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Undefined & indeterminate expressions

Video transcript
So, once again let's think of yourself as some type of ancient philosopher/mathematician, who is trying to extend mathematics as much as possible and try to make sure that you're not being lazy and leaving things undefined, when you might be able to define them. Whenever you start extending mathematics, especially in the realm of multiplication and division, there are few things that you hold dear to. You feel that if you define some type of division operation, that needs to be undone by multiplication; this is close to your heart. So you assume... You want to assume... You would like to assume that any type of division operation, if you start with some number and if you divide with a number over which... - division by that number is defined - so when I divide by some number and then multiply by that same number that this should get me this original number right over here, this should give me x right over here. And this happens when we just multiply and divide with regular numbers. If I get 3 divided by 2 times 2, that's gonna get me 3. If I say 10 divided by 5 times 5, that's going to get me 10. The other things that I want to assume... - and this is very close to my heart - I feel that any type of definitions I make have to be constant with the idea x*0 has to be 0 or any x. So these are close to my heart. I wanna extend mathematics. These two things are things that cannot be contradicted, cannot be untrue. Now, that out of the way. You wanna start exploring the divide-by-0 question. So the first thing that you say: "Well, let me just try to define it." So let's start, let's assume that I have, so this is... So let's make a further assumption... that x is some non-zero number. Let's just say, well,maybe the best way of finding out what x divided by 0 should be, how I should divide it, let's just assume there is define, and then come up with any results that there might be, there might be a resolve for. So let's say that x divided by 0 is equal to k. Well, if this is true and if we are defining what it means to divide by zero, then we are assuming that if we multiply by zero, we'll get our original number right over here. This is something that we are not willing to contradict. So let's see what happens: x divided by 0 is equal to k. On the left hand side we have a divide by zero and than multiplied by zero. Well then if two things are equal, if I do something to one thing inorder for them to stay equal, I have to do it to the other thing. This has to be equal to that. I have to multiply the left hand AND the right hand side by zero. Well, then by this assumption that I am never willing to give up, this left hand side right over here, must be equal to x. And by this assumption right over here, that I am not willing to give up, This right hand side right over here must be equal to 0. But I just hit a contradiction! I assume that x does not equal to 0, and now I am being forced to say that x=0. And I am not willing to give up the idea, I am not willing to give up either of these ideas. I am defining what it means to divide by zero. Or if I am defining what it means to divide by anything... ...that if I then multiply by that something, that I should get my original number. And I am not willing to give up the idea that anything times 0 is 0. So all of these things... The only thing that I can give up is this right over here. And I'll say, well, I guess k will have to stay undefined. This whole contradiction happened because I attempted to define what x/0 is. Now that out of the way... OK... This was a situation when x does not equal zero. But what about when x DOES equal zero. So let's think about that a little bit. And once again, I will try to define it. So I will assume... that 0 divided by 0 is equal to some number. Well once again, so let's say it is equal to k again. And so, once again... we are trying to do the same logic, so we'll write 0/0 is equal to k. Actually, let me colourcode these zeros. This will be a magenta zero and this is a blue zero right over here. And once again, I am not willing to give up the idea that if I start with a number x, I divide it by something over which division is defined, and then I multiply by that something, I should get my original x again. I can't give this up. Otherwise it doesn't seem like a good definition for the division. So what I am gonna do - I am gonna multiply the left-hand side times 0 and by this property that I am not willing to give up, the left-hand side should simplify to this magenta zero. It should simplify to this over here. But once again, anything I do to once side of the equation, inorder for the equation to hold true, I need to the other side of the equation. And these two were equal beforehand. Any operation I do to this inorder for it to still be equal , I need to do to that. So let me multiply the right-hand side by zero. So the left I get 0, I just get this magenta 0, and on the right I could just write the zero here, but I won't simplify it. I get k times 0. Well, this I see right over here... This actually is not a contradiction. This actually is true for any k, This is one of the core assumptions that I've made in my mathematics that I am not willing to give up. So this is true True for any k. It's not a contradiction. But the problem here is I wanna come up with the k, I'd like a resolve for a k. It would not be nice if this turned out to be 0. if this turned out to be one, or if this turned out to be negative one But now I see, given the assumptions right here this could be ANY... this could be absolutely any k I cannot determine what k this should be This could be a hundred thousand, this could be 75, it could be anything true for any k I cannot determine what k this should be and that's why when you get a little bit more nuance in early math people will say, well 0 divided by 0, well we don't know what that's gonna be there's no consistent answer there so we're just going to call it undefined there's no good answer that seems better than any other answer but now we see a little bit nuance here one divided by zero... you just couldn't define it it led to direct contradictions zero divided by zero... it could be anything you can't determine it and so that's why, when you do higher level math and you'll often hear this when you take a calculus course we see that zero divided by zero is indeterminate