How do computers add?
To understand how binary addition works, it is useful to first think about how addition works in general and it is probably easiest to consider how we do this in our own, familiar, decimal base-ten.
Lets start with something very easy 0 + 1
Now, still very easy. 1+1
And still easy is 1 + 1 +1
These are all very easy because they are single digit sums. We’ll come back to this simple idea again soon but in binary form.
Now lets think about what happens when we add two numbers together and the answer is more than one digit.
Lets add together 9 and 1.
9 + 1 is 10. This answer contains more than one digit!
How do we write this? We put a 0 in the ones column and then we carry the one to the next column.
So now we have to add up all the values in the next column, well there is only a 1 in that column so now we write down the 1 as well.
9 + 1 is 10.
It is important to remember that when we have more than one digit when we add up the numbers in one column then we carry the most significant (biggest bit) of the number to the next column and include it in the next addition.
Binary is just like decimal addition except there are only two digits.
The highest single digit value you can get to in binary is one!
There are four very important rules you need to remember when adding in binary. These rules are not particularly special rules. Really they are just some ordinary addition facts but we are expressing their answers in binary form:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 1 = 2 (but, remember two in binary is 10)
- 1 + 1 + 1 = 3 ( three in binary is 11)
Let’s do a binary addition, lets add five (101)and three (011) in binary. As we do this, remember that when the sum of the digits in a column added together is more than one digit, we carry the most significant digit over to the next column.
We will start with the rightmost (smallest) column, just like decimal addition.
1+1 = two.
Two in binary is 10
So we write down the smallest value (0) and carry the 1 to the next column.
Now we add up the values in the next column. We have the 1 we carried over from the previous column and the 1 that was already in the bottom row. So we have another 1 (+0) + 1.
Just like before we know that 1 + 1 is 0 carry 1
Now we add up the values in the third column. There is the 1 we carried over and there is the 1 that was already there.
1 + 1 is 0 carry 1
There is now a fourth column. This column only contains 1, so we write that down below.
So, 101 + 11 = 1000
If we convert these three numbers to decimal then we get:
5 + 3 = 8!
So we can see that this method works. You can add together two binary number with any number of bits as long as you follow these basic rules.
The Rules of Binary Addition
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 1 = 0 carry 1
- 1 + 1 + 1 = 1 carry 1