Arithmetic » Number systems (sets)

Contents

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 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 {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
3x2 + 50 = 2
3x2 = –48
x2 = –16
x = –4i or x = 4i