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So it's exactly the same thing mathematically, but it tends to confuse people. And I think the reason that it tends to confuse people is there's this assumption in the inductive step. So when you write the inductive step, you say, if you assume that I can do this for n pieces of something, n bits, n's, problem size, and software, whatever n is in your proof by induction, you have to assume that it works. | [
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And I think the reason that it tends to confuse people is there's this assumption in the inductive step. So when you write the inductive step, you say, if you assume that I can do this for n pieces of something, n bits, n's, problem size, and software, whatever n is in your proof by induction, you have to assume that it works. And it's kind of weird when you say, well, I'm trying to design something. | [
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So when you write the inductive step, you say, if you assume that I can do this for n pieces of something, n bits, n's, problem size, and software, whatever n is in your proof by induction, you have to assume that it works. And it's kind of weird when you say, well, I'm trying to design something. Well, just assume it works. But I haven't designed it yet. But you have to assume it works. | [
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But I haven't designed it yet. But you have to assume it works. So I think that throws people sometimes, especially when they get into software, everything's a little more abstract. Sometimes also in the bit-sliced design. The proof works if you assume it works and you show that you can add one more step. You don't have to solve the problem for n. All you have to solve it for is if n works, n plus 1 also works. | [
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] |
You don't have to solve the problem for n. All you have to solve it for is if n works, n plus 1 also works. Yeah. Yeah | [
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Well, just assume it works. But I haven't designed it yet. But you have to assume it works. So I think that throws people sometimes, especially when they get into software, everything's a little more abstract. Sometimes also in the bit-sliced design. The proof works if you assume it works and you show that you can add one more step. You don't have to solve the problem for n. All you have to solve it for is if n works, n plus 1 also works. | [
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You don't have to solve the problem for n. All you have to solve it for is if n works, n plus 1 also works. Yeah. Yeah. No? How would you actually apply this for the ripple carry adder? So I mean, you would have to do, I think, the actual proof would be based on something like place value. So you'd talk about the value of an n-bit number. And then you would say, well, n-bit numbers have this range. | [
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So you'd talk about the value of an n-bit number. And then you would say, well, n-bit numbers have this range. And I know that the addition works. That's the assumption, is that I can add any n-bit numbers and get the right answer. Then I can say, well, if I put one more bit on front of my two numbers, can I prove that given this worked, that I get the right answer with a full adder? And that's the thing. | [
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That's the assumption, is that I can add any n-bit numbers and get the right answer. Then I can say, well, if I put one more bit on front of my two numbers, can I prove that given this worked, that I get the right answer with a full adder? And that's the thing. You have to assume that n works. You don't have to make n work. You just have to assume it works. And then if you do the inductive step, it will work. | [
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You just have to assume it works. And then if you do the inductive step, it will work. Everything will work. So that's sort of this strange leap of faith you have to make that your answer works before you actually design it. So you have to sort of mentally get comfortable with that in order to finish your design. Because if you can't get yourself started, well, of course it doesn't work. | [
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So you have to sort of mentally get comfortable with that in order to finish your design. Because if you can't get yourself started, well, of course it doesn't work. If you don't design it, it doesn't work. But you do have to, as part of your design, assume that it will work for a smaller number of bits. And of course, don't forget the base case. If you leave the base case out, it's not going to work. | [
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And of course, don't forget the base case. If you leave the base case out, it's not going to work. All right, so I wanted to just mention that, because you will do some bit slice designs, and you will do recursive software designs in 220. So remember this. So what is bit slicing? So bit slicing is a hardware approach that is basically like induction. So it means we're going to break off a small part of the problem, say one bit, like we did for the adder, one bit of each input, or a few bits is fine. | [
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So what is bit slicing? So bit slicing is a hardware approach that is basically like induction. So it means we're going to break off a small part of the problem, say one bit, like we did for the adder, one bit of each input, or a few bits is fine. And then we're going to solve the whole problem by using the solution for the remaining part. So we'll take one little piece and solve it, and then use the solution for the remaining part. | [
