Talk:Exponentiation/Archive 2

message: I think this section is too terse, and needs more examples and explanation.  (Compare it to the previous section on integer powers: the difference is stark.)  It should aim to be mostly accessible to, say, a high-school student or freshman who has learned the basic rules of arithmetic for complex numbers, knows trigonometry, and maybe something about polar/cartesian forms, but hasn't learned any complex analysis and has never heard of branch cuts.  Euler's formula probably needs to be reiterated somewhere here (as opposed to just linked), but not rederived.
It's fine to refer the reader to the article on branch cuts and give the formal definition of the principal value, of course, but it should also give more explanation.  The hypothetical reader I mentioned above will have no idea how to make sense of "a branch cut extending from the origin along the negative real axis".  (Using z for both log(a) and the exponent of a in the same paragraph is also confusing.)  On the other hand, the same reader should be able to grasp something like:
:Because of the periodicity explained above, the equation e^x = a for x = \log a has infinitely many complex solutions x: for any solution x, the number x+2πi is also a solution.  So, the complex logarithm could be defined as a multivalued function.  In order to assign a specific value to a^z = e^{z \log a}, however, we must choose a particular value of \log a.  The most common choice, known as the principal value of the logarithm, is the complex number \log a whose imaginary part lies in the interval (-\pi, +\pi].  Such a choice is known as a branch cut, and the principal value corresponds to a branch cut extending from the origin along the negative real axis.  Although the complex logarithm is commonly used in the formal definition of complex exponentiation a^z, for computation one often first converts a to polar form, as described in more detail below.
The section "general formula" should be expanded to give an actual derivation of this formula, from Euler's formula, by explicitly converting a+bi to polar form.  (This will also simplify the formula, since it can then be written in terms of r = \sqrt{a^2 + b^2}.)  The section should again reiterate that a branch cut is involved in the choice of angle.
I would then suggest following it with a couple of examples, e.g. (-2)^{3+i} and i^i.
I hope this is helpful; thanks for your efforts.  —Steven G. Johnson 18:39, 16 January 2007 (UTC)
replies: text: Thanks for taking the time to comment on the article.  I incorporated the paragraph above and rephrased the rest of that section.  I agree it was much more terse than the rest of the article.  I agree that "general formula" would be better replaced with an explanation of how to use polar form to compute complex powers. CMummert · talk 20:09, 16 January 2007 (UTC)
thread_title: Complex Powers of Complex Numbers

message: #The graph: "Exponentiation with various bases: red has base e, green has base 10, and blue has base 1.7 " would be more instructive like this: "Exponentiation with various bases: red has base e, green has base 10, black has base 1, blue has base 1/e and magenta has base 0.1. ". And the graph belongs in the subsection Exponentiation#Real_powers_of_positive_real_numbers.
#The graph: "From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8  " would be more instructive like this: "From top to bottom: x−8, x−4, x−2, x−1, x−1/2, x−1/4,x−1/8, x0, x1/8, x1/4, x1/2, x1, x2, x4, x8  "
Bo Jacoby 21:25, 16 January 2007 (UTC)
replies: text: Also, using colour (where avoidable) to convey information discriminates against people with colourblindness.  I'm sure there's a WP: guideline about this somewhere. –EdC 21:44, 16 January 2007 (UTC)
replies: text: Yup: WP:MOS#Color coding and WP:WAI#Color. –EdC 21:46, 16 January 2007 (UTC)

text: The best thing is almost always to put the labels directly on the graph.  i.e. don't use a legend, just put text labels next to/over the curves.  Since the exponent is written in a small font as a superscript, it might be good to label the curves with the exponenent alone (i.e. add text labels "8", "7", ... to the curves, and then say in the caption that the curves are x^b where b is given by the label).  Direct labelling avoids an indirection, and also circumvents problems for color-blind readers (or black-and-white printed copies).  —Steven G. Johnson 23:54, 16 January 2007 (UTC)
replies: text: Done, thanks. –EdC 21:42, 17 January 2007 (UTC)
replies: text: Thanks Ed, but you changed a different graph.  I think we were referring to :Image:Root_graphs.png.  —Steven G. Johnson 01:38, 18 January 2007 (UTC)
thread_title:  Suggestions for improvement of the graphs 

