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4.14: Properties of Rational Numbers

Difficulty Level: Basic Created by: CK-12
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Have you ever compared numbers?

Molly has spent the day skiing. She did 12 runs on the diamond trails and was very pleased with her speed and ability. She felt that out of the 12 runs, that 9 of them were particularly good.

She wrote 912\begin{align*}\frac{9}{12}\end{align*}.

Besides being a fraction, this is another type of number too.

Do you know what it is?

Guidance

Some numbers are considered rational numbers. A rational number is a number that can be written as a ratio.

What is a ratio?

A ratio is a comparison of two numbers. For example, you might discover that the ratio of boys to girls in your class one day was 12 to 13. That same ratio could be also be expressed using a colon, 12 : 13, or as a fraction, 1213\begin{align*}\frac{12}{13}\end{align*}.

In fact, any number that can be written as the ratio of two integers is classified as a rational number. Let's take a closer look at how to identify rational numbers now.

How can we determine if an integer is a rational number?

That is a good question. Let’s look at a value and see if we can write it as a ratio.

10

This number can be written as a ratio. Each whole number can be written over 1. That means that it can be written in the form of a ratio. Notice that the fraction bar is a way to tell if the integer can be written as a ratio.

10 is a rational number.

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

This fraction is a rational number. Notice that it is written as a ratio already. We are comparing the numerator and the denominator. Yes, it is negative. But that is okay, because we can have negative fractions. We call them rational numbers.

23\begin{align*}-\frac{2}{3}\end{align*} is a rational number.

.687

This decimal can be written as a rational number over 1000. This is a rational number too.

.687 is a rational number.

Are there any others?

Yes. Terminating decimals and repeating decimals are also rational numbers.

• Terminating decimals, which are decimals with a set number of digits, are always rational. For example, 0.007 is a terminating decimal, so it is rational.
• Repeating decimals, which are decimals in which one or more digits repeat, are always rational. For example, 0.3¯\begin{align*}0.\bar{3}\end{align*} is a repeating decimal in which the digit 3 repeats forever, so it is rational.

Are there any numbers that are not rational?

Yes. Some decimals don’t terminate and they don’t repeat. They just go on and on and on forever. These are a special group of numbers called irrational numbers. They are not rational numbers. You will learn more about them in another Concept.

Determine whether each is a rational number.

Example A

4\begin{align*}-4\end{align*}

Solution: Yes

Example B

13\begin{align*}\frac{1}{3}\end{align*}

Solution: Yes

Example C

.89765....\begin{align*}.89765....\end{align*}

Solution: No, it does not terminate or repeat.

Here is the original problem once again.

Molly has spent the day skiing. She did 12 runs on the diamond trails and was very pleased with her speed and ability. She felt that out of the 12 runs, that 9 of them were particularly good.

She wrote 912\begin{align*}\frac{9}{12}\end{align*}.

Besides being a fraction, this is another type of number too.

Do you know what it is?

Molly's comparison is a rational number. It is a ratio that is written in fraction form.

Other types of rational numbers are negative numbers, fractions, terminating and repeating decimals.

Vocabulary

Here are the vocabulary words used in this Concept.

Rational Number
any number positive or negative that can be written as a ratio.
Ratio
a comparison between two quantities. Can be written using the word “to”, using a colon, or using a fraction bar
Terminating Decimal
a decimal that has a definite ending
Repeating Decimal
a decimal where some of the digits repeat themselves.
Irrational Number
a decimal that does not terminate or repeat but continues indefinitely.

Guided Practice

Here is one for you to try on your own.

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

.85\begin{align*}.85\end{align*}

We can say that this is eighty -five hundredths.

Next, we convert it to a fraction.

85100\begin{align*}\frac{85}{100}\end{align*}

This is a rational number.

Video Review

Here is a video for review.

Practice

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

1.11\begin{align*}-11\end{align*}

2. 316\begin{align*}3 \frac{1}{6}\end{align*}

3. 9\begin{align*}9\end{align*}

4. .08\begin{align*}.08\end{align*}

5. .34\begin{align*}-.34\end{align*}

6..678\begin{align*}.678\end{align*}

7. 45\begin{align*}\frac{4}{5}\end{align*}

8.19\begin{align*}-19\end{align*}

9. 25\begin{align*}25\end{align*}

10. .17\begin{align*}.17\end{align*}

11. .2347\begin{align*}.2347\end{align*}

12. 17\begin{align*}-17\end{align*}

13. 347\begin{align*}347\end{align*}

14. 87\begin{align*}87\end{align*}

15. 97\begin{align*}-97\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English Spanish

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.

improper fractions

These are rational numbers where the numerator is greater than the denominator. Improper fractions can be rewritten as a mixed number – an integer plus a proper fraction. An improper fraction represents a number greater than one.

proper fractions

Rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one.

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.

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 $8$.

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.

reduce

To reduce a fraction means to rewrite the fraction so that it has no common factors between numerator and denominator.

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