Do your eyes glaze over when you see a long string of ones and zeroes? If they do, that just means that you’re a normal member of the human race. Binary numbers are for computers, not people. But if you do need to convert binary numbers to decimal numbers, we’ll show you how to do it without a calculator or a conversion tool.
If you’re going to be working with computers on a technical level, such as electronic engineering, programming, or system administration, you will need to know the basics of working with binary numbers; fortunately, there are good online courses available in all of these subjects and more.
1 For the Money, 10 For the Show
First of all, let’s take a look at what a binary number is.
You’re used to decimal numbers — in fact, you’re probably so used to the way that they work that you probably don’t think about it. You count from 1 to 9, then after that, put a zero in the ones’ place and out a 1 in the tens’ place. In other words, once you reach the base number (10) or a multiple of it by itself (100, 1000, 10000, etc.), you put a 1 in the next place to the left: 99 + 1 = 100.
With binary numbers, you do the same thing — the only real difference is that two is the base instead of ten: 1 + 1 = 10, and 11 + 1 = 100.
11 To Get Ready, and Go, Cat, Go!
“Whaaaaat? Run that by me again!”
OK. It may look a little (as in seriously) weird, so we’ll break it down:
A binary 1, all on its own, is just 1, the same as it is in decimal numbers. But the base of the binary number system is two, and just like with decimal, once you reach the base (2) or a multiple of it by itself (4, 8, 16, etc.), you put a 1 in the next place to the left. 1 = 1 still makes two in binary, but there isn’t any numeral 2 for you to write — instead, you put a zero in the ones’ place, and a 1 in the twos’ place. 1 + 1 + 1 still makes three, just like in the Beatles’ song, but in binary, you take 10 (which is two), and add 1 — since there’s a zero in the ones’ place, you represent it by writing down 11.
Adding It All Up
What happens when you add 1 to binary 11 (three)? 1 + 1 in the ones’ place means that you put a zero there, and add a 1 to the twos’ place — but there’s already a 1 in the twos’ place, so you add them, and put zero in the twos’ place, then put a 1 in the next place to the left, which is the fours’ place. In other words, 11 + 1 = 100 in binary just means “three plus one equals four.”
Now let’s take it up another level. Add 11 (three) to 100 (which is just four), and the result is — 111, which, it should be no surprise, is seven. And if you add 1 to 111, you get 1000, which, of course, is just eight, binary-style. What’s 1111? It’s 111 (seven) plus 1000 (eight), which is fifteen. And what’s 1 + 1111? It’s 10000, which is the binary way of writing sixteen.
Stepping Through It
By now you should be getting the picture. The magnitude of each position in a binary number is twice the magnitude of the position just to the right. If you know the position of each of the 1s in a binary number, you know the value (in decimal) of each of those 1s. (The value of a zero is always zero, so you don’t have to worry about the 0s — just keep track of their positions). For example, in 1010 (binary), the 1 that’s farthest to the right is in the twos’ place, so it’s a two, and the 1 on the left is in the eights’ place, so it’s an eight. The other places have zeroes, so two plus eight equals ten. You’ve just converted a binary number to decimal.
Write It Down
“But,” you may be thinking, “binary numbers can be pretty long. How am I supposed to keep track of the position of each digit, and the magnitude of each position?”
It would be a trick to keep track of all of those things in your head with a long binary number, so we’ll leave such things to memory experts and math prodigies. But as it turns out, it’s actually pretty easy to do it on paper.
First, look at the number and count how many digits (positions) it has. (If it starts with leading zeroes on the left, ignore them, and only count 1s and 0s from the right, out to the farthest 1 on the left.)
You already know (or can quickly calculate) the magnitude of each position (it’s just two times the position to the right), and now you know how many positions you need. Write down the magnitude of each position, in order, either vertically, or horizontally, like this:
1024 512 256 128 64 32 16 8 4 2 1
Line ‘Em Up
You just did the hard part. Everything else is simple addition. What do we mean? First, let’s pick a longish-looking binary number: 1011001110. How many digits (positions) are there? There are ten, so you need the list of magnitudes to go out to ten positions: 512 256 128 64 32 16 8 4 2 1.
Now you write the binary number below the magnitudes, with each digit beneath its magnitude, and draw a line beneath the number, like this:
512 256 128 64 32 16 8 4 2 1 1 0 1 1 0 0 1 1 1 0 ----------------------------
Drop Down and Add
Now look at each magnitude. If there’s a one below it, write it below the line. If there’s a zero below it, put a zero below the line:
512 256 128 64 32 16 8 4 2 1 1 0 1 1 0 0 1 1 1 0 ---------------------------- 512 0 128 64 0 0 8 4 2 0
Now add all the numbers below the line (tossing out the zeroes):
512 128 64 8 4 2 --- 718
You Did It!
And that’s how it works. There are other ways to do it, of course (lining the numbers and magnitudes up vertically, starting from the right and multiplying by powers of two), but they all come to basically the same thing.
It’s a skill that’s very useful if you plan on going into such in-demand fields as electronic engineering, computer programming, system or network administration, or any computer-related engineering field.