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# 4.14: Properties of Rational Numbers

Difficulty Level: At Grade Created by: CK-12
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Eva just learned about rational numbers in math class. She learned that rational numbers are numbers that be written as a ratio of two integers. She understands that numbers like \begin{align*}\frac{1}{3}\end{align*} and \begin{align*}\frac{9}{5}\end{align*} are rational numbers. Eva's friend Mike says that all integers are rational numbers too, but Eva isn't sure if that is true. After all, integers don't look like fractions. How could Mike help to convince Eva that integers are rational numbers too?

In this concept, you will learn how to identify rational numbers.

### Identifying Rational Numbers

A ratio is a comparison between two numbers. Ratios can be written in words, in fraction form, or using a colon.

Here is an example.

Suppose your class has 12 boys and 13 girls. You could say that the ratio of boys to girls is:

• 12 to 13
• 12:13
• \begin{align*}\frac{12}{13}\end{align*}

All three forms are equivalent ways of expressing the same ratio.

The rational numbers are the set of numbers that can be written as a ratio of two integers. In other words, any number that can be written as a fraction where both the numerator and the denominator are integers is a rational number. Any number that cannot be written as a fraction in this way is not a rational number and is considered an irrational number.

Let's look at an example.

\begin{align*}\frac{-2}{3}\end{align*}

This number is already written as a fraction so it is definitely a rational number.

The answer is that \begin{align*}\frac{-2}{3}\end{align*} is a rational number because it is the ratio of -2 to 3.

Let's look at another example.

10

This number is not already written as a fraction; however, you could rewrite it as a fraction. Remember that any number divided by 1 is just itself. So you have

\begin{align*}10= \frac{10}{1}\end{align*}

The answer is that because 10 can be written as the ratio of 10 to 1, it is a rational number. In fact, all integers are rational numbers because they can all be written as a ratio of themselves to 1.

Numbers written in decimal form can also be rational numbers as long as they are repeating or terminating decimals.

Here is an example.

0.687

This number goes out to the thousandths place. You can read this number as 687 thousandths. Reading it in this way can help you to rewrite this number in fraction form:

\begin{align*}\frac{687}{1000}\end{align*}

Because 0.687 can be written as the ratio of 687 to 1000, it is a rational number.

### Examples

#### Example 1

Earlier, you were given a problem about Eva, who just learned about rational numbers.

Mike is trying to convince Eva that integers are rational numbers too. Eva understands that a rational number is a number that can be written in fraction form as a ratio of two integers.

If Mike wanted to convince Eva that integers are rational numbers too, he could give a specific example. For example, the integer -20.

-20 does not look like the ratio of two integers. However, it can be rewritten in fraction form.

\begin{align*}-20= \frac{-20}{1}\end{align*}

Because any integer could be similarly written as a ratio of itself to 1, all integers are rational numbers. In fact, integers make up a subset of the set of rational numbers.

#### Example 2

Show that the following number is rational by writing it as a ratio in fraction form.

.85

You read this number as eighty-five hundredths. You can convert it to a fraction:

\begin{align*}\frac{85}{100}\end{align*}

The answer is that .85 is a rational number because it can be written as the ratio \begin{align*}\frac{85}{100}\end{align*}.

#### Example 3

Determine whether or not the following number is rational. If it is rational, show how it can be written as a ratio in fraction form.

-4

This number is not already written as a fraction; however, it can be rewritten as a fraction.

\begin{align*}-4= \frac{-4}{1}\end{align*}

The answer is that because -4 can be written as the ratio of -4 to 1, it is a rational number.

#### Example 4

Determine whether or not the following number is rational. If it is rational, show how it can be written as a ratio in fraction form.

\begin{align*}0.33 \overline{3}\end{align*}

This is a repeating decimal and all repeating decimals are rational. This means you can rewrite this number in fraction form as the ratio of two integers.

\begin{align*}0.33 \overline{3}= \frac{1}{3} \end{align*}

The answer is that because \begin{align*}0.33 \overline{3}\end{align*} can be written as the ratio of 1 to 3, it is a rational number.

#### Example 5

Determine whether or not the following number is rational. If it is rational, show how it can be written as a ratio in fraction form.

\begin{align*}π \approx 3.14159 \ldots\end{align*}

\begin{align*}\pi\end{align*}, pronounced “pi” is an example of a number that is not rational. It is an irrational number. Its decimals continue on forever, but never repeat. It cannot be rewritten in fraction form.

### Review

Rewrite each number as the ratio of two integers to prove that each number is rational.

1. −11
2. \begin{align*}3 \frac{1}{6}\end{align*}
3. 9
4. 0.08
5. -0.34
6. 0.678
7. \begin{align*}\frac{4}{5}\end{align*}
8. -19
9. 25
10. 0.17
11. 0.2347
12. -17
13. 347
14. 87
15. -97

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
common denominator The common denominator is the least common multiple of the denominators of multiple fractions. Each fraction can be rewritten as an equivalent fraction using the common denominator.
Denominator The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
Equivalent Fractions Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
Irrational Number An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Least Common Denominator The least common denominator or lowest common denominator of two fractions is the smallest number that is a multiple of both of the original denominators.
Least Common Multiple The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers.
Lowest Common Denominator The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators.
Mixed Number A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.
Numerator The numerator is the number above the fraction bar in a fraction.
proper fraction A proper fraction has a numerator that is a lesser absolute value than the denominator. Proper fractions always represent values between -1 and 1.
rational number A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
reduce To reduce a fraction means to rewrite the fraction so that it has no common factors between numerator and denominator.
Repeating Decimal A repeating decimal is a decimal number that ends with a group of digits that repeat indefinitely. 1.666... and 0.9898... are examples of repeating decimals.
Terminating Decimal A terminating decimal is a decimal number that ends. The decimal number 0.25 is an example of a terminating decimal.

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