Data Representation: Two's compliment
Decimal to two's compliment binary
Two's compliment is a method that allows us to represent negative numbers in binary. It works by making the most significant bit negative. By the most significant bit we mean the bit in the column with the largest column heading
Example
An 8 bit number in two's compliment could be any number between -128 and 127 as can be seen below.
Biggest number possible.
| -128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Total |
| 0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
127 |
Smallest number possible.
| -128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Total |
| 1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-128 |
As you can see the most significant bit, in this case 128, is negative. Note that the range of numbers that can be represented is still 256, including 0, we have just changed which numbers can be represented
If we were talking about a 5 bit number then the most significant bit would be -16 with a range of -16 to 15 able to be represented as seen below.
Biggest number possible.
| -16 |
8 |
4 |
2 |
1 |
Total |
| 0 |
1 |
1 |
1 |
1 |
15 |
Smallest number possible.
| -16 |
8 |
4 |
2 |
1 |
Total |
| 1 |
0 |
0 |
0 |
0 |
-16 |
There are several ways to convert from a negative denary number to a two's compliment binary number. The first is very similar to our method for converting from positive denary to binary.
When converting a negative number to two's compliment we know the most sigificant bit must be a 1 so the next step is to calculate how much we need to add back in order to get to our number. Let's convert the number -82 to two's compliment binary.
-128 + ? = -82
? = -82 + 128 = 46
Now that we have our number we can continue as we did before. Put a 1 in the column with the largest column heading that is not bigger than our number. Place zeros in any columns that we have gone past.
| -128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Total |
| 1 |
0 |
1 |
|
|
|
|
|
|
Subtract the column heading from our number and start again. In this case 46 - 32 = 14.
| -128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Total |
| 1 |
0 |
1 |
0 |
1 |
|
|
|
|
And repeat until the number to allocate is zero then any remaining columns are zero. So 14-8 = 6.
| -128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Total |
| 1 |
0 |
1 |
0 |
1 |
1 |
|
|
|
6-4 = 2
| -128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Total |
| 1 |
0 |
1 |
0 |
1 |
1 |
1 |
0 |
-82 |
Check by using a calculator -128 + 32 + 8 + 4 + 2 = -82
Negative denary to two's compliment binary practice
Click the button to get a number to convert
Practice
Click the button below to generate a number to convert to two's compliment binary.