section_path: Properties/Addition
section_name: Addition
paragraph_text: Addition
  
  Given the set N {\displaystyle \mathbb {N} } of natural numbers and the successor function S: N → N {\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to the next one, one can define addition of natural numbers recursively by setting a + 0 = a and a + S (b) = S (a + b) for all a, b. Thus, a + 1 = a + S(0) = S(a +0) = S(a), a + 2 = a + S(1) = S(a +1) = S(S(a)), and so on. The algebraic structure (N, +) {\displaystyle (\mathbb {N},+)} is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers.
  
  If 1 is defined as S (0), then b + 1 = b + S (0) = S (b + 0) = S (b). That is, b + 1 is simply the successor of b.