# Thread: Zero divided by zero

1. ## Re: Zero divided by zero

Originally Posted by Porthos
The layman's/haven't studied calculus in 20 years way I like to put it:

You can't divide by zero (for all the reasons listed in this thread)... But what if you acted like you could (in some situations)?

Turns out, one gets a LOT of useful information.
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.

2. ## Re: Zero divided by zero

Originally Posted by Jay R
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.
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.

3. ## Re: Zero divided by zero

Originally Posted by Porthos
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.
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.

Originally Posted by Porthos
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.
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.

Originally Posted by Porthos
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.
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

Originally Posted by Porthos
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.
They do. But they have applications because they actually describe meaningful relationships.

4. ## Re: Zero divided by zero

Originally Posted by Jay R
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
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).

5. ## 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.

6. ## Re: Zero divided by zero

Originally Posted by Porthos
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.
Especially since many mathematical systems treat infinitesimals as being exactly zero

7. ## Re: Zero divided by zero

Originally Posted by Bohandas
Especially since many mathematical systems treat infinitesimals as being exactly zero
Huh? Like what?

8. ## 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...

1-0.9999=(1x10^-infinity)

0.9999...=1

1-0.9999...=0

(1x10^-infinity)=0

9. ## 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.

10. ## 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

11. ## Re: Zero divided by zero

Originally Posted by Lethologica
infinitesimals that are exactly equal to 0
There is no such thing, much as a round square or a 3-sided triangle.
Originally Posted by Bohandas
despite the fact that there's clearly an infinitesimal between them
Clearly?

There's no difference at all between them, like 6/3 and 2 and 1.999...

12. ## Re: Zero divided by zero

Originally Posted by danzibr
There is no such thing, much as a round square or a 3-sided triangle.

Clearly?

There's no difference at all between them, like 6/3 and 2 and 1.999...
I just showed the sequence where 1-an infinitesimal = 0.9999...

13. ## Re: Zero divided by zero

Originally Posted by Bohandas
I just showed the sequence where 1-an infinitesimal = 0.9999...
I remain unconvinced that we need them aside from dx. They seem to have incredibly limited utility.

14. ## Re: Zero divided by zero

Originally Posted by Rockphed
I remain unconvinced that we need them aside from dx. They seem to have incredibly limited utility.
The fields of numerical physics and deep learning beg to differ.

15. ## Re: Zero divided by zero

Originally Posted by NichG
The fields of numerical physics and deep learning beg to differ.
How are they different from the aforementioned "dx" (by which I mean an elementary difference in a variable)? What are they used for in those fields? This thread is literally the first I have ever heard of them. Going by the description given, I don't see their utility.

16. ## Re: Zero divided by zero

Originally Posted by Bohandas
I just showed the sequence where 1-an infinitesimal = 0.9999...
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...

17. ## Re: Zero divided by zero

Assume x = y

xx = xy

xx - yy = xy - yy

(x + y)(x - y) = y(x - y)

x + y = y

y + y = y

2y = y

2 = 1

And that is why we can't divide by 0.

18. ## Re: Zero divided by zero

Originally Posted by Rockphed
How are they different from the aforementioned "dx" (by which I mean an elementary difference in a variable)? What are they used for in those fields? This thread is literally the first I have ever heard of them. Going by the description given, I don't see their utility.
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 re-parameterization trick which uses noise to create differentiable extensions of otherwise non-differentiable 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'...

19. ## Re: Zero divided by zero

Originally Posted by danzibr
1 is a real number. 0.999... is a real number.
Bohandas is starting from a different premise here. Naturally you arrive at different answers.

I do dislike Bohandas' wording--he is being exceptionally imprecise by making statements about numbers without defining the number system he's working in.

20. ## Re: Zero divided by zero

Originally Posted by Lethologica
Bohandas is starting from a different premise here. Naturally you arrive at different answers.

I do dislike Bohandas' wording--he is being exceptionally imprecise by making statements about numbers without defining the number system he's working in.
... What?

The only way you can have 1 not equal to .9 repeating is to have a nonstandard definition of .9 repeating.

It's not a matter of which number system you're working in. "1=.999... is true for reals, but not for hyperreals" is nonsense.

21. ## Re: Zero divided by zero

Originally Posted by danzibr
... What?

The only way you can have 1 not equal to .9 repeating is to have a nonstandard definition of .9 repeating.

It's not a matter of which number system you're working in. "1=.999... is true for reals, but not for hyperreals" is nonsense.
That's true. 0.999... is not a clear expression in the hyperreals to begin with. It doesn't refer unambiguously to a single number, as there are many numbers that differ from 1 by an infinitesimal (as many as there are infinitesimals).

22. ## Re: Zero divided by zero

Originally Posted by Lethologica
That's true. 0.999... is not a clear expression in the hyperreals to begin with. It doesn't refer unambiguously to a single number, as there are many numbers that differ from 1 by an infinitesimal (as many as there are infinitesimals).
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.

23. ## Re: Zero divided by zero

Upon further reading, you are correct. I was being misled by a poorly constructed source.

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