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### Exercises Complex Numbers

- Convert the complex number Z = -4 - j 5 into exponential form.
- Determine the product of the complex numbers Z1= 4 - j and Z2 = 3 + j 4
- The complex number is given in normal form Z = -3 +j 4. Write it down in exponential form.

__Regarding task 1:__

r = √(4^{2} + 5^{2}) = 6,4 ¦ φ = arctan(-5/-4) = 51,34°

Since the real and imaginary parts are both negative, Z lies in the 3rd quadrant => φ = 180° + arctan(-5/-4) = 231,34 °

__Regarding task 2:__

First way to solution:

__Z__ = __Z___{1} __Z___{2} = (4 – i)(3 + 4i) = 12 +16i -3i - 4i^{2} = 16 + 13i

Second way to solution:

r_{1} = √(4^{2} + 1^{2}) = √17 => r = r_{1} * r_{2} = 5√17 = 20,62

r_{2} = √(3^{2} + 4^{2}) = 5

φ_{1} = arctan(-1/4) = -14,04° => φ = φ_{1} + φ_{2} = -14,04° + 53,09° = 39,09°

φ_{2} = arctan(4/3) = 53,09°

=> __Z__ = 20,62 e^{39,09° }

Note: Pointer Z1 lies in the 4th quadrant, because real part > 0 and imaginary part < 0, therefore φ1 remains negative and no 180° must be added to it.

Test:

__Z__ = 20,62 [cos39,09° + sin39,09°i] = 16 + 13i

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