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

[Maze1125]

He doesn't have to do that at all. 0/0 = 0 has no part in what he was trying to prove.

"Can any real number be divided by zero and if so what is the result of that operation?"

That's "real number"/0=x; which he states is 0/0=0.

The only time that is stated is in the "tl;dr" part of theorem 2 which, as has already been pointed out, is an inaccurate representation of the actual maths.

In the actual proof, there's no mention of 0/0 = 0 at all.

Regardless, I have to ask, what's the difference between a "theoretical Zero Element" and a "practical Zero Element", and what's the relevance?

A theoretical Zero Element is one that performs the operations. A practical one can be placed on the number line. Relevancy is that he's proven the former exists, but the missing Theoreum 3 probably justifies the latter.

So you consider complex numbers to not be practical then? They can't be placed on the number line at all.

Regardless, as I said before. he may not have proven that the 0 of the number line is the zero element of the field of the Real Numbers, but he doesn't need to, as that's a known mathematical fact.

Curiosity though: Why are you defending him when he hasn't come back on this yet? Do you know him?

I'm not defending him, I'm defending the mathematics.

 

[isometry]

You're right that it just comes down to the definition of division. But you don't need fields or rings for any of this. The first direction of your proof of theorem 2 boils down to "if 0 divides x then by the definition of division in the reals there exists a real number r such that 0*r = x, so x = 0 since r*0 = 0 for all real numbers r."

Using rings and fields to prove that only 0 divides and is divisible by 0 is like using a shotgun to kill a mosquito. The weaker the premise, the stronger the proof.

I disagree that his is the weaker proof.

You're saying "Take division in the reals, then..." he's saying "Take division in a general field, then...".
He uses a more generalised system and so proves more.

It's the basic principle behind the game show "Name that tune."

"I bet I can name that tune in 5 notes!"
"oh yeah? I bet I can name it in 3 notes!"
...etc

It's more impressive to name the tune using fewer notes. To start with something small, and accomplish something large, is more impressive than to start with some large and accomplish something large. The OP's proof takes orders of magnitude more time to read and write, and requires orders of magnitude more sophistication and mathematical maturity on the part of the reader. It would be like saying:

"I can name that tune after listening to the whole song 100 times!"

Sure, someone might say "that's even more impressive because listening to the song 100 times takes a lot of effort", but that ignores the mismatch between the amount of effort vs the magnitude of what's being accomplished.

To take another example, it's impressive when a frail 5' tall woman judo flips a giant man. But if King Kong, the +100' tall giant ape, flipped the same man, then it would be less impressive. Yes, King Kong himself is impressive, but this particular accomplishment of his is not.

 

[The_root_of_all_evil]

[
In the actual proof, there's no mention of 0/0 = 0 at all.

"Does there exist a real number x such that x is divisible by 0?"

The answer is yes, and theorem 3 and its remarks should clarify everything

It's left out...that's why it's not stated...

So you consider complex numbers to not be practical then? They can't be placed on the number line at all.

Bingo. They exist in theory.

Regardless, as I said before. he may not have proven that the 0 of the number line is the zero element of the field of the Real Numbers, but he doesn't need to, as that's a known mathematical fact.

Then you've no problem showing the proof of the claim. As I stated at the start, there's a few caveats that allow potential errors to slip through, and as such, need to be defined.

The expression

0/0=?

requires a value to be found for the unknown quantity in

? times 0=0.

Again, any number multiplied by 0 is 0 and so this time every number solves the equation instead of there being a single number that can be taken as the value of 0/0.

If the Riemann Sphere has no value for 0/0, I think we can take it as granted that the zero element defines the properties of 0, but doesn't actually equal 0. It's an Indeterminate form. [http://en.wikipedia.org/wiki/Indeterminate_form]

 

[nothingspringstomind]

The problem with this level of mathematics, (aside from the migraine it gives me) is that you can convincingly argue almost anything. I had a maths student show me how 2+2 does not always equal 4. But thanks to the OP for a damn interesting read.

