# 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 .

Thus,

If

where and are positive real numbers.

Example:

Find the value of

Solution:

Thus,

## 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:

If , then

If , then

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.

i.e.,

For example, if then

Again

Example:

Find the conjugate of each of the following complex numbers:

(i) (ii)

Solution:

(i) Let i then

Hence, is the conjugate of .

(ii) Let

i.e.

Then

Hence, is the conjugate of