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Signed binary

in this video we're gonna talk about how to express negative numbers in binary using the sign-magnitude method and the two's complement method and later in this video we're going to talk about how to add and subtract binary numbers using the two's compliments so let's say if we want to write negative six as a number how will we do that so we're gonna start with the sign-magnitude method in order to express a negative six first we need to find out what six is and we're going to use a 4-bit binary to do so so two to the N minus one that's going to be 2 to the 4 minus 1 which is 2 to the 3 and that's 8 so this is going to be 8 4 2 & 1 so first let's express 6 as a binary number 6 is 4 plus 2 so we're gonna put a 1 in the column under 4 & 2 and we're gonna put a zero everywhere else so that would be positive 6 in sign-magnitude the first three numbers corresponds to the magnitude of the number 6 the last number zero this corresponds to the sign so if the last number is zero that means we're dealing for positive number if it's 1 that means we're dealing with a negative number so to express negative six we just need to replace this with 4 1 so using the sign and magnitude method negative six is gonna be 1 1 1 zero and that will be the answer now let's work on some other examples using the sign-magnitude method let's express the following negative numbers in binary and we're going to use a 5 bit binary system so 2 to the N minus 1 that's going to be 2 to the 5 minus 1 which is 2 to the 4 and that's 16 so this is going to be 16 8 4 2 and 1 so let's start with positive 7 because this is positive the first one is going to be 0 7 is 4 plus 2 plus 1 so we're gonna put a 1 under those numbers so that number corresponds to 7 to get negative 7 all we need to do is replace this 0 for 1 remember that first number in the case to sign everything else is going to be the same so using the sign and magnitude method negative 7 would correspond to this 5 bit binary number now let's try negative 12 so first let's find the number for 12 so the first number is going to be a 0 and 12 is simply 8 plus 4 so we're gonna put a 1 under those two numbers so that's the binary equivalent of 12 so now let's find a negative 12 so we're just gonna replace this 1 4 1 so it's going to be 1 0 0 I mean 1 1 1 is 0 0 we're just changing that number into an 1 and so that's how we can represent negative 12 using sign magnitude for the last one negative 15 let's start with positive 15 so the first number is gonna be 0 15 is the sum of all four of these numbers 8 plus 4 plus 2 plus 1 is 15 so we're gonna put a 1 for all four of them so to express negative 15 change the sign to negative so this would be the 5 bit binary number and that will correspond to negative 15 so this method of representing negative numbers that is the sign-magnitude method it's very easy for people to understand it's very user-friendly however it doesn't work very well for computers especially when computers need to perform reasons such as addition or subtraction for instance let's say if we want to add seven and negative seven we should get zero so let's add these two binary numbers 0 0 1 1 1 and 1 1 1 0 1 so 1 plus 1 that would be 2 but we can't really write 2 so we'll just put 0 carry over the 1 so this right here would be 0 carry over the 1 but we have another 1 plus this 0 which will turn this into a 1 so this right here would put a 0 and a 1 carry over but the 0 part Plus this 1 will give us another one and then we have 1 plus 0 which is 1 and 0 plus 1 which is 1 and this is 1 2 4 8 16 so we're adding 16 8 4 & 2 so the result that we'll get is not 0 as you can see this would be 30 and that can't be right so as you could see when expressing negative numbers using the sign and magnitude method it's not very helpful if we need to perform operations such as addition or subtraction so now let's talk about express and negative numbers using the two's complement method and we're going to see why this works out so well so let's start with 7 so 7 is 0 1 1 1 now what we need to do is find the ones compliment first so we're just gonna every one that we see we're gonna replace it with a 0 and every 0 that we we've seen we're gonna replace it with 1 now our next step is to add 1 that will give us the twos complements 0 plus 1 is 1 and then we could just bring down everything else so negative 7 would correspond to the number that we see here now keep in mind the first one that you see indicates a negative value now how can we quickly check to see if this is indeed negative 7 because the binary equivalent looks like 8 plus 1 which is 9 but here's how you can check to see if it's negative 7 remember this one is associated 4 negative sign so if you make this negative 8 this becomes negative 8 plus 0 plus 0 plus 1 which gives us negative 7 so that works so that's how we can represent a negative number let's try another example so go ahead and represent negative 12 as a binary number using the two's complement so first we