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Adding Binary Numbers

Binary addition follows the same carry-over logic as decimal addition, applied to base-2 values. Two worked examples demonstrate how bits combine and carry across place values to produce correct sums.

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About this video

Binary addition operates on the same fundamental principle as decimal addition: values are combined column by column, and any result that exceeds the capacity of a single position carries over to the next higher place value. Because binary uses only 0 and 1, a carry occurs as soon as two bits in the same column combine, shifting a value leftward into the next position rather than allowing a digit greater than 1 to remain. Two worked examples illustrate the process using place-value blocks. In the first, 1101 (13) and 0110 (6) are added together, with carries moving through the 4-bit and 8-bit positions until the result resolves to a 16 and a 2, confirming the expected decimal sum of 19. The second example adds 1111 (15) and 0101 (5), where multiple carries — including a case with three overlapping 4-bit values — propagate through the columns to produce the binary result 10100, equal to 20 in decimal. Both examples reinforce how binary arithmetic, though visually different from its decimal counterpart, follows consistent and predictable rules.

What you'll learn

What's covered

Adding Binary Numbers

Aligned to

CompTIA Tech+
1.1 Compare and contrast notational systems.
1.3 Illustrate the basics of computing and processing.

Key terms

Binary Addition
The arithmetic process of adding two binary numbers using base-2 rules, where 0+0=0, 0+1=1, 1+1=10 (sum 0 carry 1), and 1+1+1=11 (sum 1 carry 1).
Carry
In binary addition, the value of 1 passed to the next higher bit position when a column's sum exceeds the maximum value representable by a single bit.
Bit
Short for binary digit, the smallest unit of data in computing, holding a single value of 0 or 1; all digital information is ultimately composed of bits.
Binary Number
A number expressed in base-2 using only the digits 0 and 1, where each position represents a power of two.

Transcript

Another way that we can get familiar with these binary numbers is by learning how to add binary numbers together. So let's take a look at adding binary numbers together.

First Example

Let's use our blocks to illustrate how we add two numbers together. Up here is going to represent one number and down here another number. This number right here is going to be 1 1 0 1, and this number down here will be 0 1 1 0.

What does this look like? We've got one block of eight, we've got one block of four, and we've got one block of one, so 8 + 4 is 12, + 1 is 13. So 13. Down here we've got a block of four and a block of two, so that's 4 + 2, so that's 6. 6 + 13 is 19, so our answer in the end should be 19.

Let's use the blocks now to add this up. This one right here is just a one, so we can move that into its spot, we're good. This is a two, so we can move it into its spot, so we're good there. We've got two blocks of four, which don't fit in here, so what we have to do is move them over to the next block right here. And now this doesn't fit in here, so now we've got to move this over. So now we've got one block of 16 plus a block of two, so 17, 18, and then one block of one, 19. So we've come up with the correct answer by doing it this method.

Second Example

Here's one more problem. We have a 1 1 1, and so this is 8 plus a 4 plus a 2 plus a 1, and so what that is equal to is 15. We're going to add that to one block of four and one block of one, so that would be a 0 1 0 1, so that's five. So 5 + 15 is 20, that should be the answer. Let's see what we come up with.

In this first column right here we're trying to add these two bits together, but there's no room here, so we've got to move it over to here. Now we can carry this down, and we see that there's no room for these either. We've got a block of four, so we can carry it over here. Now this is interesting, because we've got three blocks of four, and what we can do is we can group one of these into a block of eight and move it over, but the other one can stay right there, that's fine. So now we've got two blocks of eight, which doesn't fit in here, so we move those over to 16. Now we've got a 16 plus a four, so 1 0 1 0 0. So if we add a 16 plus a 4, that's 20. We now have come up with the solution using binary numbers.

So that's how you add with binary numbers.

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