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Thread: Zero divided by zero

20171007, 12:47 PM (ISO 8601)
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Re: Zero divided by zero

20171007, 01:08 PM (ISO 8601)
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20171007, 01:24 PM (ISO 8601)
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20171007, 01:58 PM (ISO 8601)
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Re: Zero divided by zero
True, true. I'd have a similar complaint for negatives, though.
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 nonzero 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.
pp=p, I reckon. Yeah, pp=1*p+1*p=(11)*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^(aa)=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^(a2a)=p^(a)*p^(2a)=p*0=p. Looks like p^negative reals could be either p or 0, so it's not welldefined.
Uhoh, maybe p^a is not welldefined for nonnegative 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 welldefined, 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 welldefined. 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+pp isn't going anywhere, since a is not equal to a+pp since pp=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 welldefined otherwise.
4) Likewise, division by p is not welldefined.
5) I'd be curious to see an argument for 1=2.
Yup, interesting.

20171007, 02:17 PM (ISO 8601)
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20171007, 02:29 PM (ISO 8601)
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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 wellknown 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.

20171007, 02:33 PM (ISO 8601)
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20171007, 03:20 PM (ISO 8601)
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20171007, 03:42 PM (ISO 8601)
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20171007, 03:53 PM (ISO 8601)
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20171007, 04:06 PM (ISO 8601)
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Re: Zero divided by zero
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.Last edited by NichG; 20171007 at 04:08 PM.

20171007, 04:41 PM (ISO 8601)
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Re: Zero divided by zero
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.

20171007, 05:06 PM (ISO 8601)
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Re: Zero divided by zero
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.

20171009, 11:44 AM (ISO 8601)
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20171009, 12:18 PM (ISO 8601)
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Re: Zero divided by zero
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) = x^{2} 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.
ThriKreen Ranger/Psionicist by me, based off of Rich's A Monster for Every Season

20171009, 01:18 PM (ISO 8601)
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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

20171010, 03:23 AM (ISO 8601)
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