Recently we did a video on showing why 9.999 going on forever is equal to 10. And afterwards we got a request from someone in Korea to do something related which we're going to do today. And I found a really, really nice movie clip that kind of introduces it, so let's just watch that first. A negative times a negative equals a positive. A negative times a negative equals a positive. Say it. A negative times a negative equals a positive. Say it. A negative times a negative equals a positive. Again. A negative times a negative equals a positive. A negative times a negative equals a positive. I can't hear you. A negative times a negative equals a positive. Louder! A negative times a negative equals a positive! Louder! A negative times a negative equals a positive! Why? Well first of all, I mean, what's the movie here? What's the movie? Very famous movie. You may not be familiar with it but it's famous among teachers, especially: "Stand and Deliver". You may be familiar with the actor who's the same actor who played Commander Adama on Battlestar Galactica. Anyway if you haven't seen the movie, check it out. "Stand and Deliver". Great, great movie. Great clip. So the question is "why is negative times negative positive?" And well you might ask, "well what else can it be?" Well, the only reason you ask "what else can it be?" is because you've been indoctrinated into believing that negative times negative is positive. It's actually a bit tricky when you first encounter it. You may have forgotten about when you first encountered it. So let's just remind ourselves of what the issue really is. ok And for that we'll try and put ourselves into the... shoes of some ancient civilization. So they've just introduced numbers. So maybe they've got all the positive numbers under control, they've got zero. And now they just lost their innocence, they've introduced money and now there's people owing money to other people. So there's people in debt. And so to describe this they've made up negative numbers? Now you just have to combine these positive numbers and the negative numbers. And, well, some of the things are kind of under control. Let us see where we've got so far. We've got things like, 3 + 4... obviously 7. Then somebody might be 3 in debt And now he gets 4 and so that leaves him with 1. What about 3 in debt and then he goes further 4 into debt. And so that is 7 debt. But now, "times". So what is that supposed to mean? A negative number times a negative number. So what do we do with this? Hmm.. A bit tricky, so lets just step up back a bit. So minus 3 times 4... Well that's something we can handle. So basically whenever we see something like this the way we interpret it is like 3 times 4 so we take 3, and then we do 3 plus 3 plus 3 plus 3, so 4 times and that gets twelve. So we can do the same stuff here with, minus 3 So it's (minus 3) plus (minus 3) plus (minus 3) plus (minus 3) It's minus 12. Ok good, all under control. What about this one here? Hmm That's already a little bit tricky, right? 3 times -4 What's that supposed to mean? So we've got 3, and then we're supposed to do this.. minus 4 times Well, whatever, right? So at this point in time, and it's actually a really really good approach in mathematics and just in general is we just got to do it. Just going to do it Nike approach. That's the T-shirt today. We're just going to do it. Well we know all the stuff about the positive numbers, how they behave. right? We know all the rules And we want those rules to still all work when we're finished with introducing all those extra bits about the negative numbers. So, we're just going to think positive and we think maybe it's gonna work out. And if it's gonna work out, so, what's gonna happen? Well, the first thing is, with positive numbers we know that if we've got two positive numbers, like 3 and 4, we can also switch the order, and then the result is the same. So, in this case, if things are good in the end, we should be able to just switch two numbers around and we've got (-4)*3, and we already know – that's (-12). All under control. Now, back to this one here. And again, we just continue
with our "just do it" approach. Right? So we want things to hang
together in exactly the same way as we're used to. So for a start, we don't know what that is, OK? So let's just take the "times -4" away, store it away there And now what we're going to do is we're going to turn the (-3) into something familiar. Into something positive. OK, And what we're gonna do with this is we're going to just add 3. OK? So we're gonna add 3. And what comes out? Zero. OK? So... Well, we haven't turned it to something positive but we have turned it into something
that we can control, a 0 and a 3. We know how to multiply those two numbers with... minus... negative numbers, right? And so, what we're going to do now is we're gonna just go on autopilot We're multiplying that whole equation by (-4) Alright? So let's get going. 0 times (-4) is? OPERATOR: 0!
BURKARD: Perfect :) OK, 3 times (-4) is?
OPERATOR: (-12). Very good. Alright, we're on track. And (-3) times (-4)? We still don't know but we can just put it in now. OK? So we put it in, and now we've got an equation. And there's basically one unknown, right? One thing we dont' know, and two things we do know. And we can just solve things now
for the unknown, basically. Or, other way, we just add 12 to both sides of this equation. OK? And that gives us: (-3) times (-4) is equal to 12. And so that's basically forced on us. So if we want things to work the way they used to work, with just positives numbers we don't have a choice here we have to say that (-3)*(-4) is equal to 12. OK, that's just our "just do it" approach. But, of course, now comes the checking period. So how will we check that all of
this kind of makes sense? Well, we can check it as mathematicians. As mathematicians, we can now go and say with the way we've introduced multiplication of negative numbers by negative numbers is everything consistent? Are we going to get any things that go wrong? Can we get contradictions, stuff like this? And we've checked, and everything is fine. So that.. that's good. But then, of course, the other thing we want to know is that, you know, do these things actually work in real life? So if we have something in real life and it kind of matches in with negative numbers, is this going to give us the right result? That would be good, right? Well, we could just be lazy and, you know, unleash it on humanity and wait a couple of hundred years to see
whether anything goes wrong. Well, we kind of have done this, and
nothing did go wrong, So we can probably, yeah... OK, it's OK. It's OK, right :) But let me just give you an example of the first bit that things can't go wrong within mathematics. And I've got a really, really nice application for this. And I do this quite often when I go to schools. So, when I go to schools, I've got different routines So I may talk about the mathematics of juggling, so I talk about... you know, I juggle a little bit and then I talk about the maths of it. But I've also got one routine where I...
