Types of Numbers

Integers, rationals, irrationals, and the reals — and what 'real roots' means.

The idea

The real numbers are built up in stages — integers, rationals, irrationals — and each stage answers a question the previous one cannot.

Counting numbers cannot say what $3 - 5$ is, so we take the whole numbers together with their negatives, the integers: $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}.$ Integers cannot say what $3$ divided by $4$ is, so we allow ratios: a rational number is any $\tfrac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$; together the rationals form the set $\mathbb{Q}$. Every integer is already rational — take $b = 1$ — so each family contains the one before it.

Gaps remain even so. Some lengths, such as the diagonal of a square whose side is $1$ — the number $\sqrt{2}$ — are not a ratio of integers, and neither is $\pi$. Such a number is irrational, and its decimal expansion runs forever without settling into a repeating pattern.

The rationals and the irrationals together fill the number line completely, and that collection is the real numbers $\mathbb{R}$. Every real number is either rational or irrational — never both, and never neither.

Ways to work on it

• Walkthrough. The nested families: integers, rationals, irrationals, reals.
• Practice. Classify a number as specifically as possible.
• Hardest. Count and classify the real roots of x^2 = k.

Not sure where to start? Take the ten-question placement test.