Irrational number

Rational or irrational

A rational number can be written as $\tfrac{a}{b}$ with $a$ and $b$ whole numbers. That covers more than it first appears:

Number	As a fraction	Rational?
$7$	$\tfrac{7}{1}$	yes
$0.25$	$\tfrac{1}{4}$	yes
$0.333\ldots$	$\tfrac{1}{3}$	yes
$\sqrt{2}$	—	no
$\pi$	—	no

A repeating decimal is rational. $0.333\ldots$ is $\tfrac{1}{3}$ exactly, and the repetition is what makes the fraction findable.

Irrational means no such fraction exists — not that nobody has found one.

What their decimals look like

$$\sqrt{2} = 1.41421356\ldots \qquad \pi = 3.14159265\ldots$$

Both run forever with no repeating block. Any written decimal is therefore an approximation, which is why exact answers keep the radical: $\sqrt{2}$ is exact and $1.414$ is not.

They are still ordinary lengths

An irrational number is not an unreachable one. $\sqrt{2}$ is the exact diagonal of a $1 \times 1$ square, a line you can draw with a ruler in a second:

$$1^2 + 1^2 = 2 \quad\Rightarrow\quad \text{diagonal} = \sqrt{2}$$

So the Pythagorean theorem produces irrational lengths from whole-number sides constantly — a length can be perfectly definite and still have no fraction that names it.

Where they turn up

Most square roots are irrational: the roots of the perfect squares are whole numbers, and nearly everything else is irrational. The same is true of the quadratic formula, whose answers are irrational whenever the discriminant is not a perfect square.