Categories of Numbers

Mathematicians have named five families of numbers over the centuries. Four are listed below.

Category	Symbol	Definition	Purpose	Other
natural	N	1, 2, 3,...	Counting	Some mathmeticians include 0
integers	Z	..., -3, -2, -1, 0, 1, 2, 3,...	Counting changes	Includes all natural numbers
rational	Q (quotient)	`n/d` for any integers `n` and `d` not 0	Measuring size or change in size / comparing size	Includes all integers. Can also be representated in decimals and percentages.
real	R	decimals	Measuring size or change in size, including any possible length, locating on the number line	Some numbers are represented by a symbol or letter. Real numbers that aren't rational are called irrational (never ending decimals [hence written as symbols]). Famous ones include: √2 and π

Note: Symbols are normally styled in font blackboard bold by mathematicians.

The letter symbols in this table are used to make statements about what families each number are a part of purely in mathematical notation. The symbol that is used to say whether a number is in a family is ∈.

∈: "is in"
- 2 ∈ N, 2 is in the natural numbers
- full statement: 2 ∈ N, 2 ∈ Z, 2 ∈ Q, 2 ∈ R
∉: "is not in"
- π ∉ N is not in the natural numbers
- full statement: π ∉ N, π ∉ Z, π ∉ Q, π ∈ R

N Z Q R

All N ∈ Z, Q and R
All Z ∈ Q and R
All Q ∈ R

From: MathTrackX: Polynomials, Functions and Graphs