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Thread: Zero divided by zero

20170812, 04:44 PM (ISO 8601)
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Zero divided by zero
Ok so it's been explained to me that the reason why you can't divide by zero is that the statement:
x/y=z
is equivalent to the statement
z*y=x
and if y=0 then in general there's no valid value that z can take that will make the equation consistent.
It occurs to me however that 0/0 seems like it shoukd be a special case. if y=0 AND x=0 it would seem to me that z should be the set of all numbers, as you could in fact have ANY number as z and have the equation be consistent. So what's up with that?

20170812, 05:27 PM (ISO 8601)
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Re: Zero divided by zero
NEVER DIVIDE BY ZERO!!!!! PERIOD.
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20170812, 05:35 PM (ISO 8601)
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Re: Zero divided by zero
I was pretty disappointed when my teacher said to take whatever the calculator says as the valid answer to that question.
I guess mathematicians have agreed on an answer (I think it's "don't do it!") but I feel like it's just one of these things you should decide depending on circumstances.

20170812, 06:52 PM (ISO 8601)
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Re: Zero divided by zero
AFAIK 0/0 has no meaning in mathematics, exactly because any answer could be true. I think that there is a rule saying that you can only have one result for each operation. It's possible that people who studied maths better than me (which doesn't take much) have more indepth answers.

20170812, 07:19 PM (ISO 8601)
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Re: Zero divided by zero
Numberphile has a video on the subject.
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Re: Zero divided by zero
Last edited by Grey_Wolf_c; 20170812 at 07:33 PM.
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20170812, 07:48 PM (ISO 8601)
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Last edited by Vinyadan; 20170812 at 07:49 PM.

