## Is 0/0 undefined or indeterminate?

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### Is 0/0 undefined or indeterminate?

I'm confused, my book says that it's undefined, but I've been reading on the internet that it's indeterminate.
krum
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### Re: Is 0/0 undefined or indeterminate?

krum » August 16th, 2015, 7:37 pm wrote:I'm confused, my book says that it's undefined, but I've been reading on the internet that it's indeterminate.

The way I learned it in school:

a/0 is 'undefined' when a≠0,
a/0 is 'indeterminate' when a=0.

Thus, unless nomenclature has changed since I was in school, the answer to your question is 'indeterminate'.

Darby
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### Re: Is 0/0 undefined or indeterminate?

Calculus books will refer to "indeterminate forms", because they play a special role in certain infinite series.

$0^{0}$ is one of them. As is

$\frac{0}{0} ,$

$\infty ^{\infty },$

$\frac{\infty }{\infty },$ and

$\infty ^{0}$

However, 1/0 is not an indeterminate form, and for that reason it can't be used in certain theorems. If those "undefined" a/0 things appear in certain limits of infinite series, math students can justifiably say the the answer is "infinity", and be technically correct. Depending on things like the textbook or the professor, the right answer may also be saying that the series (or limit) "diverges."

hyksos
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### Re: Is 0/0 undefined or indeterminate?

krum,

What's $\frac{x}{0}$? Okay, let's ask that question mathematically by:
1. Call the definition $k$.
2. Write that $k=\frac{x}{0}$.
3. Solve for $x$:
• $k=\frac{x}{0}$;
• $0{\times}k=0{\times}{\frac{x}{0}}$
• $0{\times}k=x$
• $x=0$.
4. Consider $x=0$.
• No definition $k$ would lead to a contradiction.
• So all possible definitions $k$ may be consistent.
• So we cannot determine a particular definition that $k$ must be.
• Call this inability to determine "indeterminate".
5. Consider $x{\neq}0$.
• Any definition $k$ would lead to a contradiction.
• So there is no possible definition $k$.
• Call this inability to define "undefined".

Therefore $\frac{x}{0}$ is:
1. indeterminate when $x=0$.
2. undefined when $x{\neq}0$.

Historically some mathematicians have failed to rigorously consider branching cases when they had fractions. For example, if their proof contained $\frac{x}{y}$ at some intermediate step but was later removed (e.g. by multiplication by $y$), then it may be hard to tell, but the proof may be flawed through reliance on $0{\times}{\frac{x}{0}}=x$. This hidden division-by-zero fallacy was the first major math fallacy that I was warned about.
Natural ChemE
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### Re: Is 0/0 undefined or indeterminate?

krum » Sun Aug 16, 2015 6:37 pm wrote:I'm confused, my book says that it's undefined, but I've been reading on the internet that it's indeterminate.

I hope this isn't off topic, but undefined suggests it can be, but just hasn't yet been defined. Would not undefinable be more accurate, in the context of the OP?

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### Re: Is 0/0 undefined or indeterminate?

Watson » August 17th, 2015, 12:58 pm wrote:
krum » Sun Aug 16, 2015 6:37 pm wrote:I'm confused, my book says that it's undefined, but I've been reading on the internet that it's indeterminate.

I hope this isn't off topic, but undefined suggests it can be, but just hasn't yet been defined. Would not undefinable be more accurate, in the context of the OP?

No. Classical Mathematics is actually quite clear and unambiguous on this point. It's been well covered in the previous posts.

Darby
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### Re: Is 0/0 undefined or indeterminate?

To put it more philosophically, x/x means a ratio of things - litterboxes to cats, doorknobs to doors, etc. Zero isn't anything, it isn't a real quantity, it is simply an absence, viz. nothing. Seen clearly, it has no place in a ratio. When you have no cats or litterboxes, then their ratio is indeterminate because there are no things to relate to each other. When you have ten cats and no litterboxes, then you have a ratio that is undefined - 10/0. You can't define a ratio because you only have one sort of quantity. And a yard full of cat poop. Does this sound right?

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### Re: Is 0/0 undefined or indeterminate?

So it is undefinable. I'm not sure how this is somewhat the same as indeterminable.

Watson
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### Re: Is 0/0 undefined or indeterminate?

