7.1 Rational and Irrational Numbers

Learning objectives

Learning objectives

• Identify rational numbers and irrational numbers

• Classify different types of real numbers

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this course! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you 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 fractions, both positive and negative, are rational numbers. A few examples are

Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this 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 all the counting numbers and all 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 7.1

Solution

1. Recall the ratio representation. A rational number is written as a fraction $\frac{a}{b}$ in which $a$ and $b$ are integers and $b$ is nonzero.
2. Convert the integer. Since $-15$ is an integer, place it over denominator $1$ to obtain $\frac{-15}{1}$.
3. Convert the decimal. Write $6.81$ as $6\frac{81}{100}$; multiplying $6$ by $100$ gives $600$, and $600+81=681$, so the improper fraction is $\frac{681}{100}$.
4. Convert the mixed number. For $-3\frac{6}{7}$, compute $3\cdot7+6=27$, so the improper fraction is $-\frac{27}{7}$.
5. Report the part answers. The ratios of integers are $\frac{-15}{1}$, $\frac{681}{100}$, and $-\frac{27}{7}$.

Takeaway: Any terminating decimal or mixed number can be rewritten as a ratio of integers, giving $-15/1$, $681/100$, and $-27/7$.

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.

$\begin{array}{lccl} \mathrm{Integer} & & & -2,-1,0,1,2,3 \\ \mathrm{Decimal} & & & -2.0,-1.0,0.0,1.0,2.0,3.0\,\mathrm{These decimal numbers stop.} \\ \end{array}$

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

$\begin{array}{lcccccccc} \mathrm{Ratio of Integers} & & \frac{4}{5}, & & -\frac{7}{8}, & & \frac{13}{4}, & -\frac{20}{3} & \\ \mathrm{Decimal Forms} & & 0.8, & & -0.875, & & 3.25, & \mathrm{-6.666…} & \mathrm{These decimals either stop or repeat.} \\ & & & & & & & -6.\overset{---}{66} \\ \end{array}$

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Irrational Numbers

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.

Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never stop and never repeat. For example,

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.

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

If the decimal form of a number

• stops or repeats, the number is rational.

• does not stop and does not repeat, the number is irrational.

Example 7.2

Solution

1. State the rational-number test. A decimal is rational if it either terminates or repeats, because then it can be written as a ratio of two integers.
2. Read the repeating bar. In $0.58\overline{3}$, the bar marks the digit $3$ as repeating forever, so the decimal has a repeating pattern.
3. Read the terminating decimal. The decimal $0.475$ stops after three places, so it can be written as $\frac{475}{1000}$ and reduced to $\frac{19}{40}$.
4. Read the decimal with an ellipsis. In $3.605551275\ldots$, the displayed digits continue without a repeating block, so the decimal neither stops nor repeats.
5. Classify each number. Parts ⓐ and ⓑ are rational, while part ⓒ is irrational.

Takeaway: Terminating and repeating decimals are rational because they can be written as ratios of integers, whereas a nonrepeating infinite decimal is irrational.

Let's think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

Example 7.3

Solution

1. Compare the radicand with a perfect square. A square root is rational exactly when its radicand is the square of an integer.
2. Evaluate $\sqrt{36}$. Since $6^{2}=36$, the square root is $6$, an integer, so $\sqrt{36}$ is rational.
3. Compare $44$ with neighboring squares. Because $6^{2}=36$ and $7^{2}=49$, the number $44$ lies between two consecutive perfect squares and is not itself a square.
4. Conclude the classification. Therefore $\sqrt{44}$ is not an integer, and its decimal form neither stops nor repeats, so $\sqrt{44}$ is irrational.

Takeaway: The square root of a perfect square is rational, but the square root of a non-square integer such as $44$ is irrational.

Classify Real Numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

Figure 7.2 illustrates how the number sets are related.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

Example 7.4

Solution

1. Test for whole numbers. The whole numbers are $0,1,2,3,\ldots$, and from the list only $8$ has this form.
2. Test for integers. The integers are the whole numbers, their opposites, and $0$, so $-7$ and $8$ qualify; also $-\sqrt{64}=-8$ because $8^{2}=64$, so it is an integer.
3. Test for rational numbers. Every integer is rational with denominator $1$, $\frac{14}{5}$ is already a ratio of integers, and $5.9=\frac{59}{10}$ is a terminating decimal.
4. Test for irrational numbers. The number $5$ is not a perfect square, so $\sqrt{5}$ cannot be expressed as a ratio of integers and is irrational.
5. Test for real numbers. Every listed value is either rational or irrational, so all of them are real.

Takeaway: The nested sets classify $8$ as whole, integer, and rational, $\rac{14}{5}$ and $5.9$ as rational, $\sqrt{5}$ as irrational, and every listed value as real.

Section 7.1 Exercises

Section 7.1 Exercises

Rational Numbers

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

In the following exercises, determine which of the given numbers are rational and which are irrational.

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

Classifying Real Numbers

In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

Everyday Math

Writing Exercises

Self Check

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

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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

1. Write each as the ratio of two integers: ⓐ $\,-19$ ⓑ $\,8.41.$

1. Identify each of the following as rational or irrational:

ⓐ $\,0.29$ⓑ $\,0.81\overset{-}{6}\,$ⓒ $\,\mathrm{2.515115111…}$

1. Identify each of the following as rational or irrational:

ⓐ $\,0.2\overset{-}{3}$ⓑ $\,0.125$ⓒ $\,\mathrm{0.418302…}$

1. Identify each of the following as rational or irrational:

1. ⓐ $\,\sqrt{81}$

2. ⓑ $\,\sqrt{17}$

1. Identify each of the following as rational or irrational:

1. ⓐ $\,\sqrt{116}$

2. ⓑ $\,\sqrt{121}$

1. 1. ⓐ $\,5$

2. ⓑ $\,3.19$

1. 1. ⓐ $\,-12$

2. ⓑ $\,9.279$

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

1. $0.\overset{---}{45}$, $\mathrm{1.919293…}$, $3.59$

1. 1. ⓐ $\,\sqrt{25}$

2. ⓑ $\,\sqrt{30}$

1. 1. ⓐ $\,\sqrt{164}$

2. ⓑ $\,\sqrt{169}$

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

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