Talk:Exponentiation/Archive 1

message: Go to Talk:Super-exponentiation. It says there that exponentiation is properly called involution and that exponentiation is just an awkward term. Any votes to move this page?? 66.245.22.210 17:46, 3 Aug 2004 (UTC)
replies: text: I think this article needs a History section; obsolete terminology (with citation) might be appropriate there.--agr 11:35, 24 May 2006 (UTC)
thread_title:  [[Exponentiation]] or [[Involution]]?? 

message: Where can I find the page that tells how to cube two variables that are being added or subtracted?  I can't find it anywhere... pie4all88 22:48, 26 Sep 2004 (UTC)
replies: text: Try binomial theorem. Revolver 04:00, 27 Sep 2004 (UTC)
thread_title:  (x+h)^3 

message: The stuff about power-associative magmas in the abstract algebra section of the article is wrong. Power associativity is not enough to get this sort of thing to work, as the following 3-element magma shows:
    | 1 a b
  --+-------
   1| 1 a b
   a| a a 1 
   b| b 1 b
The above magma is power-associative, it has an identity element, and every element has a unique inverse. Yet a2a-1 is not equal to a.
I'm going to change the article to use groups instead. It's possible to do it in somewhat greater generality than that (e.g., Bol loops), but it's probably not worth it. --Zundark 19:58, 17 Dec 2004 (UTC)
replies: text: An excellent idea.  Generality can be excessive. Zaslav 23:15, 3 February 2006 (UTC)
thread_title: Exponentiation in abstract algebra

message: It has been suggested that this article or section be merged with exponential function.
:I think this is a bad idea. The (real of complex) exponential function is just one aspect of exponentiation in general, and the variety of topics in the article suggests this. Revolver 20:48, 14 August 2005 (UTC)
thread_title:  merging 

message: This § does not in fact appear to be about powers of one. Can anybody propose a more sensible title for it? Doops |  talk 23:19, 9 November 2005 (UTC)
replies: text: Hi Doops. I wrote it and used the notation 1x meaning e2πix, because e2πi = 1. However, this nice and useful notation was deleted by other wikipedians who considered it 'original research'. Only the title survived. Bo Jacoby 15:31, 11 November 2005 (UTC)
thread_title:  "powers of one" 

message: "The inverse of exponentiation is the logarithm" is not quite true.
Exponentiation yx defines two functions: the exponential function x→yx and the power function y→yx. The logarithm is inverse to the exponential function, but not to exponentiation. The radicals are inverse functions to the power functions. Bo Jacoby 14:05, 6 March 2006 (UTC)
thread_title:  The inverse 

message: This is a cross post to user talk: Pol098#Exponentiation_and_Metric  (Section Link) on the Intro change in this article, and the need for a better expanded intro:
You refer to the introductory tying the article on exponents to the metric system as pointless. I don't believe in reverts in general, so let me state my reasoning on hopes of achieving a conversion by thou heathen sinner <G>:
If you would be so kind as to consider that these articles are tied into others by links, you will quickly realize the numbers of articles dealing with measurements (which have a generally pragmatic utility to the lay reader) far outnumber the articles that are simply dry math. Since these articles tie into this topic by exactly that corespondence I addressed the tie in the intro as an appitizer of sorts for people linking in that manner. Burying the information way down in the body makes no sense... they are reading about measurements for their own purposes, and not interested in a sub-topic of math as a general topic... unless perhaps my little sentence wets their interest thus making your article experience much more traffic. Wouldn't that be a good outcome? 
Now I cannot say that the quick and dirty change I posted was worded perfectly, I was nested deep within six to seven related edits at the time, and there are others that will always fiddle with sentence structure... but I do object to arbitrary removal of material that is certainly not off point or topic. You extended that so as to misconstrue it to multiples of other integers, so why not just reword it to qualify it better to the set of 1 X 10^x form.
 
