What is Sets of numbers?

Also known as: natural number · integer · rational number · real number

Definition 1.1 High School Mathematics · Chapter 1 — Numbers and Sets of Numbers

$\N$ is the set of natural numbers: $0, 1, 2, 3, \dots$
$\Z$ is the set of integers: $\dots, -2, -1, 0, 1, 2, \dots$
$\Q$ is the set of rational numbers: all quotients $\frac{p}{q}$ with $p \in \Z$ , $q \in \N$ and $q \neq 0$ .
$\R$ is the set of real numbers: all the numbers that can be placed on the number line.

Examples

Example 1.3

Let us place a few numbers in the smallest family that contains them.

$\frac{28}{4} = 7$ , so $\frac{28}{4} \in \N$ even though it is written as a fraction: always simplify first.
$-3.5 = -\frac{35}{10} = -\frac72$ is rational but not an integer.
$0.333\dots$ (the digit $3$ repeating forever) equals $\frac13$ , a rational number.
$\sqrt 2$ and $\pi$ are real but not rational, as we are about to see for $\sqrt 2$ ; such numbers are called irrational.