## What you need to know

Things to remember:

• A rational number is either an integer, a fraction, or a terminating decimal (one that stops), or has a recurring (repeating) digit or digits.

So, what are rational numbers? Well, a rational number is any number that can be written as a fraction with integers on the top and bottom. We have four different types of rational number:

1. Integers, such as -57, – 4, 0, 12, 1000, etc. (Put as a fraction with 1 on the bottom)

$\dfrac{-57}{1}$

$\dfrac{-4}{1}$

$\dfrac{0}{1}$

$\dfrac{12}{1}$

$\dfrac{1000}{1}$

2. Terminating decimal numbers (decimals that end), such as 0.2, 0.56, 1.234, etc. (Multiply by a power of 10 to remove the decimal and put it over the power of 10)

$\dfrac{2}{10}$

$\dfrac{56}{100}$

$\dfrac{1234}{1000}$

3. Recurring decimals (decimals that have a repeating digit or digits), like 0.11111… or 0.142857142857… These are more complicated to write as a fraction. But, we usually put a dot or line over the top of numbers to show that is being repeated.

$0.111111…=0.\overline{1}$

$0.142857142857…=\overline{142857}$

4. Fractions themselves are rational numbers.

So, why do we have a special name for these numbers? Because there is an opposite type, which we call irrational numbers!

– Irrational numbers are numbers that can’t be written as a fraction, like $\pi$ and $\sqrt{2}$.

However, not all square roots are irrational. For example, $\sqrt{9}$ can be simply written as “3”, which is a rational number!

Which of these isn’t a rational number?

7                      $\dfrac{2}{5}$                     8                      -6                     $\sqrt{17}$

We can check this against our list of 4 types of rational numbers.

7 – Integer

$\dfrac{2}{5}$ – Fraction

8 – Integer

-6 – Integer

$\sqrt{17}$ None

$\sqrt{17}$ is none of the four types of rational number, so must be the irrational number!

## Example Questions

#### Question 1: Which of the following are rational and irrational numbers?$\sqrt{11}$                          $\sqrt{36}$              $\sqrt{81}$                          $\sqrt{121}$         $\sqrt{23}$

Hint: See if you can write the square root as an integer.

$\sqrt{11}$ – irrational

$\sqrt{36}=6$ – rational

$\sqrt{81}=9$ – rational

$\sqrt{121}=11$ – rational

$\sqrt{23}$ – irrational

#### Question 2: Which of the following are rational and irrational numbers?4                      $\pi$                        -4                     $0.\overline{5}$                $\sqrt{49}$

$\pi$ – irrational
$0.\overline{5}$ – rational
$\sqrt{49}=7$ – rational