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Square roots by division method visualised

Let's understand why the square root by division method (digit by digit method) works.

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Video transcript

now if I walk up to you and ask you can you find the square root of 4096 now you have you may say I can do the factorization method or if you have a calculator you can just use it that's the shortest method but there's this other method called and it's in the book it's called finding it by the division method now that's just one of the many methods you can use so let's look at in this video what this division method is a method or an algorithm is simply a set of steps a recipe basically and if you follow that recipe perfectly you will get the dish in this case your dish is your square root of this number but my goal in this video is not to tell you what these steps are exactly and see if you can follow it but to understand why the recipe works to begin that the first thing I want you to do is imagine what square root of 4 0 9 6 really is it's exactly the same as asking if I had a square of area 4096 the side of the square will be the square root these both are the same because you multiply side and multiplied by itself you'll get 4096 now our job then is to guess what the side of the square should be such that the area is 4096 and the problem is that there isn't a great way to do this it's a guessing game that's why I'm saying we have to guess so to start guessing maybe the first thing we can do is find out how many digits would be with the would would there be in this number on this side to do that let's look at this our square is a four-digit number so what will the number of digits in our square root be you can do this by noticing that 10 square is a hundred that's three digits so we know our square root has to be more than ten and notice that 100 square though is a thousand but 10,000 and that's not four digits that's five so you know that our square root has to be less than hundred so more than ten less than hundred in other words the square root that we are trying to find is a two-digit number one digit and then one more it's a two-digit number that's awesome because now we don't have to guess all from all possible numbers after guess from the set of two-digit numbers now how do we do this we want to do this digit by digit we want to first try and guess the first is it and then the second digit now what is guessing this first digit really mean so if you can find this first digit what you've really found is what is the largest square that you can fit into this that's a multiple of ten so where is this multiple of ten come from because when you guess it is it's a three over here you're not really guessing three threes in the tens place so you're guessing that it's our square root is thirty plus something right when you see 32 you will say that number is 30 plus two what do you think is the largest square you can fit which is a multiple of ten again you have to guess 10 square is just hundred 4096 is very far away so you go you make a few jumps maybe you try four so which is basically if you if you guess for here that's they're guessing 40 square but what is four square it's 16 so 40 square is thousand six hundred you still have a lot of space maybe then you try six square that's like asking sixty square how far of this will it occupy let's look at that sixty square is 3600 that's pretty close to 4096 then maybe you go okay maybe seven squared lots of it like 70 square lots of it but then you you do that you find that 70 square is actually larger right seventy squared is 4900 so that doesn't fit and that's it you have your first digit and you're sure this is the first digit because you'd know that this side is greater than 60 but less than 70 in other words the first is it of your square root is six and then now we still have a little bit of work to do which is to find the second digit but step one finding first digit is done now what do you do after this how do you guess the second digit the second digit is basically this gap over here that's what it really is so we're trying to find what this gap is right now one way to do this is find this remaining area the area in the square that's left and then try to see if we can guess for this question mark should be so what's the remaining area over here the total area is 4096 the remaining area this area of this sixty square is 3600 so the remaining area is this minus this so I'm gonna call it remaining area remaining area it is just 4096 - 3600 which is 496 right there's 400 between them and then a 96 more so 496 that's this entire area this l-shaped area but then I don't have any formula for l-shaped figures so now I'm going to try and break this L down into some shapes why why do I want to do that because then I may be able to write that area in terms of this question mark if I can write it as maybe some length into breadth or something like that then I can guess this question mark now this will become clearer as we do it so I'm gonna break this down into maybe one rectangle here one more rectangle here and then one square over here let's see how that looks so two rectangles over here there they are and one square over here now notice that this question mark is going to be the same question mark here and in this