X / 0 = holding any major development up?

[JSK333]

I'm curious if there are major equations, theories, or the like, being held up by running into division by 0.

I once had a thought about a way to get around this problem, and wanted to try applying it to an existing problem.

 

[chroot]

Anything divided by zero is undefined. There is no room to consider any alternatives.

- Warren

 

[JSK333]

So I guess that's a "no"?

In HS I tried using a variable for 1/0 to see how it would affect equations that ended with division by zero; to see if they could proceed further and anything meaninful resulted.

It was a bit interesting, but I had nothing substantial to try it with.

What do you guys think about this idea? Say, [0] = 1/0 for example? And then when you get division by zero, use the variable [0] and continue solving until you reach an answer?

Solomon

 

[quartodeciman]

Division by zero is dead on arrival for any math system with the structure of a field. That is because x * 0 = 0 for any x in the field. Also, 0 unequal 1. So, x * 0 can't be 1.

But it isn't all lost. You can give up the field structure, but that carries its own troubles. More interestingly, it is possible to use something just about as good as 0 for scientific work. Check out these links.

link to hyperreal numbers --->
http://www.wikipedia.org/wiki/Hyperreal_number

link to surreal numbers --->
http://www.wikipedia.org/wiki/Surreal_number

link to mathematical structure of a field --->
http://www.wikipedia.org/wiki/Field

That stuff is still under research.

[corrected for my error in explanation: I treated the supposed field as if it had an order, <, which might not necessarily be the case. So, the right expression is '0 unequal 1' rather than the original '0 < 1'. quart]

 

[drnihili]

Originally posted by quartodeciman
Division by zero is dead on arrival for any math system with the structure of a field. That is because x * 0 = 0 for any x in the field. Also, 0 unequal 1. So, x * 0 can't be 1.

I'm sorry but I didn't see anything in the definition of a field that required x*0=0 for all x in the field. Did I miss something?

 

[Hurkyl]

x * 0 = x * 0 + 0
= x * 0 + (x + (-x))
= (x * 0 + x) + (-x)
= (x * 0 + x * 1) + (-x)
= x * (0 + 1) + (-x)
= x * 1 + (-x)
= x + (-x)
= 0

 

[quartodeciman]

Right! It's a theorem.

 

[drnihili]

Ok, I can see how that follows from the definition. I take it that the hyperreals and surreals substitute division by infinitessimals. What precisely do you give up by not having a field so long as you ensure that (x/0)*0 =x? I'm guessing you're about to tell me that it depends on exactly how you give up having a field. But any clues or pointers to good reading would be appreciated. I'm working on some stuff in this area but coming at it from set theory and logic rather than analysis.

 

[quartodeciman]

Let's see now.

From the above demonstration:

0 is a normal additive identity
x has an additive inverse, -x
1 is a multiplicative identity for x
x, 0, -x reassociate
x factors (undistributes) from 0,1

One of these has to break to avoid x * 0 = 0.

I suppose you want more than just the system {0} with 0+0=0, 0-0=0,0*0=0,0/0=0.

Maybe there is something in math subjects like rings and modules (instead of fields and vector spaces).

Most important: where are you headed with this? What is the goal? Why divide by zero?

 

[JSK333]

My reason is purely curiousity.

I remember in match class in HS, we would solve problems then ended with division by 0, and of course, that's where you stopped with an answer of "undefined."

So I am curious what happens if you say, ok, let's say 1 / 0 is not undefined, and assign that value to a variable, then continue solving...

So, if you ended up with (4x - 3) / 0 you would say you had (4x - 3)[0] and then continue solving for x, for example.

Solomon

 

[quartodeciman]

More interesting it would be to know why you might be dividing by something that runs the risk of going to 0. That might indicate something about the starting problem.

1. a mistake was made at an earlier step
2. a function behaves like it might blow up at some point

 

[drnihili]

Well I'm not primarily interested in division by zero, it's only come up secondarily. I've been working on an extension of some work of De Bois-Reymond regarding ordering functions that have infinite limits based on rate of growth. As part of that I've been looking at asymptotic functions and also functions that approach zero. n/x as x -> 0 comes up in this regard.

I need to take a closer look at the hyperreals and surreals to see how they compare with what I'm looking at.

So do I have a look at Robinson? Or is there somewhere else I should go first?

 

[quartodeciman]

here are two more links:

Phil Apps on nonstandard analysis

http://www.haverford.edu/math/wdavidon/NonStd.html

here is the main book I actually own:

Martin Davis, Applied Nonstandard Analysis, Wiley-Interscience(1977)

I don't know your subject and so I don't know what will help.

 

[drnihili]

Thanks.

I started off trying to develop an alternative way of comparing the size of infinite collections that would behave more intuitively than cardinality in certain cases. I've got the basics of that worked out thanks to some earlier work by Du Bois-Reymond and Pincherle. But I've still got to connect everything up to notions of infinity coming from analysis to make sure I'm not just re-inventing the wheel. I'll tell you what, I'll post a thread with an example problem and see what folks have to say about it.

Link to thread