# Complex Numbers, Subtraction of Complex Numbers, Properties with Respect to Addition of Complex Numbers, Argument of a Complex Number

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## Subtraction of Complex Numbers

Let two complex numbers and be represented by the points and respectively.

which represents a point

The difference i.e. is represented by the point .

Thus, to subtract a complex number from another, we subtract corresponding real and imaginary parts separately.

Example:

Find if:

,

Solution:

## Properties: With Respect to Addition of Complex Numbers

### 1. Closure:

The sum of two complex numbers will always be a complex number.

Let and , .

Now, which is again a complex number.

This proves the closure property of complex numbers.

### 2. Commutative:

If and are two complex numbers then

Let and

Now

[Commutative property of real numbers]

i.e. Hence, addition of complex numbers is commutative.

### 3. Associative:

If , and are three complex numbers, then

Now

Hence, the associativity property holds good in the case of addition of complex numbers.

4. Existence of Additive Identity:

If is any complex number, then

i.e. is called the additive identity for .

### 5. Existence of Additive Inverse:

For every complex number there exists a unique complex number such that . is called the additive inverse of .

In general, additive inverse of a complex number is obtained by changing the signs of real and imaginary parts.

## Argument of a Complex Number

Let represent the complex number ,, , and makes an angle with the positive direction of x-axis. Draw , Let

In right

Then can be written as ... (i)

Where and or

(i) is known as the polar form of the complex number , and and are respectively called the modulus and argument of the complex number.

## Multiplication of Two Complex Numbers

Two complex numbers can be multiplied by the usual laws of addition and multiplication as is done in the case of numbers.

Let and then,

or

or. Since

If and are two complex numbers, their product is defined as the complex number

Example:

Evaluate:

Solution: