Natural numbers: Difference between revisions
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Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]].
==Algebraic properties of the natural numbers==
The natural numbers (including zero) are closed under addition and multiplication. They satisfy commutativity and associativity of both operations, and distributivity of multiplication over addition. They form a monoid under addition, which is the free monoid with one generator. In addition, positive naturals form a monoid under multiplication -- the free monoid with countably infinite generators, which are the prime numbers.
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