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 of Electrical and electronics.
In summary, complex numbers extend the real number system to provide a complete, consistent framework for solving a wide range of problems in mathematics, physics, engineering, and other fields. They are indispensable for advanced analysis and have practical applications that make them essential in both theoretical and applied contexts.
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
(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
