Editing Multiplication (section)

==Axioms==
{{Main|Peano axioms}}
In the book ''[[Arithmetices principia, nova methodo exposita]]'', [[Giuseppe Peano]] proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
:<math>x \times 0 = 0</math>
:<math>x \times S(y) = (x \times y) + x</math>

Here ''S''(''y'') represents the [[Successor ordinal|successor]] of ''y''; i.e., the natural number that follows ''y''. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including [[Mathematical induction|induction]]. For instance, ''S''(0), denoted by 1, is a multiplicative identity because
:<math>x \times 1 = x \times S(0) = (x \times 0) + x = 0 + x = x.</math>

The axioms for [[integer]]s typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (''x'',''y'') as equivalent to {{nowrap|''x'' − ''y''}} when ''x'' and ''y'' are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
:<math>(x_p,\, x_m) \times (y_p,\, y_m) = (x_p \times y_p + x_m \times y_m,\; x_p \times y_m + x_m \times y_p).</math>

The rule that −1 × −1 = 1 can then be deduced from
:<math>(0, 1) \times (0, 1) = (0 \times 0 + 1 \times 1,\, 0 \times 1 + 1 \times 0) = (1,0).</math>

Multiplication is extended in a similar way to [[rational number]]s and then to [[real number]]s.{{Citation needed|date=December 2021}}