0 divided by 0

D H

I think you completely missed the point of my post, which was that the correct term for 0/0 is that it is indeterminate rather than undefined. 1/0 is undefined, but 0/0 is indeterminate.

0/0 cannot be given meaning, period. 1/0 can be given a meaning in various contexts. a/0 is ∞ on the projective real number line for all non-zero a. In complex analysis, a/0 (with a≠0) is sometimes treated as complex infinity, a number whose magnitude is greater than any real number but whose argument is indeterminate.

wilsonb

In layman terms, in any value at all, there can be an infinite amount of 0s. Hence there can also be infinite 0s in a single 0. hence indeterminate.

pwsnafu

That's not what "indeterminate" means. It's a technical term. It has a definition and is used only for that definition. You seem to be trying to rationalize the word choice. Definitions don't work like that.

PS: If we take your logic as is, then you are arguing all questions whose answer is "zero" is indeterminate. Or (worse) all finite answers are indeterminate. Neither is correct.

mathwonk

arrggggh.

Anti-Crackpot

Can't help but wonder if Godel would agree with this statement as opposed, for instance, to the following statement:

"Sufficiently complex mathematical systems cannot be contradiction-free."

via Wikipedia...
"Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic."
http://en.wikipedia.org/wiki/Gödel's...eness_theorems

arildno

An utter misconception.

How do you "prove" that a poker hand can have 5, and only 5 cards?

It is laid out as a rule of the game, and in like manner, maths is a game where we pick whichever rules we want to play with. Those rules are called "axioms"

Obviously, we may construct as many maths games we want. Just like we can invent new card games, by laying down some new set of rules.

pwsnafu

Nonsense. If you are going to bring up the incompleteness theorems, at least learn them. The first theorem says you can't be contradiction free and be complete at the same time (at least for anything which includes PA). It says nothing about just being consistent. We prioritize consistency over completeness.

micromass

Godel would certainly agree with DH. Godels incompleteness theorems state (more or less):

1) Our current mathematical system can never be shown to be contradiction-free.
2) A mathematical system can never be AND complete AND contradiction-free.

(1) is certainly problematic, but it does not mean that there are actually contradictions in current math. We just can't prove it. So we more or less accept it on faith.

arildno

And, to add to micromass:
Jus because there MIGHT be some contradiction deeply embedded,as yet unrecognized bu us, in our preferred mathematical system, we need not worry too much about it, since if we DO notice it, it might well be easily remedied, by, for example, adding some pedantic little detail in an axiom formulation that doesn't do anything else than preventing just that contradiction from happening.
All previous results that (1) did NOT depend upon the flawed axiom to begin with, and (2) won't depend upon the new axiom would remain unaffected, and perhaps practically all the results which DID use the flawed axiom to begin with.

The contradiction might be more lethal than that, of course, but that it should be so is no implication that follows from the fact that we do not know if our current system is free of contradictions.

D H

As others have noted, that is not what Godel's theorems say. What they do say is that in in any sufficiently complex mathematical system, (a) there will exist statements written using the constructs of the system that can neither be proved nor disproved using the constructs of that system, and (b) that the system is mathematically consistent is one of those statements that cannot be proved or disproved.

In other words, mathematical systems of sufficient complexity cannot be both consistent and complete. Consistency is essential, but completeness is not; it's just a nice thing to have. Mathematicians have given up on completeness (Hilbert's second problem) and assume consistency.

wilsonb

You can switch em using calculus.
and 0/0 is just a product statement, it might have came from some complicated f(x) = h(x)/g(x)
where one simply got 0/0, when computing the limit.

undefined is a statement, where you get R/0, where R is any real number.
0/0 or anyother form of 0^n/0^n, when u compute limits of f(x) as a whole,
which give rise to the method of L'Hopital, when computing limit of f(x), when h(x)/g(x) gives you 0/0.

its 2 different things, one is the axiom & another is taking limits and tend to 0/0.
Additional Reference: Steward J' Calculus 7th ed.

micromass

No. You are totally missing the point of limits. In limits, we never get the operation 0/0.

If we have numbers 0 and 0, then its division 0/0 is not defined. This has nothing to do with limits or with functions.

If you have functions, then you can perhaps deal with [itex]\lim_{x\rightarrow 0}\frac{\sin(x)}{x}[/itex]. But, you do NOT divide by 0 here. The limit means that if you get close to 0, then the expression sin(x)/x gets close to 1. You NEVER evaluate the function sin(x)/x in 0. So you NEVER deal with 0/0.

wilsonb

Mathematics are not just about the calculations. The calculations are to solve a particular situation, a problem statement, in which this mere question does not have.
This question lacks the problem statement we need to determine how we can solve it mathematically.

To put it in a sense where 0 = nothing, so 0 / 0 is definitely 0. There is nothing to divide it from and with, but the statement of 0 = nothing made it possible as nothing is required to divide nothing.

In short, UNDEFINED means it is undetermined UNLESS a problem statement is issued. Therefore, undefined can be anything as long as you fulfill the requisite and criteria of the conditions.

micromass

No.

No. Not in this case at least.

Mark44

No problem statement is needed, other than the main idea of this thread, which is what does 0/0 mean?

There is really nothing more complicated here than the arithmetic involved in the division of two numbers. It has been mentioned before in this thread that the division operation requires two input numbers, but an important point has been omitted: from division we require exactly one result. We require a single answer from all of the other arithmetic operations - why should division be any different?

The argument that 0/0 = 0 arises incorrectly from the fact that division and multiplication are inverse operations. If a/b = c, then a = b * c. This is true as long as b ≠ 0.

If we insist that 0/0 = 0 makes sense because 0 (the denominator) * 0 (the quotient) = 0 (the numerator), then we should also accept 0/0 = 2, because 0 * 2 = 0. Since we have gotten two different answers (and infinitely more are possible), this is a violation of the commonsense requirement that division produce a single result.

The upshot is that dividing by 0 is never defined, period.

Robert1986

I just got an idea for my dissertation: I'm going to study the patterns and cycles by which threads like this get started on these forums. I can't wait until the next .999999... thread comes along. Then perhaps a thread from an arrogant ignoramus who is convinced that there is something fundamentally wrong about the real line.

micromass