Number Systems

Binary is a number system, and it operates just the same as other number systems that you're used to working with on a day-to-day basis. Because some of us have not dealt with binary numbers before, it can get a little confusing. But as long as you keep in mind that it operates off of the same parameters as the systems you're already used to, then it can be a lot easier to understand how binary works.

Examples of number systems

Here are just a few examples of the number systems we're going to see. There are actually a lot of different number systems.

Decimal is probably the one that you're used to working with the most. You probably use it on a daily or weekly basis, where you have to calculate something or you add something up, and so you use this often. That's why we're going to correlate strongly between binary and decimal. Binary is the topic at hand, the main topic that we want to get to the bottom of. Hexadecimal is yet another system that looks quite a bit different. And Roman numerals are an example as well, which really operates quite a bit different than these other three examples here.

If you were to go and do a search for number systems or numeral systems on the web, this Wiki article would be one that would pop up. This is a list of numeral systems, and as you scroll through here — there's Roman numerals right there — you can see that there are a lot of different types of systems that are out there, a lot of ones that we probably haven't encountered before. And then there are some of these base ones, like binary, and the decimal system. You can see they go by multiple names and they're defined a little differently in here. There's hexadecimal. There is a long list of different options in here.

Roman numerals

Here we have the Roman numeral system, which is quite a bit different than the day-to-day math that we're used to. Many of you have probably encountered Roman numerals in the past and might even know how it works, with the I for one, a V is five, and X is ten, and you add up these numbers in sequence. So a V, 5, plus I, 1, equals six. Unless the I comes before: if the lower number comes before a higher number, then it's subtraction, so this actually equals four. So it's a real fascinating system.

Another thing too is that notice that there is no representation for zero in the Roman numerals. So this is just one example of a system that we don't use on a regular basis, but many of us probably have some familiarity with, or at least have seen it out there.

The decimal system

Then we have the decimal system. This is the one that you're probably most familiar with and use on a day-to-day basis. We also call it the base 10 system. That's because there are 10 digits, 0 through 9. That's an important difference to denote: zero is a placeholder in the decimal system. So 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 gives us ten digits right there. So we have 10 digits, but it starts at zero. That's going to be important when we get into binary numbers, because we're going to have to calculate some things and we're going to have to understand that zero is our first digit.

So we've got zero through nine here, and when we add things up, it's kind of like you're adding one every time. When you are counting up, you're adding a single digit every time. So zero plus one is one, plus one is two, plus one is three, four, five, six, seven, you get the point, we go all the way to nine. What's interesting is what happens when we get to nine: this actually gets reset and we carry a one.

A lot of us learned this back in elementary school, and maybe we've forgotten it, or maybe we just need a refresher, or maybe it's that you were never exposed to it. But what I want you to do is start thinking about every single one of those positions having a certain value attached to it. We've got these columns. We've got the ones column, we've got the tens, the hundreds, the thousands. Every time we do this we multiply by ten to get the next one. So we've got 1 times 10 equals ten, times ten equals one hundred, times ten equals a thousand, times ten equals ten thousand, and so on and so forth. That's where we get this base 10 from, right there: every single column is going to go up by 10 times.

Then we can write a number in here, 148. So we've got 1 hundred, we've got 4 tens, and we've got 8 ones. Some of us back in elementary school can maybe recall that when we were learning this there was just a single little cube, a small little cube, for that ones, and we would have a certain amount of cubes that we'd have in that column. And then in the tens we had a line: there were ten of those little cubes, but they were in a line all stuck together, and we'd have all of those in there, and in this example right here we'd have four of those. In the hundreds we would have a square of it, it would go across by 10 and down by 10 and all those little squares, and we'd just have a block of that; in this example right here it would just be one block. And then if we had something in the thousands, that was like an actual cube that we would have, that would have a thousand of those little blocks in it. So I don't know if you remember that, but we can think of it along those lines.

Another way that I would like you to think about this is that in here we would have a thousand times zero — there's no number here, so it's a thousand times zero, so that equals zero. In this we'd have one times one hundred, so that would equal a hundred right there. And this right here, we have four times ten, so we would have 40 right here. And then this column right here, we would have eight times one, and so we would have eight. And then what we could do is we could add these together, and that would come up for a total of 148.

