Wikipedia:Reference desk/Archives/Mathematics/2014 December 12

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December 12

Real Projective Line Page - Defined and Undefined Contradictions?

Hello. I was a little confused about why ∞+∞ is not defined but ∞*∞ = ∞. This doesn’t make sense. I believe both should be left undefined (for now) and i think 0*∞ , ∞/∞ , and 0/0 should be defined as C where C is a constant. Idk if this constant could be infinity but I certainly think these should be defined. if a/0 = ∞*b then that implies a/b = ∞*0. A similar proof could be done with the others. Am i allowed to change the page or add a note because this is more of an idea but idk if this is 100% correct (maybe a note to the right of the equation?). Jetstream5500 (talk) 05:02, 12 December 2014 (UTC)

That article is almost completely unsourced so some of the information is bound to be a bit sketchy. To me, the projective line is a geometrical construction, not an arithmetic one, so the idea of defining arithmetic operations on it at all is rather dubious. But for ∞+∞, keep in mind that the projective line makes no distinction between +∞ and -∞, so ∞+∞ could also mean ∞-∞ which isn't defined even as a limit. If you do want to take a whack at fixing the article then I'd suggest finding a good reference first. --RDBury (talk) 05:25, 12 December 2014 (UTC)

But my point is Idk if I could find an article in relation to that subject. And i also understand why they are undefined. The point of my argument is ∞*∞ is the same as adding ∞'s which is undefined. So that must be undefined. Also I am not making assumptions I am only using the idea that 1/0 = ∞ and making logical implied truths. Saying 0/0 is c not undefined is mainly for notation purposes as this number does have a value. Im not to familiar with the subject just thought it was an interesting article. — Preceding unsigned comment added by Jetstream5500 (talk • contribs) 05:33, 12 December 2014 (UTC) EDIT --Im mainly hesitant to change the article because I'm not familiar with the subject but it looks wrong.Jetstream5500 (talk) 05:37, 12 December 2014 (UTC)

The operations with ∞ are defined basically by convention, so they don't necessarily have any justification other than usefulness. The MathWorld article in the external links section can be used as a source for that section at least. ∞*∞ = ∞ doesn't have the same problem as ∞+∞ = ∞ because the limit still works; if you substitute ±∞ for ∞ in the factors, you still get ±∞ in the limit for the product, which gives a single value of ∞ in the projective line. --RDBury (talk) 06:14, 12 December 2014 (UTC)

First, I should mention that I created the article Real projective line (9 years ago) and it seems much of what I wrote back then remains intact. I'm not good with sources but I used a combination of Mathworld, discussions with other Wikipedians and common sense.

The reasons for the definitions of arithmetic operations on the RPL are quite clear - they are derived from limits of real functions. if $\lim f(x)=\infty ,\ \lim g(x)=\infty$ then there's no definite answer for what $\lim(f(x)+g(x))$ is, so $\infty +\infty$ is left undefined. But $\lim(f(x)\cdot g(x))=\infty$, and so $\infty \cdot \infty =\infty$. And so on.

The RPL is probably not as useful and ubiquitous as the Riemann sphere, so you'll likely find less sources and justifications for properties of the RPL. But the RPL is simply "Riemann sphere with the real numbers" so most of what you'll find about the sphere will apply. The arithmetic operations there, too, are inspired by limits, and our article (presumably better sourced) lists the above equalities.

Re Jetstream's comment "∞*∞ is the same as adding ∞'s" - no, we're talking about reals, not integers. Real multiplication has a life of its own. $\aleph _{0}\cdot \aleph _{0}$ (using cardinal numbers which are an extension of the integers) is essentially adding $\aleph _{0}$ to itself $\aleph _{0}$ times. But $\infty \cdot \infty$ is not; and the fact that we're talking about unsigned infinity is critical. -- Meni Rosenfeld (talk) 09:18, 12 December 2014 (UTC)

Also, the RPL with the operations as defined does not form a field. $1/0=\infty$ but $\infty \cdot 0$ is not 1, it is undefined (again, the limits are indeterminate). -- Meni Rosenfeld (talk) 09:33, 12 December 2014 (UTC)

