Wikipedia:Reference desk/Archives/Mathematics/2016 May 15

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May 15

infinity/infinity

If Infinity is divided by infinity, is the answer 1? Or can the answer be anything you want depending upon how the infiniteies were derived?--178.106.99.31 (talk) 17:54, 15 May 2016 (UTC)

Infinity is not a number, so dividing infinity by infinity is undefinable. ←Baseball Bugs ^{What's up, Doc?} carrots→ 17:59, 15 May 2016 (UTC)

No, that's not an answer. Infinity can be a number in some contexts — see for example Riemann sphere, aka the "extended complex numbers". But even in the Riemann sphere, you can't divide infinity by infinity, though you can add it to anything except infinity, and multiply it by anything except zero. --Trovatore (talk) 20:12, 15 May 2016 (UTC)

some infinite sets can be placed in one to one correspondence with other infinite sets, so they can be divided out in a sense along these lines (ie..one to one) but other infinite sets are strictly larger than others (real numbers vs integers) so cannot be divided out one to one)...68.48.241.158 (talk) 18:03, 15 May 2016 (UTC)

I think a case can be made that the Dirac delta function is one special case. I believe also that 0/0 is the same as infinity/infinity and it's apparent that x^2/x is 0 for x=0. These relate to the series expansion above. --DHeyward (talk) 18:25, 15 May 2016 (UTC)

0 divided by 0 is also undefined. ←Baseball Bugs ^{What's up, Doc?} carrots→ 18:28, 15 May 2016 (UTC)

But for continuous functions, it's important to point out that it's not indeterminate just because it evaluates numerically to 0/0. x^2/x is not indeterminant undefined for any value of x even though straight substitution is 0/0. The answer is "0". it's trivial as to why but is the starting point for evaluating series and functions that converge to 0 or infinity at different rates. --DHeyward (talk) 21:37, 15 May 2016 (UTC)

Literally speaking, you cannot evaluate x²/x at x=0. You can take the limit as x approaches 0, and that limit is 0. But that is not the same thing. --Trovatore (talk) 21:45, 15 May 2016 (UTC)

It trivially reduces to f(x)=x^/x=x . It seems odd that f(0)=x^2/x is only 0 in the limit but f(0)=x=0 is not. That seems to defy laws of equivalence/identity. --DHeyward (talk) 22:29, 15 May 2016 (UTC)

It doesn't reduce to x at x=0. The identity ax/bx=a/b assumes x≠0. --Trovatore (talk) 22:38, 15 May 2016 (UTC)

More precisely it can be evaluated in the limit and shown to be 0 and limits are tools for resolving it. That the function f(0)=x/x=1 and f(0)=x^2/x=0 shows that evaluating to 0/0 is indeterminant but not undetermined for any function. They are substitutional identities. The identity ax/bx=a/b does not have any discontinuities and is a/b when x=0. --DHeyward (talk) 22:49, 15 May 2016 (UTC)

No, sorry, that is incorrect. The identity does not hold when x=0. --Trovatore (talk) 22:55, 15 May 2016 (UTC)

I should say, there are things you could correctly mean by what you say. For example, it's true if you mean it to be interpreted in the ring of rational functions over x. In that case, you're not dividing the number ax by the number bx, but rather the function λx.ax by the function λx.bx. (Does anyone know what the html is for $\mapsto$?) But this is not the usual interpretation of the claim; if you want to be correct you have to make that clear.) --Trovatore (talk) 23:11, 15 May 2016 (UTC)

In the cases I provided, they demonstrate "indeterminant" but the different functions are differentiable. It's my understanding that in those cases, the solution is exact and the functions are continuous without exception for x=0. L'Hospital's rule seems to apply. Am I missing something? --DHeyward (talk) 08:08, 16 May 2016 (UTC)

Yes. You're missing what x²/x, evaluated at x=0, actually means. What it means is, you take 0 and square it. Then you take 0. Then you divide the first by the second; that is, 0/0. Not the "form" 0/0, but literally zero divided by zero, which is undefined.

