Rational and Irrational Numbers

Engineers work out square roots constantly — the length of a diagonal brace, the side of a square room with a given floor area. Sometimes the answer is a tidy whole number. Usually it is not, and it matters which kind you have got: one is exact, the other has been rounded.

What you were really sorting

A number is rational if you can write it as one integer divided by another. Look back at the left-hand group — every one of them hides a fraction:

$0.3 = \frac{3}{10} \qquad -7 = \frac{-7}{1} \qquad \sqrt{16} = \frac{4}{1} \qquad 0.\dot{8} = \frac{8}{9}$

That last one surprises people. “Goes on forever” is not enough to make a number irrational. $0.888\dots$ goes on forever but it repeats, and any decimal that repeats can be turned into a fraction (you will see how in the recurring decimals lesson).

An irrational number’s decimal goes on forever and never settles into a pattern. That is what $\pi$, $\sqrt{2}$ and $\sqrt[3]{6}$ do.

Three places irrational numbers come from

In Year 9 you only meet three kinds:

1. $\pi$ — and anything made from it, such as $5\pi$ or $\frac{\pi}{2}$.
2. Square roots of non-square whole numbers — $\sqrt{2}$, $\sqrt{3}$, $\sqrt{10}$ … If the number is a square ($\sqrt{25}$), the root is a whole number and rational.
3. Cube roots of non-cube whole numbers — $\sqrt[3]{2}$, $\sqrt[3]{6}$ … but $\sqrt[3]{27} = 3$ and $\sqrt[3]{-125} = -5$ are rational.

An irrational root like $\sqrt{2}$ is called a surd. Leaving an answer as $\sqrt{2}$ keeps it exact; writing $1.41$ throws a little accuracy away. That is why exam questions often say “leave your answer in surd form”.

Families inside families

The number families nest inside each other like boxes:

Every natural number is whole, every whole number is an integer, every integer is rational — but never the other way round: $\frac{1}{2}$ is rational without being an integer.