All Tools VerseBinary addition works just like decimal addition, except you carry whenever a column reaches two instead of ten. Add binary 1 and 1 and you get 10, which is a 0 with a carry of 1 into the next column. This guide shows how to add binary numbers step by step, the four simple rules behind it, and a free tool that adds binary instantly and shows the carries.
In this guide
The four addition rules
Binary has only two digits, so column addition has just four cases: 0 plus 0 is 0, 0 plus 1 is 1, 1 plus 0 is 1, and 1 plus 1 is 0 with a carry of 1. A fifth case appears once carries are in play: 1 plus 1 plus a carried 1 is 1 with a carry of 1. That is the entire rule set, and it is enough to add any two binary numbers. For how binary represents values in the first place, see our number systems guide.
Add binary step by step
Line the numbers up on the right, exactly like decimal, and work column by column from right to left. Add 1011 and 0110. Rightmost column: 1 plus 0 is 1. Next: 1 plus 1 is 0 carry 1. Next: 0 plus 1 plus the carried 1 is 0 carry 1. Next: 1 plus 0 plus the carried 1 is 0 carry 1, which becomes a leading 1. The result is 10001. The binary addition calculator shows each carry so you can follow the working.
Carries and overflow
A carry out of the leftmost column means the result needs one more bit than the inputs. On paper that is fine, you just write the extra digit. In a fixed-width register, say 8 bits, that carry has nowhere to go and the value overflows, wrapping around. This is why programmers watch the carry and overflow flags during low-level arithmetic. The related operation of combining bits without carrying is the bitwise AND, covered in our bitwise operations guide, and you can try it with the binary AND tool.
Check your answer
The quickest check is to convert both inputs and the result to decimal and confirm the sum. For 1011 and 0110, that is 11 plus 6, which should be 17, and 10001 is indeed 17. Use the binary to decimal converter for the check, which catches any miscounted carry immediately.
Why it matters
Addition is the operation a processor performs most, and every other arithmetic operation is built on it. Subtraction uses addition with a complemented operand, multiplication is repeated addition, and address calculations are additions. Understanding the carry chain also explains why wider registers compute large sums faster and why overflow bugs happen. It is the foundation of how a machine that only knows 0 and 1 does math.
Free tools used in this guide
Frequently asked questions
How do I add binary numbers?
Add column by column from the right, carrying a 1 whenever a column totals two. So 1 plus 1 is 0 carry 1, exactly like decimal but rolling over at two instead of ten.
What is 1 plus 1 in binary?
Binary 1 plus 1 is 10, which is a 0 in that column with a carry of 1 into the next.
What happens when there is a carry out of the last column?
The result needs an extra bit. On paper you write it, but in a fixed-width register it overflows and the value wraps around.
How can I check a binary addition?
Convert both numbers and the result to decimal and confirm the sum matches. A binary to decimal converter makes this quick.
Is binary subtraction done the same way?
It is usually done by adding the two’s complement of the second number, so subtraction reuses the same addition hardware rather than a separate rule set.