Dividing by zero, the truth (this is long!)

[targren]

Ah I see. Except I was taking 0/0 as a value, not a function, working on the assumption that the output of said function was, in fact, valid.

Ahh!...
Well... you can't

0/0 may look like a number, and smell like a number, but at the end of the day it represents an operator from RxR that you don't know if it's a well-defined function. You can't treat it like a number in R till you know that.

Exactly the point of the "proof by contradiction." By assuming you can as a given, and showing that the result is impossible, you prove you can't. I probably shouldn't have omitted explicitly that assumption, though. Point taken.

 

[Saulkar]

Math?! Shiiiiiiiiiiiiite. My worst subject. You lost me at the definitions thus I am still content in the knowledge that dividing by zero is fatal on a cosmic scale.

 

[CrystalShadow]

Can you demonstrate that every mathematical construct ever devised relates to something practical?

Every mayor area? Yeah, I can, skipping the applied math branches you end up with algebra, analysis, combinatory math and topology (I include geometry here, but if you want, take it out). The father of the modern version of each of this areas was a mathematician-something else hybrid that worked hard and studing a real world / other branch of sience inspired problem ended up changing the face of mathematics.

Can i demostrate that every article in every journal relates to something practical? Nop, but after some years of reading them i have yet to find one that wasn't close to a practical field

Studying higher dimensions? Besides being easier in many regards and help do a roadmap for topology in lower dimensions, many phisicist do work in higher ones.

Ohh, you meant Quaternions! Sorry, well, that one is a textbook example of math created for the sake of physics. Being a non phisicist i have only heard the story at bars from friends, but hopefully i wont lie (a lot).
You see, this dude Olinde Rodrigues studied and published about the transformation group of the quaternions, all very pure algebra, and poor ol Olinde was ignored at large, until recent historical research people didn't even knew he published that.
Tree years later comes our chap Hamilton, a physicist and mathematician, the dude was trying to fit electrodinamics into a mathematical form that allowed him to calculate electromagnetic forces at a point in the space, he knew how to sum, but sadly electrodinamics require a lot of that nasty multiplication, and suddenly (after years of busting his ass suddenly) he thought about the algebraic structure that his new born quaternions needed to have to express the magnetic forces the right way!
Instant success! Any physicist that was trendy (studying electromagnetism and optics was what cool kids did!) needed to study quaternions to be able to correctly express the laws governing the phenomena as math. And also the mathematicians started to study them! Why? Beats me, haven't study that much history of math

OK then. XD.

All I can really say is that the fundamental rules that underpin mathematics aren't meaningfully defined by mathematics itself.

It shouldn't be a huge surprise then that most of mathematics derives from some kind of practical problem, but it doesn't preclude making up completely arbitrary rules and seeing what happens as a result.
(In a purely mathematical sense, obviously.)

But, it doesn't mean much either way whether maths is an entirely abstract subject, or one grounded in mundane concerns.

You can describe things using mathematics that don't relate to reality. That in itself is enough.

 

[Tanakh]

Exactly the point of the "proof by contradiction." By assuming you can as a gicen, and showing that the result is impossible, you prove you can't.

No, no, you can't REACH the contradiction.

I'll go over it again. The mapping / that the OP defined is not a well-defined function, so even if 0/0 = r and 0/0 = r+1 that DOESN'T imply r = r+1 because you just showed that / is not a function due having two different results for the same imput and f(x,y) = u, f(x,y) = v implying u = v is not a magic step, it is true ONLY if f is a function due it's properties.

 

[Tanakh]

You can describe things using mathematics that don't relate to reality. That in itself is enough.

You CAN? Gosh... that is actually quite impossible to prove

Anyway, i do get your point, and yeah, a math dude can go and tell reality to stick it to itself and just do whatever math he wants; he is not more of a math or less of a math for that, but for the beauty of his results.

Also, i continued my ramblings after your quote

 

[targren]

Exactly the point of the "proof by contradiction." By assuming you can as a gicen, and showing that the result is impossible, you prove you can't.

No, no, you can't REACH the contradiction.

I'll go over it again. The mapping / that the OP defined is not a well-defined function, so even if 0/0 = r and 0/0 = r+1 that DOESN'T imply r = r+1 because you just showed that / is not a function due having two different results for the same imput and f(x,y) = u, f(x,y) = v implying u = v is not a magic step, it is true ONLY if f is a function due it's properties.

So then we're back to 0/0 being undefined, because there's no "return" value (to borrow a term from programming)?

 

[Tanakh]

So then we're back to 0/0 being undefined, because there's no "return" value (to borrow a term from programming)?

There is a return value, but you just showed that if we accept the OP definitions that return value is any freaking real number (you didn't need to add 1, could have added any other real number). Thus is not a function, well-defined or otherwise, it's just a series of steps redacted in math lingo that doesn't end up having any structure studied in current or past math (with good reasons, this kind of associations between the domain and the codomain kill ALL the algebra structure we could attempt to have).