message: :It doesn't matter whether the arguments in favor of the definition are strong or weak - the fact is that they are not universally accepted, and this lack of acceptance has been documented with reliable sources.   The WP policies are clear that if a consensus on an issue does not exist in the real world it is not appropriate to claim in a WP article that one side or the other is "correct" - the article should neutrally describe the situation.  To repeat: it makes no difference how mathematically compelling you feel the arguments in favor of the definition are, because in real-life these arguments have not led to a consensus on the issue.  The article currently says "In many settings, 0^0 is defined to be 1."  This seems perfectly accurate to me. CMummert · talk 14:19, 18 January 2007 (UTC)
replies: text: A lack of uniform notation is not the same thing as a "controversy", and this article is no different from other articles on mathematical subjects in Wikipedia in neutrally reporting the various notations in common use.  (Nor is there anything wrong with "social science", and indeed articles on mathematical topics should include well-sourced historical background, etc., as well as the bare mathematical theorems and definitions.  Wikipedia is an encyclopedia, not a mathematics textbook.) There are many areas of mathematics, like this, where notation is not completely uniform across subfields, but professionals rarely get excited or actively argue about such things.  (Dubbing the big yawn of 00 inconsistencies, in particular, a "controversy" is grossly overstating the attention that anyone gives that matter in modern times.) —Steven G. Johnson 15:58, 18 January 2007 (UTC)

text: (Not that anyone has ever convinced Bo that Wikipedia is not the place to promote his notational reforms, from declaring a uniform convention for 00 to introducing the convention \sqrt{1} = -1 to a new definition for the DFT to a new function notation to a new fraction notation, despite dozens of editors telling him the same things over and over again in numerous contexts.  Nor do I expect anyone to convince him now; only weight of numbers keeps him restrained.  Thanks for wasting everyone's time, Bo.)  —Steven G. Johnson 15:58, 18 January 2007 (UTC)

text: No one here is "opposing" the 0^0=1 convention (and I suspect that most of us prefer it); you are chasing shadows.  We are opposing the false implication that this convention is universal, or that the longstanding alternative convention of 0^0 as indeterminate is considered "incorrect" in all subfields of mathematics.  (And by the way, I didn't touch your inferential statistics article; other editors, including at least one professional statistician, apparently felt it was misleading and that any worthwhile content in it should be merged into other articles.)  You are clearly capable of making positive contributions; it's unfortunate that so much of your energy is spent waging pointless notational battles against multitudes of other editors.  —Steven G. Johnson 20:24, 18 January 2007 (UTC)
thread_title: Controversy

message: I just archived this discussion page, from 12/20/06 through 12/31/06. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 106kB to 62kB (measured with the "Edit this Page" button). That's 44kB in 12 days ≈ 3.67 kB per day, or roughly 750 words per day (31 per hour).  DavidCBryant 15:32, 18 January 2007 (UTC)
thread_title:  Archiving 

message: Assuming good faith, why is Trovatore reverting my edit on the definition leading to the simple rule? You cannot deny the consequence without denying the premise, as explained above. Bo Jacoby 10:31, 2 February 2007 (UTC)
thread_title:  Trovatore's revert 

message: DavidCBryant state that the old version was better. I had inserted section headers for clarity. Please discuss your point of view. Bo Jacoby 11:51, 2 February 2007 (UTC)
replies: text: Quote from User_talk:DavidCBryant: I read both versions of the article before I reverted your edit of exponentiation on 2 Feb, 2007. Before you chopped it all up, the article read fairly smoothly. After you introduced a lot of new headers, and switched the order of presentation all around, the article was jumbled and confused. I put it back in order, thereby improving it. DavidCBryant 22:23, 6 February 2007 (UTC)

text: These issues have already been discussed in the past month. CMummert · talk 13:33, 7 February 2007 (UTC)
thread_title:  DavidCBryant's revert 

message: An edit was made this morning with edit summary "After a long discussion the 0^0 case for integer exponents is settled."  No new discussion has happened recently, and the new version has reintroduced the same absolute claims about 0^0 being 1 and being the empty product that have been discussed in great depth already. CMummert · talk 12:27, 28 March 2007 (UTC)
replies: text: The recapitulation from 22 March 2007 was not argued. The definition 35 = 1·3·3·3·3·3, (5 multiplications by 3), has not been argued either. It leads unavoidably to xn = 1·x···x, (n multiplications by x). A special case of this is 00 = 1. The absolute claim that 00 = 1 is a logical consequence of the definition that xn = 1·x···x, (n multiplications by x), which has not been argued, although I have encouraged CMummert to change the definition to reflect his point of view. Within each tiny subsection we must stick to logic.  Bo Jacoby 13:07, 28 March 2007 (UTC)
replies: text: All of these concerns have already been addressed, as you must be aware. There is no requirement that others must continue to respond to your comments when you raise no new issues. CMummert · talk 13:14, 28 March 2007 (UTC)

text: You must be consistent. The definition implies the conclusion 00 = 1. You have not modified the definition, although you have had plenty of time to do so. Bo Jacoby 15:00, 28 March 2007 (UTC)
thread_title:  2007-3-28 