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And then we're going to solve the whole problem by using the solution for the remaining part. So we'll take one little piece and solve it, and then use the solution for the remaining part. And that's essentially a proof by induction. So if you can prove that given only a small part of the input along with the answer for the rest, but you can get the full answer, then you're done. So in hardware, there's a little bit of a complication in the sense that we need to be able to express that answer for the rest concisely. | [
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So if you can prove that given only a small part of the input along with the answer for the rest, but you can get the full answer, then you're done. So in hardware, there's a little bit of a complication in the sense that we need to be able to express that answer for the rest concisely. So if we can't express it in a small fixed number of bits, then the number of wires we need from bit slice to bit slice will grow. | [
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So in hardware, there's a little bit of a complication in the sense that we need to be able to express that answer for the rest concisely. So if we can't express it in a small fixed number of bits, then the number of wires we need from bit slice to bit slice will grow. And if the answer somehow takes n over 2 bits, well, then we can't design one piece of logic, because as we add more bit slices, n over 2 gets bigger. | [
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So if we can't express it in a small fixed number of bits, then the number of wires we need from bit slice to bit slice will grow. And if the answer somehow takes n over 2 bits, well, then we can't design one piece of logic, because as we add more bit slices, n over 2 gets bigger. So sometimes we might be able to do it logically, because we just know the answer | [
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So if we can't express it in a small fixed number of bits, then the number of wires we need from bit slice to bit slice will grow. And if the answer somehow takes n over 2 bits, well, then we can't design one piece of logic, because as we add more bit slices, n over 2 gets bigger. So sometimes we might be able to do it logically, because we just know the answer. We might be able to do it mathematically. | [
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So sometimes we might be able to do it logically, because we just know the answer. We might be able to do it mathematically. But in a hardware design, we need to be able to do it by expressing the answer using a small number of bits. In the adder, it's just the carry bit. The only thing you need to know about the solution for less bits is the carry. And that will give you enough information to calculate your sum. | [
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The only thing you need to know about the solution for less bits is the carry. And that will give you enough information to calculate your sum. So what kind of problems can we do? Addition, subtraction, comparison, we'll do next. Check for a power of 2, check for multiples, do pattern matching in inputs, bitwise logic operation. So a bunch of different things we can use this kind of approach. | [
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Check for a power of 2, check for multiples, do pattern matching in inputs, bitwise logic operation. So a bunch of different things we can use this kind of approach. There's probably more, but these are the ones I could think of and that I've worked on before. When can't we use it? So any time the answer depends on all of the other bits. So for example, if I say, oh, do a prime number recognizer. | [
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When can't we use it? So any time the answer depends on all of the other bits. So for example, if I say, oh, do a prime number recognizer. Well, nothing you can tell me about these last five, well, at least I don't know how to do it. Nothing I can tell me about these last five bits will tell me whether with some extra bits on the front, this is a prime number. And you need to be able to do that. | [
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Nothing I can tell me about these last five bits will tell me whether with some extra bits on the front, this is a prime number. And you need to be able to do that. You need to be able to say, well, given the other bits, is my number a prime number looking at only one bit? If you wanted to do a bit slice design for a prime number checker, for example. So that's an example of where I don't think you can use a bit slice design. | [
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You need to be able to say, well, given the other bits, is my number a prime number looking at only one bit? If you wanted to do a bit slice design for a prime number checker, for example. So that's an example of where I don't think you can use a bit slice design. So what do you want to generate? So I mean, if you're just adding two numbers, you can use an adder, right? And then get the next one out. | [
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So that's an example of where I don't think you can use a bit slice design. So what do you want to generate? So I mean, if you're just adding two numbers, you can use an adder, right? And then get the next one out. Yeah, so that would be fine for bit slicing. I mean, you can use an adder to generate that as long as you're doing the feedback logic to get back the two in the sequence and generating the next one in the sequence. | [
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Yeah, so that would be fine for bit slicing. I mean, you can use an adder to generate that as long as you're doing the feedback logic to get back the two in the sequence and generating the next one in the sequence. Yeah. Anything else? Yeah. So remember that when we designed two's complement, we designed it deliberately to use addition mod 2 to the n, which was what we got out of unsigned. | [
