IESN - Rational and Irrational Numbers Lesson

Rational and Irrational Numbers

A rational number is a number that can be written as a fraction, or a ratio, of two numbers. This, in turn, means it can be written as a decimal. However, this decimal has to be a terminating or repeating decimal.

An irrational number, on the other hand, is a number that cannot be written as a fraction, or a ratio, of two numbers. If you were to write it as a decimal, it would be a non-terminating or non-repeating decimal. The most common irrational number is $\Pi$ , Pi.

Let's look at a few numbers and decide if they are rational or irrational.

4/9
4/9 = .444444....
Since this is a repeating decimal, we say that it is rational.

1. $\sqrt[]{2}$
   $\sqrt[]{2}$ = 1.414213562... ...
   Since this decimal is non-terminating, meaning it never stops and it is non-repeating, we say that it is irrational.

Now you try... and check your answer.

Question: $\sqrt[]{16}$  Rational or Irrational?

• • • • Solution: Rational

Question:$\frac{24}{5}$  Rational or Irrational?

• Solution: Rational

Question: $\sqrt[]{17}$  Rational or Irrational?

• Solution: Irrational

Question: $\Pi$ Rational or Irrational?

• Irrational

You may or may not remember how to convert decimals to fractions and fractions to decimals, so we are going to spend a little time with that concept. Let's start with finding the decimal expansion of a fraction.

1
25

First, recall what a fraction bar means. It is a symbol for which operation? If you said division, you are exactly right! The above problem can be read as "one twenty-fifth," but it also means "one divided by 25." By dividing the denominator into the numerator, we can change this fraction to its decimal equivalent.

$\frac{1}{25}=\hspace{5mm}{25}\\ \hspace{7mm}25\overline{)1.00}$

Here are a few more for you to do to refresh your memory with this concept. 

Question: $\frac{1}{10}$ 

• Solution: .1

Question: $\frac{2}{3}$ 

• Solution: .6666....

Question: $\frac{25}{100}$ 

• Solution: .25

Now, let's consider going in the other direction. What if we have a decimal and we want it to be a fraction.

Consider how you read 0.24. We say "twenty-four hundredths," right? Then, that's how we write it as a fraction. Then all we do is reduce or simplify. Let's look....

 $0.24=\frac{24}{100}=\frac{6}{25}$

You can also put all the digits after the decimal in the numerator and then count the number of decimal places you have and using that power of ten in the denominator.

24 is what is after the decimal, so put it in the numerator. Now, there are two decimal places, so we want the power of ten that has two zeros...that would be 100, so our denominator is 100.

Either way you do this is fine. Some folks prefer one way over the other. It is personal preference!

You try!

Question: .2  

• Solution: $\frac{1}{5}$

Question: .555     

• Solution: $\frac{111}{200}$

Question: 5.24  

• Solution: $5\frac{6}{25}$

What if we have a repeating decimal? How do we convert that to a fraction?

Let's consider $0.\overline{3}$ .

Let's first let x = $0.\overline{3}$ , which also means that $x=0.\overline{33}$ . Now, multiply both sides by 10.

10x = 10($0.\overline{33}$ )10x = $3.\overline{3}$

-( x =$0.\overline{3}$ )             Subtract your first equation from the new one.

9x = 3
9     9                     Division Property of Equality

$x=\frac{3}{9}$

$x=\frac{1}{3}$                         Reduce

Practice

If you would like to practice more problems and check your work, click Here. Links to an external site.  Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

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