Powers of 2

After working with computers for a while, you'll start noticing that there are some numbers that come up very consistently, like 256, which for some reason comes up a lot. But why does it come up a lot? Well, it really boils down to those binary numbers and the powers of two.

Counting combinations

If we have one bit, it could be either a zero or one, so it can be in two different states. When we have one bit, there are two states with this.

So how many combinations can we think of when we have two bits? Well, we could have 0 0, we could have 0 1, we could have 1 0 and a 1 1. So now we have four possibilities: 1, 2, 3, 4. We've doubled the amount of possibilities that we have. And if you think about it, it really makes sense, because we still have that single bit that could be a zero or one, and now we have an extra bit which could be a zero or a one. So we've doubled the amount of combinations we have by adding an extra bit.

Let's do that again. Here we have three bits, so we've got 0 0 0, 0 0 1, 0 1 0, 0 1 1, and we've got 1 0 0, 1 0 1, 1 1 0 and 1 1 1. So 1, 2, 3, 4, 5, 6, 7, 8 — we have eight combinations here, eight different ways that we can represent this with three different bits. And this makes sense as well, because in this case right here we have the four combinations we had with two bits, but we could do that with a zero as the third bit or a 1 as a third bit, so we've doubled the amount. So now every time we do this, we double the amount.

Four bits

If we have four bits, how many combinations do we have? I'm going to let you figure this out. Go ahead and figure out what this answer is with four bits. You can do it two ways: you can do it the pattern that we've already discovered, where we double it every time, or you could also figure out what are all the combinations of ones and zeros that you can possibly have.

So if you came up with the answer of 16, then you'd be correct. It's twice as much as eight, so we know from the pattern that we're going to have 16 different combinations. But what are those combinations? Here's a list of those combinations. If you noticed, it's the same combinations here for the first three bits, but since we can do it all as zeros and then again all as ones, we have another set of those same combinations again.

The powers of two

Here are a couple of other ways to look at this. If you just have a single bit, there's two possibilities. If you have two bits, it's 2 × 2. If you have three bits, it's 2 × 2 × 2. And if you have four bits, it's 2 × 2 × 2 × 2. We can just shorten that to two to the first power, two to the second power, two to the third power, two to the fourth power. So it's two to however many bits that you have is going to be what the answer is. And so now we get, in both these cases, 2, 4, 8, 16, and so on and so forth.

Here's the rest of that. We're going to go out to 2 to the power of 8, and here are the answers: 32, 64, 128, 256. So these are common numbers that you'll see on computers. If we take this all the way out to 2 to the 16th power, we get 65,500 136.

Now, these numbers are going to show up a lot on computers. In addition to it, it can keep going, so you'll still see these numbers. The powers of two is a lot of times why we see these strange numbers pop up on our computers, because what it's doing is it's calculating a lot of the information that it has with these powers of two. So you need to be familiar with these, especially the first eight. These are going to be something that you're going to encounter a lot, especially if you're getting into subnetting, or if you're getting into working with certain types of processing. Certain programs will have limits that you'll see here, and so you'll see it a lot, and it can be applied in a lot of scenarios when it comes to working with computers.