Complex Numbers, Positive Integral Powers of I, Conjugate of a Complex Number, Properties of Complex Conjugates

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Positive Integral Powers of I

We know that

Thus, we find that any higher powers of ‘i’ can be expressed in terms of one of four values

If is a positive integer such that , then to find in , we first divide by .

Let be the quotient and be the remainder.

Then . where .



where and are positive real numbers.


Find the value of



Conjugate of a Complex Number

  • The complex conjugate (or simply conjugate) of a complex number is defined as the complex number and is denoted by .

  • Thus, if then

Following are some examples of complex conjugates:

  1. If , then

  2. If , then

  3. If , then

Properties of Complex Conjugates

If is a real number then i.e., the conjugate of a real number is the number itself.

For example, let

This can be written as

If is a purely imaginary number then

For example, if . This can be written as

Conjugate of the conjugate of a complex number is the number itself.


For example, if then



Find the conjugate of each of the following complex numbers:

(i) (ii)


(i) Let i then

Hence, is the conjugate of .

(ii) Let



Hence, is the conjugate of

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