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In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers.<ref>PlanetMath: Peano arithmetic</ref> 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 of y, or the natural number that follows y. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic including 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 integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to xy 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 numbers and then to real numbers.


Multiplication sections
Intro  [[Multiplication?section={{safesubst:#invoke:anchor|main}}Notation_and_terminology|{{safesubst:#invoke:anchor|main}}Notation and terminology]]  Computation  Products of measurements  Properties  Axioms  Multiplication with set theory  Exponentiation  See also  Notes  References   External links   

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