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.

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