1. Operations with Real Numbers

# 1.7 The Real Numbers

Learning Objectives

By the end of this section it is expected that you will be able to:

• Identify integers, rational numbers, irrational numbers, and real numbers
• Locate fractions on the number line
• Locate decimals on the number line

# 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.

Rational Number

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

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

All signed fractions, such as 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

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 could be written as the decimal . 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 , we can write it as an improper fraction, . 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 is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

EXAMPLE 1

Write as the ratio of two integers: a) b) 7.31

Solution
 a) Write it as a fraction with denominator 1. b) Write it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths. Convert to an improper fraction.

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

TRY IT 1

Write as the ratio of two integers: a) b) 3.57

a) b)

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 for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.

 Integer Decimal form -2 -1 0 1 2 3 -2 -1 0 1 2 3

These decimal numbers stop.

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

Ratio of integers The decimal form – – .

These decimals either stop or repeat.

What do these examples tell us?

Every rational number can be written both as a ratio of integers, ,where p and q are integers and ,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:

Rational Number

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

Its decimal form stops or repeats.

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

The number (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. More on irrational numbers later on is this course.

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 2

Given the numbers . list the a) rational numbers b) irrational numbers.

Solution
 a) Look for decimals that repeat or stop. The 3 repeats in . The decimal 0.47 stops after the 7. So and 0.47 are rational. b) Look for decimals that neither stop nor repeat. has no repeating block of digits and it does not stop. So is irrational.

TRY IT 2

For the given numbers list the a) rational numbers b) irrational numbers: .

a) b)

EXAMPLE 3

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

Solution

a) Recognize that 36 is a perfect square, since So therefore is rational.

b) Remember that and so 44 is not a perfect square. Therefore, the decimal form of will never repeat and never stop, so is irrational.

TRY IT 3

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

a) rational b) 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.

Real Number

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

All the numbers we use in algebra are real numbers.  Figure 1 illustrates how the number sets we’ve discussed in this section fit together.

Do you remember that the square root of a negative number was not a real number?

EXAMPLE 4

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

Solution

a) There is no real number whose square is Therefore, is not a real number.

b) Since the negative is in front of the radical, is Since is a real number, is a real number.

TRY IT 4

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

a) not a real number b) real number

EXAMPLE 5

Given the numbers list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers.

Solution

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

b) The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so So the integers are

c) Since all integers are rational, then are rational. Rational numbers also include fractions and decimals that repeat or stop, so are rational. So the list of rational numbers is

d) Remember that 5 is not a perfect square, so is irrational.

e) All the numbers listed are real numbers.

TRY IT 5

For the given numbers, list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers:

a) b) c) d) e)

# 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.

We’ll start with the whole numbers and . because they are the easiest to plot. See Figure 2.

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

Similarly, is between 0 and . After dividing the unit into 5 equal parts we plot . See Figure 2.

Finally, look at the improper fractions . 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 2.

Figure 2 shows the number line with all the points plotted.

EXAMPLE 6

Locate and label the following on a number line: .

Solution

Locate and plot the integers, .

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

Now locate the improper fractions . It is easier to plot them if we convert them to mixed numbers and then plot them as described above: .

TRY IT 6

Locate and label the following on a number line: .

In Example 5, we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• a < ba is less than b” when a is to the left of b on the number line
• a > ba 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 7

Order each of the following pairs of numbers, using < or >. It may be helpful to refer Figure 3.

a) –____ b) -3____ c) ____ – d) ____ –

Solution
 a) – is to the right of on the number line. –___-1 – > -1 b) – is to the right of on the number line. c) is to the right of on the number line. d) is to the right of on the number line.

TRY IT 7

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

a) b) c) d) .

a) > b) > c) < d) <

# 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 8

Locate 0.4 on the number line.

Solution

A proper fraction has value less than one. The decimal number 0.4 is equivalent to , 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 4.

TRY IT 8

Locate on the number line: 0.6

EXAMPLE 9

Locate on the number line.

Solution

The decimal is equivalent to , so it is located between 0 and . On a number line, mark off and label the hundredths in the interval between 0 and . See Figure 5.

TRY IT 9

Locate on the number line: .

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,

>

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

• a < ba is less than b” when a is to the left of b on the number line
• a > ba 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 6.

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.

 0.31 0.308 Convert to fractions. We need a common denominator to compare them.

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

Notice what we did in converting 0.31 to a fraction—we started with the fraction and ended with the equivalent fraction . Converting 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.

HOW TO: Order Decimals.

1. Write the numbers one under the other, lining up the decimal points.
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.
3. Compare the numbers as if they were whole numbers.
4. Order the numbers using the appropriate inequality sign.

EXAMPLE 10

Order using < or >.

Solution
 Write the numbers one under the other, lining up the decimal points. Add a zero to 0.6 to make it a decimal with 2 decimal places. Now they are both hundredths. 64 is greater than 60. > 64 hundredths is greater than 60 hundredths. > >

TRY IT 10

Order each of the following pairs of numbers, using > .

>

EXAMPLE 11

Order using < or >.

Solution
 Write the numbers one under the other, lining up the decimals. They do not have the same number of digits. Write one zero at the end of 0.83. Since > , 830 thousandths is greater than 803 thousandths. > >

TRY IT 11

Order the following pair of numbers, using > .

>

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 lies to the right of on the number line, we know that > . Similarly, smaller numbers lie to the left on the number line. For example, because lies to the left of on the number line, we know that . See Figure 7.

If we zoomed in on the interval between 0 and , as shown in Example 10, we would see in the same way that > .

EXAMPLE 12

Use < or > to order .

Solution
 Write the numbers one under the other, lining up the decimal points. They have the same number of digits. Since > , −1 tenth is greater than −8 tenths. >

TRY IT 12

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

>

# Key Concepts

• Order Decimals
1. Write the numbers one under the other, lining up the decimal points.
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.
3. Compare the numbers as if they were whole numbers.
4. Order the numbers using the appropriate inequality sign.

# Glossary

equivalent decimals
Two decimals are equivalent if they convert to equivalent fractions.
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.
rational number
A rational number is a number of the form , where p and q are integers and . A rational number can be written as the ratio of two integers. Its decimal form stops or repeats.
real number
A real number is a number that is either rational or irrational

# 1.7 Exercise Set

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

1. 5
2. 3.19
1. 9.279
In the following exercises, list the a) rational numbers, b) irrational numbers
In the following exercises, list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers for each set of numbers.

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.
1. 0.8
In the following exercises, order each pair of numbers, using < or >.
1. 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 centre to serve 40 children.
1. How many teachers will be needed?
2. Why must the answer be a whole number?
3. Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

1. -7,
2. none
 7 8 9 10
1. <
2. >
3. >
4. <
 15 16
1. <
2. >
3. <
4. <
1. 4 buses