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okay in this video we're gonna look at another way of storing signed integers and this method is called two's complement so the previous method that we've looked at sign and magnitude uses two parts a sign and a magnitude to represent a number now what two's complement attempts to do is to just use one whole value and the the sign is still recorded but it becomes part of the number which means when you're doing arithmetic you don't even have to look at the sign you just get on with a roof day and you'll see that in another video okay but let's have a look at the process of two's complement it's got weird name and because it's called two's complement because it's a two step process there is to write down positive numbers is simple it's identical Sun magnitude but to write down negative numbers you have to go through the two's complement process if you want to know why we use two's complement it's because it allows you to do addition and subtraction without having to create any subtraction code or any subtraction hardware so you can do addition by doing subtraction okay but I'll talk a little bit more about that when we do the addition work so let's just have a look at the format there so let's as we were before let's use an 8-bit number so we are let's put some column headings down so just eight column headings doubling up each time nice and simple okay just like Santa magnitude it uses the most significant bit remember this is the one with the largest column heading as the sign bit so this again is going to be our sign bit okay so for positive numbers the process is identical so let's say we wanted to write down the number Plus 45 okay we start with the sign the sign is positive so just like in sign magnitude we put a zero in the sign bit to mean positive and then we use the remaining bits in order to represent the size the magnitude so the 45 so let's do that so we can't have any six fours but we can have a 32 that will leave me with 13 so 16 isn't gonna go but 8 is 8 from 13 leaves me with 5 so I've got 4 and a 1 so that in two's complement he's plus 45 now when you read these numbers but if you're told you've got a signed number you can always look at the sign bit it's a old look that's a positive number if it was a 1 you could say it was a negative number but you must be careful ok and you must make sure because there's more than one way of writing signed integers you must read a care question carefully if it says it's in sign of magnitude then you must use the son of magnitude methods because it isn't to complement and you have to use the two's complement methods okay so that's the positive number now when we did the Sun ranked you if I wanted to write down minus 45 I would just change the sign bit and the rest of the number would stay the same but two's complement doesn't work like that what we have to do if I want to write down what minus 45 is my starting point is to say it's negative ok two's complement and I should always say cannot write down straight away that's what I must say to myself so we know how to write negative numbers there is a way of doing it but I'm not going to teach you because it's complicated if you want to know how to directly write them down go and look it up okay right so what we do if we want to write down a negative value what we do is is oh okay hang a minute let's write down the positive value and then this process two's complement will convert and positive value into negative value okay sounds a bit weird be it's actually more difficult to explain than it is to actually just do so let's have a look - 45 so I don't know how to write down minus 45 so what I'm gonna write down is pause for at five now I've already written up but I'm gonna write it again so I'm gonna write down so step one write the positive number down so using this I'm gonna write down a sign but if not which proves it's a positive number and then gonna write 45 so I'm just going to actually copy what we've got above okay and then I'm gonna do two's confident which is a two step process there is a flip and then add okay so I'm going to show you how to do that so step two so we flip the number so we've written down positive 45 and I'm gonna make a little note at the side just say right I know that's not made what I was doing that's where I'm starting right so I'm gonna flip all the bits so when they're not they become a 1 and where there are 1 they become a no so that's of the compliment and then the final step is we add 1 so that was the flip phase step three is to add one so in binary to add one it's quite straightforward we just work from this way and the first time we find a zero we change it to a one anything we've gone past which is nothing in this case we write down a zero so we can just say there's one and then everything else just stays the same so that's how you quickly had one in binary okay so that is now minus 45 not very easy to read but you can see that the sign bit is one okay so you know it's a negative number but it's really important that everything that we store in the computer is in binary so this could be a signed integer it could be an unsigned integer it could be a character it could be a pixel color could be the amount of green in an image it could be the amplitude of a channel in an audio recording okay unless we know what the format is we cannot interpret that data so it's really important so let's let's just verify to ourselves that this is actually minus 45 right two's complement it's similar to the mathematical operation of x in a value by minus 1 and you can see effects but that's what we did we wrote down positive 45 and then we've done to confirm which is sort of turned into minus will x minus 1 because it's now minus 5 but the weird thing about multiplying by minus 1 if I take minus 45 and multiply it by minus 1 I should get 45 so what I'm saying is if I perform two's complement on this number I should be able to find out if it was 45 so I'm gonna perform two's complement again on this number and I'm going to show you an easier way to do the flipping that one you can do it all in one go right so I'm gonna just verify that I've got minus 45 cents just scroll up a little bit right so what I'm gonna do this time I'm gonna flip like I did before but this time when I get to the last one I'm gonna leave that as a 1 and then anything that follows I'm gonna write down a zero so I'm sort of like doing the ad in the same step okay so I'm gonna go flip flip flip you got to try not say Philip when you doing that when your mouth goes dry you can sometimes say Philip flip flip flip flip right that is the last one so we leave that as it is and then anything else we would just write down zeros but obviously in this case there is no more to the body so that we are saying because we performed two's complement