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Thread: Zero divided by zero

20170815, 08:25 AM (ISO 8601)
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Re: Zero divided by zero
Makes sense. My desire for mathematical precision makes me want to modify that to, "You can't divide by zero (for all the reasons listed in this thread)... But you can get close. Let's examine what happens when we get really, really close."
But your general approach is a good one.Last edited by Jay R; 20170815 at 08:25 AM.

20170815, 02:22 PM (ISO 8601)
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Re: Zero divided by zero
Fair enuf on the modification. But my counterargument would be: When one is literally an infinitesimal away from dividing by zero, that's when I'm inclined to say, "Good enough for government work" and call it a day.
Just a philosophical difference in approach, I reckon.

As an aside, getting back to the 'one gets really useful information' bit. The square root of negative one is, by definition, something that can't exist in the real number set. Yet the use of complex numbers (which when it comes right down to it is nothing more than "Sure, but what if we could have something that when multiplied with itself gave a negative number. How do we do that and then what happens next?") gives some pretty elegant solutions to, ahem, complex engineering problems.
And that's just one of their uses that I'm aware of. Looking about, I see that complex numbers are used quite a bit in the sciences.
You'd be able to give many more concrete examples, of course. But as a layman I am amused that things like this have tremendous application in Real World situations.Concluded: The Stick Awards II: Second Edition
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20170815, 02:49 PM (ISO 8601)
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Re: Zero divided by zero
For me, it's a professional difference. Stating the precise truth is crucial. "Good enough for government work" won't distinguish me from the competition.
Yes. And the square root of 2 is, by definition, something that can't exist in the rational number set.
And one half is, by definition, something that can't exist in the integer number set.
And 3 is, by definition, something that can't exist in the natural number set.
It's just a different kind of number, from a different kind of number set.
Not really. Complex numbers fall directly and logically from the roots of a polynomial, or simply from the idea of a number plane rather than a number line
They do. But they have applications because they actually describe meaningful relationships.

20170815, 06:11 PM (ISO 8601)
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Re: Zero divided by zero
Historically they were ignored when the roots of a quadratic polynomial. It was when you/Cardano had to ignore/use them to get the (then acceptably real) roots of cubic equations that they became considered. Whereas the number plane is a very late development, although a much prettier starting point (using the roots of a polynomial, I would think is a form of begging the question).
Negative numbers similarly were considered fictionary, absurd (in fact even Cardano, ignored them, which means he didn't actually use his solution when it had complex numbers)
Whereas root 2, was assumed to exist (and hence be a rational), so has a slightly different story.
Not sure about fractions, it's been suggested that it's one reason the Egyptions only used reciprocals (and I guess the idea of multiply by a half as supposed to dividing it by 2 is subtly different).

20170815, 08:56 PM (ISO 8601)
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Re: Zero divided by zero
Oh my. There's some decent mathematics here, and some not so decent.
I like crayzz's first post a good bit. Like, it went in the direction I was thinking. It has to do with groups, rings, fields, and so on. Which leads into cool things like infinites, infinitesimals, cardinals, ordinals, hyperreals, surreals, and so on.
Specifically, when you have enough structure to talk about addition and multiplication (hence subtraction and division), usually we use the symbol 0 for additive identity, and 1 for multiplicative identity (even if we aren't talking about numbers). And usually they (that is, the two identities) are assumed to be different. But there's nothing in the axioms that demand this. So you can totally set up a trivial situation where 0/0=1.
I'd also suggest reading about the extended complex plane. Also, somewhat related to this, you can easily define arithmetic with division by 0 and stuff involving infinity.

20170816, 03:52 PM (ISO 8601)
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Re: Zero divided by zero

20170816, 06:31 PM (ISO 8601)
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20170817, 12:35 AM (ISO 8601)
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Re: Zero divided by zero
The real and complex number systems don't admit infinitesimals, you need the hyperreals for that.
In normal math compare this
1(1x10^1)=0.9
1(1x10^2)=0.99
1(1x10^3)=0.999
1(1x10^4)=0.9999
...
1(1x10^infinity)=0.9999...
10.9999=(1x10^infinity)
0.9999...=1
10.9999...=0
(1x10^infinity)=0Last edited by Bohandas; 20170817 at 12:36 AM.

