Binary Counting

Why counting in binary matters

Counting in binary is really quite a cool process. However, sometimes it can be a little confusing, because it's a little outside of what we're used to. It is a critical skill to get down if we're going to go further into learning about hexadecimal or IP subnetting, or whatever it is that we're getting into. You have to know the basics here, so make sure you really understand the basics. I'll cover how to count in binary in a few different lessons, so really understand it.

Counting blocks in decimal

We're going to use counting blocks to learn how to count in binary numbers. Most likely you've learned about how to count using counting blocks in the past, but just as a refresher, because it's probably been a long time, we're going to go over how to use these in the decimal system and then we'll apply it to the binary system.

What I have here is four different columns: I have the ones, the tens, the hundreds, the thousands. It does continue to count up, so I'd have the 10 thousands, the 100 thousands and so on and so forth, but for now this is going to serve our purpose right here.

So how do we use this? What I'm going to do is just start out with a single counting block, and that's one. So we went from zero to one to two to three to 4 to five, 6, 7, 8 and 9, and now something interesting happens when we hit the next one, 10. When I try to add the 10th one, there's no room here for the 10th one, so what I have to do is I have to group this up and move it over to the tens, and this becomes a 10. Now I have one 10, and in fact what I'm going to do is I'm going to replace that with a column of 10 here. So I've got 10 now.

Now I go to count up again, so this is 11, 12, 13, and it's 13 because it's one 10 plus three 1s. So 14, 15, 16, 17, 18, 19, and once again, once I try to go to this last one here, there's no room for it, so I've got to move this over and I make another row of 10. So now I've got 20, I've got two 10s, which is 20.

All right, I count up 1, 2, 3, 4, five, 6, 7, 8, 9, and then when I try to add the 10th one, no room for it here, so I group it up and I move this over to be another 10. I'll replace that with a stick of 10 here. Now I've got three tens.

Let's fast forward a little bit. Let's go ahead and jump to what's going to happen when I have nine tens here, and let's put some ones on here, so we've got 94, 95, 96, 97, 98, 99. Now I can't fit another one in here, and so I carry this over, but now there's no room for this over here either, and so what we have to do is we have to then shuffle all of this over, and I have 100. In fact, what I'll do is I'll just replace this with a block of 100. Now I have one 100.

Then I continue to count: 101, 102, 103. Once I get all the way to, let's say, 194, 95, 96, 97, 98, 99, and here again I try to add one more to this, and there's no room here, so it moves over to the 10 column, and now there's no room here again, so I move it all over to the hundreds column, and I'll do another stack of 100 there to make it easier on us. Now I've got two 100s here.

So think about it as well: I've got two 100s here, maybe I've got four tens here, and I've got three ones. So what is that? Well, two 100s is 200, a five 10 is 50, so 250, and 3 1s is 3, so I have 253. So that's how we would use these counting blocks with decimal.

Counting blocks in binary

How would I do the same thing in binary? Well, to start out with, I've got a ones column, a twos, a fours, 8, 16, 32, 64, 128.

I'm just going to add a single block, so I go from zero to 1. Now what's going to happen is I'm going to add another block, but there's no room for it here, so I've got to shuffle it over. So now I've got a two, and I'll just replace that with a two block right there, so I've got two.

Now I add one more. Now I have one of the twos and I have one of the ones, so now I have three, three blocks total.

Next I add another one. It doesn't fit here, so what I have to do is I have to group it up and bring it over, which doesn't fit here either, so then I move it over to here. Now I have one block of four.

Then I add one more. Now I've got a block of four and a block of one, so now I have five.

I add another, and that means I have no room here, so I move it over to here. I have 1 four and 1 two, so 4 plus 2 is 6, and so there's six.

I add one more. Now I have 1 four, 1 two and 1 one, so now I have four, five, six, seven. I've got seven right there.

I add one more here. It doesn't fit, so I move it over. Doesn't fit there either, so I move it over. Doesn't fit there either, so I move it over, and now what I have is eight. I'm going to replace that with this block of eight so we can reuse these.

I'm going to add one more. Now I have 8 plus 1, so that's nine. I'm going to try to add one more here, doesn't fit, so I move it over. Now I have 8 plus 2, so that's 10. I'm going to add one more here. Now I have 8 plus 2 plus 1, and so now I have 11.

I'm going to add one more. Doesn't fit here, so I group it up here. Doesn't fit there either, so I move it over, and now I have 1 eight and 1 four, which is 12. I'll replace that.

I'm going to add one more here. Now I've got 8 plus 4 plus 1, which is 13. I try to add one more, and it doesn't fit here, so it moves over. Now I've got 8 plus 4 plus 2, so I've got 14. Replace that with the double there.

Now I'm going to add one more here. Now I've got 8 plus 4 plus 2 plus 1, so now I've got 15 right there. You can see the pattern here.

Now let's see what happens next. I add another block here, groups up, moves over, and no room here, so groups up, moves over. No room here, so it groups up, moves over, and no room here, so it groups over, so I've got 16. Now I've got one block of 16.

So that's how the binary numbers work. Let's do a pattern here, let's do every other one. So what this would be is, if I have one 16, 1 14 and 1 1, the pattern here would be 1 0 1 0 1, so that's the binary equivalent 1 1 0 1, and that would be the equivalent to one block of 16 plus one block of 4, so there's 20, plus one block of one, so that's 21. So now we have 21.

We'll be also doing this in other ways, so we'll get practice on doing this multiple ways, but make sure you understand how to count in binary. It's going to be pretty critical for what we're going to be doing in the future.