Episode 143: February 22, 2013
by Jason Marshall
What’s 1+1? It’s obviously 2, right? How about 100+10? Well, that’s obviously 110…right? Believe it or not, wrong…at least not when it comes to binary numbers. It can be kind of hard to wrap your head around, but as soon as we move from our cozy decimal numbered world into a world of binary numbers, things can seem a little confusing. But the truth is that binary addition is no harder than decimal addition. And it’s also an incredibly important part of the modern world. How does it work? And why is it so important? (Hint: think computers and calculators!) Stay tuned because those are exactly the topics we’ll be talking about over the next few weeks.
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Recap: What Are Binary Numbers?
Before we get into binary addition, let’s quickly recap what we’ve learned so far about binary numbers. The binary number system uses just the symbols ‘0’ and ‘1’ to represent an infinite number of numbers. To count in binary, start at ‘0,’ continue to ‘1,’ and then—just as we add a new digit after running out of symbols at the number 9 in decimal—in binary we add a new digit to make the binary number ‘10’ that represents the same number of apples, horses, or whatever as the decimal number 2. After ‘10’ apples comes ‘11’ (or 3) apples, then ‘100’ (or 4), then ‘101’ (or 5), ‘110’ (or 6), ‘111’ (or 7), and on and on—each digit represents the next higher power of 2.
If you want more of a refresher about the basics of counting with binary numbers and converting from binary to decimal, I highly encourage you to check out Part 1 and Part 2 of our series called What Are Binary Numbers? before continuing on.
The Four Rules of Binary Addition
As I said before, binary addition is no harder than decimal addition. In fact, all you really have to do is remember and understand four simple rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 10. The first three rules shouldn’t be at all surprising since they look just like ordinary decimal addition. These rules say that adding 0 to itself gives 0 and adding 0 and 1 (in either order) gives 1. No big surprise!
Binary addition is no harder than decimal addition.
But what about the fourth rule: 1 + 1 = 10? Well, once you understand the basics of counting in binary, this addition rule makes perfect sense. By which I mean that although adding one thing to another thing (as in 1+1) does indeed give two things, in binary we write those two things as 10. With those four rules you’re ready to move on to the next step: adding 2-digit binary numbers.
Two-Digit Binary Addition
The easiest way to add 2-digit binary numbers is to start by writing them in columns…just as you did when you learned to add decimal numbers. That means that for an addition problem like 10 + 01, start by writing the number 10 above the number 01. Now, just as in decimal addition, start by adding the numbers in the far right column. In this case, that’s 0 + 1. Remembering our basic binary addition rules from earlier, we find that the far right column is equal to 0 + 1 = 1. Just as in a decimal addition problem, you should write this number at the bottom of the right column.
Next, move to the left column and add its digits: 1 + 0. Again, the simple binary addition rules tell us that 1 + 0 = 1. By writing that number at the bottom of the left column, we’ve arrived at our final answer. We’ve found that 10 + 01 = 11. Does that make sense? Well, the binary numbers 10 and 01 are equal to the decimal numbers 2 and 1, and the binary number 11 is equal to the decimal number 3. Since 2 + 1 is certainly equal to 3, we see that binary addition does indeed work!
Carrying Binary Digits
While that’s all well and good, there’s one complication that we haven’t yet dealt with. To see what I mean, remember that when you’re adding decimal numbers like 14 and 8, you start by adding the right hand column to get 4 + 8 = 12. You then write the 2 at the bottom of that column and you carry the 1 to the top of the left column. What does this have to do with binary addition? Well, sometimes binary digits need to be carried, too.
For example, in the binary addition problem 11 + 11, we start as before by writing the numbers with their columns aligned. Adding the right column gives 1 + 1 = 10. Now, as with decimal addition, write the 0 at the bottom of the right column and carry the 1 to the top of the left column. Next, we need to add the left column which now contains three 1s. Since 1 + 1 + 1 = 11 in binary, write a 1 at the bottom of that left column and carry a 1 into a new third column on the far left. Since that 1 is the only number in that far left column, we’re done. We’ve found that 11 + 11 = 110 in binary. Or, in decimal, we’ve found that 3 + 3 = 6. Once you understand these problems, you know everything you need to deal with larger problems. Just remember to work from right-to-left and to use the four basic rules of binary addition.
Okay, that’s all the math we have time for today. But that’s definitely all that there is to say about binary addition. After all, we still haven’t figured out how computers and calculators manage to do addition! So be sure to check back next time for the next part of the story. In particular, we’ll be talking about what something called Boolean algebra (it’s not as difficult as it sounds) has to do with all of this.
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