Same method, smaller digits
You already know how to add and subtract binary numbers. It's the column method from school, with one change: each column holds only 0 or 1, so you carry or borrow at 2 instead of at 10.
That one change is why binary feels strange at first. In decimal, 1 + 1 is a quiet 2. In binary, 1 + 1 already fills the column, so it becomes 10 ("one zero", which means two). Once that clicks, the rest follows.
Adding binary numbers step by step
Line the numbers up on the right. Work one column at a time from right to left, adding the two digits plus any carry. Write the last digit of the column's total and carry the rest.
Take 10110 + 1101, which is 22 + 13. Column 1: 0 + 1 = 1. Column 2: 1 + 0 = 1. Column 3: 1 + 1 = 10, write 0, carry 1. Column 4: 0 + 1 + 1 = 10, write 0, carry 1. Column 5: 1 + 1 = 10, write 0, carry 1 into a new sixth column. Reading from the left: 100011, which is 32 + 2 + 1 = 35. Since 22 + 13 = 35, it checks.
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 1 = 10 (write 0, carry 1)
- 1 + 1 + 1 = 11 (write 1, carry 1)
Subtracting with borrows
Subtraction goes right to left too. When the top digit is 0 and the bottom is 1, borrow from the next column to the left. In binary, that borrowed 1 is worth 2 in the current column, so 0 - 1 turns into 2 - 1 = 1.
Try 1101 - 1011, which is 13 - 11. Column 1: 1 - 1 = 0. Column 2: 0 - 1 needs a borrow, so take 1 from column 3 (which drops to 0) and work out 2 - 1 = 1. Column 3: 0 - 0 = 0. Column 4: 1 - 1 = 0. The answer is 0010, or 10, which is 2.
A borrow can travel. In 1000 - 1, the first three columns are all 0, so the borrow comes from the 1 on the far left and turns each 0 it passes into 1. The answer is 111, which is 8 - 1 = 7.
How computers subtract: two's complement
Processors don't usually borrow column by column. They turn subtraction into addition. To subtract a number, they add its two's complement: flip every bit, then add 1.
Take 5 - 3 with 4-bit numbers. 3 is 0011. Flip the bits to get 1100, add 1 to get 1101. That's -3 in 4-bit two's complement. Now add: 0101 + 1101 = 10010. The fifth bit doesn't fit in 4 bits, so it's dropped, leaving 0010, which is 2.
The same trick explains why a leading 1 means negative in signed binary. In 8 bits, 11111101 is -3, because 253 - 256 = -3.
Formula notes
Two's complement of x = (flip every bit of x) + 1A - B = A + two's complement of B, keeping only the word's bits
Common mistakes
Most binary errors come from habits carried over from decimal. When an answer looks wrong, convert both inputs and the result to decimal and check the math there.
- Writing 2 in a column. Binary has no digit 2. 1 + 1 is 10.
- Forgetting the triple carry. 1 + 1 + a carried 1 is 11, so write 1 and carry 1.
- Treating a borrow as 10. A borrowed 1 is worth 2 in the current column.
- Misaligning the numbers. Always line them up on the right, padding with zeros if it helps.
- Dropping the final carry in exact math. 1111 + 1 is 10000, five digits.
Checking your work
The quickest check is to convert to decimal. Add up the place values (1, 2, 4, 8, 16...) wherever there's a 1, do the sum in decimal, and compare. For a subtraction, add your answer back to the number you subtracted. You should get the number you started with.
For longer numbers or fixed word sizes, our binary calculator shows each input and the result in binary, decimal and hex, and flags overflow when the answer doesn't fit the word you picked.
Frequently asked questions
Why does 1 + 1 equal 10 in binary?
Binary only has the digits 0 and 1. Two is the first number that needs a second column, just like ten does in decimal, so it's written 10.
What happens if you subtract a bigger binary number from a smaller one?
On paper you get a negative result and write a minus sign. Inside a computer, the answer comes out as a two's complement pattern with a leading 1.
How do you find the two's complement of a binary number?
Flip every bit and add 1. For 0110 (6) in 4 bits, flipping gives 1001, and adding 1 gives 1010, which is -6.
Is binary addition the same as the OR operation?
No. OR never carries, so 1 OR 1 is 1. Addition carries, so 1 + 1 is 10. They match only when no column has two 1s.
References
These sources support the method or guidance used for Binary Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Binary Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.