Imagine you have a bunch of pet blobs. You keep your pet blobs in a little row on your desk. They look like this:
Now, obviously all of your blobs are special in their own sweet way, but it isn’t hard to notice that one of them (the pink one) is special-er than the others. Suppose you wanted to refer to the position of the pink blob in this row. How would you do it? You would count. And if you grew up almost-anywhere in the modern world, you’re going to count it like this:
1-2-3-4-5-6-7-8-9-10-11.
The pink blob is the eleventh blob using our modern Western number system.
You might notice something special about the numbers “10” and “11” here: they have two digits, instead of just one. If you graduated from elementary school then you probably know why that is: if a number has two digits then the right-hand digit is the “ones” column and the left-hand digit is the “tens” column. A number like “54” is made up of 5 tens and 4 ones.
Base 10 Math
Let’s break that down visually. First, let’s note that each of the blobs is a “1” all of its own.
We could describe the pink blob as the 1-plus-1-plus-1-plus-1-plus-1-plus-1-plus-1-plus-1-plus-1-plus-1-plus-1th blob, if we wanted to, but that’s not particularly convenient, so we probably won’t.
In order to make our lives simpler we use two little tricks. First off, we create a bunch of special names for small collections of 1’s: for example, 1-plus-1 is given the name “2”, and 1-plus-1-plus-1 is given the name “3” (and so on and so on, up to “9”). This is something we tend to take for granted, but it’s a pretty neat when you think about it that we save ourselves all that writing-things-out effort.
Second, we use the “batching” trick: once we get past “9”, instead of coming up with more special symbols, we just say “10”: that’s 1 batch of tens and 0 batches of units.
The Base-3 Number System
Now, here’s the kicker: while the batching trick is incredibly useful, there’s no particular reason why we have to use batches of the-number-we-think-of-as-10 (I’ll explain that bizarre phrasing in a second, I promise). Instead, what if we used batches of (say) 1-plus-1-plus-1?
If we wanted to describe the pink blob in this system we’d have to call it 102. Obviously this is not what we think of when we say “102” in our normal number system, so let’s talk through what it means. We have three columns of numbers here. The 1 on the left hand column means we have 1 batch of 3-batches-of-3s; the 0 in the middle column means we have 0 batches of just-3s; and the 2 on the right hand column means we have 2 batches of 1s-on-their-own.
It’s hard to get our heads around the idea that something we’ve spent our entire lives calling “11” could also be referred to as “102”, but I hope you can see why using batches of 3s (or any other number) could be a not-totally-crazy way to count.
In our new system, of course, “10” would mean “one batch of 3s and no batches of units” -- in other words, “10” would refer to the number we’re used to calling 3. In general, the number “10” just means “one batch of our batching number and no extra units” -- this is true in any “base” that we might care to use. Obviously in everyday life you can safely assume what “10” represents, because the whole world has very conveniently converged on a common base-system for everyday life. But there are some scenarios where it’s very useful to use a different base-system. Which brings us to...
Understanding the Binary System
The binary system! We could also call the binary system “base-2”, because it’s simply the same idea applied to batches of 2. Let’s try to find our pink blob under this system:
It looks a little overwhelming but it’s really just the same system we used before, applied several times over. What we have here is:
1 batch of 2 batches of 2 batches of 2’s
0 batches of 2 batches of 2’s
1 batch of 2’s
1 batch of units
So, the number we would describe as “11” in base-10 or as “102” in base-3 would be described in binary as “1011”.
And that’s all you need to know to understand the binary system. “Why do computer use the binary system,” you might be wondering, “and not the system that humans are used to?,” -- that’s a question for another day, unfortunately. For now, though, we know enough to understand exactly how the binary system works. Fantastic!
Want to check your understanding? Try out our quick little quiz below!
http://www.guidedtrack.com/programs/wqxgvlg/run
Uri Bram writes popular non-fiction books with a conceptual approach to mathematical, scientific and analytical thinking. He is the author of Thinking Statistically and Write Harder.