# Common Number Sets

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

Symbol Description

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,...}

Integers

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, ...}

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

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 ->

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 ->

Real Numbers

Any value on the number line:

• Can be positive, negative or zero.
• Can be Rational or Irrational.
• Can be Algebraic or Transcendental.
• Can have infinite digits, such as 13 = 0.333...

Also see Real Number Properties

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

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

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

## Illustration

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
x + 7 = 0 x = −7 Integers
4x − 1 = 0 x = ¼ Rational Numbers
x2 − 2 = 0 x = ±√2 Real Numbers
x2 + 1 = 0 x = ±√(−1) Complex Numbers

## Other Sets

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

• a little + to mean positive, or
• a little * to mean non zero, like this:

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

And we can always use set-builder notation.