20170817, 12:53 AM (ISO 8601)
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Re: Zero divided by zero
That's not 'normal math' (by which I suppose you mean the reals), because you've used infinity directly in an expression. What you've written would hold for the extended real line (the reals plus points for positive and negative infinity). But there are no infinitesimals in that set.
More broadly, there is a small but significant difference between not admitting infinitesimals and admitting infinitesimals that are exactly equal to 0, and as far as I know there isn't a structure that does the latter. After all, the point of infinitesimals is that they are infinitely small nonzero quantities.Last edited by Lethologica; 20170817 at 12:55 AM.

20170817, 09:42 AM (ISO 8601)
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Re: Zero divided by zero
The point I was trying to demonstrate with that math there is that it treats the difference between 1 and 0.9999..... as exactly zero despite the fact that there's clearly an infinitesimal between them

20170817, 04:02 PM (ISO 8601)
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Re: Zero divided by zero

20170817, 07:58 PM (ISO 8601)
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Re: Zero divided by zero

20170817, 08:41 PM (ISO 8601)
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20170818, 12:40 AM (ISO 8601)
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20170818, 12:53 AM (ISO 8601)
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Re: Zero divided by zero

20170818, 06:21 AM (ISO 8601)
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Re: Zero divided by zero
1 is a real number. 0.999... is a real number.
For any two real numbers a and b, exactly one of three things is true: a<b, a=b, or a>b. In this case, 1=0.999...
This has nothing to do with infinitesimals. You did not show anything
Sorry if I'm coming off as rude.
EDIT: Here's a video demonstrating many proofs of the fact that 1=0.999...Last edited by danzibr; 20170818 at 06:40 AM.

20170818, 06:54 AM (ISO 8601)
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20170818, 08:00 AM (ISO 8601)
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Re: Zero divided by zero
In numerical physics there are many kinds of finite difference approximations of spatial gradients, which are multivariable and directional in nature. For example, one issue is how to construct rotationally invariant second order infinitesimals on square lattices, which comes up in any simulation of anything on a grid. Depending on the area in physics (I'm thinking GR specifically), you can end up dealing with 'differential geometry', where you're concerned with the relationship between infinitessimal motions on surfaces.
If we go to stochastic differential equations, there's a concept of infinitesimal noise called a Wiener process. It behaves differently from just 'dx' and is necessary for getting the same results at different step sizes. The simplest form of this is just realizing that noise terms should scale as sqrt(dt) rather than dt, but if you have structured noise then you need to carefully derive a correct finite difference approximation to get convergence as dt>0.
In deep learning, there's a lot of work in finding efficient ways to take derivatives of very complex functions which might be defined procedurally, involve control flow, etc. To that extent things like dual numbers are a generalization of 'dx' that lets you do backprop with only a forward pass. There's similarly the question of how to efficiently calculate second derivative dot products without calculating the full Hessian matrix, which is huge for deep learning problems due to large parameter count. Granted, these methods are less common than making a compute graph and doing chain rule, though PyTorch has some kind of black magic to have zero compile time for gradients even for crazy tangled recurrent models that would take 5 minutes to compile in TensorFlow. I don't think PyTorch actually uses dual numbers in particular (there are problems with doing it that way if you want to take the gradient of a large number of variables). In another deep learning area, you have the reparameterization trick which uses noise to create differentiable extensions of otherwise nondifferentiable functions  not sure that would count as a different kind of infinitessimal or not.
Anyhow, I'm not sure what your standards are for being 'different than dx', but it feels a bit like saying that complex numbers can't be too useful because there's just the one 'i'...Last edited by NichG; 20170818 at 08:10 AM.

20170818, 01:59 PM (ISO 8601)
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Re: Zero divided by zero

20170818, 04:49 PM (ISO 8601)
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Re: Zero divided by zero

20170818, 06:09 PM (ISO 8601)
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20170818, 06:36 PM (ISO 8601)
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Re: Zero divided by zero
Sure it does (refer unambiguously to a single value, that is). It does so in the reals, which are a subset of the hyperreals, so why wouldn't it in the hyperreals? That'd be like saying 1 is unambiguous when you're viewing it as a natural number, but ambiguous when you're viewing it as a real number.

20170818, 08:06 PM (ISO 8601)
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Re: Zero divided by zero
Upon further reading, you are correct. I was being misled by a poorly constructed source.