List of types of numbers

From Wikipedia, the free encyclopedia

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

Main types[edit]

  • Natural numbers (): 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 called whole numbers.[1][2]
  • Integers (): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}.
  • Rational numbers (): Numbers that can be expressed as a ratio of an integer to a non-zero integer.[3] All integers are rational, but there are rational numbers that are not integers, such as −2/9.
  • Real numbers (): 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: Real numbers that are not rational.
  • Imaginary numbers: Numbers that equal the product of a real number and the square root of −1. The number 0 is both real and purely imaginary.
  • Complex numbers (): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.
  • Hypercomplex numbers include various number-system extensions: quaternions (), octonions (), and other less common variants.[4]
  • 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[edit]

Signed numbers[edit]

  • 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 integer[edit]

Algebraic numbers[edit]

Non-standard numbers[edit]

Computability and definability[edit]

See also[edit]


  1. ^ Weisstein, Eric W. "Natural Number". MathWorld.
  2. ^ "natural number",, Merriam-Webster, retrieved 4 October 2014
  3. ^ W., Weisstein, Eric. "Rational Number".
  4. ^ Sedenions (), trigintaduonions (), tessarines, coquaternions, and biquaternions.