Skill: Classify numbers within the real number system

To classify a number, check which number sets it belongs to. The sets are nested:

$$\text{natural} \subset \text{whole} \subset \text{integers} \subset \text{rational} \subset \text{real}$$

• Natural numbers: $1,2,3,\ldots$
• Whole numbers: $0,1,2,3,\ldots$
• Integers: $\ldots,-2,-1,0,1,2,\ldots$
• Rational numbers: numbers that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$. Their decimals terminate or repeat.
• Irrational numbers: real numbers that cannot be written as a ratio of integers. Their decimals do not terminate or repeat.

Every rational and irrational number is a real number.

Worked example: Classify $\boldsymbol{-\sqrt{18}}$

Step 1: Simplify the radical.

Since $18=9\cdot 2$,

$$-\sqrt{18}=-\sqrt{9\cdot 2}=-3\sqrt{2}.$$

Step 2: Decide whether it is rational or irrational.

The number $2$ is not a perfect square, so $\sqrt{2}$ is irrational. Multiplying it by $-3$ does not make it rational. Therefore,

$$-3\sqrt{2}\text{ is irrational}.$$

Step 3: Classify it in the real number system.

Irrational numbers are real numbers, so $-\sqrt{18}$ is real.

It is not:

• natural, because it is negative;
• whole, because it is negative and not a whole number;
• an integer, because it is not a counting number with no decimal part;
• rational, because it cannot be written as a ratio of integers.

Therefore, the complete classification is:

$$\boxed{-\sqrt{18}\text{ is an irrational real number}.}$$