. [latex]\begin{align}2x-7 & = 2\left(0\right)-7 \\ & =0-7 \\ & =-7\end{align}[/latex]
• Substitute 1 for [latex]x[/latex]. [latex]\begin{align}2x-7 & = 2\left(1\right)-7 \\ & =2-7 \\ & =-5\end{align}[/latex]
• Substitute [latex]\dfrac{1}{2}[/latex] for [latex]x[/latex]. [latex]\begin{align}2x-7 & = 2\left(\frac{1}{2}\right)-7 \\ & =1-7 \\ & =-6\end{align}[/latex]
• Substitute [latex]-4[/latex] for [latex]x[/latex]. [latex]\begin{align}2x-7 & = 2\left(-4\right)-7 \\ & =-8-7 \\ & =-15\end{align}[/latex]

Example: Evaluating Algebraic Expressions

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

In the following video we present more examples of how to evaluate an expression for a given value.

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 [latex]x=3[/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].

Example: 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]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\pi[/latex].

Right circular cylinder

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

Simplify 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.

Example: Simplifying 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-\dfrac{5}{4}s\right)-\left(\dfrac{2}{3}t+2s\right)[/latex]
• [latex]2mn - 5m+3mn+n[/latex]
• [latex]\begin{align}3x-2y+x-3y-7 & =3x+x-2y-3y-7 && \text{Commutative property of addition} \\ & =4x-5y-7 && \text{Simplify} \\ \text{ }\end{align}[/latex]
• [latex]\begin{align}2r-5\left(3-r\right)+4 & =2r-15+5r+4 && \text{Distributive property}\\&=2r+5r-15+4 && \text{Commutative property of addition} \\ & =7r-11 && \text{Simplify} \\ \text{ }\end{align}[/latex]
• [latex]\begin{align} 4t-\frac{5}{4}s -\left(\frac{2}{3}t+2s\right) &=4t-\frac{5}{4}s-\frac{2}{3}t-2s &&\text{Distributive property}\\&=4t-\frac{2}{3}t-\frac{5}{4}s-2s && \text{Commutative property of addition}\\&=\frac{12}{3}t-\frac{2}{3}t-\frac{5}{4}s-\frac{8}{4}s && \text{Common Denominators}\\ & =\frac{10}{3}t-\frac{13}{4}s && \text{Simplify} \\ \text{ }\end{align}[/latex]
• [latex]\begin{align}mn-5m+3mn+n & =2mn+3mn-5m+n && \text{Commutative property of addition} \\ & =5mn-5m+n && \text{Simplify}\end{align}[/latex]

Example: 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.

algebraic expression  constants and variables combined using addition, subtraction, multiplication, and division

associative property of addition  the sum of three numbers may be grouped differently without affecting the result; in symbols, [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]

associative property of multiplication  the product of three numbers may be grouped differently without affecting the result; in symbols, [latex]a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c[/latex]

base  in exponential notation, the expression that is being multiplied

commutative property of addition  two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[/latex]

commutative property of multiplication  two numbers may be multiplied in any order without affecting the result; in symbols, [latex]a\cdot b=b\cdot a[/latex]

constant  a quantity that does not change value

distributive property  the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]

equation  a mathematical statement indicating that two expressions are equal

exponent  in exponential notation, the raised number or variable that indicates how many times the base is being multiplied

exponential notation  a shorthand method of writing products of the same factor

formula  an equation expressing a relationship between constant and variable quantities

identity property of addition  there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, [latex]a+0=a[/latex]

identity property of multiplication  there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, [latex]a\cdot 1=a[/latex]

integers  the set consisting of the natural numbers, their opposites, and 0: [latex]\{\dots ,-3,-2,-1,0,1,2,3,\dots \}[/latex]

inverse property of addition  for every real number [latex]a[/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]-a[/latex], which, when added to the original number, results in the additive identity, 0; in symbols, [latex]a+\left(-a\right)=0[/latex]

inverse property of multiplication  for every non-zero real number [latex]a[/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\dfrac{1}{a}[/latex], which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, [latex]a\cdot \dfrac{1}{a}=1[/latex]

irrational numbers  the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers

natural numbers  the set of counting numbers: [latex]\{1,2,3,\dots \}[/latex]

order of operations  a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations

rational numbers  the set of all numbers of the form [latex]\dfrac{m}{n}[/latex], where [latex]m[/latex] and [latex]n[/latex] are integers and [latex]n\ne 0[/latex]. Any rational number may be written as a fraction or a terminating or repeating decimal.

real number line  a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.

real numbers  the sets of rational numbers and irrational numbers taken together

variable  a quantity that may change value

whole numbers  the set consisting of 0 plus the natural numbers: [latex]\{0,1,2,3,\dots \}[/latex]

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Real Numbers

These lessons, with videos, examples and solutions, explain what real numbers are and some of their properties.

Related Pages The Real Number System Properties Of Real Numbers More Lessons for SAT Math Math Worksheets

The following diagram shows real numbers are made up of rational numbers, integers, whole numbers, and irrational numbers. Scroll down the page for more examples and solutions on real numbers and their properties.

Introduction to Real Numbers When analyzing data, graphing equations and performing computations, we are most often working with real numbers. Real numbers are the set of all numbers that can be expressed as a decimal or that are on the number line. Real numbers have certain properties and different classifications, including natural, whole, integers, rational and irrational.

This video goes over the basics of the real number system that is mainly used in Algebra. The video covers rational numbers, and irrational numbers.

Properties of Real Numbers When analyzing data or solving problems with real numbers, it can be helpful to understand the properties of real numbers. These properties of real numbers, including the Associative, Commutative, Multiplicative and Additive Identity, Multiplicative and Additive Inverse, and Distributive Properties, can be used not only in proofs, but in understanding how to manipulate and solve equations.

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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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• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Elementary Algebra 2e
• Publication date: Apr 22, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/elementary-algebra-2e/pages/1-8-the-real-numbers

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