Binary Addition
What is binary addition?
- Binary addition involves summing numbers in base-2, which uses only the digits 0 and 1
- Like denary addition, start from the rightmost digit and move towards the left
- Carrying over occurs when the sum of a column is greater than 1, passing the excess to the next left column
Example addition
Binary addition example
Overflow errors
- Overflow occurs when the sum of two binary numbers exceeds the given number of bits
- In signed number representations, the leftmost bit often serves as the sign bit; overflow can flip this, incorrectly changing the sign of the result
- Overflow generally leads to incorrect or unpredictable results as the extra bits are truncated or wrapped around
An overflow occurring after a binary addition
Binary Subtraction
- As well as adding binary numbers, we can also subtract binary numbers
- One method of doing this is to use two’s complement
Example 1
Subtract 0011 (3) from 1001 (9)
1. Given numbers
| Label | Bit 3 | Bit 2 | Bit 1 | Bit 0 |
|---|---|---|---|---|
| Number 1 | 0 | 0 | 1 | 1 |
| Number 2 | 1 | 0 | 0 | 1 |
2. Two’s complement
- Convert the number to subtract (
0011) to its two’s complement - Invert:
1100 - Add 1:
1100 + 0001 = 1101
| Label | Bit 3 | Bit 2 | Bit 1 | Bit 0 |
|---|---|---|---|---|
| Number 1 | 1 | 0 | 0 | 1 |
| Number 2 (Converted) | 1 | 1 | 0 | 1 |
3. Addition operation
- Now add 1001 and 1101
- Binary sum:
1001 + 1101 = 1 0110 - That’s 5 bits: the leftmost
1is overflow (carry out of MSB)
| Operation | Bit 4 | Bit 3 | Bit 2 | Bit 1 | Bit 0 |
|---|---|---|---|---|---|
| Carry | 1 | 1 | |||
| Number 1 | 1 | 0 | 0 | 1 | |
| Number 2 (Converted) | 1 | 1 | 0 | 1 | |
| Addition | 1 | 0 | 1 | 1 | 0 |
4. Remove overflow
- The result is
10110with overflow - Drop the leading
1(overflow):0110= 6 (in decimal) - In two’s complement arithmetic, the overflow bit does not contribute to the actual value of the operation but is more of a by-product of the method
- Final answer = 6
- 9 - 3 = 6