Ok. That may be true that ∞*∞ = ∞. Ill give u it might be true (haven't looked into it). The point of my statement was why do we have to say the value of ∞*0 is undefined when clearly any value is possible. Undefined is basically a term in mathematics for "we don't know". The point of saying 1/0 = ∞ removes this undefined. Why not define everything. Why not say ∞*0 = c. Where c is a constant that could be any value. That is not questioned by any means. We can show this because 2/0 = ∞ = 1/0. We can show that there is a loss of precision. Its like saying in integrals instead of using C... we use undefined..."we don't know the value".Jetstream5500 (talk) 15:50, 12 December 2014 (UTC)

So you are asking why we don't define a new set RPL ∪ {₵} where part of the arithmetic is that ∞⋅0=₵? (Let's not use "c" as our RPL page is already using that as variable.) The answer is probably that it is not particularly useful. (Have you, as an exercise, worked out the complete arithmetic?) Under the reals, we don't define either 1/0 or 0/0, although when considered as forms for limits -- lim_x->a 1/f(x) where lim_x->a f(x) = 0 & lim_x->a f(x)/g(x) where lim_x->a f(x) = lim_x->a g(x) = 0 -- they are very different from one another. In the first case, the limit is guaranteed to be unbounded (infinite, in some sense). In the second case, the result is indeterminate, it could be anything, based on your choice of f(x) and g(x). With the RPL, we have added an element for the infinite. You want a new set where we include an element for the indeterminate. Again, this simply may not be useful. I believe that you should be able to complete the arithmetic where nothing is undefined, but how does ₵ fit into the topology? Moreover, it won't work well as a shorthand for the forms. You want ₵ to represent any potential value, but if it is a separate element of your set, then it is not actually equal to any other value. You would say that 0/0 = ₵, but lim_x->0 x/x = 1 and you can't say that 1 = ₵ or you break the very concept of equality since you could equally show that 2 = ₵, and thus that 1 = 2. -- ToE 16:36, 12 December 2014 (UTC)

(ec) "Undefined" can be used with two distinct meanings: (a) we have not yet defined it, and (b) defined to be excluded from the domain of the function. Defining ∞·0 = c, as you are suggesting, unfortunately moves certain caveats into difficult places. Instead of having a theorem such as "(a/b)·b=a whenever all operations are defined on their operands", you'd have "(a/b)·b=a whenever b is neither 0 nor ∞, but sometimes even then". Given the need to fine-tune theorems with no apparent benefit, to extend the definition rather than having a universally applicable "whenever each operation is defined" seems pointless, even though it can be done with logical consistency. In other words, using the definition (b) of "undefined" is actually useful. —Quondum 16:44, 12 December 2014 (UTC)

I guess I don't like the word 'undefined'. Also I guess that would mean that if 1/0 = ∞, then 1 would be a subset of ₵. Same with 2. I guess this would kinda ruin using the equals symbol. The idea is that ₵ is a set of numbers not a value. I just thought the whole concept of defining 1/0 was really important and that -∞=∞. If you use quadratic equation (with an equation of the form y=mx+b) you can show that one answer is equal to -∞=∞ and the other answer is a subset of ₵ which is true because all lines that have a slope not equal to 0 will pass through the x-axis. Guess you can use the notation for the reals excluding 0 and infinity.Jetstream5500 (talk) 19:46, 12 December 2014 (UTC)

Actually...I just realized ₵ would have to include ∞ i think because if you take a horizontal line it crosses the x-axis (according) to quad equation at 0/0 which = ₵ and theoretically in this case it should be able to cross at ∞. Also ₵/0 = ∞ thus if ∞*∞ = ∞ from what you proved above with limits that is agreed with by ₵=∞ in this case.Jetstream5500 (talk) 22:49, 12 December 2014 (UTC)

If all we know is $\lim f(x)=\infty$ and $\lim g(x)=0$, $\lim f(x)g(x)$ can be any real number, and it can be ∞. Thus if your ₵ is actually a set, then it is in fact the RPL itself.

You could define ∞·0=₵ but it's simply not really elegant or useful - it creates a lot of degenrate arithmetic, there's no obvious topology, etc. It basically ruins the whole structure.

