## Axioms::Multiplication

### ::concepts

Times::numbers Number::product ''n''::right Example::notation Other::division Group::integers**Axioms**
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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 *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 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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