By default, that is what it means. Period, end of discussion. There is no opportunity to apply L'Hospital's rule; it is completely irrelevant. --Trovatore (talk) 08:12, 16 May 2016 (UTC)

But i would not say 0/0 is undefined, rather it is indeterminate. It can have multiple solutions depending on function ans space, including undefined but it can also have very defined solutions such as 1,0,infinty, etc. x²/x doesn't have a discontinuity at x=0. It is 0. When faced with indeterminant substitutions, there are first principle derivations that lead to a solution. Plugging in numbers demonstrates "indeterminant" but it is not the same as "undetermined" or "undefined." Continuous functions are continuous even through values that substitute into an indeterminant form. f(x)=x^2/x is continuous with solutions throughout real values of x including 0. --DHeyward (talk)

You are wrong, Trovatore is right. The formula x^2/x is not defined at x=0. As a function, it has a removable discontinuity. In many contexts one automatically "fills in" values for removable discontinuities, but the "filling in" is a step that has substance, replacing something that isn't defined by something that is. --JBL (talk) 17:03, 16 May 2016 (UTC)

In general, functions are defined by both their domain and the value they give for every member of the domain. So the function $f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto x$ is a different function than $f:\mathbb {R} \backslash \{0\}\to \mathbb {R} ,\ x\mapsto x$. Also in general, given two real functions f and g, $f/g$ is defined as the function whose domain is the set of all elements x in the domain of both f and g for which $g(x)\neq 0$, and which maps every such x to $f(x)/g(x)$. If you apply this definition to the functions $f(x)=x^{2},\ g(x)=x$, you'll see that $x^{2}/x$ does not have 0 in its domain, that is, it is undefined there. So it is a different function from the identity on $\mathbb {R}$. -- Meni Rosenfeld (talk) 20:11, 16 May 2016 (UTC)

Also, "undefined" quite literally means "something which was not defined". The definition of division of real numbers is: "Let $a,\ b\in \mathbb {R}$ where $b\neq 0$. Then $a/b$ is the unique real number c such that $bc=a$." You'll note that this definition doesn't address the case that the denominator is 0, so 0/0 is undefined.

Saying that "0/0" is an indeterminate form is a high-level description of what happens in the limit of functions that approach 0 (namely, that their ratio can be anything, depending on what the functions are). It's not really a statement about 0/0 itself. You can't make statements about it, it's undefined. -- Meni Rosenfeld (talk) 20:16, 16 May 2016 (UTC)

Okay, more book cracking: Removable singularity would be the term and I suppose the proper way to include x=0 in the domain of the function x^2/x is $f(x)={\begin{cases}x^{2}/x&x\neq 0,\\0&x=0.\end{cases}}$ , however even without the explicit inclusion of x=0 in it's domain, it is still continuous. Being a removable singularity (different from a removable discontinuity because it's not discontinuous anywhere as infinitesimal changes still converge - I think that means it can be represented by a conformal map but it's been a while since I did that). It's certainly well defined whether 0 is in the domain or not. (p.s. I hate math.) --DHeyward (talk) 03:20, 17 May 2016 (UTC)

It's both a removable singularity and (as mentioned by JBL) a removable discontinuity. For our purpose it's more relevant to talk about it being a removable discontinuity.

If you don't include 0 in the domain, then sure, the function is continuous in its domain; but it's not continuous on $\mathbb {R}$ (it can't be, without being defined on all of $\mathbb {R}$).

If you hate math, why do you participate in discussions on the math reference desk? -- Meni Rosenfeld (talk) 07:44, 17 May 2016 (UTC)

There's no such thing as a discontinuity at a point not in the domain. --Trovatore (talk) 08:38, 17 May 2016 (UTC)

This seems suspicious. Wouldn't you say that $x^{2}/x$ (aka $f:\mathbb {R} \backslash \{0\}\to \mathbb {R} ,\ x\mapsto x$) has a removable discontinuity at 0? If so, isn't a removable discontinuity a special case of discontinuity? And if not, don't the terms "removable discontinuity" and "removable singularity" lose much of their applicability? -- Meni Rosenfeld (talk) 17:44, 17 May 2016 (UTC)

It's not a discontinuity. "Removable singularity" is OK. --Trovatore (talk) 17:48, 17 May 2016 (UTC)

It seems our article on this mildly supports your version (it considers my notion of removable discontinuity nonstandard but not unheard of). Let's just say that I disagree, this is incompatible with both my experience and my common sense. -- Meni Rosenfeld (talk) 20:56, 17 May 2016 (UTC)

See Indeterminate form. --Kinu ^t/_c 19:34, 15 May 2016 (UTC)

Indeterminate forms are quite common with +-infinity. With only real numbers (i.e. no infinities) there are only 4 indeterminate forms; 0/0, 0 to the 0, the zeroth root of 1, and the logarithm of 1 in base 1. Georgia guy (talk) 20:41, 15 May 2016 (UTC)

Your latter two are not actually traditional "indeterminate forms". There's a temptation to want to come up with an abstract notion of "indeterminate form" and figure out what else it includes besides the traditional ones, but personally I think it's a waste of time. "Indeterminate form" is a historical notion, pretty much superseded now that we have rigorous notions of continuity and so on. It's still useful for teaching calculus, because it serves as a sort of warning system for where common expressions are discontinouous, but I don't believe it's useful to extend it.

