x-y = x+¬y+1 problem

Question

I am currently reading a book about "bit fiddling" and the following formula appears:

x-y = x+¬y+1

But this doesn't seem to work. Example:

x = 0100  
y = 0010  
x-y = 0010  
¬y = 1101  
¬y+1 = 1110  
x+1110 = 10010  

But 10010 != 0010...

Where did I make a mistake (if any)?

(The book is "Hacker's Delight" by Henry S. Warren.)

Answer 1

You only have a four bit system! That extra 1 on the left of your final result can't exist. It should be:

x  = 0100
y  = 0010
~y = 1101
~y + 1 = 1110
x + 1110 = 0010

The other bit overflows, and isn't part of your result. You may want to read up on two's complement arithmetic.