20170812, 07:49 PM (ISO 8601)
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Re: Zero divided by zero
Huh, I was actually just talking about this earlier today.
Addition, multiplication, subtraction, division, etc, are all things that mathematicians call "binary operations on the set of all numbers." They do operations on two elements of a set and the result is an element of that same set. That last bit is actually why "set of all numbers" was specified: subtraction is not a binary operation on the set of natural numbers ^{1}; division is not a binary operation on the set of all integers ^{2}, etc. It's easy to construct sets for which these operations are not binary, but they are all binary operations on the set of all numbers.
Binary options sometimes have whats called an identity element. The identity element is a unique element where, when used as part of an operation on any other number, makes no difference. More mathematically, if the identity for addition is e, and A is a number:
A + e = A = e + A
Obviously the identity element for addition is 0. For multiplication, it's 1.
If an identity element exists, then elements of a set can have what's called an inverse. When a binary operation is done on an element and it's inverse, the result is the identity element. For addition, if A and B are numbers, and B is the inverse of A:
A + B = 0
B = A
The inverse of a number for addition is it's own negative. We can do same for multiplication:
A·B = 1
We would say B is the multiplicative inverse of A, and we'd write the inverse of A as A^{1}.
Sorry for all the preamble, down to the meat of it:
Subtraction and division are defined as "addition of the inverse" and "multiplication of the inverse" respectively. So A  B is equivalent to A + (B), and A/B is equivalent to A·B^{1}. So then, to divide by zero, we need only multiply by 0^{1}. So what is 0^{1}? Well, by the definition of an inverse, it must satisfy the following equation.
0·0^{1} = 1
If you're familiar with the multiplicative properties of 0, that last equation doesn't make sense. Anything times 0 is 0, even it's own hypothetical inverse. Yet, if 0 has an inverse, then the previous equation must hold true. Since the previous equation can't be true, 0 must not have an inverse. Since division is defined as multiplication of the inverse, dividing by 0 isn't an idea that makes any sense.
When you divide by 0, it's not just the result that is not defined: the operation itself isn't defined. It doesn't make any sense, it's "ungrammatical" to even write out: something like 1/0 it seems like it points to something meaningful, but when you break it down to it's constituent parts, it's gibberish.
The same is true for 0/0: the operation you're trying to articulate doesn't make any sense. You could define an operation where it would, but that operation would necessarily be different than the operation we understand to be division.
Originally Posted by Bohandas
x/y=z (1) x·y^{1} = z By the definition of division (2) (x·y^{1)}·y = z·y (3) x·(y^{1}·y) = z·y by association (4) x·1= z·y by the definition of the inverse (5) x= z·y (6)
(If you're being rigorous, even trivial math becomes a pain)
So, yes, your two statements are equivalent: if (1) is true, then so must be (5), and vice versa. The problem if y = 0 happens in (2): you can't have an inverse for 0; 0^{1} doesn't exist. If you ignore that then you can move onto (3). If y = 0, then from (3), you can go in 2 ways, either to (3.2) or (3.3) depending on which property we're willing to recognize:
(x·0^{1)}·0 = z·0 (3.1) x·1 = z·0 so x must equal 0 then, and z could be anything (3.2) 0 = z·0 x is eliminated entirely, and z could still be anything (3.3)
In both (3.2) and (3.3), z could indeed be anything, which is sort of interesting: (2) is a nonsense step if y = 0, but if we ignore that we'll end up at consistent results no matter what.
But the fact remains that the whole exercise is silly in the first place: division by 0 is fundamentally an undefined operation, and anything based on it is vacuous nonsense, which can be seen if you introduce limits: the limit does not exist no matter how you do it.^{3}
————
Now, in practice, 0/0 is somewhat different than 1/0. When you're doing math, and you get a result of 0/0 (and didn't make any mistakes), there's often way of approaching the problem and getting a meaningful result, either by doing some algebra to eliminate the offending factors, or taking the limit, or what have you. The same is true less often for something like 1/0: usually the best you can do is take the limit and say that it tends to ± ∞, which is still meaningful in some way, but technically undefined.
————
1: since 1  2 = 1, the result is not a natural number, thus subtraction is not a binary operation on the set of natural numbers.
2: since 1/10 = 0.1, the result is not an integer, and thus division is not a binary operation on the set of
3: the limit does exist for x/x: lim_{x>0} x/x = 1. It sorta looks like the same should hold true for lim_{x,y > 0,0 } x/y, but it not, largely because of imaginary/complex numbers. I know that my explanation of limits are vague, not really an explaining anything, but this thing is already long enough. I can't go into more detail if someone wants me to, but I need to cut this off now before it grows by another 1000 words.
ETA: to avoid double posting:
No, Vinyadin is right. An operation must have only 1 result, else it isn't an operation. (x^{^2} = 4) isn't an operation, it's a statement that implies one of two other statements (either x = 2 or x = 2).
Polynomials, as statements, can imply as many unique statements as the highest order term i.e. x^{3} = 3 implies one of 3 possible statements: one result on the real axis and 2 on the complex plane. 4x^{3} = 0 is somewhat of a bad example, since it really does only imply 1 unique statement: x = 0.Last edited by crayzz; 20170812 at 08:10 PM.
Originally Posted by crayzzOriginally Posted by jere7my

20170812, 08:06 PM (ISO 8601)
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Re: Zero divided by zero
The pure mathematician answer is "Anything you want, well sort of". The personwhodoesrealmathsintherealworld answer is "What are you doing no stop go away that's not a real calculation it doesn't mean anything just leave."
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20170812, 08:32 PM (ISO 8601)
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Re: Zero divided by zero
In contradiction to everyone else in this thread, I am going to assert that "0/0" has a meaning. Specifically, it means that there is more going on in your system than what you originally assumed.
To explain, I will have to utilize limits. A limit, in math, is the answer to the question "as x approaches the listed value, what does f(x) do". For continuous functions, the limit is simply the function evaluated at the listed value. However, not all functions are continuous. Take, for instance, the step function. When the input to a step function is below zero, the function evaluates to 0. When the input is greater than zeros, the function evaluates to 1. Depending on who you ask, a step function given 0 either evaluates to 0, or 1, or 1/2. However, when we take the limit from the positive direction at 0 of a step function, we get that the limit is 1. When we take the limit from the negative direction at 0 of the step function, we get that the limit is 0. Graphically, you just keep the function smooth to the point you care about from the direction you care about.
Now, let us go back to our question of what to do with "0/0". If we are analyzing a function, we can apply the limit to it. If the function is a rational polynomial (meaning it is a polynomial divided by another polynomial), then if the 2 polynomials are both 0 at the same place, we know that they have a common factor. Sometimes, however, polynomials are not easily factored (or we just don't feel like it), so we can instead compare their derivatives. (A derivative is an analytical function to explain the slope of another function). Since taking the derivatives of polynomials is relatively easy, we can save significant effort by using this method (called L'Hopital's Rule after the guy who paid the guy who came up with it).
Lots of functions are defined in such a way that they end up with "0/0". There in the sinc function (sin(x)/x) and I think the J bessel functions (these I can never remember). Because of how they are defined, they end up having a value at the "0/0" points, though special programming has to be added to computers to calculate it. When programmers forget to include that special programming, it tends to make their code not work very well.

