Skill: Distinguish rational and irrational numbers

A rational number can be written as

$$\frac{a}{b},$$

where $a$ and $b$ are integers and $b\ne 0$. Rational numbers include:

• integers, such as $-3$ and $8$;
• fractions, such as $\frac{2}{5}$;
• terminating decimals, such as $0.75$;
• repeating decimals, such as $0.\overline{3}$.

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

Worked example

Classify each number as rational or irrational:

$$-4,\qquad \frac{7}{8},\qquad 0.121212\ldots,\qquad \sqrt{2},\qquad \sqrt{49}$$

Step 1: Classify $-4$.

Every integer can be written as a fraction:

$$-4=\frac{-4}{1}.$$

Therefore, $-4$ is rational.

Step 2: Classify $\frac{7}{8}$.

It is already written as a quotient of two integers, with a nonzero denominator. Therefore, $\frac{7}{8}$ is rational.

Step 3: Classify $0.121212\ldots$.

The digits $12$ repeat forever. A repeating decimal is rational, so $0.121212\ldots$ is rational.

Step 4: Classify $\sqrt{2}$.

$2$ is not a perfect square, so $\sqrt{2}$ cannot be written as a fraction of integers. Its decimal is nonterminating and nonrepeating. Therefore, $\sqrt{2}$ is irrational.

Step 5: Classify $\sqrt{49}$.

Since $49$ is a perfect square,

$$\sqrt{49}=7.$$

Because $7$ is an integer, $\sqrt{49}$ is rational.

Thus, the classifications are:

$$\boxed{-4\text{ rational},\quad \frac78\text{ rational},\quad 0.121212\ldots\text{ rational},\quad \sqrt2\text{ irrational},\quad \sqrt{49}\text{ rational}}$$