# Algebra of Complex Number

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Algebra of complex numbers deals with mathematical operations such as addition, subtraction, multiplication and division on two complex numbers.

Whenever two complex numbers are added or subtracted, their corresponding real and imaginary parts are added or subtracted.

Properties of addition of complex numbers:

The closure law: For any two complex numbers z1 and z2, z1 + z2 is a complex number.

Commutative law: For any two complex numbers z1 and z2, z1 + z2 = z2 + z1.

Associative law: For any three complex numbers z1, z2 and z3, (z1 + z2) + z3 = z1 + (z2 + z3).

Existence of additive identity: z + 0 = z = 0 + z

Existence of additive inverse: z + (–z) = 0 = (–z) + z

Properties of multiplication of complex numbers:

The closure law: For any two complex numbers z1 and z2, z1 z2 is a complex number.

Commutative law: For any two complex numbers z1 and z2, z1 z2 = z2 z1.

Associative law: For any three complex numbers z1, z2 and z3, (z1 z2)z3 = z1(z2 z3).

Existence of multiplicative identity: z.1 = z = 1.z

Existence of multiplicative inverse: For any non zero complex number z = a + ib, we have a complex number a/(a2 + b2) + ib/(a2 + b2) called the multiplicative inverse of z such that z.(1/z) = 1.

The distributive law: For any three complex numbers z1, z2, z3

z1 (z2+ z3) = z1z2 + z1z3

(z1 + z2 )z3 = z1z3 + z2z3

Two complex numbers are multiplied in the same manner as two binomials are multiplied.

For any complex number z = a + ib, there exists –z = – a + i(–b) such that: z + (–z) = 0 = (–z) + z.

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