Expressing complex numbers in cartesian form

[cmcc3119]

4 Questions:

(1 + i) / (1 - i) Ans: i

(2 + 3i) / (5 - 6i) Ans: (-8+27i)/61

1/i - (3i)/(1-i) Ans: (3-5i)/2

i^123 - 4i^9 - 4^i Ans: -9i


Could someone please explain the method (detailed) as to how these answers were obatined? I understand other questions in the same field but these four I did not know how they derived the answers. Thanks for your help and your time :)

 

[Vagrant]

check complex division and multiplication
http://mathworld.wolfram.com/ComplexNumber.html

 

[Defennder]

Firstly you have to convert the denominator of the complex fraction into a real number. Multiply its numerator and denominator by its complex conjugate. Remember that i^2 = -1. The complex conjugate of a+bi is a-bi. Once you have done that you only need deal with the numerator. Group all the real terms and simplify them. Do the same for the imaginary terms (ie. terms with variable 'i' in them).