**Identify Rational Numbers** *KS3 Revision*

## 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}

4 **– rational**

\pi **– irrational**

-4 **– rational**

0.\overline{5} **– rational**

\sqrt{49}=7 **– rational**