Why? L.H.S.

R.H.S.

(iv) Arg
Why ? Let



(v)
Why ? Let



(vi) is purely real
Why ?
If

i.e. purely real.

(vii ) is purely imaginary
Why ?
If
= ri & -ri
Hence purely imaginary either ri or -ri

ILLUATRATION - 3.
Show that
[conj(arg(i)) + i arg(conj(i))] + [conj(arg(-i)) + i arg(conj(-i))] = 0


Using above four results we get



DE - MOIVRE'S THEOREM
The theorem states that for any , we have

Why ? Let
L.H.S. then
& we know taht
R,H.S. Hence

Illustration :-
Suppose
So,now find the value of

s Ans:- Now suppose some

(Using De Movire's Theorem)

Now again using De Moviec's theorem.



CUBE ROOTS OF UNITY.

Cube roots of unity means
Lrt one of cube root of unity b x ,then.


Solution of this cubic equation will give us three cube roots of unity.


Let us call
Now,

Therefore 3 cube roots of unity are 1, w, w².

ss Dumb Question:- Why there are 3 solutions ?
Ans:- By theory of equation we know that a mnth orider equation will have n solution. Here n is 3 solution

PROPERTIES OF CUBE ROOT OF UKNITY.
1. 1 + w + w² = 0
Why? By theory of equation we know that for any nth order equation, sum of roots =
Here for x³ - 1 = 0 , Sum

Why ? : Again by the theory of Equation product of roots
For x³ - 1 = 0 we have
So product of roots 1, w, w² = 1.w.w² = w³
Hence w³ = 1. Now if w³ = 1
Now if n^th the Taking nth power on both sides we get (w³)ⁿ = 1ⁿ gives w³ⁿ = 1.

3.
Why : By one of the properties we know that Hence z = w.
So
Hence
We also know that w³ = 1 & Hence

Dumb Question : How can you say taht |w| = 1
Ans : We know that w is cube root of unity . Hence Taking modulus on both side we get


If w is a cube root of unity then find the value
Ans: (i)


(ii)

(iii)

By (1).(2)& (3)
- 1 + w - w = - 1
Hence value is -1