Non Real Roots

Introduction to Non real roots:
Non real roots are the imaginary roots. A complex number is a number that consists of real part and imaginary part.   We write it as a + bi where i is the imaginary  unit.  i = v-1
If z = a + bi then a is the real part and  b is called the imaginary part.  a is denoted as Re(z) and b is denoted as Im(z)
Roots of a complex number:-
Non real roots are roots of a complex number.
A number is called the nth root of a complex number z if ?n = z  and ? = z1/n
To find the non real roots  of a complex number, we follow the following steps.
Step1 :  Write the given number in polar form.
Step2 :  Add 2kp to the argument..
Step3 :  Apply De Moivre's theorem.
Step4 :  Put k= 0,1 ...... up to infinity.

Non Real Roots-the Nth Root of Unity:

The nth root of unity:-
Take the number 1
Step1 :  1 =  1(cos 0 + i sin 0)
step 2:         1( cos (0 + 2kp) + i sin)0 + 2kp) =  cos 2kp + i sin 2kp
Step 3: nth root of unity = 11/n = (cos 2kp + i sin 2kp)1/n where k = 0,1 .......(n-1)
=  [ cos 2kp/n + i sin 2kp/n ] where k = 0,1   ... ..(n-1)
Step 4 :when k=0 the nth root is cos 0 + i son 0
when k = 1 the nth root is cos 2p/n + i sin 2p/n,...
So ? = cos 2p/n + i sin 2p/n  which is written as ei 2p/n   

Hence the nth root of unity is e0, ei2p/n, ei4p/n etc  which is written as 1,?,?2, ?3 .......... ?n-1
From the above, we note that the non real roots are in geometric progression and the sum of the non real roots  is zero.

Algebra is widely used in day to day activities watch out for my forthcoming posts on Rationalize a Denominator and Graphing Quadratic Functions in Standard Form. I am sure they will be helpful.

Non Real Roots - Cube Root of Unity:

Cube root of unity:-
Let us find the non real roots of 11/3
Let x = 1 1/3
Then x3 = 1
x3 = cos 0 + i son 0 = cos 2kp + i sin 2kp
x  =  (cos 2kp + i sin 2kp)1/3   =   cos 2kp/3  + i sin 2kp/3  ......... k = 0, 1, 2 ......
Therefore the cube roots are cos 0 + i sin 0  = 1
cos 2p/3 + i sin 2p/3 =  ?
cos 4p/3 + i sin 4p/3 = ?2
The values of the non real roots  ? =   -1 + i v3
2
The value of the non real root  ?2 =  -1-iv3
2
When the points are plotted on garland diagram, we see them lying on the circle  of unit radius.
The roots 1,?,?2 are in geometric progression and the sum of the roots = 0