[MUSIC]. So far in this section, we've been focusing on integer representations, and we've looked at both unsigned and signed integers. Now, let's turn our attention to fractional binary numbers, on our way to, to talk about floating point values. So, let's start with a typical fractional binary number. Here, we have one you'll notice that it's a little different than the number's we've dealt with before. It has a binary point. Very analogous to a decimal point. in this case just like in decimal numbers, remember, we had the 1s column and the 2s column and the 4s column and so on? Now, we have the halves column, the quarters column, and the eighths column just as we would in a decimal value number, we would have tenths and hundredths and thousandths. So, this number is an 8 Plus a 2 plus a 1. Those are the, that's the integer side, the left side of the binary point. So that's the number 11. And then, we have a half and an eighth. Not a fourth, because that's a 0. So in this case, this number comes out to be 11.625, in decimal, okay? And, we're in, in fact, going to interpret fractional binary numbers the same way we do with fractional decimal numbers. Okay? We're going to do exactly the same things. so here we see an, an extended version of the calculation that we just did. again the, integer values and the fractional values. for the numbers both to the left and to the right of the binary point. Okay? And we can have a summation expression that adds all that up. From minus j to i we can add all those values as we just did in our little example. Let's take a look, at, at a few more examples of these fractional binary numbers. let's start with 5 and 3 4ths. How would we represent that? Well, the 5 that's going to be to the left of the binary point is going to be pretty easy. That's just 1 0 1. then the 3 4ths on the right-hand side of the binary point is equivalent to a half plus a fourth. So those are going to be represented as one, one in the halves and quarters column. So we'd expect that representation to look like this, a 1 0 1 for the 5, a binary point, then one half plus 1 4th for the 3 4ths. Okay, for 2 and 7 8ths, it's a very similar thing. The 2 would be 1 0 on the left and the 7 8ths would now be a half plus a fourth plus an eighth or point 111 and that is the representation of 2 and 7 8ths. For 63 64ths there is no integer part to the number, so we have a 0 to the left of the binary point. And then a half plus a fourth plus an eighth plus a sixteenth, all the way down to get us to 63 64ths, and that's going to be the equivalent of six 1s. so 0.111111. Now some observations. remember that shifting that we were doing with binary numbers? well, we can do it with fixed point representations as well of these fractional binary numbers. If we divide by two, it's like moving that binary point 1 over to the left. And multiplying by 2 is moving the binary point 1 over to the right. Okay? one other observation to note is that numbers that are of this form as with these leading trail of 1s are just below the value 1.0, because if we had a little bit more, then they would be just equal to 1. So, if we added one half plus a fourth, plus an eighth, plus a sixteenth, and so on infinitely far down we would approach 1, but never quite get there. So often, we'll see a notation like this to avoid writing the long, long string of 1s. We just say 1 minus epsilon, minus a smidgen to to indicate that. Okay. So what is some of the limitations of binary numbers and the representable values. Remember we had limitations on integers, we could only be so large and so negative before we ran out of room in our bit representations. Okay, so we could only, first of all, represent numbers exactly if they can be written in the form x times 2 to some power of y. other rational numbers are going to have repeating bit representations. So, for example remember, like 1 3rd in decimal is 0.33333, but it never stops, it goes on forever. Well, we have an equivalent situation with binary numbers. So take a look at the representation for 1 3rd in fact, 1 3rd is actually going to be this bit pattern. And you'll notice that that 010101 is a repeating pattern, and we'll use square brackets to represent that repeating pattern. there's other values, numbers like that in in binary 1 5th, also has a repeating bit pattern that repeats 0011 on forever, and ever. And 1 10th, you notice is also has that same repeating pattern as 1 5th did, because of course, 1 10th is just a half of a fifth. It's just that same bit pattern shifted over by 1. Okay, so it's going to start it's going to repeat the same way. Alright, so, fixed point representation is when we decided to represent factional binary by picking a place for the binary point and fixing it there, always putting it in that location. So, we have to decide where to put it. Well, if we're looking at an 8-bit fixed point numbers, again, just to keep the example small. we can decide to put the, the binary point, for example, so that it has three bits to the right and five bits to the left. Well, what does that imply? That implies that we can represent up to the number 31 on the left, so we can get as high as 31.111. So that would be 31 and 7 8ths would be the largest number we could represent. Of course, we could have chosen to put the binary point elsewhere. Fixed it at a different place with only three bits over on the left and five bits to the right. well now, this only let's us go up to seven point something. seven and what well, there's five bits here, so we can represent up to 3130 seconds. And so, the largest number we can have here is 7 and 31 37th, 30 seconds. While here, we had the numbers up to 31 and 7 8ths. Okay? So, what is the difference between these? How would we choose what to use? Well one question we have to ask ourselves is how large number do we need to be able to represent. and the other is how much precision do we want the numbers to have. In other words, what is the smallest fraction null difference that we can represent. Okay. So, the range is that range of numbers, the precision is how small of a fraction and with fixed point representations we have this trade off. The more range we have, the less precision we have, the less bits we have on the other side of the binary point and vice versa if we have less range, then we get more precision. So, that's the reason that we don't end up using fixed point representations, because of these very strong cons against it, it's really hard to pick a good place for the fixed point to be. All right. Sometimes you end up wanting range. Sometimes you end up needing more precision. And the more you have of one, the less you have of the other. You can't get the best of both. so that's why we're going to turn our attention to what are called floating point notations, where we don't fix the binary point, but allow it to float as needed.