So there are exactly seven indeterminate forms: 0/0, 0⁰, ∞−∞, ∞/∞, 0×∞, 1^∞, ∞⁰. Its a closed-ended list and will never be extended, not because there aren't other expressions that are arguably similar, but because there's just not much point. --Trovatore (talk) 20:59, 15 May 2016 (UTC)

Actually, I looked up the dates, and it appears as though the notion of continuous function actually predates the notion of "indeterminate form", so I have to backpedal a little bit on the historical sequence I suggested above. I still think the basic thrust of what I said is correct, though — once you understand continuity, you don't really need "indeterminate forms", but they're still useful as a sort of reminder of things to watch out for, but not useful to extend them. --Trovatore (talk) 21:43, 15 May 2016 (UTC)

Sort of playing the devil's advocate here, but the "indeterminate forms" are precisely those indeterminate operations that can be constructed by the usual concepts of addition, multiplication, and division – with exponentiation thrown in for good measure, although I think there are probably some reasons for regarding that as less fundamental. The often cringeworthy "L'Hopital's rule" tells us that one way of making sense of expressions like $\infty /\infty$ is to change what one means by ∞ (or, for the other forms, 0). It is not the "number" infinity that one is dividing, but rather an order of growth. Thus, I believe, to make sense of limits such as these, one is in effect thinking of the "number" $\infty$ as a point of the Stone-Cech compactification of the real line. (Maybe the "Hewitt compactum"?) Sławomir Biały (talk) 21:49, 15 May 2016 (UTC)

Well, in this context, though, I don't see any good reason to exclude the complex numbers. So if you want to generalize, you probably ought to include e^i∞, or equivalently sin(∞), or i^∞. But my point is that I don't think it's a good idea to generalize. The notion is probably just barely worth learning in calculus, but to take it beyond that is to focus on the wrong things. --Trovatore (talk) 22:07, 15 May 2016 (UTC)

Oh, to be clear, I'm not saying there might not be interesting math to do along these lines. But I would avoid calling it "indeterminate forms", which is a name for a specific pedagogical aid that people need to first understand, then understand the limitations of. --Trovatore (talk) 22:27, 15 May 2016 (UTC)

This is a perennial question, and I think it is worth attempting to give a real answer. I am not trying to generalize it to complex numbers because, in most cases when someone is tempted to write ∞/∞, they are certain that they mean something, and that something usually does not involve complex numbers. For example, when a calculus student writes $\infty /\infty =1$, they need to be told gently that $\infty$ is a concept that does not behave as other numerical quantities do. The "two infinities" are not "the same infinity". Instead, the relevant concept of infinity is really that of an order of growth. For instance, although it is true that the polynomials $n+1$ and $n^{100}+1$ both tend to infinity, they do not go at the same rate, and therefore they occupy "different infinities". This is, of course, merely an intuitive explanation. Part of our task as post-modern mathematicians is to assign meaning to this sort of sensical nonsense. What is the sense in which we really mean that there are different infinities? What is the space of all infinities? Does it have a topology? What algebraic structures does it support? Etc. Also, I disagree about indeterminate forms as merely a pedagogical crutch. And in any case it is not very relevant to the original question whether indeterminate forms as such are a complete list or no.

I guess this is related to the ultrafilter construction of the hyperreals. Sławomir Biały (talk) 23:17, 15 May 2016 (UTC)

I take issue with Trovatore. Infinity can be added to infinity and the answer is infinity. This video explains how [1]. 80.44.167.65 (talk) 14:34, 16 May 2016 (UTC)

Trovatore's comment is about infinity in the complex numbers. --JBL (talk) 17:06, 16 May 2016 (UTC)

For clarification, this means that Trovatore's infinity is infinity in any direction, rather than positive infinity as defined by Extended real number system, which distinguished the 2 signed infinities. Infinity plus negative infinity is an indeterminate form (a variant of infinity minus infinity that is one of Trovatore's above list of indeterminate forms.) Georgia guy (talk) 17:38, 16 May 2016 (UTC)