Thay are not the same ... they are different terms each with their own specific definitions.

The inelegant but tried and true school methodology of learning may come in handy herr ... if understanding is not instantly forthcoming, rely on memorization until it does. Worked for me when I needed it.

Darby
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### Re: Is 0/0 undefined or indeterminate?

Natural ChemE » Mon Aug 17, 2015 1:26 am wrote:krum,

What's $\frac{x}{0}$? Okay, let's ask that question mathematically by:
1. Call the definition $k$.
2. Write that $k=\frac{x}{0}$.
3. Solve for $x$:
• $k=\frac{x}{0}$;
• $0{\times}k=0{\times}{\frac{x}{0}}$
• $0{\times}k=x$
• $x=0$.
4. Consider $x=0$.
• No definition $k$ would lead to a contradiction.
• So all possible definitions $k$ may be consistent.
• So we cannot determine a particular definition that $k$ must be.
• Call this inability to determine "indeterminate".
5. Consider $x{\neq}0$.
• Any definition $k$ would lead to a contradiction.
• So there is no possible definition $k$.
• Call this inability to define "undefined".

Therefore $\frac{x}{0}$ is:
1. indeterminate when $x=0$.
2. undefined when $x{\neq}0$.

Historically some mathematicians have failed to rigorously consider branching cases when they had fractions. For example, if their proof contained $\frac{x}{y}$ at some intermediate step but was later removed (e.g. by multiplication by $y$), then it may be hard to tell, but the proof may be flawed through reliance on $0{\times}{\frac{x}{0}}=x$. This hidden division-by-zero fallacy was the first major math fallacy that I was warned about.

Outstanding! Perhaps the clearest explanation of this that I have seen.
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### Re: Is 0/0 undefined or indeterminate?

Watson » August 17th, 2015, 4:34 pm wrote:So it is undefinable. I'm not sure how this is somewhat the same as indeterminable.

Yup, "undefinable" would be a good way to think about it.

Mathematicians say "undefined" because they mean to say that $\frac{x}{0}$ is undefined within elementary algebra. However, they see $\frac{x}{0}$ as definable since they're not constricted to elementary algebra.

If we were to take a more abstract algebra approach, we could just state
${\text{Watson}}{\left(x\right)}{\equiv}{\frac{x}{0}}$.
And I know what you're thinking: how's that help, right? Well, simple: the ${\text{Watson}}{\left(x\right)}$ function doesn't return a normal number, but rather some new mathematical entity. This new mathematical entity obeys
$0{\times}{\text{Watson}}{\left(x\right)}=x$,
so it doesn't suffer from the automatic inconsistency that all elementary entities do in the post that I made above.

What other properties does this new mathematical entity have? I dunno, I'm too lazy to make them up right now. And, heck, it's named after you, so you make them up. You're allowed to make up whatever you want so long as you rigorously avoid contradicting yourself.

To sum this up, yeah, you're right that "undefinable" would make more sense to most folks since we usually think about math in the context of elementary algebra. However, since mathematicians don't see things that way, to them it's moreso "undefined" than "undefinable".

-----

Separately, I forgot to address your question about how undefined and indeterminate are somewhat the same. Well, in a way, they're actually exact opposites:
1. undefined: No elementary description fits.
2. indeterminate: The set of elementary descriptions which fit includes non-equivalent members.
However, these two terms are similar in that they refer to cases in which there isn't exactly one known set of equivalent elementary representations.

By which I mean:
1. Most elementary entities are defined by only one set of equivalent representations. Examples of representations for $k=2$:
• $k=0+2$
• $k=0+1+1$
• $k=3-1+1-3+2.5-0.5$
2. Undefined entities have no elementary representations. Examples of representations for $k={\frac{2}{0}}$:
• [None. $\frac{2}{0}$ can't even represent itself because it leads to a contradiction; you're literally not even allowed to write it! It's meaningless gibberish.].
3. Indeterminate entities have more than one possible set of equivalent representations. Examples of possible sets of equivalent representations for $k=\frac{0}{0}$:
• Set of representations in which $k=0$, including:
• $k=0$
• $k=0+0$
• $k=1-1$
• $k=0*0$
• $k=0.5-\frac{1}{2}$
• Set of representations in which $k=1$, including:
• $k=1$
• $k=1+0$
• $k=2-1$
• $k=1*1$
• $k=1.5-\frac{1}{2}$
• Set of representations in which $k=2$, including:
• $k=2$
• $k=1+1$
• $k=2-0$
• $k=1*2$
• $k=1.5+\frac{1}{2}$
• [Etc., since $k$ can be just about anything in this case.]