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Anything else? Yeah. So remember that when we designed two's complement, we designed it deliberately to use addition mod 2 to the n, which was what we got out of unsigned. So the adder and subtractor are basically identical for two's complement and unsigned. And in fact, in order to do the subtractor, and this comes back to what we were talking about with a carry-in of 1, if you think about, well, is there a way I could trick my adder into doing subtraction? I think you've already done it by hand. | [
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So the adder and subtractor are basically identical for two's complement and unsigned. And in fact, in order to do the subtractor, and this comes back to what we were talking about with a carry-in of 1, if you think about, well, is there a way I could trick my adder into doing subtraction? I think you've already done it by hand. A lot of you have done it by hand. But think about how you can use your adder to do subtraction. | [
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A lot of you have done it by hand. But think about how you can use your adder to do subtraction. Yes, negate it. But you don't have to add 1. You can put a 1 on the carry-in. Yeah. Anything else? Might be the end. Where am I? No, I still have more slides. Killed my PowerPoint. Sorry. Oh, darn it. Oh, darn it. Oh. OK. Maybe we'll talk about the comparator today. All right | [
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A lot of you have done it by hand. But think about how you can use your adder to do subtraction. Yes, negate it. But you don't have to add 1. You can put a 1 on the carry-in. Yeah. Anything else? Might be the end. Where am I? No, I still have more slides. Killed my PowerPoint. Sorry. Oh, darn it. Oh, darn it. Oh. OK. Maybe we'll talk about the comparator today. All right. So let's spend a few minutes getting started on a comparator. | [
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All right. So let's spend a few minutes getting started on a comparator. And then we'll finish this up Monday. So next thing we'll do is compare two unsigned numbers. Now, here it's going to make a difference. And we'll figure out how later, whether it's unsigned or choose complement. But for now, let's just start with unsigned. So which one's bigger, top one or bottom one? Top. Top. | [
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So which one's bigger, top one or bottom one? Top. Top. How'd you know? Ah, OK. So you started which side? You started on the left. OK, so humans go that way. Why? Yeah, so once you get to, say, 0, 0, say that's the same, one line. Ah, so you can just stop. You don't even need to look. You're done. Yeah, so when we build hardware, they can't just stop. I mean, you've got wires. The wires don't just say, hey, I'm going to turn now because I know the answer. | [
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I mean, you've got wires. The wires don't just say, hey, I'm going to turn now because I know the answer. So they can't just stop in the middle. The information is going to flow from one end to the other. And the output wires on the end are going to give us the answer. So it doesn't actually matter for the hardware design if we go the human way or we're going to go the other way. So in our design, we're going to go the other way. | [
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So it doesn't actually matter for the hardware design if we go the human way or we're going to go the other way. So in our design, we're going to go the other way. It doesn't matter which way we go because we're going to have to go through all the bits in the hardware. We can't stop early, at least not in this kind of design. Once we talk about sequential state and things like that, you'll see there are ways for bigger designs to stop early in some cases, but not in wires. | [
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We can't stop early, at least not in this kind of design. Once we talk about sequential state and things like that, you'll see there are ways for bigger designs to stop early in some cases, but not in wires. So we're going to design from right to left. We'll look at the least significant bit first. So how many answers are there? So there are three, three possible answers. So if we have numbers a and b, we've got three outcomes. | [
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] |
So how many answers are there? So there are three, three possible answers. So if we have numbers a and b, we've got three outcomes. We can say a is less than b, a is equal to b, or a is greater than b. So in order to decide the answer for n plus 1 bits, what do I need to know? Well, I need to know what's the answer for n bits. And then I need to know one bit of a and one bit of b. So here's an example where we can use bit slice. | [
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And then I need to know one bit of a and one bit of b. So here's an example where we can use bit slice. Because if you tell me for one fewer bit, for the less significant bits, which of those sets of bits is bigger and which is smaller, and then you tell me one bit of a and b, then I can tell you with that extra bit which one is bigger, or are they equal. Everyone agree with that? So we should be able to build a bit slice design. | [
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Because if you tell me for one fewer bit, for the less significant bits, which of those sets of bits is bigger and which is smaller, and then you tell me one bit of a and b, then I can tell you with that extra bit which one is bigger, or are they equal. Everyone agree with that? So we should be able to build a bit slice design. So what do we need to do? So how many bits do I need to pass from slice to slice? Yeah, right, because there are three possible messages. | [
