You know two facts: every fraction’s decimal ends or repeats, and every repeating decimal is a fraction. So what about a decimal that never ends and never repeats? Some square roots are exactly that.
Key idea
A rational number can be written as a fraction $\frac{a}{b}$ of integers ($b \ne 0$); its decimal ends or repeats. An irrational number cannot be written that way; its decimal goes on forever without repeating. Rational and irrational numbers together make up the real numbers.
| Rational | Irrational |
|---|---|
| $-7$, $\frac{3}{8}$, $0.25$ | $\sqrt{2}$, $\sqrt{5}$, $\sqrt{10}$ |
| $0.\overline{3}$, $\sqrt{49} = 7$ | $\pi$, $2\pi$ |
| $\frac{22}{7} = 3.\overline{142857}$ | $0.1010010001\ldots$ |
Why it works
Why can’t $\sqrt{2}$ be a fraction? Squares of fractions come close to 2 but never land on it: $\left(\frac{7}{5}\right)^{2} = 1.96$ and $\left(\frac{17}{12}\right)^{2} \approx 2.007$. Here is the reason. Suppose a fraction $\frac{a}{b}$ in lowest terms had $\frac{a^{2}}{b^{2}} = 2$. Then $a^{2} = 2b^{2}$ is even, so $a$ is even (an odd number squared is odd). Write $a = 2c$: then $4c^{2} = 2b^{2}$, so $b^{2} = 2c^{2}$ and $b$ is even too. But in lowest terms $a$ and $b$ can’t both be even. So no such fraction exists.
Worked examples
Worked example: A root that is rational
Is $\sqrt{0.81}$ rational or irrational?
- $0.9 \times 0.9 = 0.81$, so $\sqrt{0.81} = 0.9$.
- $0.9 = \frac{9}{10}$, a fraction.
Answer: rational
Worked example: Close is not equal
Is $\frac{22}{7}$ equal to $\pi$?
- $\frac{22}{7} = 3.142857142857\ldots$, a repeating decimal, so it is rational.
- $\pi = 3.14159265\ldots$ is irrational, and the two differ in the third decimal place.
Answer: No; $\frac{22}{7}$ is only an approximation of $\pi$.
Common mistake
Two opposite slips. Thinking $\pi$ equals $\frac{22}{7}$ or $3.14$ exactly: those are handy approximations, and $\pi$’s digits never end or repeat. And thinking every square root is irrational: perfect squares have whole-number roots, like $\sqrt{25} = 5$.