I think you simply need to get rid of your phobia of "undefined". You will always have things that are undefined. e.g., the eigendecomposition of ${\begin{bmatrix}1&3&2\\4&1&0\end{bmatrix}}$ is undefined, because eigendecomposition is only defined for square matrices. -- Meni Rosenfeld (talk) 18:22, 13 December 2014 (UTC)

To be fair, I believe that Jetstream doesn't have an issue with that sort of undefined (eigendecomposition of a non-square matrix) since it can't be anything, but instead objects to the variety that can be anything. It is not as if we call your $\lim f(x)g(x)$ itself undefined where all we know is that $\lim f(x)=\infty$ and $\lim g(x)=0$. That is ill-defined or insufficiently defined. It is ∞·0 that is undefined. I've not been able to think of any such examples beyond indeterminate forms.

I would say to Jetstream that $\lim f(x)g(x)$ is different from ∞·0 because the former still has room for further definition. The latter may be short hand for an indeterminate form, but within the RPL (where ∞ is actually an element) it must, if defined, mean something specific since there is no room for further definition.

Also, I don't believe that it has been mentioned here that even some fully defined limits may not exist. For instance, if f(x) = x·sin(1/x) and g(x) = 1/x, then f(x)g(x) does not have a limit as x->0. -- ToE 12:37, 14 December 2014 (UTC)

Is it appropriate to say that the real projective line is a quotient of the extended real number line under -∞ ~ +∞? Is there any such action which can automatically provide the resulting algebra, or does determining what is or is not defined have to be done on an ad hoc basis, based on what makes sense and what is useful? -- ToE 12:42, 12 December 2014 (UTC)

This would give the required topology, of course. But I don't know of a general way to this quotient that, when applied to the ERNL, would result in the arithmetic that we want on the RPL. -- Meni Rosenfeld (talk) 13:22, 12 December 2014 (UTC)

As I understand it, when the real projective line is defined in terms of homogeneous coordinates in two dimensions (equivalence classes of real number pairs, much as the rational numbers can be defined, which is then a quotient), ∞ becomes well-defined as a point on the line. On this line, any element of the group of Möbius transformations is then well-defined, and will always map every point, including ∞, to and from a point on the line. One can then map (almost) any point on the line to an element of one of two normal subgroups of the group of Möbius transformations, i.e. a mapping that one might express as a ↦ (a×), i.e. onto multiplication by a constant; and b ↦ (b+). i.e. addition of a constant. There are elements that must be excluded in these mappings: 0 and ∞ for the first mapping, and ∞ for the second mapping. We can define addition and multiplication in terms of these mappings, which leaves addition and multiplication undefined for some elements on the left, but for every element, including ∞, on the right. One can show that these operations are commutative when defined for the elements in both orders, and use this property to extend the definition to all but a few problematic pairs. I would expect that this would give a rigorous (algebraic!) way to frame the whole thing. This leads to consistent behaviour of the operations, whenever they are defined. —Quondum 15:13, 12 December 2014 (UTC) PS: this construction applies to all fields.

That's not how it all happened. The reason for not defining ∞*0 for instance originally came from the study of the limits of functions. Once upon a time only continuous functions were considered to be proper and a common idea was that one could fill in holes in a function with values that made it continuous. Nowadays people aren't so squeamish about discontinuities but continuity is still about the most important idea in calculus and how most of the world's population would think of a function. And about the most important rule for finding a limit is L'Hôpital's rule. And L'Hôpital's rule would fail if you gave actual values to indeterminates, if one would get a different value at 0 if 0/0 was 1 for 2x/x and 2/1. Brahmagupta originally defined 0/0 as 0 and later Mahavira defined x/0 as x, and there's versions where they added extra numbers like 5/0 and then cancelled zeroes at the end. There's a long history of such stuff and we even have a modern version with non-standard arithmetic. None it works without quite a bit of extra work despite the claims of some of the people pushing it that it is suitable for schoolchildren. Dmcq (talk) 21:05, 12 December 2014 (UTC)

I wasn't trying to present this as anything historical, or even standard (but I was hoping that its beauty would appeal to some). What I've given is essentially OR, but it is a purely algebraic approach that happens to agree with the approach of continuity and indeterminacy. I find it satisfying that it extends the same results rigorously to contexts that do not include continuity: a point at infinity can be added consistently to any field, with the same results. Not every mathematician works in analysis; I guess that this might be how a geometer, group theorist or algebraist might see it. —Quondum 23:10, 12 December 2014 (UTC)

