# 9.6: Properties of Rational Numbers versus Irrational Numbers

**At Grade**Created by: CK-12

**Practice**Properties of Rational Numbers versus Irrational Numbers

Have you ever tiled a room? Miguel's sister Kelly is working on that very project. Take a look.

Kelly wants to use this tile to finish the floor in her room. The room is a square room and the area of the floor is 324 square feet. She wants to put 8” tiles along the edge of the floor.

What is the length of one side of the room? How many tiles can she fit along one side?

**This Concept will teach you how to solve a problem just like this one.**

### Guidance

When you were finding square roots using a calculator, you probably noticed that each time you found the square root of a number that was not a perfect square that there were many decimal places following the decimal point.

**This means that the square root of that number was not a whole number. It was a whole number and some fraction of parts.**

The key thing to notice about each of the numbers that we worked with in the last section is that they each had many decimal places. When we find the square root of a number that is not a perfect square it brings another type of number into our work.

\begin{align*}\sqrt{16} = 4\end{align*}

Sixteen is a perfect square. So the square root is a whole number.

\begin{align*}\sqrt{50}\end{align*}

We use a calculator and here is our answer: 7.071067812...

**Yes. There are a lot of digits after the decimal point, and you can see that the dots after the last digit tell us that the numbers go on and on and on. We can round to the nearest tenth to see that 7.1 is a likely approximation for this square root.**

**This square root is a decimal that does not end. Numbers like this one are called** *irrational numbers***.** Anytime that you have a decimal that does not repeat and does not end; it is a member of the group of numbers called irrational numbers.

*Often the little dots at the end of the decimal will let you know that you are working with an irrational number.*

Not all numbers are ** irrational numbers**, some are

**. You have learned about rational numbers in another section of this book. Let’s review what defines a rational number for just a minute.**

*rational numbers*
*Rational Numbers***are numbers that can be written in the form of \begin{align*}\frac{a}{b}\end{align*} ab. They can be whole numbers, fractions, negative numbers, decimals and repeating decimals.**

Here are some rational numbers.

\begin{align*}.56 \qquad \frac{1}{2} \qquad .6\bar{6} \qquad 33\frac{1}{3} \qquad 9 \qquad -23\end{align*}

*Irrational numbers***complete the group of numbers that are not rational numbers.** This means that any decimal that does not end is irrational. The square root of any number that is not a perfect square is irrational.

**There is one famous irrational number. Do you know what it is?**

**The most famous irrational number is a number called pi or \begin{align*}\pi\end{align*} π. We use the number 3.14 to represent pi, but it is really an irrational number without an end point. It just goes on and on and on. Pi is a special ratio that you will learn about when we work with circles very soon.**

**Rational and Irrational Numbers make up the set of** *Real Numbers.***All numbers are considered real numbers whether they are rational or irrational.** So, you can look at a number, know that it is a real number and then classify it further as rational or irrational.

Identify each of the following numbers are rational or irrational.

#### Example A

**4.567....**

**Solution: Irrational number**

#### Example B

\begin{align*}\sqrt{25}\end{align*}

**Solution: Rational number**

#### Example C

\begin{align*}\sqrt{144}\end{align*}

**Solution: Rational Number**

Here is the original problem once again.

Kelly wants to use this tile to finish the floor in her room. The room is a square room and the area of the floor is 324 square feet. She wants to put 8” tiles along the edge of the floor. What is the length of one side of the room? How many tiles can she fit along one side?

We have to look at what information we have been given.

We know that the area of the room is a square and that it is 324 square feet.

We know that she is using 8” tiles along the border of the room and that she wants to figure out how many she will need.

**First, we need to figure out the length of one side of the room.**

This room is a square, so we can find the square root of the area of the room and that will tell us the side length.

\begin{align*}\sqrt{324} = 18 \ feet\end{align*}

Each side of the room is 18 feet.

**Now we can figure out the tiles.**

The tiles are measured in inches. The length of the room is in feet, so we have to first convert feet back to inches. There are 12 inches in 1 foot, so we multiply \begin{align*}18 \times 12\end{align*}

\begin{align*}18 \times 12 = 216''\end{align*}

Now each tile is 8” so we divide 216 by 8

\begin{align*}216 \div 8 = 27 \ tiles\end{align*}

She will need 27 tiles for each side of the room.

### Vocabulary

Here are the vocabulary words in this Concept.

- Square Root
- A number multiplied to find a product. The number squared is the square root of the product.

- Perfect Square
- a number whose square root is a whole number.

- Tabular Interpolation
- using a table to find approximate square roots.

- Irrational Numbers
- the set of numbers whose decimal digits do not end, they continue indefinitely. A square root that is not a perfect square is irrational.

- Rational Numbers
- the set of numbers that can be written in \begin{align*}a/b\end{align*} form.

- Real Numbers
- the set of numbers that includes all numbers whether they are rational or irrational.

### Guided Practice

Here is one for you to try on your own.

Is this value a rational or irrational number? Why?

### Video Review

Here is a video for review.

This is a James Sousa video on the set of real numbers.

### Practice

Directions: Identify each of the following numbers as rational or irrational.

1. .345....

2. 2

3. -9

4. -122

5. 3.456....

6. \begin{align*}\sqrt{25}\end{align*}

7. \begin{align*}\sqrt{16}\end{align*}

8. \begin{align*}\sqrt{12}\end{align*}

9. \begin{align*}\sqrt{38}\end{align*}

10. -4.56

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

12. \begin{align*}- \frac{4}{5}\end{align*}

13. 9.8712....

14. -19

15. 2,345

approximate solution

An approximate solution to a problem is a solution that has been rounded to a limited number of digits.Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.Perfect Square

A perfect square is a number whose square root is an integer.principal square root

The principal square root is the positive square root of a number, to distinguish it from the negative value. 3 is the principal square root of 9; -3 is also a square root of 9, but it is not principal square root.### Image Attributions

Here you'll learn to recognize irrational numbers and the set of real numbers.