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Everyone agree with that? So we should be able to build a bit slice design. So what do we need to do? So how many bits do I need to pass from slice to slice? Yeah, right, because there are three possible messages. You need to know a greater than b, a equal to b, or a less than b | [
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Everyone agree with that? So we should be able to build a bit slice design. So what do we need to do? So how many bits do I need to pass from slice to slice? Yeah, right, because there are three possible messages. You need to know a greater than b, a equal to b, or a less than b. I can't condense that into two answers. Because if your bits are equal, well, then you need to know what the answer used to be because it's the same. | [
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I can't condense that into two answers. Because if your bits are equal, well, then you need to know what the answer used to be because it's the same. So if your a and b for one bit are 0, 0, for example, then the answer depends on the lower bits. And you need to pass along any of those three answers. And how many bits do we need to encode three answers? Two, right? So you need two bits. | [
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And you need to pass along any of those three answers. And how many bits do we need to encode three answers? Two, right? So you need two bits. So here's a figure showing an abstract slice model of our bit slice. So we're going to have two bits coming in from a less significant bit. We'll call them c1 and c0 m minus 1 bits. This will be the mth bit slice. So they'll get a sub m and b sub m for the numbers a and b. | [
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This will be the mth bit slice. So they'll get a sub m and b sub m for the numbers a and b. And then they'll produce outputs c1 m and c0 m, which internally, just to differentiate these two, we're going to call z1 and z0. So the inputs of what's happening from the less significant bits, we'll call c1 and c0 internally. And then we'll produce z1 and z0 to tell the next bit or to give the final answer, well, is a less than, equal, or greater than b. | [
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So the inputs of what's happening from the less significant bits, we'll call c1 and c0 internally. And then we'll produce z1 and z0 to tell the next bit or to give the final answer, well, is a less than, equal, or greater than b. Well, so if the only thing we need to know is a less than, equal, or greater, we don't need n bits of output for anything, right? We just need one of those three answers. | [
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And then we'll produce z1 and z0 to tell the next bit or to give the final answer, well, is a less than, equal, or greater than b. Well, so if the only thing we need to know is a less than, equal, or greater, we don't need n bits of output for anything, right? We just need one of those three answers. So in this case, yeah, it's a good question. In general, for something, your bit slice design may or may need some kind of outputs at the bottom. | [
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So in this case, yeah, it's a good question. In general, for something, your bit slice design may or may need some kind of outputs at the bottom. For a comparator, you don't need anything. For an adder, you need one bit for the sum. If, for example, as we sometimes do on homework or as we might sometimes do on even exams, we can say, well, out of the bottom, we want you to produce the minimum or maximum of a and b. | [
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For an adder, you need one bit for the sum. If, for example, as we sometimes do on homework or as we might sometimes do on even exams, we can say, well, out of the bottom, we want you to produce the minimum or maximum of a and b. So if we say minimum only, then you need one output wire. If we say minimum and maximum, you need two separate output wires. Ah, very good question. Good question. | [
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Ah, very good question. Good question. I thought of that, too. How do we represent the answers? Got three answers, right? a less than b, a equal to b, a greater than b. They're a natural representation? They're not, right? It's just three answers. Pick any representation we want. So our choice will affect the amount of logic we need. So here is a pretty good choice. Actually, after I designed the one, this is from the notes. | [
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So here is a pretty good choice. Actually, after I designed the one, this is from the notes. After I designed this, I realized, well, I probably should have thought of this beforehand. But I went back and considered all the different choices I could have made and made sure I wasn't somehow doing something that was bigger than it should have been. So there's a pretty good representation. | [
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But I went back and considered all the different choices I could have made and made sure I wasn't somehow doing something that was bigger than it should have been. So there's a pretty good representation. If you're doing the extra C exercises, the optional ones, you'll get to design your own with your own representation and your own direction in software | [
