Arithmetic » Number systems (sets)

Natural numbers
Integers 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 {N} .

There is discussion about whether zero is part of {N} .
Nowadays zero is mostly part of the natural numbers.
When people do not begin the natural numbers with zero, they use {N} for the natural numbers excluding zero and {N} ₀ for the natural numbers including zero. {N} 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 {Z} .
The Z comes from the German word 'Zahl', which means number.
The set {Z} includes set {N} .
{N} is therefore a subset of {Z} .
You may write this as {N} ⊂ {Z} .

Rational or fractional 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 {Q} .
As every integer can be written as a fraction, {Q} includes {Z} .
You may write this as {N} ⊂ {Z} ⊂ {Q} .

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 {R} .
Because {Q} is a subset of {R} , we can write: {N} ⊂ {Z} ⊂ {Q} ⊂ {R} .

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 i² = –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 {C} .
Because {R} is a subset of {C} , we can write: {N} ⊂ {Z} ⊂ {Q} ⊂ {R} ⊂ {C} .

Watch out: x² = –1 DOES have solutions with complex numbers.
As i² = –1, the equation x² = –1 has solutions x = –i or x = i.
This is because (–i)² is also –1.

Example