I think you should revert and revise if you like to incorporate the sentence under the principle of the most utility to the most people. If we aren't striving for that, why are we bothering to donate our time when the media is so perfect for such a cross-link of knowledge. 
Moreover, WP:MOS wants introductions to articles to recap a sense of the article as a whole. This phrase did that, though obviously, the whole article needs such a recap so the whole is more reader friendly. That is a expansion that is worthy of your time, not chopping down the seedling of knowledge I planted. It is afterall for the benefit of the housewife, child, or businessman that we write, not solely for the someones with a technical background like an engineer such as myself, or whatever field gainfully employs you. If you are writing for a technical audience alone, I submit you need a professional journal, not this venue. FrankB 17:53, 31 March 2006 (UTC)
replies: text: I added a short paragraph on the importance of exponentiation to the intro. --agr 22:37, 31 March 2006 (UTC)
thread_title:  Keep the Reader in mind

message: The article states that "any number to the power 0 is 1". Should we write something about 0^0 which isn't generally defined (but sometimes, I've been told, defined as 0 or 1)? – Foolip 14:27, 27 April 2006 (UTC)
replies: text: I think not. There is a fine discussion on the subject in  Empty Product. Bo Jacoby 12:46, 28 April 2006 (UTC)
thread_title:  any number to the power 0 is 1 

message: Does anyone know how you could x^y = z when you know both x and z?
Example:
5^y = 390625
y can only be 8, but is there any way to figure that out without simply guessing?  It may also be near impossible to guess for solving things such as
15.3^y = 401.3 or something like that.
You can actually find a use for this in probability, if you were finding the number of occurances necessary to create a certain odd
e.g., if the probability of rolling a one on a die = \frac{1}{6}, the probabilty of it not happening = 1 - \frac{1}{6}, or \frac{5}{6}, so the number of times you must roll a die to make the odds of rolling a one in a round of rolls \approx \frac{1}{4} = x when \left(\frac{5}{6}\right)^x = \frac{1}{4}
...but how do you find x?
:That's what the log button on your calculator is for: if x^y = z, then y = log(z)/log(x). Try it.--agr 21:02, 21 May 2006 (UTC)
replies: text: Solve the equation x log(x)=log(4812) by some root-finding algorithm. For example by substituting x:=log(4812)/log(x). Starting with x=5, ending with the solution x=5.1644 . Bo Jacoby 14:24, 22 May 2006 (UTC)
replies: text: Re "what does "log" exactly do?": do us a favor and take a look at the logarithm article and let us know what isn't clear. (Asking those questions is good, by the way.)--agr 19:57, 23 May 2006 (UTC)
thread_title: Basic question

message: I agree that the intro needed to be simplified, but I think it has gone a bit too far by saying "exponentiation is repeated multiplication". The reader shouldn't have to wade through 3-4 screens full of math to learn that exponents do not have to be integers. --agr 13:23, 19 June 2006 (UTC)
replies: text: The beginner should have a chance to read about positive integer exponents in peace. The advanced reader should eventually read the Taylor formula in exponential form, eD=F, where the exponent is the differentiation operator and the result is the displacement operator. The reader can click inside the table of contents to skip the elementary screens. But of course it can be improved. Be bold and do it. Bo Jacoby 10:35, 20 June 2006 (UTC)
thread_title: Simplified intro

message: In the "Fractional exponent" section of the article, the following...
:For a given exponent, the inverse of exponentiation is extracting a root.
:If \ y is a positive real number, and n is a positive integer, then the positive real solution to the equation
::\ x^n = y
:is called the nth root of \ a
:: x=y^{\frac{1}{n}}
:For example: 81/3 = 2.
This may sound ignorant, but...there's no "a" in either of those equations.  Does it mean to say "the nth root of x"?  --zenohockey 20:47, 20 June 2006 (UTC)
replies: text: Thanks. I'll correct it immediately. It means to say the n'th root of y. Bo Jacoby 07:35, 21 June 2006 (UTC)
thread_title:  "Fractional exponent" section: nth root of a? 

message: If the expression for 10^3=10 x 10 x 10, then what is the expression for 10^{3.2}, or how do we calculate it without using the x^y button on a calculator?  
As a consequence, do non-integer exponents also deserve a slightly more detailed sub-section?
:Answer to Lars-Erik: Using the general formula ex=limn→∞(1+x/n)n, first find b as the solution to the equation eb = 10 , then compute 103.2 = (eb)3.2 = e3.2b . There are faster methods, but this one shows the principle. Bo Jacoby 10:09, 20 July 2006 (UTC)
thread_title:  Non-integer exponent 

message: Why does the Exponents page redirect to some band?  I think most people searching for exponents will be expecting math.  If no one objects, I'm going to fix it. Alex Dodge 10:25, 3 September 2006 (UTC)
replies: text: Perhaps it should redirect to List of exponential topics and maybe mention the band there.--agr 11:21, 3 September 2006 (UTC)
replies: text: That sounds good, and I'm doing it now.  (Sorry about the delay.  I just moved into college.)  However, wouldn't a disambiguation page be more standard?  I've never witnessed this "List of X Topics" construct before. Alex Dodge 18:23, 18 September 2006 (UTC)