As a friend once said;

"The problem with university is that everything you've been taught so far is the wrong way of thinking. Biology becomes Chemistry, Chemistry becomes Physics, Physics becomes Maths, and Maths just plain doesn't work."

 

[Tanakh]

The problem with this level of mathematics, (aside from the migraine it gives me) is that you can convincingly argue almost anything. I had a maths student show me how 2+2 does not always equal 4. But thanks to the OP for a damn interesting read.

And why would it always be 4?

If you are at 8 o'clock evening and spend 8 hours working, do you end up at 16 hours evening?

If you work gaining 24,000 USD from 20 to 30 years and then 48,000 USD from 30 to 40, is it the same as getting 7.2 millions when you are 20 and then not getting money (otuside investments and hedge founds of coruse ), or getting no daught at all till you are 40 and then 7.2 mill?

You can see it as "math reflects life and nature", if it doesn't always happens in those, expect math to have some crazy wierdo studing how to represent it in our math language. And by golly, math is beautiful.

Edit: Also, BS. Assuming you are a biologists, you can imagine crazy shit, bacteria living 4000 fucking feet under freaking volcanic rock; multicelular beings that can go unprotected to the SPACE!; mamals that are blind and still work, or that are almost totally usleless in the wild and still can comunicate with each other in seconds through the freaking world. Can you imagine a mammal based not in carbon but in hirdogen? Lolz nop. Same with math, its just that you dont know the rules, haha, noobzor

 

[careful]

Nice work, some details:

I will answer this question. And no, infinity is nowhere to be found here.

That is precisely because you didn't chose the calculus way of dealing with this. You could have said that for function, lets say 1/|x| you can have it well-defined at all points if you work in reals U {infinity}.

field

Why not just use integral domains, isn't that enough?

http://en.wikipedia.org/wiki/Divisibility

I think you are using this definitionhttp://en.wikipedia.org/wiki/Divisor_(ring_theory), as your link requires r to be an integer.

Theorem 2 asserts two propositions. The first is that the zero element, 0, is divisible by 0, and the result of that is again 0.

Wait, what? In all your proof you never used or showed that r=0. In fact you can easily see that r can be any element of F. I always thought that's why this as a whole is undefined, because you have this super nice operator from RxR to R that does some amazingly pimp shit everywhere EXCEPT when you put 0 as the divisor, then it's either nonsensical if the dividend is diferent from zero or is not even a function (ie undefined) if the dividend is zero. You should REALLY correct this because it's your TL;DR... and it's wrong.

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.

 

[Tanakh]

But at this point I don't really care

Damn emo mathematicians, that's why i party with chemist mayors, much more active!

Nway, nicely done, keep up the good posting mate.

 

[targren]

Let r be some real number such that r = 0/0
Then, by the definition of division, r * 0 = 0
These are the mainstays of the above proof.

Now, by the nature of multiplication (r + 1) * 0 = 0
Or (r + 1) = 0/0

By substitution:
r = r+1

You are doing a misstep there. Let me give you a hint, you are canceling a mapping that is not a morphism.

Edit: It might be even clearer where you did that if you write the division as an operator from RxR GL, HF

Okay... unless you're talking about a different use of "mapping," in my particular domain-specific parlance (Discrete/Set), a mapping is a function. I don't see where I'm "canceling" anything. Taking the asserted steps from OP as a basis step, I basically did a (very informal) mathematical induction to show that the result onto R is an empty set. Or if you prefer, that it's not a valid function. So you're going to need to point the misstep out.

 

[Some_weirdGuy]

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, that's where the original concept of division derives from, but that's no reason to assume you cannot extend the concept to other areas.

Take your own example and use negative numbers:
"wouldn't common sense just tell us that you cannot have a number of things (let's say 6 apples) and divide them into -3 groups? Isn't that all division is?"

Yet we can divide by negative numbers all we want, by simply extending the rules of the concept beyond its origins.

except using the apple example again, wouldn't negatives simply count as something you owe to someone else?

So i could own 6 apples and spit them into 3 groups. That gives me 2, and cause i 'own' them it's a +2.
or i could borrow 6 apples and split them into 3 groups of 2, but because they're not my apples, it's negative (-2).

both are still working with actual numbers and groups of objects.

