Rational and irrational numbers

Lesson

Yes you heard that right, the numbers that we use can be separated into rational and irrational numbers, although they might not mean what you think they mean! In short, rational numbers are those that can be expressed as fractions, and irrational numbers are those that can not. If you have forgotten what counts as a fraction, it can be expressed as below:

$\frac{p}{q}$pq, where $p$p and $q$q are integers, $q$q not being $0$0

Rational Numbers and Playing With Fractions

Of course fractions are already expressed in this way so they are rational. Most of the numbers we have encountered so far are also rational, so let's take a look at how they can be expressed as fractions.

All integers are rational as they can be transformed into fractions by thinking of them as having an invisible denominator of $1$1. For example, $8$8 can be expressed as $\frac{8}{1}$81 and $-45$45 can be expressed as$\frac{-45}{1}$451. Easy!

Decimals can usually also be expressed as fractions. Terminating decimals (the ones that don't go on forever) can be thought of as having a denominator that is a multiple of $10$10, such as $100$100, $1000$1000, $1000000$1000000 etc. For example, $6.4$6.4 can be expressed as $\frac{64}{10}$6410, $0.503$0.503 as $\frac{503}{1000}$5031000, and $-0.0006$0.0006 as $\frac{-6}{10000}$610000.

Recurring decimals are rational as well, such as $7.6666$7.6666 ... is expressed as $7\frac{2}{3}$723 = $\frac{23}{3}$233, and $0.142857142857$0.142857142857... is actually $\frac{1}{7}$17! Don't worry, you won't be expected to know these special long ones, phew!

Some decimals are quite sneaky and belong to the next group of numbers...

Irrational Numbers and Absurd Surds

I'm sure that you must have come across the famous never-ending decimal $\pi$π that is loved by mathematicians everywhere by now. Well guess what, this is an irrational number! No matter how hard you try, you won't be able to transform it into a fraction.

Besides $\pi$π a large part of irrational numbers are made up of surds, a special group of numbers that involve roots. Can you try and express $\sqrt{2}$2, $\sqrt[3]{5}$35 or $-\sqrt{3}$3 as fractions? It's impossible! These are all examples of surds, roots that can not be simplified down to a rational number. So for example, $\sqrt{4}$4 is not a surd as it can be simplified down to $2$2.

Can you think of any other examples of surds? Try putting them in your calculator to see if they can actually be simplified!

Examples

Use a calculator to try and work out if the following numbers are rational or irrational. If they're rational, express them in fractional form.

Evaluate if $\sqrt{6}$6, $-0.0047$0.0047, $-\sqrt[3]{27}$327, $\frac{9}{2}$92 are rational or not

Think whether these numbers are fractions, terminating decimals, recurring decimals, integers, or surds

Do: $\sqrt{6}$6 = $2.4494$2.4494...

This does not simplify down to a rational number, so it is irrational.

 $-0.0047$−0.0047 $=$= $\frac{-0.0047}{1}$−0.00471​ $=$= $\frac{-47}{10000}$−4710000​

This is a fraction so it is rational.

$-\sqrt[3]{27}$327 = $-3$3
This simplifies down to an integer so it is rational.

$\frac{9}{2}$92 is already a fraction, so this is rational.

More Examples

Question 1

Determine if the following number is rational or irrational: $\frac{3}{13}$313

Question 2

Determine if the following number is rational or irrational: $-\sqrt{64}$64

Question 3

Determine if the following number is rational or irrational: $\frac{5}{8}+\sqrt{19}$58+19

Did you know?

The number $\pi$π has currently been calculated to 10 trillion digits! Wow!

Outcomes

NA5-2

Use prime numbers, common factors and multiples, and powers (including square roots)