# Arithmetic » Number systems (sets)

## Contents

Natural numbersIntegers or whole numbers

Rational or fractional numbers

Real numbers

Complex numbers

## Natural numbers

A natural number is a number from the set 0, 1, 2, 3, 4, ...

The set of all natural numbers is indicated with symbol .

There is discussion about whether zero is part of .

Nowadays zero is mostly part of the natural numbers.

When people do not begin the natural numbers with zero, they use for the natural numbers excluding zero and _{0} for the natural numbers including zero. _{0} is then called the non-negative numbers.

## Integers or whole numbers

An integer is a number from the set ..., –4, –3, –2, –1, 0, 1, 2, 3, 4, ...

The set of all integers is indicated with symbol .

The Z comes from the German word 'Zahl', which means number.

The set includes set .

is therefore a *subset* of .

You may write this as ⊂ .

## Rational or broken numbers

A rational number is the quotient of two integers.

Therefore every number that can be written as a fraction (fractional or broken number) is a rational number.

The set of all rational numbers is indicated with symbol .

As every integer can be written as a fraction, includes .

You may write this as ⊂ ⊂ .

## Real numbers

Besides rational numbers, there are also numbers that cannot be written as a fraction. For example and `π`. Numbers that cannot be written as a fraction are called **irrational numbers**. The rational numbers and irrational numbers together form the set of **real numbers**.

The real numbers make up all numbers on a number line.

The set of all real numbers is indicated with symbol .

Because is a subset of , we can write: ⊂ ⊂ ⊂ .

## Complex numbers

Above you read that the real numbers form the numbers of a number line.

However, in some parts of science, there was a need for numbers in a plane. In other words: Two number lines. Complex numbers are invented because of this. A complex number is a combination of two real numbers `a` and `b` where a number is written as: `a` + `bi`. Here `i` is the **imaginary unit** where `i`^{2} = –1.

In this way you can show every number `a` + `bi` in a plane, where `a` is the number on the horizontal axis and `bi` is the number on the vertical axis.

The set of all complex numbers is indicated with symbol .

Because is a subset of , we can write: ⊂ ⊂ ⊂ ⊂ .

Watch out: `x`^{2} = –1 DOES have solutions with complex numbers.

As `i`^{2} = –1, the equation `x`^{2} = –1 has solutions `x` = –`i` or `x` = `i`.

This is because (–`i`)^{2} is also –1.

*Example*

3x^{2} + 50 | = 2 |

3x^{2} | = –48 |

x^{2} | = –16 |

x = –4i | or x = 4i |