need to write 12 so this is going to be we're going to use a 5 bit system so 16 8 4 2 1 so the 0 would indicate that this is going to be a positive number just like before so that's why we using the 5 bit binary system instead of a 4 bit because we need the 8 to make 12 a plus 4 is 12 so 12 would be 0 1 1 0 0 now let's find the complement of this number so we place a 0 with 1 and replace one with 0 next add 1 so one plus one is going to be zero carry over the one one plus one is zero carry over the one let me just separate these numbers from what we have below 1 plus 0 is 1 and then we could just bring down the other numbers so negative 12 is 1 is 0 1 0 0 so keep in mind the first number is to tell us that it's a negative number so now let's check our work so we have a 16 here and a negative value so that's going to be negative 16 we have a 0 in the a column so we're not going to add anything we have a 1 in the 4 column so we're gonna add 4 and here we have nothing so becomes negative 16 plus 4 and that checks out that is negative 12 so that is negative 12 as a binary number using the two's complement method so now let's talk about how we could subtract two binary numbers we're going to go over two methods the first one involves straight for subtraction and the second one involves the two's complement so let's do straightforward subtraction 1 minus 1 is 0 1 minus 0 is 1 now 0 minus 1 if we do that that's going to be negative 1 and put in the negative 1 here just doesn't look right so let's not do that what we need to do is borrow something from this one if we borrow a 1 this becomes 0 and we get 2 ones here now we're gonna have a lot of ones here so that just doesn't look right I mean you could do it this way it works but the decimal equivalent of two ones is 2 so I like to represent this as 2 that's just my unconventional method of subtracting binary numbers now we need to borrow a one from this too and so this is going to become a 2 2 minus 1 is 1 1 minus 1 is 0 and then 0 minus 0 is just 0 so now let's see if this actually works so this is 1 2 4 8 16 32 64 so now let's convert these numbers into decimal numbers so originally we had a 1 here the decimal value for that is 32 plus 4 that's 36 plus 2 38 plus 1 39 so the original number at the top was 39 and then we have a minus sign this is 16 plus 8 that's 24 plus 1 that's 25 so 39 minus 25 that's 14 now let's see if this number actually represents 14 so this is 8 for best 12 plus 2 that is 14 so this technique works it's a very simple way of subtracting two numbers now let's use the two's complement method in this example we subtract the two positive numbers we had positive 39 minus positive 25 now when using the two complements method if you need to do subtraction what you want to do is basically add a negative number so this is still going to be positive 39 or I'm just gonna write 39 to represent that and then it's gonna be plus negative 25 we're still going to get 14 but instead of having a positive 25 value here we're gonna have a negative 25 so I'm going to rewrite that 39 value as a binary number so that's what we have in this room now I'm gonna change this part which is positive 25 into negative 25 using the two's-complement method so first let's rewrite it now let's find the ones complement of that number so replace one with zero and then replace zero with one our next step is to add 1 so this is 1 1 1 0 0 1 1 so this number should represent negative 25 and let's just check it to make sure that it does so keep in mind the first one represents a negative value so that's going to be negative 64 so we have negative 64 plus 32 this corresponds to a 4 so that's plus 4 plus 2 plus 1 negative 64 plus 32 plus 4 plus 2 plus 1 so that is indeed negative 25 so now I'm gonna take this binary number and put it beneath this one so that's 1 1 & 0 0 1 1 1 and now we're not going to subtract but we're gonna add so remember the top number is positive 39 the bottom number is negative 25 and we're adding which is the nice way to do subtraction so 1 plus 1 that's going to be 0 carry over the 1 now this part here 1 plus 1 that is 0 carry over the 1 but when you add this 1 and this 0 that becomes a 1 now we're gonna do the same thing here one plus one is zero carry over the 1 and then 1 plus of this 0 is 1 now we have 1 plus 0 plus 0 which is 1 0 plus 0 is 0 1 plus 1 is 0 carry over the 1 1 plus 1 is 0 carry over the 1 now this is a carry over because we didn't have that many digits to begin with so now let's see what answer we get so with this extra carry over you can just get rid of it because that's an overflow that's gonna be out of the range so our real answer is what we see here ignoring the overflow value and you can see this is 8 plus 4 plus 2 8 plus 4 plus 2 adds up to 14 so that's how we can do binary subtraction using the two's complement so what you want to do is instead of subtracting a positive number you make that a negative number and then just add it but you have to get both of these numbers in binary form so you can keep this number the same but