Rational number

The definition

$$\frac{a}{b} \quad \text{where } a \text{ and } b \text{ are integers and } b \neq 0$$

The name comes from ratio, not from being reasonable — a rational number is one expressible as a ratio of two whole numbers.

What it includes

Number	As a fraction	Rational?
$7$	$\tfrac{7}{1}$	yes
$-4$	$\tfrac{-4}{1}$	yes
$0.75$	$\tfrac{3}{4}$	yes
$-2.5$	$\tfrac{-5}{2}$	yes
$0.333\ldots$	$\tfrac{1}{3}$	yes
$\sqrt{2}$	—	no
$\pi$	—	no

Every integer is rational, since any whole number sits over a denominator of $1$. So the integers are contained inside the rationals rather than sitting alongside them.

Terminating and repeating decimals

A decimal is rational when it either stops or repeats:

$$0.25 = \tfrac{1}{4} \qquad 0.333\ldots = \tfrac{1}{3} \qquad 0.142857142857\ldots = \tfrac{1}{7}$$

A repeating decimal looks endless and is exact all the same — the repetition is what makes the fraction findable.

A decimal that runs forever without repeating is irrational. That is the whole distinction: $\sqrt{2} = 1.41421356\ldots$ never settles into a pattern, so no fraction produces it.

Why the category matters

Arithmetic stays inside the rationals. Add, subtract, multiply or divide two rational numbers and the answer is rational again, with the single exception of dividing by zero.

That is not true of the integers — $7 \div 2$ leaves them — which is why the rationals are the natural place to do ordinary arithmetic. The sign rules hold across the whole set, so nothing new has to be learned when the numbers stop being whole. See operations with rational numbers.