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[Simmura McCrea]

Even then, you get change of an unknown amount.

[ed boy]

I would simplify it down to the statements that

a. 0 is defined s.t. X*0 = 0*X = 0 for all X in ring R (axioms of ring)
b. m/n = p when np = m
c. X/0 = p, X =/= 0 has 0p = X; contradiction, the construction does not exist for n = 0, given the existence of a unique unity.

This way is most beautiful to me, though you may feel free to disagree =)

Technically, your first point is incorrect. X*0=0 is not a standard axiom of rings. Instead, 0 is defined as the additive identity and this property is deduced from the distributivity of multiplication.

And to adress the OP, you have to think of dividing as the inverse of multiplication - that is, the operation "divide by two" is the inverse of the operation "multiply by two". For any non-zero X, the operation "multiply by X" is injective (if A*X=B*X, then A=B) and surjective (You can get any real number by multiplying something else by two). This means that an inverse exists (as it is injective) and is defined for all real numbers (as it is surjective).

However, note that the operation "multiply by 0" is not injective (1*0=0=2*0), and it is not surjective (0*X=0 for all X). If an inverse were to exist, it could be be defined for any non-zero X (as there would be no Y such that Y*0=X). Furthermore, an inverse cannot be defined for zero, since it is not injective as zero (There are multiple numbers X such that X*0=0, so how do we know which one to give for 0/0?), and therefore an inverse to dividing by zero does not exist anywhere.

[Vector]

Hmm, I'll maintain that it's correct because it the development follows directly from the axioms of a ring, but you're right, I was thinking about it all wrong.  Should have fixed the wording.  And been less stupid.


Also, that is slick.  Not my preferred explanation, but slick.

[MonkeyHead]

Personally, I am a fan of renormalization - typical Physicist right?

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

[Siquo]

And to adress the OP, you have to think of dividing as the inverse of multiplication - that is, the operation "divide by two" is the inverse of the operation "multiply by two". For any non-zero X, the operation "multiply by X" is injective (if A*X=B*X, then A=B) and surjective (You can get any real number by multiplying something else by two). This means that an inverse exists (as it is injective) and is defined for all real numbers (as it is surjective).

However, note that the operation "multiply by 0" is not injective (1*0=0=2*0), and it is not surjective (0*X=0 for all X). If an inverse were to exist, it could be be defined for any non-zero X (as there would be no Y such that Y*0=X). Furthermore, an inverse cannot be defined for zero, since it is not injective as zero (There are multiple numbers X such that X*0=0, so how do we know which one to give for 0/0?), and therefore an inverse to dividing by zero does not exist anywhere.

From that reasoning, for any X/0, you'd just get the collection "All real, imaginary, and other numbers/scalars/vectors/whatevers we haven't invented/discovered yet".

*reads stuff* Oh yay I made up one of the correct answers (unsigned infinity/Riemann sphere), except that it's totally useless as you can't do anything with it (like, multiplying it by zero is undefined again, for instance)

[ZetaX]

One can construct something called nonstandard/Conway/surreal numbers and that's the closest you will get to division by zero. Those implement the concept of "infinitesimal small positive numbers" ( >0 but < ever positive real number) and "infinite numbers" ( > every real number), but still give a ring (thus division by zero is still not possible) and even a field. This gives you a setting where you can do some intuitive things (but counterintuitive if you are used to standard analysis) stuff.
Example: We write x~y if x and y are infinitesimal close together, i.e. if x-y is infinitesimal small (positive or negative) or zero. Then a function f is continuous iff x~y implies f(x)~f(y) [in layman's terms: if x and y are close, then so are f(x) and f(y)].
Other example: If f is differentiable at x, then one has f'(x) ~ (f(x+h)-f(x))/h, where h~0 but h is not 0.

[MonkeyHead]

A Thermodynamics Prof once showed me a mathematicl proof that linked absolute zero to an infinite temperature. I wish I could remember the details as I seem to recall it would fit quite well in this thread...

[Starver]

Hella expensive cookie, bro. Could buy a lot of cookies for a tenner over here, but you'd still end up with change.

If I may be numismatistical, for a moment, until a couple of days ago when I spent some of them I could have paid for £20-worth of goods with the ten largest coins in my pocket.  And that's just the largest of the commonly-circulated coins.  A few years back I also accepted a £5 coin (the Diana commemoration one) in change, and I bought (for that aforementioned tenner!) the Dome's version of the Millenium £5 coin when I went there[1], just because.  They are more expensive, these days, but I probably would have been better investing the money in more standard markets.  Well, maybe not, given all the booms and busts since.


If it hasn't already been said, Simmura's "you can put into anything but zero piles" explanation trivially fails when you try and put them into two-and-a-half piles (even though we know there's four whole coins in each, with the "half a pile" having half of that).  But that's not actually an argument that having solved the non-integer number piles issue you can handle null piles.


[1] Yes, I went there.  Depending on which press reports you believed, I was one of the few people who bothered.  Only on the day I went there were far too many people in there to be able to see any of it...

[Pnx]

In my functions stats and trig class, it was said that dividing by zero gives you one...
And that's about as much as my math knowledge can contribute...

I'm typing completely one handed.

[DJ]

If it hasn't already been said, Simmura's "you can put into anything but zero piles" explanation trivially fails when you try and put them into two-and-a-half piles (even though we know there's four whole coins in each, with the "half a pile" having half of that).  But that's not actually an argument that having solved the non-integer number piles issue you can handle null piles.

Change it to equal piles and it works.

[Nadaka]

In my functions stats and trig class, it was said that dividing by zero gives you one...
And that's about as much as my math knowledge can contribute...

I'm typing completely one handed.

Power of 0 gives you 1, not division.

[Kay12]

In my functions stats and trig class, it was said that dividing by zero gives you one...
And that's about as much as my math knowledge can contribute...

I'm typing completely one handed.

Power of 0 gives you 1, not division.

Yep. That's another thing I've had to explain to people: "Whaddya mean X^0 equals one? Why not <X or 0 or anything in between>?"

[alway]

Oh hey, wikipedia has a page for 'mathematical fallacies' with an entire section on dividing by zero and why it's a bad thing. http://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero

Essentially, if you divide by 0, you can make any number equal any other number.

[Darvi]

To me dividing by zero means cutting it into bits with an oval chakram.

[MonkeyHead]

Essentially, if you divide by 0, you can make any number equal any other number.

Yea, but cant you do this anyway by approximating certain fractions to a terminating decimal?