you got to find the two's complement of this number which takes a lot of work so you have to choose you know which method is easier for you to do but that's how you would do it using the two's complement method let's try another example of subtracting two binary numbers so this is gonna be 0 1 0 0 0 1 1 1 feel free to try this example now before I subtract it I want to get the decimal equivalent of each number so this is 1 2 4 8 16 32 64 128 so this is 32 plus 8 that's 40 42 43 and then we're gonna subtract that by this is 64 plus 4 that's 68 plus 2 that's 70 71 notice that the second number is larger than the first now this is positive 71 as we can see the last number is 0 and this is positive 43 so when we subtract these two numbers we're going to get a negative result 43 minus 71 that's going to be negative 28 so we'll both methods give us a negative result or should we use the two's complement method or should we use the straightforward method what would you say it turns out that both methods will work let's start with the straightforward method so we have 1 minus 1 which is 0 this is 0 as well here we have one mine I mean 0 minus 1 the 0 minus 1 is negative 1 but you don't want to put a negative number when dealing with binary so we need to borrow a 1 this is gonna become 2 2 minus 1 is 1 and this is 0 1 minus 0 is 1 now 0 minus 1 we have that situation again but notice that we can't borrow a 0 here and also it's the last one so in this case I'm gonna put negative 1 here or rather I'm just gonna put one in seems to indicate that this is a negative value now let's check the work so we have 64 but this corresponds to a negative value so that's going to be negative 64 and keep in mind if the first number is a 1 it indicates a negative value if it's a 0 it indicates a positive value next we have 32 this one corresponds to 4 and then let's add what we have here so negative 64 plus 32 plus 4 so this adds up to negative 28 so as we could see it does work so I'm just going to write the answer on the bottom so this is negative 28 as a binary number just keep in mind the first number 1 corresponds to a negative value now let's see if we can get the same answer using the two's complement method so we are going to instead of subtracting we're going to add but this is going to be negative 71 as opposed to positive 71 and this should give us the same result negative 28 so the first number 43 we're going to keep the same but we need to find the negative equivalent of this number so let's get rid of this let's rewrite the top number 0 0 1 0 1 0 1 1 just had to make sure I copied that correctly so now let's change this value and I'm running out of space so I'm going to rewrite it here let's find the compliments of all the numbers so let's replace one with zero and zero with one and then we're going to add one so this is gonna be 1 0 0 1 1 1 0 1 so this should be negative 71 and let's just do a quick check to make sure that's the case so this is 1 2 4 8 16 32 64 128 now the first number is negative so that corresponds to negative 128 and then we have a 32 of 16 this one corresponds to 8 and the last one corresponds to 1 so negative 128 plus 32 plus 16 plus a plus 1 and that adds up to negative 71 so that's the right number we're gonna put that here so 1 0 1 1 1 0 0 1 and now let's add these two numbers using this process so 1 plus 1 that's going to be 0 carry over the 1 and then we have 1 plus 1 again which is 0 carry over the 1 1 plus 0 is 1 and then 1 plus 1 is 0 carry over the 1 then we have 1 plus 1 again which is another 0 carry over the 1 now we have three ones so the first two ones will give us a 0 and a 1 and then remain in 1 plus a 0 will give us a 1 so let's turn that into a 1 and then we have 1 plus 0 which is 1 and then 0 plus 1 which is 1 now our answer looks similar but not exactly the same so this part here is exactly what we see here one one zero zero one zero zero but we got this extra one so why is it different we'll keep in mind that this is gonna be our negative value it turns out that even though we have an extra digit so to speak one extra bit the value is still the same so this is 1 2 4 8 16 32 64 128 so this corresponds to negative 128 and then we have a 1 in the 64 column that's going to be positive 64 and then this is positive 32 and then we have positive 4 so let's do the math negative 128 plus 64 plus 32 plus 4 still gives us the same answer of negative 28 and this will also give us negative 28 as well this will be negative 64 plus 32 plus 4 which is the same result so the decimal equivalent of these binary numbers are the same so we still get the same right answer it just it looks a little different with the extra one so that's the end of this video now you know how to express a negative number as a binary number using the sign-magnitude method and the two's complement method and you know how to perform binary addition as well as binary subtraction using the two's complement method and the straight forward subtraction method so thanks again for watching and don't forget to subscribe