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Actually, after I designed the one, this is from the notes. After I designed this, I realized, well, I probably should have thought of this beforehand. But I went back and considered all the different choices I could have made and made sure I wasn't somehow doing something that was bigger than it should have been. So there's a pretty good representation. If you're doing the extra C exercises, the optional ones, you'll get to design your own with your own representation and your own direction in software. | [
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So there's a pretty good representation. If you're doing the extra C exercises, the optional ones, you'll get to design your own with your own representation and your own direction in software. But software is not so bad, because even if your logic is a little bigger, it's a few extra characters of typing, as opposed to working out lots of area and things like that. But here's a pretty good representation. | [
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But software is not so bad, because even if your logic is a little bigger, it's a few extra characters of typing, as opposed to working out lots of area and things like that. But here's a pretty good representation. So we just say, well, OK, the a equals b, we'll call 0, 0. a less than b, we'll call 0, 1. a greater than b, we'll call 1, 0. And then the last pattern, we've only got three messages. | [
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a greater than b, we'll call 1, 0. And then the last pattern, we've only got three messages. So we're not going to use it. So you have actually a fair number of choices. You have these three messages, and you have four different bit patterns. So how you assign them is completely up to you. Yeah, exactly. So as Mohamed points out, now that I've said not used here, I can assume safely this is not a human. | [
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Yeah, exactly. So as Mohamed points out, now that I've said not used here, I can assume safely this is not a human. They're not going to go pushing lots of buttons or anything like that. The lower bit slice will never generate this pattern. So I can put don't cares whenever I see 1, 1 coming in from the lower bit slice. Good point. OK, so let's see. I'm not sure we'll get through this one. | [
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OK, so let's see. I'm not sure we'll get through this one. Let's see if we can do one bit. So let's solve one bit. So in this case, there are no less significant bits. So let's just think about A and B. So if A and B are the same, then what's the answer? You already know the, you already know the. All right, fine, I'll just skip ahead. You know the representation. So the meaning, though, are A equals B for 0, 0 and 1, 1. | [
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You know the representation. So the meaning, though, are A equals B for 0, 0 and 1, 1. And then if 0 and 1 for A, B, then A is less than B. And 1, 0 is A greater than B, because these are unsigned. So the encoding is 0, 0. What about the 0, what about this row? 0, 1, good. What about the next one? 1, 0. And the last one? 0, 0, good. So this is one bit. So then we can go solve that. I want you to notice that these are minterms. | [
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So then we can go solve that. I want you to notice that these are minterms. So if you look at Z1, this is a minterm, just a 1 in one row. Z0 is also a minterm. So if I draw a circuit for that, all I need to do is generate a minterm. I don't need to go do k-maps or stuff like that. This structure is going to be kind of the core of our comparator logic. So here we're generating A0B for Z1 and A0, I'm sorry, AB prime, you said that way, and A prime B for Z0. | [
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This structure is going to be kind of the core of our comparator logic. So here we're generating A0B for Z1 and A0, I'm sorry, AB prime, you said that way, and A prime B for Z0. If all you wanted to do was compare A and B and say equal or not equal, you could OR these two together. That would be an XOR. And so these are the two minterms that you need for XOR | [
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This structure is going to be kind of the core of our comparator logic. So here we're generating A0B for Z1 and A0, I'm sorry, AB prime, you said that way, and A prime B for Z0. If all you wanted to do was compare A and B and say equal or not equal, you could OR these two together. That would be an XOR. And so these are the two minterms that you need for XOR. And if you XOR A and B, or one bit of A and B, that'll say equal if the XOR gives you a 0, and not equal if the XOR gives you a 1. | [
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And so these are the two minterms that you need for XOR. And if you XOR A and B, or one bit of A and B, that'll say equal if the XOR gives you a 0, and not equal if the XOR gives you a 1. So if you OR these two outputs together, it's just like an XOR. But this will be the thing that lets us know is A greater than B or A less than B on a one-bit basis in the full design. | [
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] |
So we're going to pick up where we left off, talking about expressions and operators in C, talk about basic IOs, so printf, scanf, and then talk about statements. Hopefully, you had a chance to take a look at the email that I sent out, I think it was yesterday. So we have an online automatic feedback tool for programming. And what we're hoping is people will play with it. It's used in 220, so if you get familiar with it, it'll help you in 220 also. | [
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And what we're hoping is people will play with it. It's used in 220, so if you get familiar with it, it'll help you in 220 also. But it's basically a computer tool that will help you find the bugs in your code. So I can tell you more about it if you're interested in office hours. But we have an undergrad who's doing his thesis, creating exercises for 120. And so those are available to you. | [