I'm not exactly clear what is being said is non-historical here. The real projective line is the "line at infinity" in projective geometry. It goes back to Kepler and Desargue. Sławomir Biały (talk) 15:58, 13 December 2014 (UTC)

@Sławomir Biały: You seem to refer to the geometric concept: a space of points upon which a group of transformations acts. Do you know where the interpretation as a algebra/ring (well, almost) with an added element (∞ with the operations partially extended to it) fits into the historical picture? —Quondum 16:27, 13 December 2014 (UTC)

There is a case to be made that the projective line does not have a ring structure of any kind, just a cross ratio. The article that currently resides at real projective line is about something like "projective completion of the real numbers", which is the one-point compactification introduced in analysis. This is rather different, and shouldn't (in my opinion) be identified with the projective line at all. For example, the projective completion described in our article has three distinguished points, {0,1,∞}, unlike the projective line. The projective line, if it is equipped with an ordered triple of marked points (denoted {0,1,∞}), will carry a ring-like structure defined by: x/y=[0,∞;x,y], xy=x/(1/y), x−y=x[∞,x;0,y], x+y=x−(0−y). This isn't entirely satisfactory and requires tweaking at the marked points themselves. Sławomir Biały (talk) 17:10, 13 December 2014 (UTC)

This suggests that the article should be renamed, a move that I would support. —Quondum 19:18, 13 December 2014 (UTC)

Yes, I agree with that. Incidentally, the definition of the ring operations in terms of cross ratios is probably better than the ad hoc definitions that appear in the current article, since it gives a clear reason why there are several exceptional cases. Also, here is something rather interesting: deprojectivizing things suggests that multiplication and division are more natural than addition and subtraction. If we identify the marked point ∞ with the point (1,0) of R², 0 with the point (0,1), and 1 with the point (1,1), then (using these cross-ratio formulae) the product is given by $(x_{1},x_{2})(y_{1},y_{2})=(x_{1}y_{1},x_{2}y_{2})$ and quotient by $(x_{1},x_{2})/(y_{1},y_{2})=(-x_{1}y_{2},-x_{2}y_{1})$. Subtraction is then cubic in the variables, and addition does not seem to be given by a polynomial in the variables at all (we need to allow rational functions). Is this some standard algebraic system? Sławomir Biały (talk) 21:19, 13 December 2014 (UTC)

FWIW:

• We have an article Projective line which seems to better correspond to what Slawomir discussed.
• The article on Mathworld that corresponds to what is currently at Real projective line is Projectively Extended Real Numbers. If Slawomir is correct, this might be a better name. Also, I seem to remember the article used to be called that in the past (perhaps before being created), but someone suggested the shorter name "real projective line". I mostly took their word for this being consistent with other notions of "projective line".

-- Meni Rosenfeld (talk) 00:27, 14 December 2014 (UTC)

Unfortunately in this case it seems to be "a close correspondence" rather than "consistent with". I think there is a case to be made to reserve the name "Real projective line" to belong to the family of geometric articles Real projective plane, Projective line, Projective plane, etc. "Projectively extended real numbers" seems to me to be a good name for the article.

Sławomir, as may be seen from Projective line#Line extended by a point at infinity, addition is only quadratic – actually bilinear in the representation's components – as is subtraction; one might want to include the harmonic sum (the reciprocal of the sum of the reciprocals) as one of the "natural" operations. I don't quite see where you're going with the question – it would seem to me to be merely the arithmetic on a projectively extended field. —Quondum 01:58, 14 December 2014 (UTC)

I was trying just to use cross ratios to define the various operations. If we do this, then subtraction appears to be cubic. It's certainly true that we can define various operations in some other way to make them consistent, but not using just the cross ratio. Sławomir Biały (talk) 13:27, 14 December 2014 (UTC)

Sorry, I hadn't looked properly at your definition of the operations using cross ratios, and am still trying to figure that out. To paraphrase what you I think you have given:

• We represent points of our algebraic structure as equivalence classes (a,b) ~ (λa,λb), λ ≠ 0 of R² ∖ {(0,0)}.
• We have the cross-ratio of for (w,x;y,z), for which the definition at definition in homogeneous coordinates seems appropriate.
• Three of these points we identify and label as 0 ≝ (1,0), 1 ≝ (1,1), ∞ ≝ (0,1), noting that these are distinct from the elements of R with the same labels.