I'm going to bring up one other way that we can look at this, and I don't want you to get too confused with this. If you don't understand it, just move on from it, but I'm going to relate it also to binary, so I'm going to put it out there in case this resonates with some people. In this example right here, we could also say that this is eight times ten to the zeroth place, and 10 to the zero, actually, if you were to calculate that out, that is the equivalent of one right there. And then we have 4 times 10 to the one, so essentially that's four times ten. And then we have 1 times 10 to the second, so essentially that's 1 times 100. And then here we have 0 times 10 to the third, so that continues to count up.

The point that I'm going with is that you don't necessarily need to know all the mathematics of all of this, but this is the base 10. This is where that base 10 comes in: we've got 10 to the zero, one, two, three, so each column counts up. And then if you were to calculate this all out, it would all add up to 148, and it works out magnificently.

Counting and carrying

There is one other way that I also want you to look at this, and that is that when we're counting, we start out with zero, we add one, and then another one, and then another one, and we can continue on until we hit nine. And something interesting happens once we hit nine: when we go to the next one, we reset the ones column and then we carry the one over into the next column, and now we've got 10. And then we can do the same thing over again: once we hit 9 and then we go to the next number, that column resets and we carry a 1 over into the next column, making that two. And then we do it all over again, go all the way up to nine, and once we hit 10 this column resets, we carry a 1, and that becomes three. So on and so forth, until we get to 99, and then it carries over into the hundred, and then we get to 999 and it carries over into the 1000.

So you can see where I'm going with this. Kind of think of it like an odometer that's clicking up with each one of these columns and increasing with each one of these columns.

Time

Now let's correlate that to time. Time is pretty unique as well, because what happens is we start out at zero seconds, and then there's one second, and then there's two seconds, and we count all the way up to 59 seconds. And then what happens is this gets reset and we carry a one over into the minute. And then we count up again all the way to 59, and then the next one we reset that column and carry a 1 over into the next column, and now we've got two minutes. We can do that all the way up to 59 minutes and 59 seconds, and when we add one more second to this, this resets, carries the one over, this gets reset, we carry another one over, and now we're in the hour.

What does the hour do? It counts up to 23, and then once we get to 24, then we get an extra day in there. So it's unique in the way that there's a different base system for each one of these, for each one of the second, minute, hour and day.

Binary

So that brings us to binary, and I want you to think about it in those same concepts. Once we understand that same concept, then it becomes a lot simpler. It's just really strange to initially think about, because we've just got two numbers: we've got a zero and a one, and that's all we have to work with. But it works the same exact way as the systems we're already used to.

So what do we do? We have a zero, and then it goes up to one, and then we add another one, but we don't have two, two is not representative, so we've got to reset this and then we carry a one into the next. So what do we have? We've got 0, 1, and then we have 10, which is a really odd concept here, but that's the way it works. And then we add another one, and now we have 11. And then we add another one: this column resets, we carry the one; we can't have two here, so that column resets and then we carry the one right there. And then we add another one, so now that's one, and this would be 101. We add another one and now this resets and we carry a one. And then we add another one, so we end up 111. So that's how binary works and how binary counts.

Here again I can fill in some numbers here, 1101, and I can start figuring out what does this equate to. Similar to how we expressed it in base 10, it's going to work the same way. What we have here is we have one eight, we have one four, we have zero twos and one ones. If you were to add all that together, eight plus four plus one, you would get 13. So that's the answer there.

You don't need to memorize this right now or go over this right now. I just want you to understand the concept that we have with binary numbers is exactly the same as with the decimal system that you're so used to working with.

Hexadecimal

I just wanted to point out that hexadecimal works in the same way. It's base 16, so it looks a little different and it can be a little confusing at first, but it's really the same thing. What do we have? We have a ones column. What do we have? We have a sixteens column. What do we have? A 256, a 4096, so 65,536. We can have our different numbers that are in here.

If you want to figure out what the answer is to these, you do it the same process: four times 4096, plus 5 times 256, plus 7 times 16, plus 1 times 1, and then that would come out with the answer. Or, if that relates best to you thinking of it that way, you can think of it as that last example that I have on here as well. So there's hexadecimal.

I know there are some students out there where some of this is really foreign concepts and it might be harder to grasp. Some of you may have picked this up very quickly because you're really good with numbers. It doesn't matter either way, we're going to get you to really understanding this. So if you're not as strong with numbers and math, not a problem. We're going to take a look at this from every single angle and get you up to speed and make sure that you're enjoying this process, because like I say, it's a fun little game that we're going to be able to do once we understand these binary numbers. But we really have to get really familiar and really comfortable with working with these binary numbers.