Multiplying with Negative Numbers in 2’s Complement
Preliminary insights.
Fixed number of bits
2’s complement is used to represent negative numbers in systems where the number of bits is fixed. The number of bits in a number remains the same regardless of the value attributed to the number eg in 4-bit binary arithmetic 2dec is 0010, the leading 0s are required to keep to the 4-bit format.
Negative numbers
Negative numbers can be represented by their complement.
Since 2 + 14 = 16, -2 = 14 - 16
14 is the complement of 2 and the binary equivalent of 14 can represent -2.
The decision was made to let binary numbers beginning with a 1 represent negative numbers using the complement format.
Numbers beginning with 0 represent positive numbers. 0000 is regarded as positive.
|
Negative Number |
Complement |
4-bit |
|
-1 |
15 |
1111 |
|
-2 |
14 |
1110 |
|
-3 |
13 |
1101 |
|
-4 |
12 |
1100 |
|
-5 |
11 |
1011 |
|
-6 |
10 |
1010 |
|
-7 |
9 |
1001 |
|
-8 |
8 |
1000 |
Subtracting 16
What does it mean to subtract 16 from the complement?
Let’s take 12 16
The first thing we notice is 16 in binary is outside the 4-bit format. We need a higher 4-bit word.
Points to notice.
1. The lower 4-bit word is unchanged. Subtracting 1 from 0000 requires “borrowing” a 1 from a even higher 4-bit word. IMHO a more accurate description would be “begging” since there is no obligation to repay the borrowed 1.
2.
Subtracting 1 from a string of 0s produces a string of 1s.
This happens because The
represents the overflow.
Performing multiplication when one multiplicand is negative
The correction rule is, “For each negative number in the multiplicand, subtract the other number from the word where the overflow occurs.”
It might look like 235 but subtract 162 = 256 and -21 is revealed.
Hope you are as delighted as I am to have been shown how to use 2 complement for perform multiplication.
Why does it work?
Best wishes,
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