Why we cannot divide 3+5i by 1+i?

The answer is
easy!

The divisor 1 + i cannot divide 3 + 5i because it is
not a real number. The division of complex numbers follows the next procedure. First,
we'll have to multiply both numerator and denominator by the conjugate of
divisor.

`(3+5i)/(1+i) =
((3+5i)(1-i))/((1+i)(1-i))`

We notice that the special
product from denominator returns a difference of two
squares:

`((3+5i)(1-i))/(1- i^2) = ((3+5i)(1-i))/(1+1) =
((3+5i)(1-i))/2`

Since the divisor is a real number we can
perform now the division.

We'll remove the brackets from
numerator:

`((3+5i)(1-i))/2 = (3 - 3i + 5i -
5i^2)/2`

`((3+5i)(1-i))/2 = (3 + 2i +
5)/2`

`` `((3+5i)(1-i))/2 = (8 +
2i)/2`

`((3+5i)(1-i))/2 = (2(4 +
i))/2`

`` `((3+5i)(1-i))/2 = 4 +
i`

The result of division of the complex
numbers 3 + 5i and 1 + i is the complex number 4 +
i.