  Identification of points: identifying the the three special points with the extended real numbers of the same label, we are defining a map a ↦ (1,a), ∞ ↦ (0,1), where we with pairs are referring to points (equivalence classes of pairs). In particular, this allows us to bijectively map a cross-ratio (an extended real number) to a point, though I guess we could rigorously define the cross ratio to yield a point (its inputs are already points) without making such an identification.

• We define (partial?) operations on our space by:

  Division: x / y ≝ [0,∞;x,y]

  Multiplication: x ⋅ y ≝ x / (1 / y)

  Subtraction: x − y ≝ x ⋅ [∞,x;0,y]

  Addition: x + y ≝ x − (0 − y)

The defined multiplication division (and by implication, inversion) appear to be the same as with the extended real line, but we are defining new and different addition and subtraction operators. Before I go further, how come you chose these particular definitions (within the constraint of only using the cross-ratio)? —Quondum 18:16, 14 December 2014 (UTC)

Ok, so a slight variant would be where we allow addition as defined in that section of projective line. Then we get an "algebra", where both addition and multiplication are defined everywhere (!). This algebra doesn't satisfy the usual rules, though (e.g., addition is bilinear). The question is, what is it? Sławomir Biały (talk) 11:44, 15 December 2014 (UTC)

It is, of course, a Quondum space YohanN7 (talk) 12:05, 15 December 2014 (UTC)

Is it really defined everywhere? The cross-ratio is not defined when two of the points coincide. And the bilinearity is in a sense a sham: there is still the quotient by the equivalence relation to be considered. —Quondum 00:09, 16 December 2014 (UTC)

No, don't projectivize. You indeed get that everything is well-defined everywhere, but may be zero in some places. It would make sense to assign the points $0,1,\infty$ projective weights (so the identification with (0,1), (1,1), and (1,0) becomes more canonical). Then everything in the algebra also carries a projective weight, and the for all operations of "addition", "multiplication", and "division" defined by $(x_{1},x_{2})+(y_{1},y_{2})=(x_{1}y_{2}+y_{1}x_{2},x_{2}y_{2})$, $(x_{1},x_{2})(y_{1},y_{2})=(x_{1}y_{1},x_{2}y_{2})$, $(x_{1},x_{2})/(y_{1},y_{2})=(-x_{1}y_{2},-x_{2}y_{1})$, the weights are additive. Sławomir Biały (talk) 00:40, 16 December 2014 (UTC)

Okay, at a glance, it is a real algebra in two dimensions, with three operations, ignoring the weighting (of which I haven't quite yet discerned the intention). "Addition" and "multiplication" are associative and commutative, (0,1) is the "additive" identity, (1,1) is the "multiplicative" identity, (0,0) is an absorbing element for all three operations (and which also is the result for the quirky cases ∞⋅0, 0/0, ∞/∞, ∞+∞, making it sort of a defined replacement for "undefined" in all cases). (0,0), (1,0), (0,1), (1,1) are multiplicative idempotents, (0,0), (0,1) are additive idempotents. The distributive law does not apply without modification (a "scalar" multiplier: an element of the form (k,k)). "Division" is the inverse of "multiplication" but for a similar modification. It looks a lot like the real numbers, except that certain identities require a function of one of the variables in the identity to be included as a multiplier; the function maps every variable onto a corresponding "scalar" multiplier. This algebra is so similar to the reals that there is a homomorphism from it onto the projectively extended reals, further extended with an element called "undefined"/"indeterminate", which we could label ₵. (déjà vu? ) —Quondum 05:11, 16 December 2014 (UTC)

More or less my thoughts as well, that an actual element (0,0) plays the role of "undefined". This would probably settle any uneasiness about some operations in the extended reals being undefined, except that the algebra is rather quirky in the other ways that you mentioned. The "homomorphism" to the projectively extended reals is only a partial homomorphism (a "rational homomorphism", or something like that). The purpose of the weight would be that the points 0,1,∞ are not actually elements of R^2, but rather one-dimensional linear subspaces, so they really have values in the canonical bundle, which means we need to twist with a weight (but I think this is perhaps optional). Sławomir Biały (talk) 15:17, 16 December 2014 (UTC)

