Undefined and indeterminate Why 0/0 is considered to be indeterminate
Undefined and indeterminate
- 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't 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 sat that zero divided by zero is indeterminate
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