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But we have an undergrad who's doing his thesis, creating exercises for 120. And so those are available to you. Has nothing whatsoever to do with your grade. If you don't want to do it, don't do it. Signing up doesn't commit you to anything. So if you think you might want to play with it, sign up. We encourage you to do it because it's supposed to help you. But if you don't want to, that's OK too. | [
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We encourage you to do it because it's supposed to help you. But if you don't want to, that's OK too. But I do encourage you to try. All right, so let's see. Daniel was somewhere here usually. There you are. You asked this question about Unicode in C identifiers. So I went and looked it up. Apparently, it's been in the standard since 1999, but compilers still aren't really quite well supported. | [
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] |
So I went and looked it up. Apparently, it's been in the standard since 1999, but compilers still aren't really quite well supported. So if you use these flags in the lab, standard C99 and F extended identifiers, you can use this type of code. So this one everyone understands. How many Spanish speakers are there? OK, do you know this word? Manana, yes, I think I guess. And how about Chinese speakers? Anyone speak Mandarin? OK, can you read this identifier? Yeah, it's ming tian. | [
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How many Spanish speakers are there? OK, do you know this word? Manana, yes, I think I guess. And how about Chinese speakers? Anyone speak Mandarin? OK, can you read this identifier? Yeah, it's ming tian. So not exactly what you might want. So if you want to use it to change regular text like the commented versions into universal character notation, which are the things you can compile, there are tools that can do that. | [
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] |
So not exactly what you might want. So if you want to use it to change regular text like the commented versions into universal character notation, which are the things you can compile, there are tools that can do that. There's some instructions on this web page using Perl. But in 120 assignments, use ASCII. Please don't put UCNs. All right, so do a little bit of a review before we dive in since it's been five days since we talked. | [
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Please don't put UCNs. All right, so do a little bit of a review before we dive in since it's been five days since we talked. So we're going to cover four types of operators in our class. Arithmetic operators, which we already finished, bitwise Boolean operators, relational comparison operators, and the assignment operator. We're also going to take a look at logical operators briefly. | [
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Arithmetic operators, which we already finished, bitwise Boolean operators, relational comparison operators, and the assignment operator. We're also going to take a look at logical operators briefly. A couple of the online exercises that use them. So if you want to do all of the online exercises, you'll need to remember what they mean. But you shouldn't need to use them otherwise in the class. | [
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] |
So if you want to do all of the online exercises, you'll need to remember what they mean. But you shouldn't need to use them otherwise in the class. So these are the bitwise operators, bitwise and, or, not, xor, and then the shifts we hadn't talked about yet. So let's go ahead and I wanted to put actually one slide with and to show you these things written out in bits. So we did the and last time, but it's maybe easier if you translate the hex into bits, or maybe harder since you have to write 32 bits down. | [
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So let's go ahead and I wanted to put actually one slide with and to show you these things written out in bits. So we did the and last time, but it's maybe easier if you translate the hex into bits, or maybe harder since you have to write 32 bits down. But essentially all the bitwise and is doing is going one at a time through each of the pairs of bits, taking the and bitwise for each pair, and producing that and, and then that's the answer. | [
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] |
So we did the and last time, but it's maybe easier if you translate the hex into bits, or maybe harder since you have to write 32 bits down. But essentially all the bitwise and is doing is going one at a time through each of the pairs of bits, taking the and bitwise for each pair, and producing that and, and then that's the answer | [
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So let's go ahead and I wanted to put actually one slide with and to show you these things written out in bits. So we did the and last time, but it's maybe easier if you translate the hex into bits, or maybe harder since you have to write 32 bits down. But essentially all the bitwise and is doing is going one at a time through each of the pairs of bits, taking the and bitwise for each pair, and producing that and, and then that's the answer. | [
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So we did the and last time, but it's maybe easier if you translate the hex into bits, or maybe harder since you have to write 32 bits down. But essentially all the bitwise and is doing is going one at a time through each of the pairs of bits, taking the and bitwise for each pair, and producing that and, and then that's the answer. So if you expand 120 in bits, that's the top line. 42 in bits is the second line here. | [
