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What is the difference between the zero property of multiplication and the identity property of addition?

Updated: 4/28/2022

Wiki User

11y ago

Usually, the identity of addition property is defined to be an axiom (which only specifies the existence of zero, not uniqueness), and the zero property of multiplication is a consequence of existence of zero, existence of an additive inverse, distributivity of multiplication over addition and associativity of addition.

Proof of 0 * a = 0:

0 * a = (0 + 0) * a [additive identity]

0 * a = 0 * a + 0 * a [distributivity of multiplication over addition]

0 * a + (-(0 * a)) = (0 * a + 0 * a) + (-(0 * a)) [existence of additive inverse]

0 = (0 * a + 0 * a) + (-(0 * a)) [property of additive inverses]

0 = 0 * a + (0 * a + (-(0 * a))) [associativity of addition]

0 = 0 * a + 0 [property of additive inverses]

0 = 0 * a [additive identity]

A similar proof works for a * 0 = 0 (with the other distributive law if commutativity of multiplication is not assumed).

Wiki User

11y ago