Binary Addition Calculator

What the binary addition calculator does

This adds two binary numbers together and shows you the sum in both binary and decimal. Enter the two numbers, press the button, and read off the result.

Adding in binary works on exactly the same idea as the column addition you learned for ordinary numbers. The only thing that changes is how soon a column fills up and carries, and once you see that, the rest follows.

How to use it

1. Choose a bit size. This sets how wide your numbers are, with 8 as the default. Pick a size at least as long as your longest number.
2. Enter the two binary numbers. Type each one using only 0s and 1s.
3. Press Calculate. The sum appears as both a binary result and a decimal result.

Press Reset to clear the inputs.

The rules of adding bits

You add binary column by column from the right, just like ordinary addition, carrying when a column overflows. The difference is that binary only has two digits, so a column fills up far sooner. There are really only four cases:

• 0 + 0 = 0
• 0 + 1 = 1, and 1 + 0 = 1
• 1 + 1 = 10, which is write 0 and carry 1

That third case is the whole trick. In decimal, 1 + 1 is just 2, but binary has no digit 2, so 1 + 1 rolls over to 10, the same way 9 + 1 rolls over to 10 in decimal. And when a carry lands in a column that already holds 1 + 1, you get 1 + 1 + 1 = 11, which is write 1 and carry 1.

A worked example with real numbers

Let us add 1011 and 0110. Work right to left, column by column.

• Rightmost column: 1 + 0 = 1. Write 1.
• Next column: 1 + 1 = 10. Write 0, carry 1.
• Next column: 0 + 1, plus the carried 1, is 10. Write 0, carry 1.
• Next column: 1 + 0, plus the carried 1, is 10. Write 0, carry 1.
• That last carry has nowhere to sit but a new column on the left. Write 1.

Reading the result gives 10001. To check it, 1011 is 11 and 0110 is 6, and 11 + 6 is 17, which is exactly what 10001 comes to. The carry rippled all the way across, which is common in binary because columns fill so quickly.

When the result grows a digit

As that example showed, adding two numbers can produce a sum that is one digit wider than either of them, because the final carry spills into a brand new column on the left. That is completely normal and is the binary version of how 50 + 50 grows from two digits to three in decimal. The practical thing to remember is to give yourself enough room: if you expect a large sum, pick a bit size a little wider than your numbers so the result has space to land. The decimal result always shows the true total whatever happens.

Questions people ask

How does binary addition work?

You add column by column from the right, carrying when a column overflows, exactly like ordinary addition. Since binary has only two digits, 1 + 1 already overflows to 10, so you write 0 and carry 1.

What is 1 + 1 in binary?

10. Binary has no digit 2, so 1 + 1 rolls over to the next column, giving 10, which you read as write 0 and carry 1.

Why is the answer sometimes longer than the numbers I added?

Because the final column can produce a carry that needs a new column of its own. The sum then has one more digit, just as adding two large decimal numbers can gain a digit.

What happens when a carry meets another 1 + 1?

You add all three: 1 + 1 + 1 = 11, which is write 1 and carry 1. So that column shows a 1 and still sends a carry onward.

References

1. Weisstein, Eric W. "Binary." From MathWorld, A Wolfram Resource. https://mathworld.wolfram.com/Binary.html