1.3 Sets and numbers

We shall be concerned with calculations involving real numbers. We begin by introducing some sets (collections) of numbers:

${\mathbb{N}}=\{1,2,3,\dots\}$, the set of natural numbers, as used in counting;

${\mathbb{Z}}=\{0,\pm 1,\pm 2,\pm 3,\pm 4,\dots\}$, the set of integers, positive and negative;

${\mathbb{Q}}=\{a/b:a,b\in{\mathbb{Z}},b\neq 0\}$, the set of rational numbers;

${\mathbb{R}}$ the set of real numbers; $x$ is a real number;

${\mathbb{C}}=\{z=x+iy:x,y\in{\mathbb{R}}\}$ the set of complex numbers.

These sets of numbers are contained in one another; so that,

The rational numbers arise from the integers by division. The set of real numbers also includes irrational numbers, some of which arise from geometry.

Examples of rational numbers are $1/3$ or $1.275$; examples of irrational numbers are $\sqrt{2}$. Lambert showed that $\pi$ is irrational.

Archimedes showed that $223/71<\pi<22/7.$