Could you elaborate on the homomorphism only being partial? What I had in mind is a full homomorphism (essentially the projectivisation with one added point) from this algebra to a specially defined algebra on the set R∪{∞,₵}. The operations on the latter are exactly those of the RPL, extended to incorporate the new element ₵. Subtraction and division are treated as first-class operations, being extended similarly instead of being simply defined as inverse operations with holes in their domains where the inverse does not exist. This would be a fully consistent algebra along the lines that Jetstream and ToE were discussing above, with every operation being complete. —Quondum 16:49, 16 December 2014 (UTC)

Yes, ok, if you include the funny symbol in the algebra as a sort of absorbing element, then it becomes total. Incidentally, the idea of adding 0 back into a projective space is rather appealing. It is, I believe, related to the notion of "blowing up" in algebraic geometry. Sławomir Biały (talk) 18:14, 16 December 2014 (UTC)

Given the backdrop of the homomorphism from a "complete" algebra (yours), I rather like this extension of the RPL. It suggests that there might be a more systematic way to complete other rings, removing "undefined" cases. There must be explicit exclusions for some identities (e.g. (x·y)/y=x, y∉{0,∞,₵}), but I see this as more regular than the qualification "applies subject to domain constraints on each operation".

I don't know what you mean by 'the notion of "blowing up" in algebraic geometry', but I can imagine a kind of projection (a homomorphism between rings or geometries) that collapses the number of dimensions, in which a point of this nature may be similarly useful and meaningful. —Quondum 19:47, 16 December 2014 (UTC)

Blowing up is a way of resolving certain kinds of singularities on plane curves. To do this we must "blow up" the projective plane at the singularity of the curve. This involves replacing the "bad" point with the projective line of tangent directions through the point. So it is a kind of "point with projective line structure" that this has very much the vibe of. Sławomir Biały (talk) 21:09, 16 December 2014 (UTC)

We have an article on blowing up and a stub for blowing down too. --Modocc (talk) 21:51, 16 December 2014 (UTC)

Analysts may dislike it though: it breaks the comforting maxim that if an elementary function is "defined", it must be equal to its limit. This could perhaps be replaced by its being non-₵, though. —Quondum 19:57, 16 December 2014 (UTC)

Took Calc 3 last year in 10th grade so most of this talk is blowing over my head but Im glad everyone liked my question...Jetstream5500 (talk) 22:15, 16 December 2014 (UTC)

Yes, it did stimulate a bit of discussion, didn't it? I certainly found the exploration enjoyable. Quite a bit of it is over my head too, but I learned things that I did not know before. I hope you got something from it too. —Quondum 07:09, 17 December 2014 (UTC)

---

This is a separate question but what happens at y = 2^x. It implies infinity is at 2 locations same time right? Doesn't seem to make sense.Jetstream5500 (talk) 04:08, 15 December 2014 (UTC)

You'll need to be clearer. Do you mean the two one-sided limits lim_x↑∞ 2^x = ∞ and lim_x↓∞ 2^x = 0? It means that the function 2^x on the projectively extended real numbers has an essential discontinuity at x = ∞. —Quondum 04:39, 15 December 2014 (UTC)

Yeah thats what I thought. Doesn't make "logical" sense but yeah. Are there a lot other examples similar to this where lim --> -∞ != lim --> ∞ or is it mainly exponentialJetstream5500 (talk) 05:10, 15 December 2014 (UTC)

By extending the real numbers, we have made all rational functions (functions which are ratios of polynomials) continuous everywhere. This doesn't mean there can't still be discontinuous functions. The exponential functions have a jump discontinuity at ∞ (limits of 0 on one side and ∞ on the other; since ∞ is now a first-class citizen, it's only a jump. Things get trickier on the Riemann sphere). Trigonometric functions like sin, cos and related have an essential discontinuity at ∞. Tan, being the ratio of the functions sin and cos which are continuous on $\mathbb {R}$, is now continuous on $\mathbb {R}$ too, but discontinuous at ∞.