I'm a "lightning calculator". So, a person who can calculate really-really quickly just using his mind. Right? And.. So I would, maybe, ask you for your date of birth and I immediately tell you on which day (of the week) you're born. Or I square numbers. OK? So I ask "Give me a number", and I square it, quickly. And I just want to show you what I do there. Let's say, you say "square 48", OK? Then the way I do it for this particular routine is I rewrite 48 as (40+8). OK? And then I use this rule: (a+b)^2 is equal to (a^2+2ab+b^2). OK? And, of course, here, what's "a"? "a" is 40, and "b" is 8. So let's just do it, right. So we need a^2, which is 40^2 4 times 4 is 16, then we add two zeros – 1600. Cool. Then we need 2*4*8. So 2*4*8, that's 8*8 = 64. And another zero, so it's 640. OK? And then the last one is b^2. "b" is 8, so 8^2 is another 64. OK? And quickly add it up, and you get 2304. Very quick. I mean, that doesn't even take a you know, half a second. It's just there. Alright. But to be honest, I actually don't do this one here. I do something different. I mean that if this 8 was a 2, I would do it like this. But since it's a 8, and this number is actually pretty close to 50, what I actually really do is I use (50-2). And I use the same rule. Now, (50 - 2) is (50 + (-2)). So the "b" is now (-2). So in the end we're actually going to do (-2) times (-2). So let's just see whether we get the same result, right? So 50^2 is 2500. Then we need 2*50*(-2), which is (-200). And now comes the tricky one, right?
(-2) times (-2) is 4, right? Nothing else. It's 4. So we have to actually add 4, and you can see at a glance: we get the same result. All is good. And just in general no matter what kind of, you know, acrobatics you try to do, to get into something that doesn't work, which works one way and doesn't work the other way, you're never going to get into any trouble if you introduce negative numbers like this. So that's really-really good, right? OK! But in real life, does this thing actually come up in real life? You know, (-2) times (-2)? Well, actually, to be absolutely honest, most people will be able to lead a perfectly fine life without ever using this :) Without ever using this rule. And it's actually been... historically, it's been you know, a fairly late development. And there have actually been lots of fights over it, whether it should be like this or some other way. So, it's a fairly late thing that negative times negative equals positive made an appearance and became accepted. But there are actually instances in real life where you can have an interpretation of all this stuff, which kind of makes sense, which kind of fits in what we've introduced. So let's just give a one example here. So, we're multiplying two numbers, There's the first number,
there's the second number. The first number we're going to interpret as credit (like money) if it's a positive number, and as debt
if it's a negative number. OK? The second number we're going to interpret as gaining something [if positive]. So here we're gaining something four times. Or losing something (four times) if
it's a negative number. OK? And then these are all kind of combinations
of positives with negatives. So, 3 credit, and we have this 4 times then, of course, we've got 12 credit, OK? Or 3 credit: when we lose 3 credit 4 times, well, then we have a debt of 12. OK? Or if we've got a debt of 3 and we do this 4 times, so we gain a debt 4 times, then we also have -12. So we've got a debt of 12. And what about the debt of 3, and we lose the debt... so losing the debt means gaining, right? So we lose a debt of 3 four times. That means we've got a credit of 12. Makes sense, right? So it fits in. And whatever else you can think of in real life, also makes sense. OK, so, it's all good. Right? "Why?" So why? Why? Well, because if we introduce it like this, it's the only way in which we can get really-
really nice mathematics happening, which extends all the nice things we
are used to from the positive numbers. But also, it works. It works. And, I mean, we've we've been trialing it for a couple of hundred years it really works :) So it's really good. Now just an afterthought. Within mathematics there's a few places where you're dealing with screwed up multiplications. And, just from my personal history, there was an instant when I actually was getting into research where it was getting very important to define (-2) times (-2) is (+8). Well, maybe, I'll tell you about
that one in another video :)
Stand and Deliver is great.
One thing I wish this video would have covered is the scaling perspective of multiplication.
A number can be represented as a point on a number line... but you can also represent a number as an arrow on the number line, emanating from zero.
Timesing a number by another is the same as scaling the length of the first number, taken as an arrow, by a factor of the second number. So 4 x 2 would double the length of the arrow spanning from 0 to 4, giving us an arrow from 0 to 8.
But an arrow is more than just a magnitude. It has an orientation, too. The number 4 is an arrow pointing to the right (by convention at least). And scaling its length by 2 preserves this orientation.
Multiplying by a negative number is then just what you get when you reverse the orientation. Multiplying by -1 just flips the arrow around.
I haven't actually tried teaching this perspective. (For whatever reason, multiplying negative numbers doesn't come up in the curriculum we use at my job), but it's something I would like to try some day. Kids often don't have a strong grasp of the difference between negative numbers and fractions, and I think this gives an intuitive explanation for the role of the various kinds of numbers:
And what's cool is this generalizes very nicely when kids are introduced to vector geometry and complex numbers.
Does anyone know about the (-2)(-2) := +8 thing he mentions at the end?
Because we want to retain the axiom of distributivity. If the convention were -1 * -1 = -1 instead, then
(-1)(1 + (-1)) = (-1)(1) + (-1)(-1) = (-1) + (-1) = -2.
-1*-1=ei*pi*ei*pi=e2i*pi=1
Easy!
this too. I recommend the rest of the videos in that playlist.
The amazing moment when your math professor is on reddit.
It would be weird if y=ax was the equation of a line for positive a but not for negative a.