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So if you expand 120 in bits, that's the top line. 42 in bits is the second line here. If you and those together, you get the bottom line, which evaluates to 40 or 28 hex. So that's what it's doing for you. All right, so left shift. Left shift is basically shifting all of your bits to the left. So what does that mean? It's like multiplying by 2 to the n. So if you shift by n bits, it's like multiplying by 2 to the n. | [
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So what does that mean? It's like multiplying by 2 to the n. So if you shift by n bits, it's like multiplying by 2 to the n. So for example, if you declare a equals 120, and b equals this hex number, and you shift left by 2 for a, then let's see, that would be 2 to the 2 is 4, so you should get 480. And in fact, you get 480. So with this pattern b, if we shift by 4, that will shift 4 bits to the left, so we'll get four 0's on the right side. | [
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And in fact, you get 480. So with this pattern b, if we shift by 4, that will shift 4 bits to the left, so we'll get four 0's on the right side. And then the f on the left will fall off. So the shift will overflow. So it turns out we get kind of what we'd expect. Instead of 6 f's, we have 5 f's. There's an extra 0 on the right. And we've lost the high bits by shifting them off to the left, because we can only have 32 bits in the answer. | [
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There's an extra 0 on the right. And we've lost the high bits by shifting them off to the left, because we can only have 32 bits in the answer. That number is actually smaller than the original b value. And so that's an overflow. It's not multiplied by 16, as you'd like it to be. So shifts can overflow. Shifts by 32, by the way, are not defined in C, so be careful how you shift things. | [
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So shifts can overflow. Shifts by 32, by the way, are not defined in C, so be careful how you shift things. I forgot to mention that, but it shouldn't come up. So what bits appear on the left when you're shifting right? So shifting right just means take your bits, move them down towards the small end. But what bits should I put in the high part? 0's? Yeah, so in this case, I want to get divide by 2 to the 2, or 4. | [
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So what bits appear on the left when you're shifting right? So shifting right just means take your bits, move them down towards the small end. But what bits should I put in the high part? 0's? Yeah, so in this case, I want to get divide by 2 to the 2, or 4. So I should put 0's in. That'll give me 30 in 2's complement. What if I want to shift this bit pattern 4 to the right? Should I put 1's or 0's on the left? You sure? So which one is it equal to? Is that negative 256, or is it this big number here? I didn't tell you the type, right? I just wrote some bits. | [
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That'll give me 30 in 2's complement. What if I want to shift this bit pattern 4 to the right? Should I put 1's or 0's on the left? You sure? So which one is it equal to? Is that negative 256, or is it this big number here? I didn't tell you the type, right? I just wrote some bits. So it depends on the type. Well, it's bits. I didn't tell you the type, right? I can make that an unsigned or a 2's complement. | [
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Well, it's bits. I didn't tell you the type, right? I can make that an unsigned or a 2's complement. So if it's a 2's complement number, it represents negative 256, in which case, if we wanted to divide, we'd like it to divide by 16 and get negative 16. We'd insert 1's for that. If it equals this big 4 billion number, divide by 16, you've got 268 million some odd left. We should insert 0's. | [
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] |
If it equals this big 4 billion number, divide by 16, you've got 268 million some odd left. We should insert 0's | [
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Well, it's bits. I didn't tell you the type, right? I can make that an unsigned or a 2's complement. So if it's a 2's complement number, it represents negative 256, in which case, if we wanted to divide, we'd like it to divide by 16 and get negative 16. We'd insert 1's for that. If it equals this big 4 billion number, divide by 16, you've got 268 million some odd left. We should insert 0's. | [
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If it equals this big 4 billion number, divide by 16, you've got 268 million some odd left. We should insert 0's. So how does the compiler decide what looks at the type? So if this were not just a bit pattern, but were a type, and yeah, it's true. If you type that bit pattern in, the compiler will assume 2's complement. But if you put it in a variable, the compiler will use the type of the variable. | [
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If you type that bit pattern in, the compiler will assume 2's complement. But if you put it in a variable, the compiler will use the type of the variable. So if you have a 2's complement representation, it will do what's called an arithmetic right shift, which means copy the signed bits. If you have an unsigned int, unsigned representation, it will insert 0's when you do a right shift. | [
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So if you have a 2's complement representation, it will do what's called an arithmetic right shift, which means copy the signed bits. If you have an unsigned int, unsigned representation, it will insert 0's when you do a right shift. Yeah, question? AUDIENCE 2. Yes, int is 2's complement. Yeah, if you want unsigned, say unsigned int, as I've written down here. I'm going to do the unsigned int. | [
