How to convert a complex number into desired form?

Question

For example, if I have a complex number like:

$$\dfrac{a+i\cdot b}{c-i\cdot d}$$

suppose $a, b, c, d\in \mathbb{R}$ and $c^2+d^2\ne 0$,

how can I use mathematica to convert it into the following form with no $i$'s in its denominator:

$$\dfrac{(ac-bd)+i\cdot(ad+bc)}{c^2+d^2}$$

Additionally, if I use many such $a_i+i\cdot b_i$ complex numbers in some expression, how can I convert the expression into the desired form with no $i$'s in its denominator?

e.g.

$$\dfrac{\dfrac{a}{1+i b}-\dfrac{i}{c}+\dfrac{d}{1+i e}}{-\dfrac{i}{a \omega }+a-\dfrac{i}{\alpha }+\dfrac{t}{x+i s \omega }+\dfrac{w}{u+i v}+\dfrac{\beta }{y+i \gamma }-\dfrac{i}{\omega }}$$

The mathematica code:

(a/(1 + I b) + 1/(I c) + d/(1 + I e))/(1/(I \[Alpha]) + 1/( I \[Omega] ) + a + \[Beta]/(y + I \[Gamma]) + 1/(I \[Omega] a) +  t/(x + I \[Omega]  s) + w/(u + I v))

And, if we suppose variables $a, b, c,\cdots$ and so on in an expression, are real matrices not scalars, how to Simplify them and obtain desired simplest form result since the commutative law will not hold for multiplication and how we let Mathematica follow such rules?

Answer 1

For the first part

ClearAll[a, b, c, d]
conv[expr_] := Module[{den = Denominator[expr], num = Numerator[expr]},
  ComplexExpand[num*Conjugate[den]]/(ComplexExpand[den*Conjugate[den]])
  ]

And now

 expr = (a + I b)/(c - I d);
 conv[expr]

"if I use many such ai+i⋅biai+i⋅bi complex numbers in some expression, how can I convert the expression into the desired form with no ii's in its denominator?"

It will help to provide example of input that one can use.

And, if we suppose variables a,b,c,⋯a,b,c,⋯ and so on in an expression, are real matrices not scalars, how to Simplify them and obtain desired simplest form result since the commutative law will not hold for multiplication and how we let Mathematica follow such rules?

I can't follow. May be you should make the above separate question and provide more clear example of input and desired output.