natural number

The natural numbers are the common counting numbers, starting with 1. They are the numbers that virtually all human beings learn at about the same time they learn language.

To a mathematician, the natural numbers are a set \(ℕ=\{1,2,3,\ldots\}\) that is closed under addition (and so also multiplication). Mathematicians sometimes for convenience will add 0 to this set, and then it is sometimes called the set of whole numbers.

Arithmetic on the natural numbers is defined by the Peano axioms, including addition, multiplication, and induction.

In set theory the natural numbers (including 0) are identified with the set \(\omega\) of finite ordinals. They are a well-founded linear order with no largest member, and are countably infinite.


  • B. Sidney Smith, author

Citation Info

  • [MLA] Smith, B. Sidney. "natural number." Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 21 Feb 2013. Web. 21 Feb 2013. <>
  • [APA] Smith, B. Sidney (21 Feb 2013). natural number. Retrieved 21 Feb 2013 from the Platonic Realms Interactive Mathematics Encyclopedia:


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