More generally, there are many, many kinds of infinity. I don't mean many sizes, I mean many kinds. Some such kinds have a further classification of infinity to different sizes, or directions, or whatever. You can't talk about what "infinity + infinity" is without first specifying what kind of infinity you mean. If we're talking about extended real numbers then yes, $(+\infty )+(+\infty )=(+\infty )$. In the real projective line or Riemann sphere, no, $\infty +\infty$ is undefined. With cardinal numbers, the sum of infinities is their maximum.

https://www.youtube.com/watch?v=23I5GS4JiDg does a very good job of covering the different kinds of infinity that are out there. -- Meni Rosenfeld (talk) 07:51, 17 May 2016 (UTC)

It's true that there are different infinities. But most of these are not relevant in the description of the classical "indeterminate form" ∞/∞, which is a comparison of the order of growth. What (if anything) is the concept of infinity that most closely models this phenomenon? Cardinals and ordinals and general concepts of infinity are not relevant. Clearly the infinities of classical indeterminate forms correspond to certain specific points of the Stone-Cech compactification of R. The question is, which points? Sławomir
Biały 17:54, 17 May 2016 (UTC)

I'd say the Extended real number line (aka two-point compactification of $\mathbb {R}$) and its point $+\infty$ are the relevant concepts here. -- Meni Rosenfeld (talk) 20:56, 17 May 2016 (UTC)

No, that's completely wrong. It fails to clarify the "indeterminate form" $\infty /\infty$. For example, the two functions $n+1$ and $n^{2}+1$ must tend to different points of the compactification. One can detect this easily by the limit $\lim _{n\to \infty }{\frac {n+1}{n^{2}+1}}=0$. If there is only one point at infinity, we cannot (toplogically) distinguish between the limits of these two sequences. They are both "infinity", one is not a "bigger infinity" than the other. Sławomir Biały (talk) 21:31, 17 May 2016 (UTC)

I guess I didn't understand your original question. What I meant is $\infty /\infty$ is indeterminate precisely because we are using only a single size of infinity. We have different rates at which functions can tend to infinity, and only one infinity, so the ratio of functions can tend to anything. Just like $0/0$ is indeterminate - we have different rates of approaching 0, and only one 0, so the ratio can be anything.

You can use an extension of the reals that has different sizes of infinity, like the hyperreal or surreal numbers. But then there's no such thing as "$\infty /\infty$", there are things like ${\frac {\omega }{\omega ^{2}}}={\frac {1}{\omega }}$, which is just a specific size of infinitesimal. The concept of "indeterminate form" becomes moot.

If you mean, what is the correspondence between functions of different growth rates and different points in the extended numbers - if we're talking about Surreals (which is what I'm most familiar with, even though they're not a set), I'd say that the identity function corresponds to $\omega$, and arithmetic operations behave normally (e.g., the function $x^{2}+1$ corresponds to $\omega ^{2}+1$. -- Meni Rosenfeld (talk) 07:27, 18 May 2016 (UTC)

OP here. I thought that x^2/x = x, so why are you talking about the complications of evaluating x^2/x at x=0? Its evident to me that simplifying the function leads to an unambiguous answer. If f(x)=x, then f(x)=0 when x=0. Or am I missing something?--178.106.99.31 (talk) 22:41, 16 May 2016 (UTC)

x^2/x = x is true when x is not equal to 0. If x=0 in that equation, then it says 0/0=0, which is not correct. Gap9551 (talk) 23:02, 16 May 2016 (UTC)

So every time we see x^2/x, we can only simplify it to x if we are sure that x will never be 0? Take the function f(x)=x. This can be evaluated at x=0. Therefore, you are impling that x^2/x != x. Get out of that!--178.106.99.31 (talk) 00:05, 17 May 2016 (UTC)

$x^{2}/x=x$ is only true if $x\not =0$. There are some famous mathematical fallacies that rely on the exceptional case. For example, you can use this to prove that 1=0, which I think we can agree is not true. Sławomir Biały (talk) 00:13, 17 May 2016 (UTC)

However, x=0 is a removable singularity for $x^{2}/x$ however as stated above by someone more knowledgeable than me, x=0 is not within the domain of the function. I don't think that means it's not well-behaved or discontinuous, though. --DHeyward (talk) 03:20, 17 May 2016 (UTC)

Sure, it's well-behaved. It's continuous on its entire domain, which does not include the point 0.

