Numbers can be classified according to how they are represented or according to the properties that they have.

Main types

  • Natural numbers (N {\displaystyle \mathbb {N} }): The counting numbers {1, 2, 3, ...} are commonly called natural numbers; however, other definitions include 0, so that the non-negative integers {0, 1, 2, 3, ...} are also called natural numbers. Natural numbers including 0 are also sometimes called whole numbers. Alternatively natural numbers not including 0 are also sometimes called whole numbers instead.
  • Integers (Z {\displaystyle \mathbb {Z} }): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}.
  • Rational numbers (Q {\displaystyle \mathbb {Q} }): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as −2/9.
  • Real numbers (R {\displaystyle \mathbb {R} }): Numbers that correspond to points along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true.
  • Irrational numbers (R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} }): Real numbers that are not rational.
  • Imaginary numbers: Numbers that equal the product of a real number and the imaginary unit i {\displaystyle i}, where i 2 = − 1 {\displaystyle i^{2}=-1}. The number 0 is both real and imaginary.
  • Complex numbers (C {\displaystyle \mathbb {C} }): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.
  • Hypercomplex numbers include various number-system extensions: quaternions (H {\displaystyle \mathbb {H} }), octonions (O {\displaystyle \mathbb {O} }), sedenions (S {\displaystyle \mathbb {S} }), trigintaduonions (T {\displaystyle \mathbb {T} }), and other hypercomplex numbers of dimensions 64 and greater. Less common variants include as bicomplex numbers, coquaternions, and biquaternions.
  • p-adic numbers: Various number systems constructed using limits of rational numbers, according to notions of "limit" different from the one used to construct the real numbers.

Number representations

Signed numbers

  • Positive numbers: Real numbers that are greater than zero.
  • Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used:
  • Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
  • Non-positive numbers: Real numbers that are less than or equal to zero. Thus a non-positive number is either zero or negative.

Types of integers

Algebraic numbers

Non-standard numbers

Computability and definability

See also