1.8 The Real Numbers

Learning objectives

Learning objectives

• 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

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,

Similarly, 121 is the square of 11, because ${11}^{2}$ is 121.

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 ${10}^{2}=100,$ we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is $m$ is called 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.

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.

Example 1.108

Solution

Table

1.32

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 100.”

Example 1.109

Solution

Table

1.33

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

We have already described numbers as counting numbers, whole numbers, 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.

All signed fractions, such as $\frac{4}{5},-\,\frac{7}{8},\frac{13}{4},-\,\frac{20}{3}$ 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 $\frac{3}{1},\frac{6}{2},\frac{9}{3},\frac{12}{4},\frac{15}{5}\ldots$

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

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 $7\frac{3}{10},$ we can write it as an improper fraction, $\frac{73}{10}.$ 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. Simply write the decimal as a mixed number.

Example 1.110

Solution

Table

1.34

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

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 $a=\frac{a}{1}$ 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, $(\frac{p}{q},$ where p and q are integers and $q\ne{}0),$ 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:

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.

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.

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

Example 1.111

Solution

Table

1.35

Example 1.112

Solution

1. ⓐ Recognize that 36 is a perfect square, since ${6}^{2}=36.$ So $\sqrt{36}=6,$ therefore $\sqrt{36}$ is rational.

2. ⓑ 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.

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 numbers.

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 $\sqrt{-25}?$ Is there a number whose square is $-25?$

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.

Example 1.113

Solution

1. ⓐ There is no real number whose square is $-169.$ Therefore, $\sqrt{-169}$ is not a real number.

2. ⓑ 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.

Example 1.114

Solution

1. ⓐ Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.

2. ⓑ 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 $-7,8,-\sqrt{64}.$

3. ⓒ Since all integers are rational, then $-7,8,-\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5}\,\mathrm{and}\,5.9$ are rational. So the list of rational numbers is $-7,\frac{14}{5},8,5.9,-\sqrt{64}.$

4. ⓓ Remember that 5 is not a perfect square, so $\sqrt{5}$ is irrational.

5. ⓔ All the numbers listed are real numbers.

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 fractions and decimals on it.

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

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

The proper fractions listed are $\frac{1}{5}\,\mathrm{and}\,-\,\frac{4}{5}.$ We know the proper fraction $\frac{1}{5}$ has value less than one and so would be located between $\mathrm{0 and 1.}$ 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 1.16.

Similarly, $-\,\frac{4}{5}$ is between 0 and $-1.$ After dividing the unit into 5 equal parts we plot $-\,\frac{4}{5}.$ See Figure 1.16.

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 1.16.

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

Example 1.115

Solution

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: $\frac{6}{5}=1\frac{1}{5},-\,\frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$

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

Solution

Table

1.36

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

Solution

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 1.18.

Example 1.118

Solution

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 1.19.

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.

• 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

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 $\frac{310}{1000}>\frac{308}{1000}.$ 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!

We say 0.31 and 0.310 are equivalent decimals.

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

Example 1.119

Solution

Table

1.37

Example 1.120

Solution

Table

1.38

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 1.21.

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

Example 1.121

Solution

Table

1.39

Section 1.8 Exercises

Section 1.8 Exercises

Simplify Expressions with Square Roots

In the following exercises, simplify.

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

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

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

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

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

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

Locate Fractions on the Number Line

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

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

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 >.

