Numbers come in families, and giving each family a name makes it much easier to talk about them. The three we meet first are the natural numbers, the integers, and the rational numbers.

The natural numbers are the plain counting numbers: 1, 2, 3, \dots We write this family as \mathbb{N}. They are the very first numbers anyone learns — the ones you use to answer "how many?".

The integers stretch the counting numbers in both directions: they add zero and the negative numbers, giving \dots, -2, -1, 0, 1, 2, \dots We write this family as \mathbb{Z} (from the German Zahlen, "numbers"). Every natural number is also an integer — we have just added the whole numbers on the cold side of zero.

The rational numbers are any number you can write as a fraction of two integers, like \frac{1}{2} or \frac{-7}{4}. We write this family as \mathbb{Q} (for quotient). This includes every decimal that stops (like 0.75 = \frac{3}{4}) or repeats (like 0.333\dots = \frac{1}{3}). And since 5 = \frac{5}{1}, every integer is rational too.

See how they nest

The families are nested, like boxes inside boxes: every natural number is an integer, and every integer is rational. Watch the boxes grow from the inside out, then see where a few example numbers belong. A number always lives in its smallest home — and automatically in every bigger box around it.

See it explained

Sal Khan sorts numbers into these families and shows how each one fits inside the next.