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Yeah, if you want unsigned, say unsigned int, as I've written down here. I'm going to do the unsigned int. OK, so right shift then will use the type and end up dividing by 2 to the n. Now, of course, that can overflow or underflow too. It's always going to round down. So if I take a up here, negative 120, and I right shift that by 2, divide by 4, I should get what? Negative 120 over 4? You get negative 30, right? So you can see it's put some extra 1's in up at the top. | [
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It's always going to round down. So if I take a up here, negative 120, and I right shift that by 2, divide by 4, I should get what? Negative 120 over 4? You get negative 30, right? So you can see it's put some extra 1's in up at the top. And if you look at the bit pattern, you'll see that it's basically just shifted it by 2. What if I shifted by 10? It's like dividing by 1,024. So I'll be getting, it's a 2's complement number, and it's a negative 1. | [
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What if I shifted by 10? It's like dividing by 1,024. So I'll be getting, it's a 2's complement number, and it's a negative 1. So I'll be getting 1's in at the left. So what am I going to end up with? Negative 1, right? So it'll be all 1 bits, which is negative 1. So you can think of that as divide by 1,024. That gives you negative 120 over 1,024. Round that down. That gives you negative 1. | [
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Round that down. That gives you negative 1. So I didn't wait for you to tell me. OK, if I shift this bit pattern 2 to the right, I put 0's in at the left because it's unsigned for b. And I get this bit pattern here, where I've just taken the bits and shifted them 2 to the right. If I shift 10 to the right, I get more 0's on the front. So it's just moving bits left or right in the bit pattern. | [
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If I shift 10 to the right, I get more 0's on the front. So it's just moving bits left or right in the bit pattern. Yeah, Eric? AUDIENCE 2 When you're getting those, you're looking at the dust, and it's not supposed to be. They're just binary form. Yeah. Yeah, so to know why these are the right answers, you will have to mentally translate from hex to binary, and then do the shift, and then translate back from binary to hex. | [
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] |
Yeah. Yeah, so to know why these are the right answers, you will have to mentally translate from hex to binary, and then do the shift, and then translate back from binary to hex. Yeah, yeah. So I don't expect you to be able to do this in your head. The reason I put the decimal values up here is to make sure you understand it is dividing by 2 to the n | [
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They're just binary form. Yeah. Yeah, so to know why these are the right answers, you will have to mentally translate from hex to binary, and then do the shift, and then translate back from binary to hex. Yeah, yeah. So I don't expect you to be able to do this in your head. The reason I put the decimal values up here is to make sure you understand it is dividing by 2 to the n. Had I written these as decimal, it would also make sense, except that would be a huge number. | [
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The reason I put the decimal values up here is to make sure you understand it is dividing by 2 to the n. Had I written these as decimal, it would also make sense, except that would be a huge number. So at least I wouldn't be able to do it in my head. Good question. OK. Usually with shifts, we're thinking about bits. We're using our numbers to represent bits. You can use it for multiplication. | [
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We're using our numbers to represent bits. You can use it for multiplication. It's slightly different than right shift. It's slightly different from division, because most division will round towards 0, and right shift will round down. Yeah, question? Yeah. Yeah, that's right. It doesn't matter. So if you think about the representations, it never matters. To multiply by 2, whether you have a positive or negative number for a 2's complement, you put a 0 on the right side. | [
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So if you think about the representations, it never matters. To multiply by 2, whether you have a positive or negative number for a 2's complement, you put a 0 on the right side. And the same for unsigned. I mean, there's no negative unsigned. OK, so that's it for the shifts. We also have six relational operators. So less than, lesser equal to, equal to, not equal to, greater equal to, greater than. | [
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We also have six relational operators. So less than, lesser equal to, equal to, not equal to, greater equal to, greater than. You can't put spaces in these. One thing that's a little strange in C, in order to do a comparison for equality, you have to put two equal signs. So you'll see later, one equal sign means something different. It means the assignment operator in particular. So you need to not put spaces in these, and remember that to compare for equality, you need two equal signs. | [
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It means the assignment operator in particular. So you need to not put spaces in these, and remember that to compare for equality, you need two equal signs. In C, anything that's 0, 0 bit pattern is false. Anything that is not 0 bit pattern, anything else is true. The relation operators evaluate to 0 if they're false, and 1 if they're true. But in C, anything that's not 0 is considered true. | [
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