1.1.1 · The Number System

Kinds of numbers

Numbers did not arrive all at once. People first needed to count goats and cassava, then to say "none left", then to owe someone money, and then to share a loaf fairly. Each need gave us a new kind of number.

The families of numbers

Name	Symbol	What it holds	Example
Natural numbers	$\mathbb{N}$	Counting numbers	$1, 2, 3, \dots$
Whole numbers	$\mathbb{W}$	Natural numbers and zero	$0, 1, 2, \dots$
Integers	$\mathbb{Z}$	Whole numbers and their negatives	$\dots, -2, -1, 0, 1, \dots$
Rational numbers	$\mathbb{Q}$	Fractions $\frac{a}{b}$ with $b \neq 0$	$\frac{3}{4}$ , $-2.5$ , $7$

Every natural number is a whole number, every whole number is an integer, and every integer is a rational number.¹

A number line from minus 3 to 5 with the points minus 2, 0, one half and 4 marked. — Natural numbers, integers and fractions all live on the same number line.

A number that is not rational

Some numbers cannot be written as a fraction of two integers. The length of the diagonal of a square with side 1 is $\sqrt{2}$ , and no fraction equals it exactly. We call numbers like this irrational. Together, the rational and irrational numbers make up the real numbers.