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BodhiAI
Nov. 11, 2019

Conjugate complex numbers and its properties:

Conjugate of Complex Number

Let z = a+ any complex number, then the conjugate of z is a-. It is generally denoted by or z', thus

If then

Important

It follows from the definition, that the conjugate of a complex number is obtained by replacing by . Thus the conjugate of a purely real number is the number itself.

Properties of Conjugate

Let and be any three complex numbers, then

P-1

Reasoning: Since,

P-2 z+ = 2 Re(z)

Reasoning: Since, z = a+

P-3 2Im(z)

Reasoning: Since, z + a+

P-4 If then Im (z) = 0 or z is purely real

Reasoning: Since,

2lm(z) = 0+0

lm(z) = 0 and hence z is a Purely Real Number.

P-5 If then Re (z) = 0 or z is purely imaginary

Reasoning: Since,

2Re (z) = 0

z is purely imaginary.

P-6 =

Reasoning: Now,

P-7

Reasoning: Now,

P-8

P-9

Reasoning: Here, =

P-10 where

P-11

P-12 =

P-13 Re Re

P-14 Im - Im =