Binary addition follows four rules: 0+0=0, 0+1=1, 1+0=1 and 1+1=10 (write 0, carry 1). Subtraction works column by column with borrowing, or by adding the two’s complement.
Addition
| A | B | Sum | Carry |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Example: 1101 + 0111 = 10100 (13 + 7 = 20).
Subtraction
Example: 1101 – 0101 = 1000 (13 – 5 = 8). When a column needs to subtract a 1 from a 0, borrow from the next column to the left, which adds 2 to the current column.
Try it
Check any sum in the binary calculator, which shows the working.
Worked example: addition with carries
Add 1011 (11) and 0111 (7). From the right: 1 + 1 = 0 carry 1; 1 + 1 + 1 = 1 carry 1; 0 + 1 + 1 = 0 carry 1; 1 + 0 + 1 = 0 carry 1; the last carry gives 1. Result 10010, which is 18.
Worked example: subtraction with borrowing
Subtract 0101 (5) from 1100 (12). Work from the right, borrowing from the nearest 1 to the left whenever a column needs to subtract 1 from 0. The result is 0111, which is 7. Verify with the binary calculator.
Multiplication
Multiply 101 (5) by 11 (3): write 101 shifted by 0 (101) and shifted by 1 (1010), then add: 101 + 1010 = 1111, which is 15.
Practice problems
| Problem | Answer |
|---|---|
| 110 + 11 | 1001 |
| 1010 + 1010 | 10100 |
| 1111 – 101 | 1010 |
| 11 x 11 | 1001 |
FAQ
What is 1 + 1 + 1 in binary?
11, which means write 1 and carry 1.
Can binary subtraction be negative?
Yes. Computers store the result in two’s complement.