20170812, 09:16 PM (ISO 8601)
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Re: Zero divided by zero
0/0, by itself, is indeterminate: it could take any value.
However, obtaining 0/0 as the result of a calculation could mean there is a definite answer (or no answer), since there is more information (the nature of the calculation).

20170812, 09:44 PM (ISO 8601)
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20170812, 10:12 PM (ISO 8601)
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Re: Zero divided by zero
0 is stranger than that. One way you can define division is something like, for A/B, "how many times must I subtract B from A to reach 0" (this is mentioned in the Numberphile video Grey Wolf linked). In other words:
If A/B = C, then A  B·C = 0
So what happens for A = B = 0? Well,
0/0 = C, so 0  C·0 = 0
There are infinite solutions for C. You write "How many zeroes does zero contain? One zero." And that's true, 0  1·0 = 0. But zero also contains no zeros: 0  0·0 = 0. It also contains 6.3 zeros, and even negative zeros: 0  6.3·0 = 0; 0  (1)·0 = 0. Hell, it also contains imaginary zeros. "How many zeros does zero contain" doesn't have a coherent answer. Which makes it really strange, since it's the only number to not have an answer to that question.Originally Posted by crayzzOriginally Posted by jere7my

20170812, 10:20 PM (ISO 8601)
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Re: Zero divided by zero
Zero divided by zero exposes information about the generating process of those zeros. So it basically says 'more information required'.
For example, if I take lim x>0 sin(x)/x, that's 1. But if I take lim x>0 sin(2*x)/x, it's 2. Both become 0/0 at that limit however.
One way you can see what's going on formally is to use something called dual numbers, which is an extension of real numbers that (among other things) can answer questions like 0/0, the same way that complex numbers allows you represent things like x in x^2 = 1
The main idea of dual numbers is that you add 'a tiny bit' to each number, which represents the derivative of that number with respect to the process by which the number was obtained. So that gives us a number z = x + eps*y, where 'eps' here is like the 'i' in complex numbers. The 'x' bits behave like real numbers, and everything just adds and subtracts normally, but 'eps' has a particular multiplication rule: eps*eps = 0.
We should then extend all the mathematical functions to return the proper 'eps' bits, which are just their derivative at that point. So for example: sin(x+eps*y) = sin(x) + eps*cos(x). But also we have to keep in mind that where we see 'x' alone if we're only using real numbers, it should become 'x + eps*1' since the derivative of x is 1.
Now, if we compute lim z>0 sin(z)/z, its no longer ambiguous. Writing that out in terms of x+y*eps form, it becomes: lim x> 0 (sin(x) + eps*cos(x))/(x + eps) = eps/eps = 1
If we have a situation where both the x and y component of the dual number in the numerator and denominator are both zero, we need to extend further to second, third, etc derivatives.

20170813, 12:54 AM (ISO 8601)
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Re: Zero divided by zero
Why do people always seem to involve trigonometry to explain this? I prefer to go with this explanation: we all agree that x/x would be 1, right? And that 2x/x would be 2. But if x is 0, in both cases you end up with 0/0 = 1 or 0/0 = 2. You can use the same logic to prove 0/0 is *any* number.

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Re: Zero divided by zero

20170813, 01:53 AM (ISO 8601)
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Re: Zero divided by zero
The reason I chose that is that it avoids the 'well, thats just because you didn't simplify the expression and divide out the x's' reading, which seems like a likely confounding factor to understanding. But if you use sin(x)/x then it doesn't simplify like that, so it feels more to the point.

