7.3 Subtracting 2's complement integers
You will probably have carried out subtraction of denary numbers using rules for subtraction that include the process of ‘borrowing’
whenever you need to subtract a larger digit from a smaller one. It is possible to perform binary subtraction in a very similar
way, but that is not what happens in computers. The processor contains the circuits needed to perform addition, and it is
much more efficient to use these circuits also to perform subtraction than it is to build in extra circuits to perform subtraction.
But how can subtraction be converted into addition? The answer is by first converting the number to be subtracted into its
additive inverse. For example, the denary subtraction
7 − 5
can be converted into addition provided the additive inverse of 5 is used. As I mentioned in Section 3.4, the additive inverse of 5 is −5, and so the equivalent addition is
7 + (−5)
In 2's complement binary arithmetic, the additive inverse of a number is known as its 2's complement. I'll start, therefore, by showing you how to find the 2's complement of any binary number.
7.3.1 Finding the 2's complement
In Section 2.4 you saw how to find the 2's complement representation of any given positive or negative denary integer, but it is also useful
to be able to find the additive inverse of a 2's complement integer without going into and out of denary. For instance, 1111 1100
(−4) is the additive inverse, or 2's complement, of 0000 0100 (+4), but how does one find the additive inverse without converting
both binary integers to their denary equivalents?
The answer is that the additive inverse, or 2's complement, of any signed binary integer can be found by a two-step process:
first find the complement (1's complement) of the given number and then add 1. 1's complement or complement means that all the 1s are changed to 0s and all the 0s to 1s.
An example should make this clear.
Example 8
Find the 2's complement of the signed integer 0001 1011.
Answer
First find the complement of the given integer (change all the 1s to 0s and all the 0s to 1s), getting:
1110 0100
and then add 1 to get:
1110 0101
So the 2's complement of 0001 1011 is 1110 0101. (Check: the given integer is+27, and 1110 0101 is −27.)
Activity 24 (Self assessment)
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Write down the complement of 1010 0101.
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Find the 2's complement of the signed integer 1011 0111. Check your answer by converting both integers to denary.
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Find the additive inverse of the signed integer 0000 1111.
Now read the answer
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The complement of 1010 0101 has all the 1s changed to 0s and all the 0s changed to 1s. Hence the complement is 0101 1010.
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The 2's complement is found by first finding the complement and then adding 1. So it is
0100 1000 + 0000 0001 = 0100 1001
0100 1001 is equal to 73 in denary and 1011 0111 is equal to (−128 + 55) = −73, which checks.
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Additive inverse is just another name for 2's complement, so the method is as above, giving:
1111 0000 + 0000 0001 = 1111 0001
7.3.2 Subtraction
As I indicated at the start of this section, subtraction is converted to addition by replacing the number to be subtracted
by its additive inverse, which in the case of binary arithmetic is its 2's complement. An example should make this clear.
Example 9
Subtract the signed integer 1010 1010 from the signed integer 0001 0110.
Answer
The additive inverse of the number to be subtracted, 1010 1010, is 0101 0101 + 1 = 0101 0110. Using this additive inverse
transforms the computation to an addition:
(Check: 1010 1010 is −86 and 0001 0110 is 22, so the calculation is 22 − (−86), which is 108, and 0110 1100 is indeed 108.)
Activity 25 (Self assessment)
Carry out the following subtraction by first finding the additive inverse of the number to be subtracted:
1100 1010 − 0000 1110
Now read the answer
The additive inverse of the integer to be subtracted is its complement plus 1:
1111 0001 + 0000 0001
which is 1111 0010.
So the calculation is now:
The ninth bit can be ignored here, as mentioned in Section 7.2. So the result is 1011 1100.
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