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What is the difference between the zero property of multiplication and the identity property of addition?
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).
What is the additive inverse of 78?
Two numbers that are the same distance from zero on the number line but are on opposite sides of zero are opposite numbers, or opposites. The opposite of a number is called its additive inverse. The opposite of 78 is -78.
What is the additive inverse of 0?
Zero!
What is the additive inverse of eighteen?
The additive inverse of 18 is -18. The additive inverse of any number is the opposite of that number, such that the sum of the original number and the additive inverse is zero.
What do you call two opposite numbers whose sum is zero?
These are often called "opposite numbers". The more precise term is "additive inverse". For example, the additive inverse of 5 is minus 5.
