# Thread: Zero divided by zero

1. ## Re: Zero divided by zero

Originally Posted by DavidSh
This could be fun. I think there was already a proof that the only ring for which 1/0 is defined is the ring with only a single element, but let's see how far we can get.

Do you add Peeleean (SUPER-complex) numbers in the natural way, as if they were 2-dimensional vectors?

How do you multiply Peeleean numbers?
Peeleean numbers are algebraically similar to imaginary numbers in most respects. With the most notable exception* being that exponents do not alter its identity.

*That is, the only one I can think of at the moment.

2. ## Re: Zero divided by zero

Originally Posted by Peelee
Peeleean numbers are algebraically similar to imaginary numbers in most respects. With the most notable exception* being that exponents do not alter its identity.

*That is, the only one I can think of at the moment.
Should I understand you that P * P = P ?

3. ## Re: Zero divided by zero

Originally Posted by DavidSh
Should I understand you that P * P = P ?
Well, the very bad math seems to work out there, so sure.

P is elegant in its simplicity.

4. ## Re: Zero divided by zero

Originally Posted by georgie_leech
I think it's more along the lines of how you can't have a physical square with a side length of i units.
True, true. I'd have a similar complaint for negatives, though.
Originally Posted by Peelee
Devil's advocate: I propose a Peeleean set that contains one number, 1/0, expressed as P. Thus, any number that is divived by zero can be expressed as being multiplied by P, and is now valid (and called a SUPER-complex number, with the standard form being a + bP).

Better?
Heeeeeeey that is interesting. Like how i:=(-1)^.5, p:=1/0. Gee, I wonder what we can do with this.
Spoiler: Rambling

Well, first, 0*p=1, right? Like, that's how we're defining it. I bet this makes a*p=p for any non-zero real number a (or maybe *any* real?). Fix a real number a, put on the old spectacles for p=0/1, multiply both sides by a, a*p=a*0/1, a*p=0/1=p, so yeah, a*p=p. So really, a/0=p for any real a.

I'll take the usual properties of + and * for granted.

p-p=p, I reckon. Yeah, p-p=1*p+-1*p=(1-1)*p=0*p=p.

How about... (a+bp)(c+dp)=ac+(b+d)p+bdp^2=ac+p+p^2, where p^2=(1/0)(1/0)=1/0=p. So p^2=p. So (a+bp)(c+dp)=ac+p. Huh, it just gobbles stuff up. What you'd expect of something which you can think of as infinity, I suppose.

Also, related to p^2=p, I'd wager p^a=p for any a>0. Intuitive enough.

For negative exponents, p^(-1)=1/p=1/(1/0)=0/1=0. Likewise, p^a=0 for any a<0.

Huh, I don't know how to define p^0. p^0=(0^0/1^0), but 0^0 is undefined. Or I guess you could try (for some fixed a>0) p^0=p^(a-a)=p^a*p^(-a)=p*0=p. Looks like p^0=p. Gee, that's weird. Well, maybe not.

Actually... fix a>0, p^(-a)=p^(a-2a)=p^(a)*p^(-2a)=p*0=p. Looks like p^negative reals could be either p or 0, so it's not well-defined.

Uh-oh, maybe p^a is not well-defined for non-negative a also. Well, actually, we're probably fine, for p^a you simply can't meaningfully rewrite it as p^(2a)*p^(-a). That'd be like writing 2=2/0*0. lolwhoops, j/k :P

Waaaaaaait a minute. I was about to contemplate "rationalizing" these numbers, like (a+bp)/(c+dp), but if p^negative real is not well-defined, you really shouldn't have p's in any denominators.

Waaaaaaaaaaaait another minute. My p^0 argument actually doesn't work. Guess we'll have to settle for p^a=p for a>0, and otherwise is undefined. And a*p=p for any real a. And division by p is not well-defined. This reminds me a little bit of uncountable ordinals. ... Just a little.

I wonder more about a+p. a+p=a*0/0+p=0*a*p+p=p+p=p. Yup, gobbles up numbers in this manner too.

So really... it's like you have numbers without p, and then p, since p just gobbles everything up.

I don't think (maybe?) you can prove 1=2 though, or something of the like. You can't divide by p, so a*p/p isn't going anywhere. Likewise, a+p-p isn't going anywhere, since a is not equal to a+p-p since p-p=p.

Yeah, that's what I'll settle for. For these Peeleean numbers a+bp, it reduces down to good ol' real numbers, then p. You could probably define some equivalence relation, and get two equivalence classes, real numbers, then p. I guess I could write a summary.

(this is assuming the usual + and * properties)
1) a+b*p=p for any real numbers a and b.
2) p^a=p for any positive real a.
3) p^a is not well-defined otherwise.
4) Likewise, division by p is not well-defined.
5) I'd be curious to see an argument for 1=2.

Originally Posted by DavidSh
This could be fun. I think there was already a proof that the only ring for which 1/0 is defined is the ring with only a single element, but let's see how far we can get.

Do you add Peeleean (SUPER-complex) numbers in the natural way, as if they were 2-dimensional vectors?

How do you multiply Peeleean numbers?
Yup, interesting.

5. ## Re: Zero divided by zero

Ya know, this is pretty interesting. Brb off to get my Fields Medal.

6. ## Re: Zero divided by zero

P and equivalent constructions create a bunch of problems and solve little, which is the opposite of elegant simplicity.

I'm going to appeal to (ugh) the Efficient Market Hypothesis and suggest that given 1/0 is such a basic and well-known quantity, if there were any simple and elegant fixes that led to anything even remotely interesting, they would have been applied by now.

