## Rotation of Complex Number

- Multiplying i is a rotation by 90 degrees counter-clockwise
- Multiplying by -i is a rotation of 90 degrees clockwise

__Example__
z=1

If we multiply it by i, it becomes

z=i so that it has rotated by the angle 90 degrees

## What is the significance of Complex Numbers? Why they are required?

- Real numbers such as natural number,rational number , irrational number are invented in the history as and when we encounter various mathematical needs. Same happen with the complex numbers.

We had no solution for the problem

x^{2}=-1

Eular was the first mathematicain to introduce solution to this problem,he introduced the symbol i

$ i=\sqrt{(-1)}$

So i^{2}=-1

So solution to the problem becomes

x=i or -i

- He called the symbol i as imaginary unit.
- Just like all the other number ,this number was added to our Number vocabulary. This like other numbers is useful in explaining where physical explanation.
- It is very useful in the field Electrical and electronics.

## Power of i

We know that

i

^{2} =-1

i

^{3}=-i

i

^{4}=1

i

^{5}=i

i

^{6}=-1

i

^{7}=-i

This can be generalized as

For any integer k, i

^{4k} = 1, i

^{4k + 1} = i, i

^{4k + 2} = - 1, i

^{4k + 3} = -i

**Example**

Find the value $(\frac {i}{2})(\frac {-2i}{3}) (\frac {i^3}{4})$

**Solution**

$(\frac {i}{2})(\frac {-2i}{3}) (\frac {i^3}{4})$

$=\frac {-i^5}{12} = \frac {-i}{5}$

## Identities for Complex Numbers

Here are some the identities for Complex numbers

For all complex numbers z

_{1} and z

_{2}
- (z
_{1} + z_{2})^{2} = z_{1}^{2} + z_{2}^{2} + 2z_{1}z_{2}
- (z
_{1} - z_{2})^{2} = z_{1}^{2} + z_{2}^{2} - 2z_{1}z_{2}
- (z
_{1} + z_{2})^{3} = z_{1}^{3} + z_{2}^{3} + 3z_{1}z_{2}^{2}+ 3z_{1}^{2}z_{2}
- (z
_{1} - z_{2})^{3} = z_{1}^{3} - z_{2}^{3} + 3z_{1}z_{2}^{2} -3z_{1}^{2}z_{2}
- z
_{1}^{2} - z_{2}^{2} = (z_{1} + z_{2} )(z_{1} - z_{2})

## Solved Example

**Example 1**

Find the non-zero integral Solution for this equation

$|1-i|^x =2^x$

**Solution**

$|1-i|= \sqrt {2} =2^{1/2}$

So

$2^{x/2}=2^x$

2^{x-x/2}=1

2^{x/2} =1

or x=0

So there is no non-zero integral solution for this equation

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