Basically, if $f(1/x)$ is discontinuous at 0, then $f(x)$ is discontinuous at ∞. -- Meni Rosenfeld (talk) 09:08, 15 December 2014 (UTC)

On a personal note, I have thought up the notion of projectively extending the real numbers independently and been interested in it for a long time, since I was 14 maybe (of course, back then I didn't know it would be called that). I couldn't get enough of how elegant the whole thing is. One thing that did bother me is precisely this point that it doesn't seem to "add up" for exponential functions. I've since come to terms with this discontinuity. -- Meni Rosenfeld (talk) 09:14, 15 December 2014 (UTC)

Complex Exponential False “Proof” That All Integers Are 0

Question by Ram nareshji deleted as probable copyvio [1] Nil Einne (talk) 14:42, 16 December 2014 (UTC)

See Exponentiation#Failure of power and logarithm identities and Exponentiation#Complex exponents with positive real bases 2 which explains this exact problem. Also consider the related problem of $e^{-4\pi ^{2}n^{2}}=\left(e^{i2\pi n}\right)^{i2\pi n}=1^{i2\pi n}=1$, which forgets that, in this scheme, the power of 1 is involves a multivalued log of 1 in an exponent.Phoenixia1177 (talk) 08:54, 12 December 2014 (UTC)

Equivalence of two relations in Braid groups.

Question by Ram nareshji deleted as probable copyvio [2] Nil Einne (talk) 14:42, 16 December 2014 (UTC)

Expressing the probability that A is related to B

Can you say it's P(A=B)? (given that A and B are two different things, imagine that A is a person, and you want to know if an object B belong to this person).--Senteni (talk) 13:29, 12 December 2014 (UTC)

There is a technical sense of relation_(mathematics). The thing to note is that many things can be thought of as 'relations' if they are defined as subsets of the Cartesian product. One common notation for "a is related to b" is 'aRb', but sometimes other notations are used, e.g. $a\thicksim b$. It is usually best practice to define the relation and notation before using it in some statement about probability. But, once you've done that, an expression like $\mathbb {P} (aRb)$ is perfectly sensible and reasonable. I should also point out that you should probably not use the equal sign '=', as that is usually reserved for some type of equivalence relation, if not strict equality in the normal sense of identity. Does that make sense? SemanticMantis (talk) 16:18, 12 December 2014 (UTC)

Yes, that makes perfect sense. Thanks.Senteni (talk) 18:01, 12 December 2014 (UTC)

I will note that the meaning of "relation" in mathematics is quite different from the meaning of "relation" in computer science. Robert McClenon (talk) 20:42, 14 December 2014 (UTC)

Good point. Within computer science there is Relation (database) and also Is-a and Has-a type object relations from object-oriented programming. -- ToE 11:44, 15 December 2014 (UTC)

@Robert McClenon:, @Thinking of England:The notions of relation are not so different, Ted Codd defined the relational model of databases explicitly in terms of the mathematical sense of "relation" that I linked above. To me the only difference is that not all computer scientists know the mathematical concept of relation, even if they are (mis)using the term ;) Some of the terminology is a little different, but the core concept of a relation is the same in math and CS. Both 'is-a' and 'has-a' can be described as mathematical relations in a very natural and straightforward manner. If someone wants to use the term "relation" for something that is not a mathematical relation and is working in a field where mathematical relations are used to unambiguously define key concepts, then that person has left the path of wisdom. SemanticMantis (talk) 19:36, 16 December 2014 (UTC)

Medians of a Triangle are concurrent

I have a basic geometry proof and I want to make sure my thought process is correct or if there is a better approach to the problem. I need to prove that the medians of a triangle are concurrent (the three medians of a triangle all meet at the same point, the centroid). This is what I have so far-

Consider the triangle ABC. A unique circle also contains the point ABC and center O. Since O is equidistant from A and B, it must lie on the perpendicular bisector of AB. The same for BC and AC. So, the three perpendicular bisectors are concurrent at O.

Is this correct? Can you assume the median is a perpendicular bisector? How do you know the distance from O to AB is the same distance as O to BC and O and AC? — Preceding unsigned comment added by Pinterc (talk • contribs) 22:51, 12 December 2014 (UTC)

You have correctly proved that the three perpendicular bisectors (also called right bisectors) are concurrent, but after that you go wrong. The medians are generally not the same as the right bisectors, and the distances of O from the three sides are generally unequal (and irrelevant). --65.94.50.4 (talk) 23:05, 12 December 2014 (UTC)