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
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
- (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
$2^{x/2}=2^x$
2x-x/2=1
2x/2 =1
or x=0
So there is no non-zero integral solution for this equation
Related Topics
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Class 11 Maths
Class 11 Physics
