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Rotation of Complex number and Power of i

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
x²=-1
Eular was the first mathematicain to introduce solution to this problem,he introduced the symbol i
$ i=\sqrt{(-1)}$
So i²=-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
i² =-1
i³=-i
i⁴=1
i⁵=i
i⁶=-1
i⁷=-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 z₁ and z₂

(z₁ + z₂)² = z₁² + z₂² + 2z₁z₂
(z₁ - z₂)² = z₁² + z₂² - 2z₁z₂
(z₁ + z₂)³ = z₁³ + z₂³ + 3z₁z₂²+ 3z₁²z₂
(z₁ - z₂)³ = z₁³ - z₂³ + 3z₁z₂² -3z₁²z₂
z₁² - z₂² = (z₁ + z₂ )(z₁ - z₂)

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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