Binary Subtraction with 2's Complement

Question

I need help subtracting with binary using 2's representation and using 5 bits for each number:

1) -9 -7 = ? Is there overflow?

-9 = 01001 (2's complement = 10111) and -7 = 00111 (2's complement = 11001)

Now we need to add because we're using 2's complement

10111 +11001 = 100000 But this answer doesn't make sense. Also, I'm assuming there's overflow because there are more than 5 bits in the answer.

2) 6 - 10, same process as before. Negative binary numbers don't make sense to me

What is the expected answer? How would it be represented in 2s complement? Does it fit in the provided digits? — dmckee --- ex-moderator kitten, Jan 20 '13 at 22:01
Use 8-bits rather than 5-bits, to avoid confusion. This approach will surely help. `-9(binary) : 0000 1001` & `-9(2's complement) : 1111 0111`. `-7(binary) : 0000 0111` & `-7(2's complement) : 1111 1001`. Now on adding these two: `1111 0111 +1111 1001` will yield `1` as carry & `1111 0000` as the output. The leftmost `1` in `1111 0000`, indicates the output is `-(negative)`. So now take the complement of the output & add `1` to it. That is your answer in binary. Your Question can be inferred as: `(-9)-(+7)`. So you don't have to waste time in deducing 2's complement of `-7`. — phougatv, Nov 21 '14 at 07:20

score 7 · Accepted Answer · answered Jan 20 '13 at 22:07

7

1) -9 - 7

-9 - 7 = -9 + -7

9 (binary) = 01001
-9 (2's complement) = 10111
7 (binary) = 00111
-7 (2's complement) = 11001

 10111 +
 11001 =
110000

This doesn't fit into 5 bits. Removing the overflow we get 10000, which is -16 (binary).

2) 6 - 10

6 - 10 = 6 + -10

6 (binary) = 00110
10 (binary) = 01010
-10 (2's complement) = 10110

 00110 +
 10110 =
 11100

This fits into 5 bits and is -4 (binary).

answered Jan 20 '13 at 22:07

melpomene

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I understand computer store negative numbers by their 2's complement. I'm wondering what if `6 - a, where a == -10`. How does this process of `-(-10)` happen? Does it try to get the 2's complement of -10 (10110) ? – Weishi Z Mar 25 '17 at 00:19
@WeishiZeng I don't know how real processors do it. I like to transform `x - y` into `x + -y` because then I don't need to remember a separate algorithm for subtracting binary numbers. If you have `00110 - 10110`, that becomes `00110 + -10110`, `00110 + 01010`, `10000`, which is -16. So yes, I simply take the 2's complement of `10110` by flipping all bits and adding `1`. – melpomene Mar 25 '17 at 08:59

score 0 · Answer 2 · answered Jan 20 '13 at 22:05

0

10111 + 11001 is not 100000 but 110000.

answered Jan 20 '13 at 22:05

m0skit0

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score 0 · Answer 3 · edited Dec 17 '15 at 07:20

Answer to the 1st question is wrong. To find -9-7 using two's complement, we need follow these steps:

STEP:1 Convertion of first  number

1st the binary conversion of 9:     01001

2nd find the complement of binary:  10110

Add 1 to the binary complement:     10110  
                                       +1
                                    -----
                                    10111  
STEP 2: Convertion of second number

1st the binary conversion of 7:     00111

2nd find the complement of binary:  11000

Add 1 to the binary complement:     11000 
                                       +1
                                   -------
                                    11001
STEP 3: Adding

Now add the two outputs -9 + (-7):  10111
                                   +11001
                                  --------
                                   110000
The most important thing is checking the answer whether it is correct or not.
you may use index for the binary digits:       7 6 5 4 3 2 1 0
                                                   1 1 0 0 0 0

find the 2 raised to the power of each index having 1 digit.
                                                (-)2^5 + 2^4

*Note (-) is used because in two's complement, the most significant bit (the bit with the highest index) is a sign bit -2^5 + 2^4 = -32 + 16 = -16 which is the correct answer for -9-7=-16. For this reason 2's complement become a popular way of representing negative number. For sign magnitude we need to assume a sign bit, which is hard to implement in a computer, and for 1's complement we need to add 1 to find the correct answer.

Binary Subtraction with 2's Complement

3 Answers3

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