Magical Squaring

hi I'm Professor Arthur Benjamin I teach mathematics at Harvey Mudd College in Claremont California today I'm at the carriage house of the mathematical Association of America and I'm going to teach you how to square numbers in your head faster than a calculator now with me today are my assistants Mari and Catherine and Catherine's going to call out some numbers for me to square mari is going to verify them on the calculator now first let me tell you what the square of a number is if you want to square a number like five that means taking five times itself which gives you 25 now the way you do it up do that on the calculator is in one of two ways on most calculators you could hit five then times then equals and that will give you the square or if you have a fancier calculator like this one you could press a number let's say like 6 and hit this x squared button and that will give you 36 by only having to press the button once so what I'm going to try and do now is to square some numbers in my head faster than Mari can do on her calculator even using the shortcut method ok so Catherine get us warmed up give us a two-digit number to square 773 squared is 5329 correct how about another two-digit number 51 is 2600 one how about a three-digit number this time two hundred two fifty six is 65,536 good one more three-digit number for the fun of it seven sixty two squared that's a harder one is five hundred eighty thousand six hundred forty-four good would you like to know how that's done let me show you okay so let's start with a guess give us a small example as our first our first number 25 oh you know what if it ends in five that's almost too easy I'll get to that in a minute but give me a let's say 23 instead okay that's a good first example now to square the number twenty-three I will either go I go up or down to the nearest easy num in the case of 23 that nearest easy number is 20 so I'm going to go down 3 to 20 now whatever comes up must come down or whatever comes down must go up if I go down 3 to 20 I balance it by going up 3 to 26 so the first part of my calculation of 23 squared is do is I do 26 times 20 now that's the same as 26 times 2 with a friendly 0 attached let's do that one in our heads now if I do 2 times 26 and we do it left to right right 2 times 20 is 40 2 times 6 is 12 40 plus 12 from left to right is 52 so 20 times 26 is 520 almost done all we have to add to this is the square of the number that we went up and down we went up and down 3 3 squared is 9 and that's your answer 529 pretty easy right here let's try another example let's say the number you asked me earlier 51 let's square the number 51 okay so this time the nearest easy number would be 50 so I'm going to go down 1 to 50 up 1 to 52 now let's do 52 times 50 now there are two ways we can do this problem one is we do 5 times 52 and attach a 0 let me show you that way first so 5 times 50 is 250 5 times 2 is 10 250 plus 10 is 260 attach the friendly zero to get 2,600 almost done all we have to add to that is the square of 1 which is 2 1 giving us 2601 by the way there was another way we could have done the 50 times 52 problem we could take advantage of the fact that 50 is half of a hundred right and so if I do 50 times 52 I can take half of 52 which is 26 then multiply that by a hundred to get 2600 either way we reduce to twenty-six hundred plus one which is two thousand six hundred and one here let's do another example let's try the square of 97 this time all right 97 this is an example where we're going to go up instead of down because the nearest easy number here would be a hundred so I'll go up three to a hundred down three to 94 now you tell me ninety-four times a hundred is ninety four hundred almost done all we have to add is the square of three not the square of seven by the way we only traveled three to get up to a hundred three squared is nine and that's your answer nine thousand four hundred and nine now I mentioned earlier that we would do problems that ended in five so let me show you if any of this was confusing to you if you got if you got a little bit lost here's something that you might might find even easier to do if the number ends in five like let's say twenty five squared then you can go up five or down five it doesn't matter you'll get the same multiplication if you go down five to twenty you then have to go up five to thirty twenty times 30 is that's easy that's two times three with two zeros attached is six hundred add to that the square of five which is twenty five six hundred plus 25 is 625 and that's the answer by the way there's an another way of doing these problems that end in five that you might find even easier still if you square a number that ends in five there's only two things you need to know first of all the answer will always always end in 25 just like here also how does the answer begin it begins by taking the first digit which is 2 multiplying that by the next higher digit which is 3 2 times 3 is 6 and that's your answer 625 all right let's try another example let's say let's say the problem was 35 squared okay so again 35 squared is going to end with 25 not 35 25 how does it begin it begins by taking the first digit 3 times the next higher digit for 3 times 4 is 12 and there's your answer 1225 okay you do this one okay you tell me how about squaring 75 okay think about it think of it I let's just give me the answer how do you how do we do this 7 times 8 is 56 so the answer is 56 25 let me show you why this works okay I mean here we are at the mathematical Association of America and we want to see reasons why does the mathematics why does the why do these shortcuts work the way they do here's the algebra that I that justifies the method that you've seen here I don't think about this algebra as I'm doing the calculation but it tells me that it's always going to work here's the formula a squared equals a plus D times a minus D plus d squared okay first of all let me show you that this is a true equation right if you take a plus D a minus D if you've had an algebra course you know that's equal to a squared minus d squared when you add the d squared the DS go away they disappear and you're just left with a squared which is what we have here now this is exactly what we were doing when we were squaring these numbers let's say for instance when we squared 97 97 squared what was our D what was the distance we traveled we traveled three and this formula says that 97 squared is equal to 97 plus three times 97 minus three plus three squared that is it's a hundred times 94 plus nine and that's exactly what we did here okay by the way this will also work for this formula shows it'll also work for three digit numbers and higher let's say give us a three-digit number and I'll square it and show you how it works how about a three-digit number this time 762 squared is five hundred eighty thousand six hundred forty four good let me show you how that works okay so to square a number like seven sixty two here's what we do now seven 62 is not a bad number to multiply but let's move it up to eight hundred so I'm going to go up 38 to 800 down 38 actually I double seven 62 to get seven 24 I do 800 times seven 24 let's see eight times 700 is 5,600 eight times 24 is 192 5,600 plus 192 is fifty-seven 9250 72 so that's five hundred seventy nine thousand two hundred add to that the square of 38 and since I've been doing these for a very long time I know 38 squared is 1444 add those together from left to right to get five hundred eighty thousand six hundred forty-four and that's exactly what I was doing in my head when I was solving that problem by the way another reason I work my calculations from left to right is because it allows you to say your answer while you're still calculating thereby giving the illusion that you're doing the problem even faster here let me wrap things up by doing one more calculation for you give us a four digit number this time I'm not going to write it down on the board I'm just going to do it in my head I'll think out loud so you can hear what's going on but this is just for your amusement how about a four digit number four nine one three okay so I double that to get five times four eight two six that's 24 million hold on to the 130-thousand turn that into dimes add to that the square of eighty-seven which is seven thousand five hundred sixty-nine so I get one hundred thirty seven thousand five hundred sixty-nine good thank you very much you

Channel: Mathematical Association of America

Views: 547,258

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Keywords: Mathemagician, MathAssociationofAmerica, MAA, ArthurBenjamin, f22VideoSolutions

Id: _FoONg5Meyc

Length: 12min 7sec (727 seconds)

Published: Mon Nov 18 2013