Why is negative times negative positive? PART I (James Tanton)

welcome let's address the age-old question that drives students and many teachers nuts why is negative times negative positive let's answer this a couple of ways but first of all let's make sure we understand what multiplication is at least at a very basic level for example if I write five times three that can be thought of as repeated addition at least at the level of counting numbers many mathematicians will disagree with me but at the very beginning this can be thought of as five groups of three so it's 3 plus 3 plus 3 plus 3 plus 3 of course given the answer 15 what's very curious here not many people seem to every recognizes is curious that 3 times 5 which is actually completely different object it's 3 groups of 5 actually has the same answers 15 we're so used to this that we don't question anymore but there's a lovely geometric understanding of why this would be case why is a times B whoops a times B sure to be the same as B times a at least for counting numbers well let's have a look at it 5 groups of 3 let's actually councilman let's draw dots here's one group of 3 dots there's a second group of 3 dots here's a third group of 3 dots a fourth group of 3 dots and finally a fifth group of 3 dots so if I look this way I am indeed seeing five groups of three the 15 dots what's interesting is I changed my perspective and look this way I'm really seeing this problem three groups of five it's the same picture the same number of dots therefore the answers must be the same of course there's matter which numbers I use in particular which accounting numbers whose in particular a times B is sure to be B times a by this geometric representation alright so there we go for at least four positive counting numbers we know what multiplication means and we know that multiplication is commutative let's start bringing negative numbers into the mix so let's look at this here's a positive number times a negative number can I make sense of this well if I think of as addition of multiplication as repeated addition it's fine this is saying five groups of negative three so here's two of them here's a third negative three and negative three that doth negative 15 make so five groups negative three not a problem negative Dean positive times negative in this case is negative now comes something curious well why would I ask for with negative three times five in my model this typically does not make sense negative three groups of five no idea what that means now I have to go come to a choice this commutative law a times B equals B times a dewy feel so comfortable this law that we think it should be true for all types of numbers and particular negative numbers most people to answer yes but I'd like to recognize that really is a choice if we do say yes then we could answer this problem now you three times five should be no different for negative five times negative three which will here the answer is negative fifteen here it is the previous problem so if we choose to believe that numbers are cumulative with regard to multiplication even negative ones then we can handle negative times positive now comes the juicy one let's get rid of this it goes and let's examine now negative times positive for example let's look at negative three times negative five well if I switch the order around make this negative positive negative three I'm still stuck I do not know which what to assign to this alright let's go back a step again let's look at more complicated multiplication let's look at something like 27 times the fifteen I can think of this really is a geometry problem again I can say this is he goes a rectangle with one side being 27 inches long another side being 50 inches long and I'm asking for the area of this rectangle well these numbers 27 15 are Ted awkward let's split the 15 to two easier numbers say 10 and 5 and let's also split that 27 into two easier numbers let's make a twenty and seven so now means my multiplication problems really comes to for smaller problems what's the area of this space is twenty by ten rectangle or the answers 200 what's the area of this piece it's 20 by five areas one hundred area this part is 70 and xerath's fourth part is thirty-five therefore I can say the 27 times 15 is the same as 200 plus 100 plus seventy plus 35 I believe that's 405 all right fabulous we've just seen that 27 times 15 is 405 though what may be a little sneaky and redo the problem in quite a strange way I'll still split the 10 to the 15 into 10 and 5 Plus this time split 27 into 30 and negative 3 if we're to believe that arithmetic of all numbers applies to negative numbers as well then even though I do not know what a side link negative 3 just means the mathematics behind this diagram we might say is still valid all right according to this diagram what is 27 times 15 well this piece is 330 times 10 the space is 30 times our 550 this piece is negative 3 times 10 we can deal with negative times positive just done done that that's negative 30 and this piece is negative 15 so this is telling me the asset of 15 times 27 is 300 450 minus 30 is 420 minus 15 lo and behold 405 starting the problem this way again gives me 405 now let's be very strange let's solve the same problem again a third way again I'll draw a rectangle but whoops let's get back to my brush mark here draw a rectangle but instead of doing 27 is 20 plus 7 let's again do it as 30 minus 3 and so during 15 is 4 10 plus 15 as two it's 20 and negative 5 now what positive times positive here no doubt that's 600 negative 3 times 20 we've dealt with names times positive that's negative 60 negative 5 and 30 we've dealt with thats negative hundred and 50 but this one negative 3 times negative 5 that is my mystery question I know we've been trained to say negative 15 but let's say I want the math tell me what should be right now it's a mystery well we know the answers problem has to be 405 so right now we have 600 we have negative hundred 50 we have negative 60 and a mystery number and that and we know the answer must be 405 can the math tell us what this mystery answer must be well yes 600 takeaway hundred 50 that's 450 takeaway 60 that is 390 plus something mysterious must be 405 well there it is the math is telling me this mysterious object must be 15 that is positive 15 I am forced to say that negative 3 times negative 5 is positive 15 in fact you use this argument that any school student can use this argument justify about any pair of numbers negative numbers when multiplied together must be positive for example if you want to show that no negative 4 times negative 8 is positive I'd probably look at no say 16 times 22 and think of that two different ways there we are well let me be a little more mathematical and for those that want the true mathematical reasons why negative times negative is positive it's hidden behind the scenes in this argument let me give it to you one begins by listing the rules of arithmetic that one thing should be true whoo

Channel: DrJamesTanton

Views: 24,027

Rating: 4.47541 out of 5

Keywords: WEB, negve, times, PART

Id: nJqyr8h22nY

Length: 7min 46sec (466 seconds)

Published: Tue Sep 22 2009