You might already be familiar with binary for example. This is 1 0 1 is equivalent to 5 in decimal that's because this is the ones place. This is the twos place fours place eights place 16s 32s 64s and 128s and we have a 1 in the fourth place and a 1 in the ones place 4 plus 1 is is 5 So this is fine, but how might we represent a negative number, so let's say we wanted to [represent] negative 5 Well, there's a couple ways we can do that one way is to take this 128s place and instead of using that as the 128s place use that as a sign so change this to a 1 here? To indicate [that] this is negative, and then the rest of it is the same 1 0 1 and so this would be 5 here and then instead of this representing 128 it represents that the number is negative, so [negative] 5 now of course. It's important to know how many bits You're working with right because if we're only using 4 bits then a 5 would be 0 1 0 1 that's equal to 5 [but] then we're going to use this this top bit here in this case We're only using 4 bits to the top that is this fourth bit and so negative 5 might be a 1 1 0 1 And now instead [of] this being the eights place This is actually representing a sign So this would be negative 5 and this would be 5 so it's important to know how [many] bits you're working with so [that] you know which bit is the first bit and therefore which Bit indicates what the sign of the number is so to keep going with this example. We can look at just regular counting and binary here of course if we have all 0 0 0 0 0 that's equal to 0 And then if we have a 1 in the ones place that's equal to 1 we have a 1 in the twos place That's equal to a 2 and a 1 that's equal to 3 and then of course a for is for a 4 and a 1 is 5 a 4 and a 2 is 6 For a 2 plus a 1 is is 7 so that's simple enough, and then if this this first Bit here is is indicating our negative sign? then we can go backwards to so if we have a negative 1 that's negative [one] a negative to a Negative you know [2] and a 1 is 3 and then a negative negative 3 negative 4 negative You know 4 and a 1 is 5 with a negative is negative 5 4 2 2 is 6 so this is negative 6 and then a 4 plus 2 plus 1 is 7 and? That's a negative 7 and so this is our sign bit And so this first bit is our sign bit and then these other bits are just our 1 2 & 4 place simple enough Couple weird things about this though one is that you'll notice there's a negative 0 right because you can have 0 0 0 and then It can either be 0 in the sign bit place or a 1 in the sign bit place So you can have you know there's a difference between 0 and negative 0 so that's that's kind of weird [the] other thing that is maybe a little bit inconvenient We'll look at some some other approaches that that don't have this problem is If you try to add these things together things get kind of weird so let's say we want to add a 5 and a negative 5 so normally 5 plus negative 5 you would expect to get 0 simple enough, but here if we look at 5 0 1 0 1 and negative 5 is 1 1 0 1 If we add these together [1] plus 1 is 2 which be a 0 and then carry the 1 1 plus 0 plus 0 is 1 1 plus 1 again is to the but will that is a 0 and carry the 1 and then here 1 plus 1 again is 2 so 0 and carry the 1 So in this case what we're seeing is 5 plus a negative 5 is not 0 it's 0 0 1 0 which Well, and we have a carry so we have a carry coming out of this this this one Bit that we don't you know if we're working with 4 bits? we're going to ignore this this carry bit and so we have 0 0 1 0 which is which is 2 Well, that's kind of weird We're adding 5 and negative 5 we wouldn't expect to get [2] and you can try adding some other things here It doesn't it doesn't work So let's take a look at another scenario here. This is called one's complement. What this is Is again everything from zero to seven is the same same same as we saw before and this first bit here is all zeros So we know that these are all positive But what we do for the negative numbers is we actually just flip all of the bits We take the compliment of all of the bits, so you know [2] here is 0 1 0? Well 0 0 1 0 negative 2 is 1 1 [0] 1 so we're just flipping each of those bits so we flipped the 0 401 We Flipped the 1 4 0 with the 0 [4] a 1 and, so on if you look at each of these numbers So 5 is 0 1 0 1 negative 5 is 1 0 1 0? So what happens with with this so we still have this kind of strange thing where we have negative 0 and we'll come back to That in a minute [but] what we can do is we can try to you know add some numbers, so if we add 5 So 0 1 0 1 that's 5 and negative 5 1 0 [1] 0 We add those adding negative 5 here [what] we end up getting is 1 plus 0 is 1? 0 plus 1 is 1 1 plus 0 is 1 0 plus 1 is 1 we get 1 1 1 1 which is negative zero So that's closer, right? You know 5 minus 5 you expect to get 0 negative 0 is I guess the same thing. [so] that's pretty good And and that actually works for any of these you know we can do 3 minus 3 or 3 so 3 is? 0 0 1 [1] negative 3 is 1 1 0 0 so this is 3 plus negative 3 if we add these together [so] [1] plus 0 is [1] 1 plus 0 is 1 0 plus 1 is 1 0 plus 1 is 1 we get 1 1 1 1 again [and] [so] again 3 minus 3 is negative 0 so this is definitely doing doing a lot better than then here where we were saying a 5 minus 5 was was 2 That's definitely not right So negative 0 definitely a lot closer. Let's try some other some other things kind of see if we can see a pattern here so if we did let's Say 5 Plus negative three so five minus three let's see what we get there five is 0 1 0 1 and negative 3 is 1 1 0 0 so 1 plus 0 is 1 0 plus 0 is 0 1 plus 1 is [2] so we'll put a 0 and carry a 1 and then 1 Plus 1 again is 2 so we'll put a 0 and carry a 1 And so if we ignore this this first bit here because we're working