CXC CSEC Mathematics

## The number sets Numbers are grouped into sets, each one containing the one before it. | Set | Symbol | Members | |---|---|---| | Natural numbers | $\mathbb{N}$ | $1, 2, 3, \dots$ | | Whole numbers | $\mathbb{W}$ | $0, 1, 2, 3, \dots$ | | Integers | $\mathbb{Z}$ | $\dots, -2, -1, 0, 1, 2, \dots$ | | Rational numbers | $\mathbb{Q}$ | $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$ | | Real numbers | $\mathbb{R}$ | every rational and irrational number | A **rational number** can be written as a fraction of two integers, which means its decimal form either stops or repeats. So $0.25 = \dfrac{1}{4}$, $-7 = \dfrac{-7}{1}$ and $0.\dot{3} = \dfrac{1}{3}$ are all rational. An **irrational number** cannot be written as such a fraction, and its decimal never stops and never repeats. Examples are $\pi$ and $\sqrt{2}$. A square root is irrational unless the number under it is a perfect square, so $\sqrt{49} = 7$ is rational while $\sqrt{50}$ is not. The sets are nested, and the relationship is written with the subset symbol: $$ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} $$ Every natural number is therefore also a whole number, an integer, a rational number and a real number. The irrational numbers sit inside $\mathbb{R}$ but outside $\mathbb{Q}$, so $\mathbb{R}$ is the union of the rational and irrational numbers. To place a number, choose the smallest set that holds it. For example $-3$ is an integer, $0$ is a whole number…