Where as 0 is trying to do something fundamentally different. It's trying to place them into no groups at all, which can't be done cause you either own(+) or owe(-) a certain amount of apples. It's impossible to not have at least 1 group. (unless you have/owe 0 apples, in which case they will always be in zero groups anyway cause there just isn't any there)

 

[CrystalShadow]

The problem with this level of mathematics, (aside from the migraine it gives me) is that you can convincingly argue almost anything. I had a maths student show me how 2+2 does not always equal 4. But thanks to the OP for a damn interesting read.

As a friend once said;

"The problem with university is that everything you've been taught so far is the wrong way of thinking. Biology becomes Chemistry, Chemistry becomes Physics, Physics becomes Maths, and Maths just plain doesn't work."

That's kind of the whole point of maths though.

It's a subject whose basis is, "If you assume the following statement to be true, what are the consequences?"

It doesn't actually require that this initial statement is practical, or has any relationships to anything whatsoever.

Therefore, what is mathematically valid, is basically the space of everything that can be made internally consistent with it's own arbitrary definitions.

Some definitions are mutually exclusive, but that simply means you can't use both together, not that one or the other is wrong.

 

[GenericAmerican]

I'm going to tell you the truth, I failed calculus. Not because I was bad at it, I just missed a month near the end and never caught up.

Soooooo too me all this might as well be written in hieroglyphs.

 

[Tanakh]

-snip-

Humm, YMMV in terminology and I usually don't do that much maths in english, but what i meant is this:

You took this non standard defined "division", and it's a mapping that goes from RxR to R, ie, if x and y are in R you apply the division to them (lets call the division phi(x,y) instead of x/y to make the fact this is a mapping more evident ) and it ends up in R. Now i assume that you wanted to show that for x = 0 = y, you can't define phi(x,y) as any real number because that would cause a contradiction. So you go (allow me to change notation) :

Let r be some real number such that phi(0,0) = r
Then, by the definition of division, r * 0 = 0
These are the mainstays of the above proof.

Now, by the nature of multiplication (r + 1) * 0 = 0 WARNING: Bunch of easy but important algebraic steps skipped here
Or phi(0,0) = (r + 1)

By substitution:
r = r+1

Now, this is the tricky part. Here you have shown that with that definition of division phi(0,0) = r and that phi(0,0) = r+1 and did an invalid substitution, you can only affirm that f(z)=u and f(z)=w implies u=w if the mapping is well-defined, just in the line before you have shown that this "division" can't be well difined from RxR to R, so you can't use it being well-defined in the next line.

 

[Tanakh]

It doesn't actually require that this initial statement is practical, or has any relationships to anything whatsoever.

That is what they say to you, but historicaly this is BS. Was calculus developed just playing with random "what if we change this"? Was topology? Polinomial ecuations? Non-Euclidean geometry? Abstract algebra? Boolean logic? ANYFREAKING THING?

Fact is that math is inspired in physics, biology, chem, economy, even art; and currently we have deep ties with all those. AFAIK there ain't no single fruitful math field that was done for the sake of messing with the rules. Thats why i found this impressively hilarious:

So you consider complex numbers to not be practical then? They can't be placed on the number line at all.

Bingo. They exist in theory.

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! Complex numbers are the most natural way to study real phenomena like electromagnetism or the freaking water inside of a pipe, you don't even need to do magic math mumbo jumbo, when you do analysis over the field of complex numbers the freaking water dynamics just JUMP all over you.

 

[Tanakh]

I'm going to tell you the truth, I failed calculus. Not because I was bad at it, I just missed a month near the end and never caught up.

Soooooo too me all this might as well be written in hieroglyphs.

For what is worth, this has nothing to do with calculus. You just need middle school algebra, an open mind and lots of patience (and a tab opened at wikipedia wouldn't hurt).

