Simplify the complex number z = ( 2-4i)/ (1-i) + (3+i)/2

To simplify the complex number z = (2-4i)/(1-i)
+(3+i)/2.

We bring the given complex number to the form x+yi, where
x and y are real.

First term =  (2-4i)/(1-i) =
(2-4i)(1+i)/(1-i)(1+i). We multiplied numerator and denominator by
1+i.

(2-4i)/(1-i) = (2+2i -4i -
4i^2)/(1-i^2)

(2-4i)/(1-i) = (2-2i+4)/(1+1) , as i^2 =
-1

(2-4i)/(1-i) =
(6-2i)/2

(2-4i)/(1-i) = 3-i = (3 -
i).

2nd term = (3+i)/2 = 3/2
+i/2.

adding the two terms , we get: (3-i) +(3/2+i/2)  =  (3+3/2)
+(-i+i/2) = 9/2 -i/2

Therefore  ( 2-4i)/ (1-i)  + (3+i)/2 =
(3-i)+(3/2+i) = (3+3/2)+(-i+i/2) =  9/2 - i/2.

Therefore the
simplified form of complex number =  4.5 -0.5i.