Numbers can be classified according to how they are represented or according to the properties that they have. Contents 1 Main types
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Main types[edit] 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.[1][2] 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.[3] All integers are rational, but the converse is not true. Real numbers ( R displaystyle mathbb R ): Numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true. Irrational numbers ( I displaystyle mathbb I ): 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 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 ), tessarines, 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.
^ Weisstein, Eric W. "Natural Number". MathWorld. ^ "natural number", Merriam-Webster.com, Merriam-Webster, retrieved 4 October 2014 ^ [1] v t e
Countable sets Natural numbers ( N displaystyle mathbb N ) Integers ( Z displaystyle mathbb Z ) Rational numbers ( Q displaystyle mathbb Q ) Constructible numbers Algebraic numbers ( A displaystyle mathbb A 𝔸) Periods Computable numbers Definable real numbers Arithmetical numbers Gaussian integers Division algebras Real numbers ( R displaystyle mathbb R ) Complex numbers ( C displaystyle mathbb C ) Quaternions ( H displaystyle mathbb H ) Octonions ( O displaystyle mathbb O 𝕆) Split Composition algebras over R displaystyle mathbb R : Split-complex numbers Split-quaternions Split-octonions over C displaystyle mathbb C : Bicomplex numbers Biquaternions Bioctonions Other hypercomplex Dual numbers Dual quaternions Hyperbolic quaternions Sedenions ( S displaystyle mathbb S 𝕊) Split-biquaternions Multicomplex numbers Other types Cardinal numbers Irrational numbers Fuzzy numbers Hyperreal numbers Levi-Civita field Surreal numbers Transcendental numbers Ordinal numbers p-adic numbers Supernatural numbers Superreal numbers C |