Skill: Classify numbers as natural, whole, integer, rational, irrational, or real

To classify a number, first identify what kind of number it is, then use the containment chain:

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

• Natural numbers: $1,2,3,\ldots$
• Whole numbers: $0,1,2,3,\ldots$
• Integers: positive and negative whole numbers, including $0$
• Rational numbers: numbers that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$
• Irrational numbers: real numbers that cannot be written as a ratio of integers
• Real numbers: all numbers on the number line

Worked example: Classify $-\frac{6}{3}$

Step 1: Simplify the number.

$$-\frac{6}{3}=-2$$

Step 2: Check whether $-2$ is natural or whole.

It is not natural because natural numbers are positive. It is not whole because whole numbers are $0$ and positive numbers.

Step 3: Check whether it is an integer.

Yes. $-2$ is a negative whole number, so it is an integer.

Step 4: Check whether it is rational.

Yes, because it can be written as a ratio of integers:

$$-2=\frac{-2}{1}$$

So it is rational.

Step 5: Check whether it is real.

Every integer and rational number is real, so $-2$ is real.

Therefore, $-\frac{6}{3}$ is an:

$$\boxed{\text{integer, rational, and real number}}$$

It is not natural, whole, or irrational.