 

[targren]

So then we're back to 0/0 being undefined, because there's no "return" value (to borrow a term from programming)?

There is a return value, but you just showed that if we accept the OP definitions that return value is any freaking real number (you didn't need to add 1, could have added any other real number). Thus is not a function, well-defined or otherwise, it's just a series of steps redacted in math lingo that doesn't end up having any structure studied in current or past math (with good reasons, this kind of associations between the domain and the codomain kill ALL the algebra structure we could attempt to have).

Gotcha. So it's essentially returning R.

So essentially, the rest of the proof would be to show that the OPs definition/mapping of division is wrong. Probably via axiom.

This has been a good mental workout in an otherwise boring day of coding, and now it's got me several new concepts to study (including fields and rings). Thank you.

 

[Matt Simon]

this whole thread is really just a semantics argument, and semantics is inherently a language issue, languages being wholly inefficient things that aught to be replaced by a much simpler numerical system. in essence every argument here fails because it uses something as imprecise as language to describe something as absolute as mathematics.
example. when language is used infinity can not have a value but
using E in place of the symbol for the sum of all numbers n between 0 and infinity we find that E(2^n) = infinity
E(2^n) = (1 + 2 + 4 + 8 + ...)= infinity
x*1= x
2-1=1
x*(2-1)=x
x*2 - x*1=x
E(2^n)*2 - E(2^n)*1= infinity
E(2^n)*2 - E(2^n)*1= (2 + 4 + 8 +...) - (1 + 2 + 4 + 8 + ...)
E(2^n)*2 - E(2^n)*1= -1 + ( 2 -2 + 4 - 4 + 8 - 8 + ...)
E(2^n)*2 - E(2^n)*1= -1 + ( 0 )
E(2^n)= -1
infinity = -1
can anyone disprove that... no, because unlike most attempts to showcase mathematical anomalies i actually understand the order of operations so the argument is based entirely in mathematics which is impervious to language.
had to go back and edit this because it was pointed out i was using E(n^2) instead of E(2^n)

So in the case of dividing by zero
x/0=r
x=r*0
x=0
but nobody likes to see zeros move around so lets instead
h=0
x/h=r
x=r*h
x=r*0
x=0
so the division by zero is thus possible because division is just a different way to write a multiplication function.
Unfortunately r=? because all available data was used up before is value could be determined
thus r is undefined not because the division is impossible but because the data is incomplete, the mathematical equivalent of having a sentence that doesn't have a noun for its verbs. Bad structure but not technically wrong.
So instead of using a variable lets use a function
h=0
F(x)-F(x)=0
F(x)=x^2
(F(x)- F(x) )/h=r
x^2-x^2/0=r
0/0=r
so still not enough information but
F(x+0)=0 & h=0
F(x+0)= (x+0)^2
so
( F(x+h) ? F(x) )/h = r
( (x+h)^2-x^2 )/h=r
(x^2 +2xh +h^2 -x^2)/h=r
(2xh + h^2)/h=r
2xh/h + h^2/h =r
2x +h = r
2x+0=r
r=2x
now we still don't have a definite value of r but that doesn't make it undefined, the fact that x also does not have a determined value means that r is a function of x but this is a new function of x, a better function of x so we shall call it f prime of x or
F'(x)
now since mathematics is a numerical system that also equates to a graphical representation it should be noted that the zero acts like a limit, the line will approach it but the rate of change as you approach becomes so small that if you followed the line you may or may not actually get to the 0 line, but you certainly won't go past it. So lets denote it as
lim h->0
Then I could write
F(x)=x^2
F'(x) = lim h->0, (F(x+h) ? F(x))/h
and it would contain all necessary information to find this F'(x)
now the thought occurs, if a variable represents a number that is either unknown or undetermined couldn't a function be treated as a variable so the whole defining F(x) becomes unnecessary leaving us with
F'(x) = lim h->0, (F(x+h) ? F(x))/h
for those of you who haven't figured it out yet I just created calculus. It worked only because it is possible to divide by zero its just useless unless you are willing to hold back and manipulate the equation to get a more complete conclusion.

 

[CrystalShadow]

You can describe things using mathematics that don't relate to reality. That in itself is enough.

You CAN? Gosh... that is actually quite impossible to prove

Anyway, i do get your point, and yeah, a math dude can go and tell reality to stick it to itself and just do whatever math he wants; he is not more of a math or less of a math for that, but for the beauty of his results.

Also, i continued my ramblings after your quote

If you did, you must've edited your post... Or the forum code is faulty. XD

And, OK, yes, you can't actually prove that there is any mathematical statement that doesn't relate to some part of reality.

Of course, you also can't prove physics is a real description of anything...

