Real number sets and how to classify numbers

When we count, we usually start with 1, 2, 3, and keep going. Once we add zero, negative numbers, fractions, and numbers like π, we need a few more names for the groups they belong to. Let’s see how those groups fit together.

Split the real numbers into rational and irrational numbers

The real numbers are the numbers on the number line. A rational number can be written as a fraction $a / b$ , where $a$ and $b$ are integers and $b \neq 0$ . In decimal form, rational numbers either terminate or eventually repeat. For example, $- 0.25 = - 1 / 4$ , and $1 / 3 = 0.333 \dots$ .

An irrational number is real but cannot be written as that kind of fraction. Its decimal goes on without an eventually repeating pattern. Both $\sqrt{2}$ and $π$ are irrational. The two groups together make up all the real numbers.

Every natural number belongs to each larger set that contains it.

Find the sets inside the rational numbers

Every integer is rational because we can put it over 1. The integers include the negative integers, zero, and the positive integers. Within the integers, the whole numbers are zero and the positive integers. The natural numbers here start at 1, so they’re the counting numbers. Some books include zero in the natural numbers and state that convention separately.

Common symbols and examples
Set	Symbol	Examples
Real	$ℝ$	$- 4, \sqrt{2}$
Rational	$ℚ$	$- \frac{1}{4}, \frac{3}{2}$
Irrational	$ℝ ∖ ℚ$	$\sqrt{2}, π$
Integers	$ℤ$	$- 2, 0, 3$
Whole	$W$	$0, 1, 2, \dots$
Natural	$ℕ$	$1, 2, 3, \dots$

Classify a number and write a set

The smallest set in the diagram that contains $- 5$ is the integers. That makes $- 5$ rational and real as well. The smallest set containing $0$ is the whole numbers under the convention we’re using. Since $\sqrt{9} = 3$ , that number is natural and belongs to each larger set that contains the natural numbers.

To list the members of a set, put them in curly braces. The dots show that the list of counting numbers continues.

ℕ = {1, 2, 3, 4, \dots}

Real number sets and how to classify numbers

Split the real numbers into rational and irrational numbers

Find the sets inside the rational numbers

Classify a number and write a set

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