Complement Calculations - Why do I get different answers with ones' and twos' complement?

Question

I am finishing an assignment related to complement calculations. The question is as follows:

Convert the following decimal numbers to eight-bits binary and perform the subtraction by using ones' and twos' complement arithmetic.
28 - 41 

By using 1s complement I can get the answer -13 with the workings below

  0 0 0 1 1 1 0 0
+ 1 1 0 1 0 1 1 0
------------------
  1 1 1 1 0 0 1 0

Signed-magnitude in Binary: 10001101
Decimal representation: -13

For 2s complement, I understand that I should add 1 after replacing the digits but when I do the calculation, the answer doesn't make sense at all.

00101001 ---> 11010110 (Ones complement) ---> 11010111 (Twos complement)

  0 0 0 1 1 1 0 0
+ 1 1 0 1 0 1 1 1 
-----------------
  1 1 1 1 0 0 1 1

Signed-magnitude in Binary: 10001100
Decimal representation: -12

This is most likely caused by the addition of 1. But referring to the lecture notes and the online tutorials I don't think I should make a subtraction afterwards. Did I do anything wrong during the calculation? :-(

Answer 1

You need to consider your result as twos complement as well, so subtract $1$, getting $11110010_2$, then complement, getting $1101_2=13_{10}$ Arithmetic works staying withing twos complement.