Editing Natural number (section)

===Relationship between addition and multiplication===
Addition and multiplication are compatible, which is expressed in the [[distributivity|distribution law]]: {{math|''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 <math>\mathbb{N}</math> is not [[closure (mathematics)|closed]] under subtraction (that is, subtracting one natural from another does not always result in another natural), means that <math>\mathbb{N}</math> is ''not'' a [[ring (mathematics)|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 {{math|''a'' + 1 {{=}} ''S''(''a'')}} and {{math|''a'' × 1 {{=}} ''a''}}. Furthermore, <math>(\mathbb{N^*}, +)</math> has no identity element.