question mark this length whatever this gap is the same gap over here so if I write the same gap in terms of the question mark how do I do that it's equal to this area plus this area plus this area just pause and see if you can write this in terms of question marks okay so I'm gonna I'm gonna do this now this length is 60 this length is question mark so area of this rectangle is 60 into question mark right 60 times question mark 60 into question mark no but I have one more of that these two are exactly the same this is 60 and question mark again so that's 2 times 2 times 60 into question mark that's these two rectangles but I need this square that square side is this question mark so I have a plus question mark square and this sum has to be equal to 496 because that's the remaining area right now I can't simplify because I can take this question mark out before I start guessing I want to make it as simple as it is for me to guess so then I just do that I take my question mark out I get 120 plus question mark 120 over here plus 1 question mark from the square and maybe this is as simple as it gets I may have to start guessing now I shouldn't put equal to actually because it may not exactly be equal to so I'm gonna say maybe less than or equal to I'm actually trying to find that question mark such that this area will be less than 496 but as big as possible I want to find it as close as possible but then it may not exactly be 496 so less than or equal to 496 so the real question is what's the biggest question mark such that this thing this area will still be less than 496 because if it becomes more then we know that we've already overshot just like we did in the first digit case so let's start guessing I'm gonna start with 1 so 1 into 1 21 that's my first guess even as I write it I can see that's probably not gonna give me my answer because that's just that's very close 1 21 but I have to space still 496 I can definitely go bigger so I'm going to try and jump to 123 now so that'll be 3 the question mark I'm guessing this gap I'm guessing to be 3 so 3 into a 123 123 now if I do that now that's equal to 369 I still feel like I have a space of about a hundred so I'm gonna try this again I'm going to try four so four into one 24 now let's calculate this 4 into 1 24 you can do this I'm gonna try and do it in my head 4 times 120 is 4 12-hour 480 plus 4 times 4 is 16 496 that's exactly matching what the gap is it need not have it we just were trying to make it lesser if this had been 495 we were still been really happy but it's exactly 496 and what does that mean making this gap let's let's uh now fill that gap in making this gap equal to 4 right equal to 4 what it does is exactly fill his remaining area in other words 60 + 4 this side being 64 makes the area exactly 4096 in other words 4096 is a perfect square we did not know this when we started but now we know that it is a perfect square and we are happy and now to get even happier let's look at the recipe as it's given in the book but this time when we see it maybe we can understand what each step is as we do them so let's do that 4096 and step 1 of the recipe in the book says group these into digits to do digits at a time if it's odd start from the right if it's decimal start from the decimal and go either way you can check this out in the book now the reason we're doing this is now you know that we're trying to group it into multiples of hundred you take to two digits at a time that's multiples of hundred so that you can guess our square root one digit at a time so in this case you get 2 bars that basically tells you you have two digits to guess then what do you do you go here and ask what's the numbers whose square is less than 40 and then you say 6 is that number and you write 6 here and you say 36 is that you may be seeing a pattern over here and then you subtract and you get 4 over here and you take the 96 down that's the remaining area as we did it but here you're just subtracting now at this time what you have done is guessed the first digit now this this is the part of the recipe that really confused me which was take six x to put a question mark next to it and then put another question mark over here now guess what this question mark should be such that 120 question mark 120 if this third digit is a question mark into this question mark will be less than 496 now we can try one two three and all that but you already know four works so we can directly do that so I can just go ahead and mark this as four and write four over here when I saw this I had no idea it felt like magic but now I understand why I'm trying to find the area of this this remaining area such that this gap can be guessed and 124 is this part this question mark part is over here so 4 into 124 is what we wanted you may be noticing how these two are exactly the same and you get zero in other words we get the area to match exactly so we stop and then we say our square root is 64 now this recipe is what we call the division method and I think it's called the division method because it looks a lot like long division but as you may have seen it's not got much to do with division maybe one other way to name this would be a digit by digit method of finding the square root digit by digit and that tells you a lot more about what you're really doing so you have the recipe now but you also know why this recipe works