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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, i4k = 1, i4k + 1 = i, i4k + 2 = - 1, i4k + 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
  1. (z1 + z2)2 = z12 + z22 + 2z1z2
  2. (z1 - z2)2 = z12 + z22 - 2z1z2
  3. (z1 + z2)3 = z13 + z23 + 3z1z22+ 3z12z2
  4. (z1 - z2)3 = z13 - z23 + 3z1z22 -3z12z2
  5. z12 - z22 = (z1 + z2 )(z1 - z2)
  6. 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
    $2^{x/2}=2^x$
    2x-x/2=1
    2x/2 =1
    or x=0
    So there is no non-zero integral solution for this equation



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