But what you need to get, with no doubts and no half-measures, is that it is not defined at x=0. That's just how terms are evaluated in math. When you evaluate it at x=0, you get a numerator of 0 and a denominator of 0, and at that point you can forget everything else; it's just plain 0/0 and remembers nothing at all about the expression it came from. And 0/0 is not defined. --Trovatore (talk) 04:51, 17 May 2016 (UTC)

I get that, but there is also the observation $f(x)=x^{2}/x$ has a holomorphic function $g(x)=x$ that is a continuous replacement for f(x) over f(x)'s entire domain and removes the singularity at x=0. As I understand it, they are equivalent functions making the singularity meaningless/removeable. --DHeyward (talk) 06:46, 17 May 2016 (UTC)

That is true, but that is in a context with lots more assumptions. The literal interpretation of the notation, in a general context, makes x²/x undefined at x=0.

Here's why I think it's important to emphasize it: I think a lot of times, calculus students learn to do this sort of limit by cancelling stuff out of the numerator and denominator and then plugging in the value where they want to take the limit. Then they get a well-defined value, and lo and behold that is the value they want. It's not a coincidence — but it's not what it means to take the limit. It actually takes a couple of steps to justify that it gives you the correct limit. They aren't hard steps, but they are steps, and more importantly they remind you what's actually going on, which is not just formal algebra on the numerator and denominator. --Trovatore (talk) 07:18, 17 May 2016 (UTC)

I agree with you. My mathematics is mostly focused in engineering rather than theory and it took me a bit to find the terms to to speak to theory. 0/0 is undefined and the functions that return it are indeterminate. My last mathematics course was complex variables many moons ago (post graduate math for optics and electromagnetics) and I like to get refreshed on theory over application so I appreciate the discourse and enlightenment. Engineering would have ended in either a Taylor series or algebra, neither of which address the fundamental singularity that 0/0 is not defined. My first comment was Dirac delta function which, I learned through this discourse is not a function in mathematics theory. Nevertheless, it's practical to see an infinite impulse with zero width has an integral of 1 from -infinity to +infinity. That's more engineering saying that 0/0=1 but it has no mathematical function theory to support it. It's a construct to describe a distribution. I'm not qualified to say where removable singularities exist, only that it's practical to use it. It took me a while to find the theoretical terms mathematicians use to describe it. Engineering, too often, borrows from theory without understanding its limits. --DHeyward (talk) 07:40, 17 May 2016 (UTC)

Technically, one can't speak of "the" function $x^{2}/x$ because specifying the domain is part of defining the function. Having x = 0 in the domain or not amounts to choosing between different functions even if they agree where their domains overlap.--Jasper Deng (talk) 04:17, 17 May 2016 (UTC)

There are conventions for what the domain of a function should be when it is specified by merely a formula. Namely, "the largest domain that makes sense". So, unless stated otherwise (or context suggests otherwise, e.g. we're not talking about real numbers), it's clear that when we say "The function $x^{2}/x$" we mean the function $f:\mathbb {R} \backslash \{0\}\to \mathbb {R} ,\ x\mapsto x$. -- Meni Rosenfeld (talk) 07:57, 17 May 2016 (UTC)

The assumption that computations and limits can be exchanged,

$\lim(f(x))=f(\lim(x))$

is not always valid. It is true that

$\lim _{x\rightarrow 0}{\frac {14+x}{7+x}}={\frac {\lim _{x\rightarrow 0}(14+x)}{\lim _{x\rightarrow 0}(7+x)}}=2$

but it is not true that

$\lim _{x\rightarrow 0}{\frac {14x}{7x}}={\frac {\lim _{x\rightarrow 0}(14x)}{\lim _{x\rightarrow 0}(7x)}}={\frac {0}{0}}$

The function f(x,y)=x/y is not continuous around (0,0), no matter whether 0/0 is defined or not. Nor is x/y continuous in (∞, ∞), no matter whether ∞/∞ is defined or not. Nor is xy continuous in (0, ∞), no matter whether 0∞ is defined or not. Nor is x^y continuous in (0, 0), no matter whether 0⁰ is defined or not. Bo Jacoby (talk) 09:26, 17 May 2016 (UTC).

So, just to be clear, the function f(x)= x is not the same as f(x)= x^2/x??--178.106.99.31 (talk) 20:54, 17 May 2016 (UTC)

I think the interpretation is that $f(x)=x^{2}/x$ can be replaced by $g(x)=x$ over the entire domain of f(x) and since g(x) is defined at the removable singularity f(0), they are equivalent. Equivalence isn't derived from algebra but by properties of functions that are more complicated than substituting $x/x=1$ into f(x) because that identity is undefined for x=0. --DHeyward (talk) 23:16, 17 May 2016 (UTC)

But to answer 178's question directly and literally, no, the two functions are not the same. --Trovatore (talk) 23:32, 17 May 2016 (UTC)