Or that reality even exists to begin with.

But then you really end up going into the definitions of what actually constitutes proof.
And I've certainly annoyed more than enough people with that issue already. XD

 

[The_root_of_all_evil]

Saying complex numbers exist in theory is like saying 1, 100, 3 or 0 exist in theory. Maze1125 put a (very obvious) trap and Root sprung it!

Point me to the number 1. The original number 1. Just because I define complex, imaginary and infinite numbers as theoretical doesn't mean anything more than they're not real. Theory still has a practical use, just not a practical form.

Semantic traps are hardly welcome in discussions though.

 

[Tanakh]

Point me to the number 1. The original number 1. Just because I define complex, imaginary and infinite numbers as theoretical doesn't mean anything more than they're not real. Theory still has a practical use, just not a practical form.

Semantic traps are hardly welcome in discussions though.

I know, but they are fun! Then again... maybe it's an acquired taste.

 

[Tanakh]

But then you really end up going into the definitions of what actually constitutes proof.

Proof: The sublte art of talking your teachers or peers into accepting that the stuff you say happens indeed happens! Mora akin to talking that gal you like into a drink with you than to discovering the "UNIVERSAL TRUTH".

Ur turn.

 

[The_root_of_all_evil]

I know, but they are fun! Then again... maybe it's an acquired taste.

Perhaps, but it's like shooting fish in the barrel. You get wet, covered in scales and left with unusable bits.

Wait, maybe it's not quite like that...

 

[Tanakh]

example. when language is used infinity can not have a value but
using E in place of the symbol for the sum of all numbers n between 0 and infinity we find that E(n^2) = infinity
E(n^2) = (1 + 2 + 4 + 8 + ...)= infinity
x*1= x
2-1=1
x*(2-1)=x
x*2 - x*1=x
E(n^2)*2 - E(n^2)*1= infinity
E(n^2)*2 - E(n^2)*1= (2 + 4 + 8 +...) - (1 + 2 + 4 + 8 + ...)
E(n^2)*2 - E(n^2)*1= -1 + ( 2 -2 + 4 - 4 + 8 - 8 + ...)
E(n^2)*2 - E(n^2)*1= -1 + ( 0 )
E(n^2)= -1
infinity = -1
can anyone disprove that... no, because unlike most attempts to showcase mathematical anomalies i actually understand the order of operations so the argument is based entirely in mathematics which is impervious to language.

Lolz mate, you are funny! Holes everywhere, but for starters:

- Summation is an binary operator, sure, by induction you can extend it to any natural number of summands, but then you go and "sum from 0 to infinity" OMG!!! Both your calculus and your algebra teacher should be ahsamed! That is a series, and you are not using the sum, but finding a limit.

- OMFG mate, Sigma (n^2) with n goig from 1 to infinity is a non convergent series! Thus you can't do E(n^2)*2 - E(n^2)*1!!!

I can tell you like this stuff, and you might be good at it if you try! But stop doing make belive and actually study math.

Edit: About the other stuff. It does sound like we are using "language", but actually function, well-defined, real number, divisor, field, ring, and all that crap are math concepts defined with logic rigor; we could translate that to simbolic logic and use only those signs, and by we i don't meant me, because i never took a simbolic logic class, but a dude that has could

 

[Tanakh]

You get wet, covered in scales and left with unusable bits.

Sounds like my last relationship!

I know, i know, saw that coming, and bite me low content

 

[DiMono]

You're right; my OP has holes in it. I didn't take enough time to edit it. But at this point I don't really care, I am relinquishing ownership to whoever wants to take up the position of custodian for this topic, if this thread doesn't die shortly anyways.

So your course of action is to admit that your first post has problems, and then rather than fixing them, say "Your problem now, suckers!"? Poor form, dude. Poor form. If you're going to start a topic like this, you kind of don't have a choice but to follow it up and correct yourself where errors are found. I'm not impressed.

 

[zombie711]

Forgive my apparent ignorance but wouldn't common sense just tell us that you cannot have a number of things (let's say three apples) and divide them into zero groups? Isn't that all division is?

No becuase when your dividing your splitting. Dividing by 0 would mean you split it until it no longer exist, which you cant do with division allow because your not taking anything away your splitting.

 

[Evil Teddie]

If someone were to post a video on youtube explaining all of this on a blackboard while explaining it I could understand this, I need to know the methodolgy behind this, operators and sets? I have no idea...

 

[Schtoobs]

It might be of interest that 0! = 1 (zero factorial equals one) source: wiki - "factorial". So x/0! = x for all real numbers. Not totally irrelevant. Funny that I should see this on the escapist the same week I finish writing a solution to a puzzle, the syntax of which had me stumped for a while as I was stuck dividing by 0. An interesting (and baffling) topic.

Oh and hello. It's my first post! (very long time lurker)