20170813, 03:21 AM (ISO 8601)
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Re: Zero divided by zero
The truth is, 0 is just something that humanity doesn't fully understand yet. There are all kinds of mathematical things you can to do zero, and some of them make perfect sense, some make no sense, some make sense, but have infinite answers, and we don't know what to do with it all.
It's hard to think about with a realworld analog. How do you do stuff with absolutely nothing?
In the real world, having nothing, means either you can't do anything, (Bake an apple pie with x apples, x=0. That results in the inability to bake any pies) or it means you're actually doing something to something else. (There are x doors in your way, x=0. Time to invade the castle!)
But in math, it's a number just like any other, and you can do math to it. Even negative numbers once felt like this to humanity. ("How can you possibly have 5 cups of tea?!" said a skeptical guy right after negative numbers were thought up.) But now we understand them, and can do math to them like anything else.
The same happened with imaginary numbers. ("You can't take the square root of 1! A number times itself gives a positive number!" said every mathematician at the time.) but then we thought up the idea of imaginary numbers, and decided i^2=1. And now, because of that, we have the Mandelbrot Set.
Even irrational numbers were something we thought didn't make any sense back in Pythagoras's time. And look at all we can do now, with pi, and e, and the square root of 2.
But unlike negative, imaginary, and irrational numbers, we're still "figuring out" zero. Like we're still figuring out the placement of prime numbers. Like we're still figuring out if the Riemann hypothesis is true. Like we're still figuring out if the number of real numbers is aleph one, or if it's more than that.
One day, some guy or girl will be born somewhere, who'll grow up to be the next Ramanujan, and he/she will think really hard one day about the nature of zero, and have an epiphany about how to use it, and we'll all figure out just what 0/0 really MEANS, and that will lead to some kind of new awesome thing, that we, as of this moment, don't even know exists.
Until then, all we can do is put up the mathematical equivalent of "Danger: Sharp edges, do not touch" sighs around it.

20170813, 11:55 AM (ISO 8601)
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Re: Zero divided by zero
Important qualifier: you have to take lim(x>0) x/x and lim(x>0) 2x/x, as NichG did with sin(x)/x. For all the reasons we've discussed, x/x does not take on a definite value for x=0.
People use sin(x)/x because it has additional value beyond simply demonstrating the indeterminacy of 0/0. It's one of the classic examples used to illustrate the squeeze theorem, which finds a limit that isn't amenable to direct calculation by proving that the function must lie between two other functions which converge to the same point. (The most classic example of that is determining the value of pi by squeezing a circle between circumscribed and inscribed polygons with increasing numbers of sides.)

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Re: Zero divided by zero
There's often cases where it's useful for whatever would be evaluated as '0/0' to be something depending on context (particularly when there's real world consequences).
But when it needs to be done rigorously it's phrased very carefully.

20170814, 09:19 AM (ISO 8601)
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Re: Zero divided by zero
You are just touching on the beginnings of a large branch of advanced mathematics called "limit theory". Much of calculus is based on, not dividing by zero, but watching what happens when you get closer and closer to zero.
For instance consider the following question:
What is (2x^2 + 3x)/x, as x approaches zero?
Consider x = 1: x^2/ = 1, so (2*1 + 3*1) /1 = 5/1 = 5
Consider x = 0.1: x^2/ = 0.01, so (2*0.01 + 3*0.1)/0.1 = 0.32 / 0.1 = 3.2
Consider x = 0.01: x^2/ = 0.0001, so (2*0.0001 + 3*0.01)/0.01 = 0.0302 / 0.01 = 3.02
Consider x = 0.000001: x^2/ = 0.000000000001, so (2*0.000000000001 + 3*0.000001)/0.000001
= 0.000003000002 / 0.000001 = 3.000002
X can never reach zero, or the division is undefined. But it's clear to see that the limit as x approaches 0 is 3.
There's a lot of very deep and important math here, and it starts with calculus.