with 4 bits we get 1 So 5 minus 3 is [1] that's close. It should be 2 but you know let's let's see. What happens if we do Say 6 minus 2 or 6 plus negative 2 So 6 is 0 1 1 0 negative 2 is 1 1 0 1 1 1 0 1 [0] plus 1 is 1 1 [plus] 0 is 1 1 plus 1 is 2 0 and carry the 1 1 plus 1 is 2? put a 0 and carry the 1 again ignore that first [bix] [we're] working with four bits so 0 0 1 1 is 3 so [we're] saying [sex] minus 2 is 3 close But but we're definitely we're definitely seeing an interesting Pattern here in fact if we look You know these places where we're getting this negative 0 if we just go down 1 we see zero Which is what we want here 5 minus 3 we'd expect to see two, but we're seeing one so if we just go down 1 we'd actually see a 2 6 minus 2 we'd expect to see 4 but we're actually getting 3 but if we go down 1 we get the right answer So we could just always add [1] To each [of] these of course it's kind of weird to say add 1 2 to negative 0 to get to 0 but If you kind of bear with me a minute You'll see that if we add 1 to the binary value in it and it flips over You do get to the right thing and here you get to the right thing So the ones compliment. You know just flipping all of these bits is Close it's very close. It's off by 1 on all on all of our arithmetic And it also still has that nice property that we saw with the you know with [the] sign bit Which is you know you can look at this first bit and you can tell these are all negative numbers And if the first bit is [zero], then they're positive numbers so it seems like the ones complement has a lot of the You know a lot of nice properties. [you] know that you'd have in the sign, bit the one thing. That's you know well I guess it's not too bad is It to get from to inverter to negate a number You know if you're going from five to negative five with a sign bit all you'd have to do is flip one bit You just flip that first bit to a one and you get to negative five Well with the ones complement you have to invert each bit so it's a little bit more work But of course inverting bits is is pretty easy to do in hardware if we're building hardware to do this for building a computer I'm going to be doing some math, so [once] [compliment] seems pretty good, but it is off by by one So what we can do is look at another scenario Which is two's complement and if we compare ones complement to two's complement? It's basically the same except what it does is it gets rid of this negative zero? [so] instead of having negative zero we just go [right] to negative one Negative two three four five six seven and the the kind of weird thing here is we have a negative eight And we don't have an eight, but the nice the nice property about this is that all [of] our math works out very nicely so if we have [five] minus five 5 here is zero one zero one and So that's 5 and negative 5 is 1 0 1 1 1 0 1 1 and if we add these things if we add negative 5 1 plus 1 is 2 so we'll put a 0 and carry the 1 1 plus 1 is to put a 0 and carry the 1 1 plus 1 is 2 so we'll put a 0 and carry the 1 And 1 plus 1 is to put a 0 and we have [a] one that we're carrying and so you'll notice that each when we're when we're looking at the the Negated version of something when we add each of these Place values we get a 2 each time, [and] that's why it's called two's complement But in any event you see that we end up with 0 which is just what we would want when we add a number Or to its its inverse and in fact you can try it with any of these numbers and you'll find out [that] you you get all [zeroes], so [Let's] try you know 6 minus 2 so 6 is 0 1 1 0 that's 6 and Minus 2 is 1 1 1 0 1 1 1 0 and? if we add these together we add our negative 2 oops Negative 2 adding negative 2 there 0 plus 0 is 0 1 plus 1 is 2 so we'll Put a 0 and carry the 1 1 plus 1 plus 1 is 3 so [that's] a 1 and a carry a 1 And the 1 plus 1 is 2 so 0 and carry a 1 and again we ignore that fifth bit there and so 0 1 0 0 is 4 and sure enough 6 minus 2 should be equal 4 so this two's complement works works pretty well [and] it also has this nice property again that that first bit is a sign bit right because all of these negative numbers Have a 1 there And what's even more interesting about two's complement is if you look at the place values. It actually makes sense This is our ones place. This is our twos place. This is our forest place and this place here It's you know. It looks like a sign bit, but really it's a negative 8 place and You can see when [we] have a negative 8 and all zeros We have a negative 8 but if you have a negative 8 and a 1 you get negative 7 right? if you have negative 8 and a 4 you get negative 4 if you get negative 8 Plus 4 Plus 2 plus 1 you end up at Negative 1 So this this this bit value actually has has a mathematical meaning. Which is why our math works out, so it's pretty cool The one thing that's a little bit harder with two's complement Is is negating a number? So if we want to go from a 5 to a negative 5? it's kind of a two-step process right so 5 is 0 1 0 [1] The first thing you have to do is take the ones compliment 1 0 1 0 so flip all the bits So you're just inverting everything that's that's easy enough, but then you have to add 1 So you invert? and Then add [1] so 2 add 1 2 0 [1] 0 1 is just 1 0 1 1 and then this Invert so this is 5 then we invert it and then we add 1 and that gets us [to] Negative 5 you see negative 5 1 0 1 1 [1] 0 1 1 so it's a little bit more work to go from a positive number to [2] its inverse Because you have to invert it and then add 1 But what you'll see you know in the next video when we when we build a circuit that can add and subtract You'll see that it's it's actually not too [hard] to build to build this ain't hardware
