Rational and Irrational Numbers

Rational Numbers

A rational number can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

Examples: $\frac{3}{4}$, $-2$, $0.75$, $0.\overline{3}$

Key property: Rational numbers have decimal representations that either terminate or repeat.

• $\frac{1}{4} = 0.25$ (terminates)
• $\frac{1}{3} = 0.333...$ (repeats)

Irrational Numbers

An irrational number CANNOT be written as a fraction. Its decimal never terminates and never repeats.

Examples: $\pi \approx 3.14159...$, $\sqrt{2} \approx 1.41421...$, $e \approx 2.71828...$

Square Roots

$\sqrt{n}$ is the number that, when multiplied by itself, gives $n$.

Perfect squares have rational square roots: $\sqrt{1} = 1, \; \sqrt{4} = 2, \; \sqrt{9} = 3, \; \sqrt{16} = 4, \; \sqrt{25} = 5, ...$

Non-perfect squares have irrational square roots: $\sqrt{2}, \; \sqrt{3}, \; \sqrt{5}, \; \sqrt{7}, \; \sqrt{10}, ...$

Approximating Irrational Numbers

$\sqrt{7}$ is between $\sqrt{4} = 2$ and $\sqrt{9} = 3$.

Since 7 is closer to 9: $\sqrt{7} \approx 2.6$

More precisely: $\sqrt{7} \approx 2.646$

The Real Number System

$\text{Real Numbers} \begin{cases} \text{Rational} \begin{cases} \text{Integers} \begin{cases} \text{Whole Numbers} \begin{cases} \text{Natural Numbers} \end{cases} \end{cases} \end{cases} \\ \text{Irrational} \end{cases}$

Every number on the number line is a real number — either rational or irrational.

Quick test: Can you write it as a fraction? Yes → rational. No → irrational.