Skill: Recognize irrational numbers

A rational number can be written as a fraction of integers, such as $\frac{3}{4}$ or $-2=\frac{-2}{1}$. Its decimal either terminates or repeats.

An irrational number cannot be written as a fraction of integers. Its decimal goes on forever without repeating. Familiar examples include $\sqrt{2}$, $\sqrt{3}$, and $\pi$.

For square roots, check whether the number inside the radical is a perfect square:

• If it is a perfect square, the square root is rational. For example, $\sqrt{25}=5$.
• If it is not a perfect square, its square root is irrational.

Example: Is $\sqrt{18}$ rational or irrational?

1. List nearby perfect squares:

$$4^2=16 \qquad\text{and}\qquad 5^2=25.$$

2. Since $18$ is not a perfect square, $\sqrt{18}$ cannot simplify to an integer or a fraction.

3. Therefore, $\sqrt{18}$ is irrational.

It is located between $4$ and $5$ on the number line because

$$4<\sqrt{18}<5.$$