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Re: Zero divided by zero

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Re: Zero divided by zero
When the expression is defined, yes. But only when the expression is defined.
The truth will make more sense to you if you go back to the original, elementary school definition of division.
For X / Y, start with X things. Take Y of them away at a time until you can't take away a set of Y anymore.
So for 6 / 2, you start with 6 things.
Take 2 things away, and there are 4 left.
Can you take 2 things away again? Yes, so do it a second time, and you have 2 left.
Can you take 2 things away again? Yes, so do it a third time, and you have 0 left.
Can you take 2 things away again? No. Since you did it 3 times, 6 / 2 = 3.
Now with 0 / 0, you start with 0 things, and you take 0 things away.
Can you take 0 things away again? Yes, so do it again.
Can you take 0 things away again? Yes, so do it again.
Can you take 0 things away again? Yes, so do it again.
Can you take 0 things away again? Yes, so do it again.
Can you take 0 things away again? Yes, so do it again.
...
Dividing by zero is undefined, by which I mean straightforwardly that the definition of what we call "dividing" can't apply to doing it by zero.
Yes, you can come up with clever tricks like the above to pretend that you divided by zero, but the real answer, if there was one, would be the unique number of zeroes that add up to that number, and that doesn't mean anything. It's undefined  just as undefined as asking how many pounds an idea weighs, or which direction you take to get from Hobbiton to Never Land.Last edited by Jay R; 20170814 at 11:25 AM.

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Re: Zero divided by zero
In the fundamentals of algebra, numbers are an artificial way of communicating informations. The binary internal operators that we normally use and that normally we apply to the group of real numbers, are only two: one additive operator (+) and one multiplicative operator (•).
A division like x/y is just a comfortable way to write
x • y^1, where "y^1" stands for the inverse of y, another artificial construct that helps to tie our calculations together. In the end, you can always look at 0/0 as 0 • 1/0, which is not a real (doesn't belomg to the real camp) number, because by definition of null element, 0 • x will always be 0, at the same time, 1/0 can't be represented among the real numbers, and since (•) is a binary internal operation, which means that the two elements with which it operates have to both belong to the same group, this means that 0/0 can't be displayed using the common gropus of numbers that we know (N Z Q R C).
I guess that this problem can be discussed in many ways that all are correct, I wanted to approach from the basics of algebra, which to me is like triyng to correct a typo with knowledge of grammar. Similarly to languages, maths have its laws, grammar disposes of language and algebra disposes of maths, we can see numbers like words and groups of numbers like languages: "gissjb" may mean nothing in your language, but there may be a language, maybe a forgotten one, or one that will be invented, for which that word will have a meaning, similarly, 0/0 has no meaning among the common groups of numbers, but there may be a group of numbers that has been discovered (that I'm not aware of) or that will be invented for which 0/0 has a meaning.English isn't my first language, so I will likely express myself poorly.
Please assume that I'm arguing in good faith, and that I mean no offense to anybody.

20170814, 12:06 PM (ISO 8601)
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Re: Zero divided by zero
Not exactly on topic, but as someone who had to learn matrix algebra last semester for a statistics class, hearing about inverses and identity <things> like mentioned here is helpful to understand what I saw there. It's neat to see how subtracting is adding the inverse, division multiplying the inverse, etc., and makes sense with taking the inverse of a matrix.
Thanks y'all for giving some mathematical theory background.