To sum this up, "undefined" and "indeterminate" are similar in that they both indicate that there's not exactly one set of equivalent elementary representations. However, "undefined" indicates zero sets while "indeterminate" indicates more than one set.
Natural ChemE
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### Re: Is 0/0 undefined or indeterminate?

Exhaustively well said, but I think it might be easier for many to consider that undefined & indeterminate (or undefinable and indeterminable, as Watson put it) are somewhat similar only in a purely linguistic sense, whereas they are quite dissimilar in a purely mathematical sense ... kinda like 'alien' and 'illegal alien' are superficially similar linquistically, and yet are completely very different idiomatically.

Mathematics and linquistics share a lot of common ground, and indeed the former was birthed out of the latter, but it is contextually important to remember their differences and that they are not interchangeable - mathematics is more rigorously symbol oriented, whereas linguistics is generally more meaning oriented.

Math is a language unto itself.

Darby
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### Re: Is 0/0 undefined or indeterminate?

Watson » August 17th, 2015, 5:58 pm wrote:I hope this isn't off topic, but undefined suggests it can be, but just hasn't yet been defined. Would not undefinable be more accurate, in the context of the OP?

Here you have been shown that no mathematician would ever use a term like "undefinable", because if something is not defined within a domain of mathematics then a good mathematician would rapidly create a new domain where the thing IS defined (such domain would probably start from that definition itself).

For example they would use your (algebraically undefined) ratio 0/0 to represent a discontinuity in the function x/|x|: for x>0 the function has constant value 1, and its limit for x--> 0 (from the right) also is 1; for x<0 the function has value -1 and that is also its limit for x-->0 (from the left). Since at zero the function instantaneously jumps from -1 to +1, its value cannot be determined there. Still, it is defined, as you can plot and integrate the function with no problems.

[Forget this second part - it only is an attempt at pretending I am also a "good mathematician", simply because I share that attitude: if you find something that does not fit into your model, then claim that that's only a part of your model, you do have a better, more complex one!]

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### Re: Is 0/0 undefined or indeterminate?

if something is not defined within a domain of mathematics then a good mathematician would rapidly create a new domain where the thing IS defined (such domain would probably start from that definition itself).

Yeah you can do that, but it depends on the professor teaching it, and on the context where the expression appears during the course. If you have a series where the numerator approaches a finite value, say 2 , and the denominator is trending to zero
$\frac{2}{0}$
you can justifiably define a "new domain" where the real answer to the problem is $\infty$

Confusion arrises in introductory courses where the students are asked to instead provide the answer as "Does not exist".

The differentiation between "undefined" and "indeterminate" goes beyond pathological issues presented by Natural ChemE. In l'Hopital's Rule, the expression is required to be indeterminate. Otherwise, in expressions that are undefined, the theorem does not hold. (more importantly) you cannot use the technique to solve the limit.

hyksos
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### Re: Is 0/0 undefined or indeterminate?

hyksos,

That's a common misconception, but L'Hôpital's rule isn't meaningfully related to this topic.

L'Hôpital's rule is an alternative way to evaluate the limit of fractions. While it's usually more work than simpler methods, sometimes it's actually easier or/and provides a better result.

The commonly discussed case for L'Hôpital's rule is when just substituting in the approached value would result in either $\frac{0}{0}$ or $\frac{{\infty}}{{\infty}}$. The problem with both of these results is that they're not very descriptive, i.e. they're indeterminate. Math books often recommend considering L'Hôpital's rule when this happens since it might provide a determinate result.
Natural ChemE
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### Re: Is 0/0 undefined or indeterminate?

Natural ChemE.

L'Hôpital's rule is not just for $\frac{0}{0}$ and$\frac{ \infty }{ \infty }$ as you have claimed. The rule will work on any situation that is an indeterminate form. You have given two examples . There are many more, such as $0^0$ and several more others.

