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 -> Examples: 3/2 (=1.5), 8/4 (=2), 136/100 (=1.36), -1/1000 (=-0.001) Q is for "quotient" (because R is used for the set of real numbers): the result of dividing one number by another. It comes from the Italian "Quoziente".
	Irrational Numbers Any real number that's not a rational number. Read More ->
	Algebraic Numbers Any number that's a solution to a polynomial equation with rational coefficients. Includes all rational numbers, and some irrational numbers. Read More ->
	Transcendental Numbers Any number that's not an algebraic number Examples of transcendental numbers include π and e. Read More ->
	Real Numbers Any value on the number line: $Continuous real number line showing fractions, decimals, square roots, and pi$ 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 aren't 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 number is √(-1) (the square root of minus one), and its symbol is i, or sometimes j. i² = -1 Read More ->
	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 Read More ->

rational, real

Illustration

Nested Venn diagram showing Natural numbers inside Integers, Rationals, Reals, and Complex numbers

Natural numbers are a subset of integers

Integers are a subset of rational numbers

Rational numbers are a subset of real numbers

Combinations of real and imaginary numbers make up the complex numbers.

Number Sets In Action

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

Equation	Solution	Number Set
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, ...}

And we can always use set-builder notation.