20170814, 02:09 PM (ISO 8601)
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Re: Zero divided by zero
(I know I'm being a bit extra with my explanation of back ground stuff: it's not because I think you probably don't know, I just feel I can explain things better this way. It's mostly to explain my reasoning thoroughly and make sure we're on the same page.)
In math and logic, there's what we call "implication": If proposition/statement A being true means proposition/statement B must also be true, we say that A implies B, usually denoted with an arrow: A ⇒ B. For example, (4x = 2) ⇒ (x = 1/2). We say that x = 1/2 is a solution to the original statement: that just means it's a value of x where (4x = 2) is true.
Nonlinear equations, as statements, don't always have straight forward implications. (x^{2} = 1) does not imply (x = 1). There are two solutions to the original statement: x = 1, and x = 1. In terms of implications, we'd write (x^{2} = 1) ⇒ ((x = 1) OR (x = 1)). We don't actually know what x is, but we can narrow it down. That we don't actually know what x is is important.
Now, if I'm reading you correctly, you're trying to say that (z = 0/0) ⇒ (z ϵ C)^{1}: that the "solution space"^{2} for z is every single possible number. But that really doesn't mean anything. I could, for every single mathematical statement, write the same thing. (z = 15.34) ⇒ (z ϵ C): 15.34 is in the complex set. It tells us nothing about what z actually is. It's meaningless.
Take a look at a simpler example that a few people threw around: z = x/x at x = 0. It's been mentioned a bunch of times, but the solution here is z = 1. That is to say ((z = x/x) AND (x = 0)) ⇒ (z = 1). There's a reason for this, and it is not because we're actually doing any division: we're using limits, as previously explained.^{3} We write out lim_{x→0} x/x to show we're doing limits, and do some rearranging:
lim_{x→0} x/x (1) = lim_{x→0} 1 (2)
In this case, the limit as x approaches 0 is one, no matter how we approach. And so we can say z = 1. Now, again, this isn't a contradiction of (z ϵ C): 1 is a part of the complex set. But it does massively reduce the solution space: z can't be just anything anymore, it MUST be 1.
You can do the same thing for z = 2x/x as factotum explained:
lim_{x→0} 2x/x (3) = lim_{x→0} 2 (4)
And so now z MUST be equal to 2. You can start with different conditions and make z equal anything. And I know that sounds like I'm agreeing with what you're arguing, but I'm not. At no point do I actually work with a statement like z = 0/0. And I wouldn't be able to: remember that, through this method, we get a definitive answer: 1 specific number. If we accept reasoning like z = x/x = 0/0 = lim_{x→0} x/x = 1, we get contradictory answers. Using this reasoning, I can make z = 1 and 2 simultaneously. In more rigorous terms, (z = 0/0) ⇒ ((z = 1) AND (z = 2)), but ((z = 1) AND (z = 2)) is necessarily a false statement. A true proposition cannot imply a false one.
Tl;dr
Even if we accept your reasoning, the idea that z could be anything is pretty meaningless, mathematically speaking. And if we take your reasoning a bit further, we end up with contradictions, meaning some initial premise (i.e. that z = 0/0 in the first place) must be false. And this really isn't surprising, since 0/0 is mathematically meaningless to begin with (though, again, a result of 0/0 might mean something to you personally, in how you approach the problem to avoid that result in the first place).
————
1: (z ϵ C) is notation that tells you what set z is an element of the set C. C is the usual notation for the complex numbers, which is any number that can be expressed as a + b·i, which includes all real numbers. I think there are numbers outside this set: at least I vaguely remember reading about this at some point, but for most purposes C includes every number you're ever going to come into contact with.
2: I don't know if "solution space" is technical terminology basically means the set of values for which the original statement is true. Notation wise, we'd write {z: z = 0/0}, which we'd read as "the set of all values z for which (z = 0/0) is true." I think you're asserting {z: z = 0/0} = C: that the set for which z = 0/0 is true is equivalent to C.
3: Rockphed already explained this, but: a limit is a value towards which a function tends at some point, <i>from both the positive and negative direction</i>. That's important, because sometimes functions are defined at some point, i.e. they return a value, but they approach two different values depending on which direction you approach from.
For example, look at the floor function. If you approach x = 0 from the negative direction, y tends to 1. If you approach x = 0 from the positive direction, y = 0. The limit is different depending on your approach, and thus the limit does not exist. Something like y = x trivially has a limit: no matter how you approach x = 0, y always tends to 0. And thus the limit exists. It's important to remember that, while a function may be defined at some point without a limit, a function with a limit (that approaches the same value at the same point from every direction, including the complex plane) must be defined. This is how using limits gets us around something like 0/0: we can't actually do the operation as written, but if we can show that whatever is giving us 0/0 approaches the same value regardless of direction, that's enough to show what the function evaluates too at that point.
———
Originally Posted by JeenLeenOriginally Posted by crayzzOriginally Posted by jere7my

20170814, 03:05 PM (ISO 8601)
 Join Date
 Jan 2007
Re: Zero divided by zero
Concluded: The Stick Awards II: Second Edition
Ongoing: OOTS by Page Count
Coming Soon: OOTS by Final Post Count II: The Post Counts Always Chart Twice
Coming Later: The Stick Awards III: The Search for More Votes
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No matter how subtle the wizard, a knife between the shoulder blades will seriously cramp his style  Jhereg Proverb