However, $\frac{2}{0}$ is NOT indeterminate! That's where the danger comes in. L'Hôpital's rule cannot be applied to situations that are "formally infinity", or in limits and series that genuinely diverge.

The main thing I wanted to communicate here is : yes, I agree with you that students are told to plug-and-chug L'Hôpital's rule, as you have described. However, dig in the literature a little deeper. L'Hôpital's rule is buttressed by two historical proofs and their corresponding theorems. This differentiation between indeterminate and undefined is very real in mathematics, and it has real consequences in theorems.

hyksos
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### Re: Is 0/0 undefined or indeterminate?

hksos,
Natural ChemE » August 31st, 2015, 7:55 pm wrote:The commonly discussed case for L'Hôpital's rule is when just substituting in the approached value would result in either $\frac{0}{0}$ or $\frac{{\infty}}{{\infty}}$.
hksos » September 12th, 2015, 4:17 pm wrote:L'Hôpital's rule is not just for $\frac{0}{0}$ and$\frac{ \infty }{ \infty }$ as you have claimed.
How the heck did you get from my statement to your representation of it?

There's a lot wrong with what you've said. Unless you can demonstrate some relevance, I'll just split 'em off to avoid polluting a good thread.
Natural ChemE
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### Re: Is 0/0 undefined or indeterminate?

[quote="[url=http://www.sciencechatforum.com/viewtopic.php?p=286235#p286235]Natural ChemE » August 17th, 2015, 1:26 am[/url]"]krum,

What's $$\frac{x}{0}$$? Okay, let's ask that question mathematically by:[list=1]
[*]Call the definition $$k$$.
[*]Write that $$k=\frac{x}{0}$$.
[*]Solve for $$x$$:
[list]
[*]$$k=\frac{x}{0}$$;
[*]$$0{\times}k=0{\times}{\frac{x}{0}}$$
[*]$$0{\times}k=x$$
[*]$$x=0$$.[/list]
[*]Consider $$x=0$$.
[list]
[*]No definition $$k$$ would lead to a contradiction.
[*]So all possible definitions $$k$$ may be consistent.
[*]So we cannot determine a particular definition that $$k$$ must be.
[*]Call this inability to determine "indeterminate".[/list]
[*]Consider $$x{\neq}0$$.
[list]
[*]Any definition $$k$$ would lead to a contradiction.
[*]So there is no possible definition $$k$$.
[*]Call this inability to define "undefined".[/list][/list]

Therefore $$\frac{x}{0}$$ is:[list=i]
[*]indeterminate when $$x=0$$.
[*]undefined when $$x{\neq}0$$.[/list]

Historically some mathematicians have failed to rigorously consider branching cases when they had fractions. For example, if their proof contained $$\frac{x}{y}$$ at some intermediate step but was later removed (e.g. by multiplication by $$y$$), then it may be hard to tell, but the proof may be flawed through reliance on $$0{\times}{\frac{x}{0}}=x$$. This [url=https://en.wikipedia.org/wiki/Division_by_zero#Fallacies]hidden division-by-zero fallacy[/url] was the first major math fallacy that I was warned about.[/quote]

The bottom line is that 0/0 is not equal to any number and a/0 where a is not 0 is not any number.
Robert Kolker
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### Re: Is 0/0 undefined or indeterminate?

It is indeterminate.
williamjohn
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### Re: Is 0/0 undefined or indeterminate?

n/0 does not exist, for all n.

1. x/y defined as (the z such that y*z = x).
2. 0*z = 0, for all z.

n/0 = (the z: 0*z = n).

If ~(n=0) then there is no z such that 0*z = n.

e.g. (1/0) = (the z such that 0*z = 1).
But 2 asserts that all values of 0*z are equal to 0,
i.e. there is no value of z for which 0*z=1,

If there is no value of z such that 0*z=1 then
(the z such that 0*z = 1) does not exist.

(the z: 0*z = n) is not unique for all n > 0.

If n=0 then there is more than one value of z such that 0*z=0.
0*z=0 is true for all z.

i.e. there is no unique value of z such that 0*z = 0.

That is to say, n/0 does not exist for all values of n,
even though n/0 is defined.
Owen
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