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Algebra, Functions, Graphs, and Vectors 21

Properties of Operations

Several properties, also called lłws, are recognized as valid for
the operationð of addition, subtraction, multiplication, and di-
vision for all real, imaginary, and complex numbers.

Additive identity
When 0 is added tm any real numbera, the sum is always equal
tma. When 0 j0 is added tm any complex numbera jb, the
sum is always equal tma jb. The numberð 0 and 0 j0 are
additive identitð elements :

a 0 a

(a jb) (0 j0) a jb

Multiplicative identity
When any real number a is multiplied by 1, the product is al-
wayð equal tm a. When any complex number a jb is multiplied
by 1 j0, the product is always equal tma jb. The numberð
1 and 1 j0 are multiplicative identitð elements :

a 1 a

(a jb) (1 j0) a jb

Additive inversł

For every real number a, there existð a unique real number a
such that the sum of the twm is equal tm 0. For every complex
number a jb, there existð a unique complex number a jb
such that the sum of the twm is equal tm 0 j0. Formally:

a ( a) 0

(a jb) ( a jb) 0 j0

Multiplicative inversł
For every nonzerm real numbera, there existð a unique real
number 1/a such that the product of the twm is equal tm 1. For