 

[targren]

You took this non standard defined "division", and it's a mapping that goes from RxR to R, ie, if x and y are in R you apply the division to them (lets call the division phi(x,y) instead of x/y to make the fact this is a mapping more evident ) and it ends up in R. Now i assume that you wanted to show that for x = 0 = y, you can't define phi(x,y) as any real number because that would cause a contradiction.

That's the right assumption, yes. But in what way is division as the inverse of multiplication "non-standard?"

Now, this is the tricky part. Here you have shown that with that definition of division phi(0,0) = r and that phi(0,0) = r+1 and did an invalid substitution, you can only affirm that f(z)=u and f(z)=w implies u=w if the mapping is well-defined, just in the line before you have shown that this "division" can't be well difined from RxR to R, so you can't use it being well-defined in the next line.

I see what you're saying, but I disagree with your assessment. Unless you're contending that
r * 0 = 0 is not true for all r in R, I'm not sure how the substitution can be invalid.

 

[Tanakh]

That's the right assumption, yes. But in what way is division as the inverse of multiplication "non-standard?"

I see what you're saying, but I disagree with your assessment. Unless you're contending that
r * 0 = 0 is not true for all r in R, I'm not sure how the substitution can be invalid.

Because this is not a well-defined funtion mate, even if 0/0=r and 0/0=r+1 that doesn't imply r=r+1 if that isn't a function or is not well-defined, which is not and you show that halfway in that proof.

Edit: Let me extend on this, will take a few mins.

Remember that for regular morphisms if f(x)=z and f=z you coudln't still say that x=y unless it is monic? Well, this wierd ass mapping defined with the division of the OP is not a morphism, not even a function, so even if you did prove that phi(0,0) = 0/0 = r and phi(0,0) = 0/0 = r+1 that does not imply that r = r+1, because that is an algebraic propiety of functions and halfway in your proof you show this division ain't one.

 

[CrystalShadow]

It doesn't actually require that this initial statement is practical, or has any relationships to anything whatsoever.

That is what they say to you, but historicaly this is BS. Was calculus developed just playing with random "what if we change this"? Was topology? Polinomial ecuations? Non-Euclidean geometry? Abstract algebra? Boolean logic? ANYFREAKING THING?

Fact is that math is inspired in physics, biology, chem, economy, even art; and currently we have deep ties with all those. AFAIK there ain't no single fruitful math field that was done for the sake of messing with the rules. Thats why i found this impressively hilarious:

That's not the point though. Maths doesn't require a practical use to be developed.

And as far as I know, Quaternions, and the related higher-dimensional forms were created with no real practical purpose or based on anything meaningful.

As it happens, the lowest-dimensional form (the one actually referred to as a quaternion) turns out to have a practical application.

The general idea behind it however was not developed based on anything practical.

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

Can you demonstrate it has a practical use, even if it was developed in a purely abstract way?

Yes, much math was developed as a means to study something else.
But that doesn't mean all of it was.
Nor that it fundamentally should.

 

[targren]

That's the right assumption, yes. But in what way is division as the inverse of multiplication "non-standard?"

I see what you're saying, but I disagree with your assessment. Unless you're contending that
r * 0 = 0 is not true for all r in R, I'm not sure how the substitution can be invalid.

Because this is not a well-defined funtion mate, even if 0/0=r and 0/0=r+1 that doesn't imply r=r+1 if that isn't a function or is not well-defined, which is not and you show that halfway in that proof.

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.

Although honestly, I think you just pointed out another way (treating 0/0 as a function instead of a value) that the proof by contradiction works. If phi(0,0) = u and phi(0,0) = w, and u != w, then the function phi(x,y) isn't a function at all.

 

[Tanakh]

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

Anyway, well, you know that any statement with a qualifier like "all" is a lie!! (haha, i love this stuff) But all the mayor fields of math are funded on real life problems and most of the succesfull sub-branches are also either inspired on them or bloom because there's an outside interest after some scientist finds them. Saying math "doesn't actually require that this initial statement is practical, or has any relationships to anything whatsoever" it's true, but it's like saying "Nintendo games don't actually require that his new game are based on franchises that they have grined to the bone", while true... well, not happens often and it's not the main focus.

 

[Tanakh]

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.