Common Number Sets

There are sets of numbers that are used so often that they have special names and symbols:

Symbol Description

set natural

Natural Numbers

The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics). Read More ->

The set is {1,2,3,...} or {0,1,2,3,...}

set integer


The whole numbers, {1,2,3,...} negative whole numbers {..., -3,-2,-1} and zero {0}. So the set is {..., -3, -2, -1, 0, 1, 2, 3, ...}

number line

(Z is from the German "Zahlen" meaning numbers, because I is used for the set of imaginary numbers). Read More ->

set rational

Rational Numbers

The numbers you can make by dividing one integer by another (but not dividing by zero). In other words fractions. Read More ->

Q is for "quotient" (because R is used for the set of real numbers).

Examples: 3/2 (=1.5), 8/4 (=2), 136/100 (=1.36), -1/1000 (=-0.001)

(Q is from the Italian "Quoziente" meaning Quotient, the result of dividing one number by another.)


Irrational Numbers

Any real number that is not a Rational Number. Read More ->

set algebraic

Algebraic Numbers

Any number that is a solution to a polynomial equation with rational coefficients.

Includes all Rational Numbers, and some Irrational Numbers. Read More ->


Transcendental Numbers

Any number that is not an Algebraic Number

Examples of transcendental numbers include π and e. Read More ->

set real

Real Numbers

All Rational and Irrational numbers. They can also be positive, negative or zero.

Includes the Algebraic Numbers and Transcendental Numbers.

Also see Real Number Properties

A simple way to think about the Real Numbers is: any point anywhere on the number line (not just the whole numbers).

Examples: 1.5, -12.3, 99, √2, π

They are called "Real" numbers because they are not Imaginary Numbers. Read More ->

set imaginary

Imaginary Numbers

Numbers that when squared give a negative result.

If you square a real number you always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 also, so "imaginary" numbers can seem impossible, but they are still useful!

Examples: √(-9) (=3i), 6i, -5.2i

The "unit" imaginary numbers is √(-1) (the square root of minus one), and its symbol is i, or sometimes j.

i2 = -1


set complex

Complex Numbers

A combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary.

The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers.

Examples: 1 + i, 2 - 6i, -5.2i, 4



number sets


Natural numbers are a subset of Integers

Integers are a subset of Rational Numbers

Rational Numbers are a subset of the Real Numbers

Combinations of Real and Imaginary numbers make up the Complex Numbers.

Number Sets In Use

Here are some algebraic equations, and the number set needed to solve them:

Equation Solution Number Set Symbol
 x − 3 = 0 x = 3 Natural Numbers set natural
 x + 7 = 0 x = −7 Integers set integer
4x − 1 = 0 x = ¼ Rational Numbers set rational
x2 − 2 = 0 x = ±√2 Real Numbers set real
x2 + 1 = 0 x = ±√(−1) Complex Numbers set complex

Other Sets

We can take an existing set symbol and place in the top right corner:

set of positive integers   Set of positive integers {1, 2, 3, ...}
set of positive integers   Set of nonzero integers {..., -3, -2, -1, 1, 2, 3, ...}

And we can always use set-builder notation.