Need help for calcul:

( 1/2 ( -1 + i√3 )) ^2
thanks for your help

That can be rewritten

(1/2)^2 * (-1 + i√3)^2

= (1/4)[1 -2i√3 -3]
= (-1/2)[1 +i√3]

Note that you are squaring a complex number that has a magnitude (modulus) of 1.

The number that is being squared can be written
e^(i*2*pi/3) = cos(2*pi/3) + i sin(2*pi/3)

The square of that number is e^(i*4*pi/3)
which agrees with the previous result.

To simplify the expression (1/2(-1 + i√3))^2, follow these steps:

Step 1: Simplify the inside parentheses.
The expression inside the parentheses is (-1 + i√3). Distribute the 1/2 to both terms: (-1/2 + i√(3)/2).

Step 2: Square the expression.
To square the expression, multiply it by itself:
(-1/2 + i√(3)/2) * (-1/2 + i√(3)/2).

Now, let's expand this expression using FOIL (First, Outer, Inner, Last):
FOIL stands for:
- Multiply the First terms: (-1/2 * -1/2 = 1/4)
- Multiply the Outer terms: (-1/2 * i√(3)/2 = -i√(3)/4)
- Multiply the Inner terms: (i√(3)/2 * -1/2 = -i√(3)/4)
- Multiply the Last terms: (i√(3)/2 * i√(3)/2 = -3/4)

Combining the terms, we get:
1/4 - i√(3)/4 - i√(3)/4 - 3/4.

Simplifying this expression:
1/4 - 2i√(3)/4 - 3/4.

Step 3: Combine like terms.
Combine the real and imaginary parts separately:
(1/4 - 3/4) - 2i√(3)/4.

Simplifying further:
-2/4 - 2i√(3)/4.

Step 4: Reduce the fraction.
Divide both the numerator and denominator by 2 to simplify the fraction:
-1/2 - i√(3)/2.

Thus, the final simplified expression is -1/2 - i√(3)/2.