on a negative value that is now hopefully plus 45 so let's check so let's put some column endings on 1 2 or 8 16 32 64 1 2 8 so we've got 32 plus 1/8 plus a 4 plus a 1 now I know and this how I add these up in my head that's for a that makes 545 okay so the two's complement process takes a positive number and gives you it's negative equivalent and it can take a negative number and give you a positive equivalent now this is important if I am given a value in two's complement so let me just write down I'll do a six bit value doesn't matter so I'm gonna do six bits and I say this is a two's complement value what is that value so I'm going to put column headings down remember that's the sign bit whenever I'm looking at a number and as soon as I know what the format is so someone said to me this is two's complement I'll look at the sign bit and it got right two's complement the sign bit is zero so this is a positive number so I'm going to write down plus and because it's a positive number I can read it I can read positive numbers just like sign of magnitude so I'm gonna look at this number and there's a 16 plus 4 which is 20 plus 2 to 22 plus 1 so this is 23 now if I was given another number in two's complement so this one I'm gonna go backwards this time right and again I'm gonna be told it's a two's complement value so I look at this and I go right okay looking at that sign there it's negative so I'm gonna write down there's my answer - but I don't know what number is not 9 is 4 okay he's completely unreadable so what I'm gonna do in order to be able to work out what that number is I'm gonna do two's complement and find out what the positive version is so what I'm gonna do two's complement so let's go for it let's go flip flip flip flip flip that's the last one to leave that as a 1 anything else we would write down as zeros now that if we look at this it's now a positive number so it's a 4-1 so that is plus 5 which means our original number was minus 5 so you've got to be careful just because a question says two's complement does not mean you automatically flipping out which is the most common mistake students make what you do is you look at the Sun but if it's positive just read it like we did with this one if it's negative ie there's a 1 so okay then let's just do two's complement on it make sure we wrote down a minus first so that remembered we were starting with the minus five which is another common mistake students make the much right plus five is the answer but it isn't so we can say right I can't read it but if I do two's complement I can find it well the positive version of that number was and then that gives me the answer - five okay just as a summary of all these integers if you get something like this let's just do an 8 bit 8 16 32 64 1 to 8 I'm not doing this with a mouse by the way I've got a pen I will be a genius if I could write like this using the mouse nobody can okay right so if I had this number oh let's write down this number and I said a it's an unsigned number or B it's on a monkey or C is two's complement or to see as we often write that tells me how to interpret this pattern of bits okay if it was an unsigned number I just add up all the columns where there's a 1 now in this case that's the biggest unsigned number in 8-bit which is remember next column add n minus 1 so that will be the number 2 ID 5 for sanam magnitude how goes straight to this and I say Oh sign bit it's a negative number and then I read of the remainder of the bits because they contain the magnitude this bit now that would be the largest negative number in 8-bit son monkey which we know is minus 120 one less than the sign bit column in two's complement we look at it we go oh it's negative but we then got what I mean I can't read it so what I'm gonna do is I'm gonna perform two's complement on it now let me just write this out again because I need to be able to do the complement at the bottom let's put some column headings on and I'm gonna do two's complement so remember that's flip all the bits until you get to the final one and leave it as a one anything else after that just write down as a zero right so we're gonna flip flip flip flip flip flip flip we get to the final one and we leave it as a 1 so that is plus 1 which means the negative version of that was minus 1 minus 1 is one of the most beautiful numbers in two's complement it doesn't matter how many bits you have it will always be all the ones so you can spot it really easily but can you see what we've got there we've got three different outcomes just because we interpreted this pattern of bits in three different ways and that's the real essence of data representation bits are just bits until we put our spin on it and say that's what they represent okay and everything we do in debt a representation comes back to that that what we say they represent matters okay you can't just take a pattern of bits and know what it represents you need to you need to you need to know that it's an integer it's a character it's a pixel color okay it's any part of an encryption key whatever it is you need to know he might be that these eight bits are part of a larger number a 64-bit number or a 2048 bit number but if we we know what it represents then we can interpret it properly and the interpretation can be wildly different okay so with two's complement let's just recap sign bit is the most significant bit if you're writing down a positive number just put a sign bit ignore and then use the remaining bits to write the number down if you've got to write down a negative value so write down of the positive version and perform the two's complement they're flipping out okay if we just think about the range of numbers so the biggest number per to compliment he's gonna be a zero followed by all the ones that's gonna be positive so this is eight bits positive 127 but actually the largest negative number is minus 128 and it is the only thing that's a bit weird about two's complement so if I've got this value in two's complement and I go oh my what is that now I know it's negative so I can't read it so I'm going to perform flipping out and remember the flipping out you flip all the bits until you get to the last one when you get to the last one you leave it as a one everything else remains zero so when you start this one you go oh that's the last one so leave it as a one everything else remains zero and it stays the same so that's 128 so this is minus 128 so the you can get one extra value in two's complement as opposed to sign a magnitude remember sign a magnitude if we read that as Sun among cued that will be minus zero in Sun among to be nonsense it's sign size sign size in to confirm it's all embedded okay so in the next video lesson we're going to look at arrhythmia so how we add and subtract numbers and we're going to be using two's complement okay