- In this video, we're going to take a look at the binary number system, we're also going to see how to convert from binary to decimal. First, a brief look at the fundamentals of number systems. In all number systems, the first digit begins with zero, so whatever base we're working in, that tells us how many digits, or how many numbers, are in that number system. For example, decimal, Base-10 number system has 10 digits, starting at zero, through nine, that's our 10 digits. Binary, binary is Base-2, Base-2 has two digits, starting at zero, zero and one are our two digits. So the first column in our number system is always the column of ones, whatever base we're working in, the next column is that times the previous column. Let's take a look. In Base-10, the first column is the column of ones, our right most column, Base-10, the next column would be 10 times one, or 10. The next column would be 10 times that, 10 times 10, or 100, the next column, 10 times 100, 1,000 and so on. In Base-2, we also start with a column of ones, Base-2, the next column would be two times the previous column, or two. The next column after that would be two times two, four, the next column, two times four, eight, and so on. So in Base-10, we said the first column was the column of ones, the next column is 10 times that 10, then 10 times that 100, et cetera. What digits can go in each of those columns? Well any digits in our number system, zero through nine. So whatever I put in the ones column is how many ones I have, like one is one, one. Two, would be two ones, three would be three ones, nine would be nine ones, and that's actually the last digit in our number system. So 10 would be one in the 10s column, and zero in the ones column, that gives us our 10. So the one in the 10s column means the number of 10s, as opposed to the one in the ones column, which is the number of ones. 99 would be nine 10s, and nine ones, 100 would be one 100, zero 10s and zero ones. Now this may seem really basic, but understanding Base-10, the decimal number system, you can understand any number system. Let's take a look at the binary number system, the binary system, the first column, is our column of ones, the next column, it's Base-2, is two times this column, so that would be two, the next column, Base-2, two times two is four, the next column would be eight, and we could keep going, but we're going to leave it as this. Okay, so what can go in each one of these columns? Well we're using Base-2, binary, so we have a choice of two digits, a zero, or a one. Let's take a look, so a zero in binary, zero ones, that's equivalent to a zero in decimal, a one in the ones place is equivalent to one in decimal, a one. Now remember, in binary, we can only use a zero or one, so one zero, what this means is, how many twos do I have, I have one two, and zero ones, that equates to two. One one, in binary, is one two, plus one one, two plus one is three, one zero zero, in binary, is one fours, zero twos, zero ones, so we have four. Wherever we have a one, we just add up that column, if we have a zero, we don't add that column. It's that simple. So as we can see, five would just be one zero one, six would be one one zero, one four, one two, zero ones. Seven, eight would be one in the eights, and zero in the fours, twos and ones. Nine, 10, 10 would be one eight, plus one two, eight plus two is 10. 11, 12, 13, so 13 is one eight plus one four is 12, plus one is 13. 14, 15, 15 is one eight, plus one four, plus one two, plus one one, eight plus four, plus two, plus one is 15. So what would 16 be? You're right, 16 would be, we'd have another column over here of 16s, two times eight is 16, so we'd have another column over here, and it would be one in the 16s column, followed by zero, zero, zero, zero. So let's look at how to convert from binary to decimal, it's very simple. So given this binary number, what would be the decimal equivalent? You're right, all we have to do is add up wherever we have a one, add up that column, we have one two, zero ones, that gives us two. What would this be? You're right, one eight, plus one two, that's 10. How about this? You have one in the 16s, one in the ones, what's that give us? You're right, 16 plus one, 17. How about this? Notice that each column is two times the previous column, so one 64, plus one four, let's see, that's 68, plus one in the twos, that's 70, I have zero everywhere else, 70 is the answer. How about this one? Oh, one in the 128s, what else do I have? One in the twos, 128 plus two, that gives us 130. And let's take a look at this last one here, oh, that looks tough, a lot of adding to do, isn't it? So we'd have to add 128, plus 64, plus 32, plus 16, plus eight, plus four, plus two, plus one. What does that give us? 255. And there's actually an easy way to figure that out, when we have all ones in this column. What would be the next value after 128? The next column? Well that would be 256, so 256 would be one in the 256 column, and then all zeros, so the next column is 256, and we have all ones here, we just subtract one from 256, and that's 255. If that doesn't make sense, don't worry about it, all you have to do, again, is add up wherever we have a one in that column, add those column headings up, and we get the answer.