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Hey guys I'm still trying to get the hang of twos complement arithmetic and I can get the correct answer, since I'm working on practice problems with solutions.

When I take the answer that's in binary, I can't seem to equate it out to the decimal answer before applying twos complement and adding the numbers.

000100-111001 In decimal it's 4 - 57= -53

Becomes

0001000+000111 which would be 4 + (-57)?

Giving a solution of 001011

How can 001011 be proven as equaling -53?

Thanks!

ModdedLife
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1 Answers1

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To answer your question, first 001011 is not equal to -53. This is the wrong answer. We know it must be positive since the highest order bit is a 0, not a 1. 001011 is actually equal to 11 (in base ten).

Let's do 4 - 57 now as an example. This is the same as 4 + (-57). Converting to binary (I will use just a byte for this example) we get: 4 is 0000 0100, 57 is 0011 1001. To convert 57 to negative 57 use two's complement:
1. Negate it: 1100 0110
2. Add one: 1100 0111
So now we get the following equation: 

  0000 0100
+ 1100 0111
------------
  1100 1011

We achieve the answer by simply adding down the rows. The answer we have gotten is 1100 1011. We know it is negative because the highest order bit (which here is leftmost) is a 1. To find its magnitude, we apply two's complement:
1. Negate: 0011 0100
2. Add one: 0011 0101
And this is equal to 53 in base ten.

Another way to see if it is correct is to add it with the positive version of the number. 

  1100 1011
+ 0011 0101
------------
 10000 0000

Since two's complement is defined as the number subtracted from 2^n where n is the number of bits, you will always get this result for the sum. Knocking off the 1s digit it's interesting how what remains is just 0 - and any number plus its negative is zero.

mage
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