Rational & irrational numbers

A recipe asks for 3/4 of a litre of milk, the receipt says €2.50, your room is 4 m wide. All of these numbers have something in common: you can write them as a fraction of two whole numbers. That is exactly what maths calls rational. Surprisingly, there are numbers where this never works, no matter how long you search. The diagonal of a square tile with sides of 1 m is one of them.

Rational means: writable as a fraction

A number is rational if you can write it as a fraction $\frac{a}{b}$ of two whole numbers, where the denominator must not be 0. That works far more often than you might expect. 7 is rational, because 7 = 7/1. And −3 is rational, because −3 = −3/1. Every decimal with finitely many decimal places can be rewritten too: 0.25 is 25/100, simplified 1/4, and 2.5 is 5/2. So whole numbers, negative numbers and the decimals of everyday life are all rational.

What the decimal places reveal

When you divide numerator by denominator, one of two things always happens. Either the decimal stops, as in 1/4 = 0.25 or 3/8 = 0.375. Or a group of digits repeats forever, as in 1/3 = 0.333… or 1/7 = 0.142857142857… That repetition is called the period, and it is written with a bar above the repeating digits. Both cases belong to the rational numbers, because 0.333… is a fraction too, namely 1/3.

Numbers no fraction can reach

And now the exception. You work out the diagonal of our tile with the Pythagorean theorem, which says $a^2 + b^2 = c^2$ in every right-angled triangle. From $1^2 + 1^2 = 2$ it follows that the diagonal is $\sqrt{2}$, that is 1.41421356… These decimal places never stop and never become periodic. The Greeks already proved that no fraction of whole numbers gives exactly $\sqrt{2}$. Numbers like this are called irrational. The most famous one is $\pi$ = 3.14159…, the ratio of a circle's circumference to its diameter. 22/7 = 3.142857… comes close to $\pi$ but does not hit the value, and no other fraction does either.

On the number line both kinds sit peacefully side by side. Between 1 and 2 there are infinitely many fractions such as 3/2 or 7/5, and right among them, at 1.414…, sits $\sqrt{2}$. Together the rational and irrational numbers make up the real numbers, and they fill the number line without any gaps. Watch out for a common mistake: not every root is irrational. $\sqrt{9}$ is exactly 3 and therefore rational, while $\sqrt{10}$ is irrational.