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Zero Is the Identity for Addition 0+a=a and a+0=a
Existence of Additive Inverses For each number a, there is another number, denoted -a, such that a+-a=0 and -a+a=o. Each of a and -a is the additive inverse of the other.
Commutativity of Multiplication a⋅b=b⋅a
Associativity of Multiplication a⋅(b⋅c)=(a⋅b)⋅c
One Is the Identity for Mulitplication 1⋅a=a and a⋅1=a for any number a.
Existence of Multiplicative Inverses For each nonzero number a, there is another number b, such that a⋅b=1 and b⋅a=1. Multiplicative inverses are also reciprocals.
Distributive Property a⋅(b+c)=(a⋅b)+(a⋅c) and (b+c)⋅a=(b⋅a)+(c⋅a) for any numbers a, b, and c.
Unnamed Additive Property of Equalities If a=b, then a+c=b+c
Unnamed Subtractive Property of Equalities If a=b, then a-c=b-c
Unnamed Multiplicative Property of Equalities If a=b, then a⋅c=b⋅c
Unnamed Divisive Property of Equalities If a=b, then a÷c=a÷b