Obviously we can define anything to be anything in math if we really want to because nothing about axiomatic logic prevents us from doing so, but that doesn't mean we should.

7. ## Re: Zero divided by zero

Originally Posted by Lethologica
P and equivalent constructions create a bunch of problems and solve little, which is the opposite of elegant simplicity.

I'm going to appeal to (ugh) the Efficient Market Hypothesis and suggest that given 1/0 is such a basic and well-known quantity, if there were any simple and elegant fixes that led to anything even remotely interesting, they would have been applied by now.

Obviously we can define anything to be anything in math if we really want to because nothing about axiomatic logic prevents us from doing so, but that doesn't mean we should.
So no Fields Medal, but instead a Nobel Prize in Mathematics, then? Yes, I know. That's the joke.

8. ## Re: Zero divided by zero

Originally Posted by Peelee
So no Fields Medal, but instead a Nobel Prize in Mathematics, then? Yes, I know. That's the joke.
Depends on what base you reckon your age by.

9. ## Re: Zero divided by zero

Originally Posted by Lethologica
Depends on what base you reckon your age by.
Aren't all bases technically base ten?

10. ## Re: Zero divided by zero

Originally Posted by Peelee
Aren't all bases technically base ten?
I mean, I'm personally of the opinion that my base is #1 because it is the best base, but I'm pretty biased.

11. ## Re: Zero divided by zero

Originally Posted by Lethologica
P and equivalent constructions create a bunch of problems and solve little, which is the opposite of elegant simplicity.

I'm going to appeal to (ugh) the Efficient Market Hypothesis and suggest that given 1/0 is such a basic and well-known quantity, if there were any simple and elegant fixes that led to anything even remotely interesting, they would have been applied by now.

Obviously we can define anything to be anything in math if we really want to because nothing about axiomatic logic prevents us from doing so, but that doesn't mean we should.
Eh, what is being called P here is basically the coefficients of a Laurent series (the same way the dual numbers I mentioned earlier are basically Taylor expansions), and its very useful there in that if you have harmonic functions with some particular set of poles, lots of things about that function are specified strictly by the poles and you can basically ignore everything else - the same way that a static electric field is specified only by the locations of the charges and contained no other independent information. That has lots of useful consequences for applied math purposes and provide efficient ways to solve inhomogeneous Laplace equations, which crop up everywhere in physics.

I wouldn't be surprised if something like this would show up if you squinted at matched asymptotics as well, since there you need a way to say 'these two functions behave the same at infinity' or 'these two functions behave differently at infinity'.

So IMO they actually solve a lot of things.

12. ## Re: Zero divided by zero

Originally Posted by NichG
Eh, what is being called P here is basically the coefficients of a Laurent series (the same way the dual numbers I mentioned earlier are basically Taylor expansions), and its very useful there in that if you have harmonic functions with some particular set of poles, lots of things about that function are specified strictly by the poles and you can basically ignore everything else - the same way that a static electric field is specified only by the locations of the charges and contained no other independent information. That has lots of useful consequences for applied math purposes and provide efficient ways to solve inhomogeneous Laplace equations, which crop up everywhere in physics.

I wouldn't be surprised if something like this would show up if you squinted at matched asymptotics as well, since there you need a way to say 'these two functions behave the same at infinity' or 'these two functions behave differently at infinity'.

So IMO they actually solve a lot of things.
Being basically the same thing and being the same thing are different.

It is fair, though, that I forgot to give a fair hearing to the projectively extended real line despite that I definitely looked it up earlier in this thread (I don't remember if I actually posted about it). That one actually defines an infinity term such that a/0 = infinity for nonzero a.

13. ## Re: Zero divided by zero

Originally Posted by Lethologica
Being basically the same thing and being the same thing are different.

It is fair, though, that I forgot to give a fair hearing to the projectively extended real line despite that I definitely looked it up earlier in this thread (I don't remember if I actually posted about it). That one actually defines an infinity term such that a/0 = infinity for nonzero a.
Being precise, what's being called P here is the coefficient of the 1/z term in the Laurent series, as 'epsilon' in dual numbers is the coefficient of the z term in the Taylor series. So for example, if I'm doing a contour integral on some function f(z), the value of that contour integral is proportional to the integral over the interior of the P part of f(z). This is usually written in a different way, as Cauchy's integral formula.

14. ## 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 in-depth answers.
IIRC, something is a function if, and only if, each set of inputs results in exactly one result.

"Solve for x" isn't trying to be a function.

15. ## Re: Zero divided by zero

Originally Posted by kyoryu
IIRC, something is a function if, and only if, each set of inputs results in exactly one result.
By that definition, sine, cosine, and tangent are not functions, which they quite clearly are. We're going to need a different definition. F(x) = x2 also would not be a function, and ... wait. You said if and only if each input results in one result, not that no result can have more than one input. Okay, so according to your definition F(x) = a and/or b is not a function, while G(x) = a and G(y) = a is a function. I can get behind that.

16. ## Re: Zero divided by zero

There are things called branch points that complicate this definition. For example, the complex function log(z) = log(|z|) + theta(z) * i

If you do a contour around zero, every time you go around you pick up a 2 pi i

17. ## Re: Zero divided by zero

Originally Posted by georgie_leech
I'd be careful about arguing from what makes physical sense. After all, we can divide by negative numbers
Not physical, per se, but negative numbers make perfect sense with division and multiplication if you consider them as debts, for example.

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•