message: :Many languages, including Java, Python, Ruby, Haskell, ML, Scheme, MATLAB, Microsoft Windows' Calculator, and others (especially when using IEEE floating-point arithmetic, but also for integer arithmetic), evaluate 00 to be 1. [7]
:7. For example, see 
Not only does the C99 standard not define pow(0,0) (except to say that it may raise a domain error, and allowing implementations to define __STDC_IEC_559__ if they follow IEC 60559), but there's nothing in the C Rationale about pow at all, and what there is about cpow seems unenlightening to me. I've reverted the incorrect inclusion of C in the list, but can anyone think of a reason that that reference to the Rationale should remain, or should I get rid of it too? --Quuxplusone 22:33, 8 March 2007 (UTC)
replies: text: In that case, we should have "By contrast, C allows implementations to choose whether pow (0, 0) should return a value or raise a domain error; implementations may define __STDC_IEC_559__ if they follow IEEE 754.[7]"  What does IEEE 754 say about 0^0? –EdC 22:41, 8 March 2007 (UTC)
thread_title:  Reason for linking to C Rationale? 

message: I was noticing that the "Powers of i" and the "Powers of e" sections are both under the heading "exponential with integer exponents." I propose that "powers of e" gets moved under "Real Powers of Positive Real Numbers", and "powers of i" goes to "Complex powers of complex numbers". Does anyone have a problem with this? --shaile 16:03, 16 March 2007 (UTC)
replies: text: We are talking about integer powers of i and e. So they are correctly placed where they are under exponentials with integer exponents. Bo Jacoby 01:03, 18 March 2007 (UTC)
thread_title:  Location of Powers of i and e 

message: I've removed the image of x^y as I thing it is missleading. I've had a bash at drawing x^y for positive x. It clearlt shows that taking limits along curves will give a limit of 0.--Salix alba (talk) 09:05, 3 April 2007 (UTC)
replies: text: I've now replace Sam's image and added some more curves to it. What you find is that its only curves which are tangent to the y-axis which give a limit different to 1. The image of any curve in the x-y plane with a different tangent will have a limit of 1. --Salix alba (talk) 11:48, 3 April 2007 (UTC)
replies: text: Yes, this is currently footnote 7. CMummert · talk 13:25, 3 April 2007 (UTC)
replies: text: Thinking about this in singularity theory terms is amusing. Consider the space of all (smooth/analytic) functions f(t)=(f_x(t),f_y(t)) from R to R^2 and compose it with the exponation function. We can consider the 1-jet of these fuctions, which give a two dimensional set of functions f(t)=(a t,b t). Those funtion for which the limit is not 1 form a sub-space of dimension 1, that is co-dimension 1. We can then expand this to consider the full space of germs and again find that its a co-dimension one set of functions with a non-unitray limit. --Salix alba (talk) 17:02, 3 April 2007 (UTC)
thread_title:  Image of x^y 

message: is there a separate article on the exponential map in differential geometry?  if not, should information on it be added here or should I write an article on it?
SmaleDuffin 16:35, 4 April 2007 (UTC)
replies: text: See the article Exponential map. Bo Jacoby 20:57, 5 April 2007 (UTC)
thread_title:  Exponential map in differential geometry? 

message: This article has gone from B-class to 'Good'. Well-written and explained. Attractive presentation with some kewl formulae and graphs. No math errors there I could find. Neutral and not given to controversy.  Stable: um, yeah. Images: has graphs that really help the exposition. Well done and take a step up. Gifir2007 11:28, 7 April 2007 (UTC)
thread_title:  GA-status granted 

message: The third sentence of the article is Exponentiation can also be defined for exponents that are not positive integers. But it doesn't go on and say what the definition is. Obviously,
: a^{-n} = \frac{1}{a^n} = \frac{1}{\underbrace{a \times \cdots \times a}_n}, 
but leaving the sentence on its own can create some confusion for those who don't know basic index laws. Thanks GizzaChat © 12:00, 7 April 2007 (UTC)
replies: text: The sentence in question covers more than negative integers. That definition is given later in the article. Indeed most of the article is dvoted to explaining the thired sentence. It could be worded more clearly, however.--agr 00:02, 8 April 2007 (UTC)
replies: text: I expanded it this morning (before the comment signed agr was left here). If it is still too vague, be bold and rewrite it. Please keep in mind that the lead is supposed to summarize but not duplicate what is located in the rest of the article. CMummert · talk 00:34, 8 April 2007 (UTC)
thread_title:  Definition for negative integers 

message: The article says that 2 to the 3^4 is different from 2^3 to the 4, but it doesn't mention order of operations.  Is 234 2^81 or 8^4?
:In traditional math typesetting, 234 is 2^81, and the 4 would be in a smaller font than the 3, which would again be smaller than the 2. CMummert · talk 11:13, 9 April 2007 (UTC)
replies: text: 2^3^4 = 2^(3^4) ≠ (2^3)^4 = 2^(3·4). Yes, the article could be explicite about it. Bo Jacoby 19:34, 9 April 2007 (UTC)
thread_title:  Order of Operations