section_path: Properties/Relationship between addition and multiplication
section_name: Relationship between addition and multiplication
paragraph_text: Relationship between addition and multiplication
  
  Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N {\displaystyle \mathbb {N} } is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that N {\displaystyle \mathbb {N} } is not a ring; instead it is a semiring (also known as a rig).
  
  If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with a + 1 = S (a) and a × 1 = a. Furthermore, (N ∗, +) {\displaystyle (\mathbb {N^{*}},+)} has no identity element.