1.2 Forms of Rational Numbers

Learning Objectives:

You will learn to write a rational number as a ratio of two integers or as a terminating or repeating decimal.
You will recognize that pi is not rational but can be approximated.
You will understand that division of two integers will always result in a rational number.
You will recognize and generate the equivalent representations of positive and negative rational numbers including equivalent fractions.

Introduction

The Scuba Descent

Uri and his new diving partner Ariel are going to be buddies on a 40 foot dive. Ariel is a new diver and is still learning to make a descent. Uri can make a free descent quite easily. This means that he doesn’t hold onto anything as he descends to the appropriate depth. Ariel will hold onto the anchor line as she descends. Then they will meet on the bottom.

Uri has decided to go down with Ariel. He will descend freely next to her, while she descends holding onto the rope. He looks at his watch and sets the timer before they descend.

When they reach the bottom, Uri looks at his watch. He sees that the descent took them 2 minutes. Not bad at all considering that Ariel is a beginner. Uri and Ariel meet up with the group and check in with the Dive Master. Then they are off for a beautiful dive!!

How far did Uri and Ariel descend per minute?

To answer this question, you will need to understand how to divide integers.

Descent by Rafael Robayna / CC BY-NC-SA

Guided Learning

Identify a Rational Number as the Ratio of Two Integers

Example A

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 is 12 to 13. That same ratio could be also be expressed using a colon, 12 : 13, or as a fraction, $\frac{12}{13}$ .

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

How can we determine if an integer is a rational number? That is a good question. Let’s look at an example and see if we can write it as a ratio.

Example B

10 is a number that can be written as a ratio. Every 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. In other words, if it can be written as a fraction, it is a rational number.

10 is a rational number.

Example C

$-\frac{2}{3}$ is a fraction that is a rational number and is written as a ratio already. We are comparing the numerator and the denominator. Yes, it is negative and that is okay. We can have negative fractions and they are still considered rational numbers.

Example D

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

Example E

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, $0.\bar{3}$ 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 a later lesson.

Determine whether each is a rational number.

-4
$\frac{1}{3}$
0.89765....

Check your answers with a peer.

Real Life Example Completed

The Diving Descent

Here is the original problem once again. Reread it and underline any important information.

Uri has decided to go down with Ariel. He will descend freely next to Ariel, while she descends holding onto the rope. He looks at his watch and sets the timer before they descend.

How far did Cameron and Ariel descend per minute?

First, we need to write integers to represent the depth and the time.

-40 feet is the depth

2 minutes is the time

We divide the depth by the time to find out the number of feet per minute.

$-40 \div 2 = -20$

Uri and Ariel descended -20 feet per minute.

Review

Rational number

A number that can be written as a ratio is a rational number.

Irrational numbers

Decimals that don’t terminate and don’t repeat are a special group of numbers called irrational numbers.

Ratio

A ratio is a comparison of two numbers. A ratio can be expressed as 12 to 13, using a colon, 12 : 13, or as a fraction, $\frac{12}{13}$ .

Terminating decimals

Decimals with a set number of digits, are always rational. For example, 0.007 is a

terminating decimal, so it is rational.

Repeating decimals

Decimals in which one or more digits repeat, are always rational. For example, $0.\bar{3}$ is a repeating decimal in which the digit 3 repeats forever, so it is rational.

Practice Set

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

1. -11

2. $3 \frac{1}{6}$

3. 9

4. .08

5. -.34

6. .678

7. $\frac{4}{5}$

8. -19

9. 25

10. -7

Video Resources

Rational and Irrational Numbers

Terminating and Repeating Decimals