Everyday Math

Writing Exercises

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

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

1. Simplify: ⓐ $\sqrt{36}$ ⓑ $\sqrt{169}.$

2. Simplify: ⓐ $\sqrt{16}$ ⓑ $\sqrt{196}.$

3. Simplify: ⓐ $-\sqrt{4}$ ⓑ $-\sqrt{225}.$

4. Simplify: ⓐ $-\sqrt{81}$ ⓑ $-\sqrt{100}.$

1. Write as the ratio of two integers: ⓐ $-24$ ⓑ 3.57.

2. Write as the ratio of two integers: ⓐ $-19$ ⓑ 8.41.

3. For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: $0.29,0.81\overset{-}{6},2.515115111\ldots.$

4. For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: $2.6\overset{-}{3},0.125,0.418302\ldots$

1. For each number given, identify whether it is rational or irrational: ⓐ $\sqrt{81}$ ⓑ $\sqrt{17}.$

2. For each number given, identify whether it is rational or irrational: ⓐ $\sqrt{116}$ ⓑ $\sqrt{121}.$

3. For each number given, identify whether it is a real number or not a real number: ⓐ $\sqrt{-196}$ ⓑ $-\sqrt{81}.$

4. For each number given, identify whether it is a real number or not a real number: ⓐ $-\sqrt{49}$ ⓑ $\sqrt{-121}.$

1. For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: $-3,-\sqrt{2},0.\overset{-}{3},\frac{9}{5},4,\sqrt{49}.$

2. For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: $-\sqrt{25},-\,\frac{3}{8},-1,6,\sqrt{121},2.041975\ldots$

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

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

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

ⓐ $-\,\frac{1}{3}\mathrm{\_\_\_}-1$ ⓑ $-1\frac{1}{2}\mathrm{\_\_\_}-2$ ⓒ $-\,\frac{2}{3}\mathrm{\_\_\_}-\,\frac{1}{3}$ ⓓ $-3\mathrm{\_\_\_}-\,\frac{7}{3}.$

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

ⓐ $-1\mathrm{\_\_\_}-\,\frac{2}{3}$ ⓑ $-2\frac{1}{4}\mathrm{\_\_\_}-2$ ⓒ $-\,\frac{3}{5}\mathrm{\_\_\_}-\,\frac{4}{5}$ ⓓ $-4\mathrm{\_\_\_}-\,\frac{10}{3}.$

3. Locate on the number line: 0.6.

4. Locate on the number line: 0.9.

1. Locate on the number line: $-0.6.$

2. Locate on the number line: $-0.7.$

3. Order each of the following pairs of numbers, using $<\,\mathrm{or}\,>:\,0.42\mathrm{\_\_\_}0.4.$

4. Order each of the following pairs of numbers, using $<\,\mathrm{or}\,>:\,0.18\mathrm{\_\_\_}0.1.$

1. Order the following pair of numbers, using $<\,\mathrm{or}\,>:\,0.76\mathrm{\_\_\_}0.706.$

2. Order the following pair of numbers, using $<\,\mathrm{or}\,>:\,0.305\mathrm{\_\_\_}0.35.$

3. Order the following pair of numbers, using < or >: $-0.3\mathrm{\_\_\_}-0.5.$

4. Order the following pair of numbers, using < or >: $-0.6\mathrm{\_\_\_}-0.7.$

1. $-\sqrt{4}$

2. $-\sqrt{1}$

3. ⓐ 5 ⓑ 3.19

4. ⓐ $-12\,$ ⓑ 9.279

1. $0.75,0.22\overset{-}{3},1.39174\ldots$

2. $0.4\overset{-}{5},1.919293\ldots,3.59$

3. ⓐ $\sqrt{25}$ ⓑ $\sqrt{30}$

4. ⓐ $\sqrt{164}$ ⓑ $\sqrt{169}$

1. ⓐ $-\sqrt{81}$ ⓑ $\sqrt{-121}$

2. ⓐ $\sqrt{-36}$ ⓑ $-\sqrt{144}$

3. $-8,0,1.95286\ldots,\frac{12}{5},\sqrt{36},9$

4. $-\sqrt{100},-7,-\,\frac{8}{3},-1,0.77,3\frac{1}{4}$

1. $\frac{3}{4},\frac{8}{5},\frac{10}{3}$

2. $\frac{3}{10},\frac{7}{2},\frac{11}{6},4$

3. $\frac{2}{5},-\,\frac{2}{5}$

4. $\frac{3}{4},-\,\frac{3}{4},1\frac{2}{3},-1\frac{2}{3},\frac{5}{2},-\,\frac{5}{2}$

1. $-1\mathrm{\_\_\_}-\,\frac{1}{4}$

2. $-2\frac{1}{2}\mathrm{\_\_\_}-3$

3. $-\,\frac{5}{12}\mathrm{\_\_\_}-\,\frac{7}{12}$

4. $-3\mathrm{\_\_\_}-\,\frac{13}{5}$

1. $-1.6$

2. $0.37\mathrm{\_\_\_}0.63$

3. $0.91\mathrm{\_\_\_}0.901$

4. $-0.5\mathrm{\_\_\_}-0.3$

1. $-0.62\mathrm{\_\_\_}-0.619$

2. 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?

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