text: That's a horrible idea.  Make a disambiguation page at Exponents instead of a redirect. --Raijinili 07:39, 23 September 2006 (UTC)
replies: text: Yes.  I agree.  And, as such, I have made a simple disambiguation page.  Does this look acceptable to everybody?  Alex Dodge 21:24, 23 September 2006 (UTC)
thread_title:  Exponents redirect 

message: in eX, when X is a vector {x1, x2, ...}, then 
eX is a vector (ex1,ex2,...} Am I right?
I think that we had better list some formula for this such as:
eX*eXT=eX*XT?
...
Jackzhp 20:09, 3 September 2006 (UTC)
thread_title:  when the power is a vector 

message: There is a discussion at the ref desk about whether raising to a different power expresses a different dimension. If you want to contribute, be quick, because these discussions die out in a few days. DirkvdM 08:59, 4 September 2006 (UTC)
thread_title:  Dimension and exponentiation 

message: I would like some clarification on terminology, especially the word "power".
In the expression: 
p = b^x; 
is p the "power", or is x? Some dictionaries define the "power" as the exponent, others as the result of multiplying a number by itself a specified number of times.
The latter conforms to a common usage, e.g., if we list the "powers of two", the answer would be 2, 4, 8, 16, ...rather than 1, 2, 3, 4...  So I would prefer to call p the power.  But if x is the "power", what is the term for p?
In any case, a complete list of terms would be helpful. Given the expressions:
p = b^x; b = \sqrt[x]{p}; x = log_b {p} ;
is the following correct? 
b is the BASE
x is the EXPONENT
p is the POWER [(xth) power of base b ] 
b is the ROOT [(xth) root of p] 
x is the LOGARITHM of p to base b 
Are there any other words used to describe the elements of this equation?
Drj1943 02:06, 26 November 2006 (UTC)
thread_title:  Terminology 

message: I was recently presented with this interesting 'proof':
:e^{ix} = e^\frac{i2\pi x}{2\pi} = (e^{i2\pi})^\frac{x}{2\pi} = 1^\frac{x}{2\pi} = 1
The poser of this problem told me the invalid part was that e^{ab} = (e^a)^b only holds for real exponents, but this is contradicted by this article. Can anyone find the flaw and explain it to me? I think it might be a useful addition to this article and to Invalid proof... plus it'll stop the stupid thing bugging me! -- Perey 19:05, 11 December 2006 (UTC)
replies: text: 1^\frac{x}{2\pi} is multivalued. One of the values are 1 and another of the values are e^{ix}. Bo Jacoby 23:31, 11 December 2006 (UTC)
replies: text: Thank you. I can't see anywhere where this information is present on the current page. The long-since deleted section Powers of one kind of hinted at it in a roundabout way, where it said that for the xth powers of one, e^{i2\pi x} (after stating e^{i2\pi} = 1), if x is integer then the result is 1, if x is rational then the result is a root of unity, and if x is real then the result is a 'direction' (meaning according to prior definition that it's a complex number on the unit circle). While this does suggest special cases for irrational exponents of one, it doesn't say that they're multivalued. Should it have? I really think a 'Powers of one' section should be reintroduced—without the confusing start from e^{i2\pi} = 1. -- Perey 06:17, 12 December 2006 (UTC)

text: The subsection on Multivalued power says: "If eb = a, then e(b+2πi·n)x are the values of ax  ".  As e0 = 1,  then e(0+2πi·n)x/2π are the values of 1x/2π. For n=0 you get 1. For n=1 you get eix. I am pleased that you recall and miss the section of powers of one. Alas the notation 1x was very strongly opposed by other WP editors. The formula e^{i2\pi} = 1 should not confuse you. It merely says that the circumference of the unit circle is 2π. Bo Jacoby 08:44, 12 December 2006 (UTC)
replies: text: Oh, yes, of course. Somehow I missed that whole section, seeing only the 'Multivalued logarithm' section—so I guess I inferred that logarithms could be multivalued (log(e^x) = x+2\pi n), but not the reverse. And it's not so much that e^{i2\pi} = 1 is confusing; but then, in my studies Euler's formula is bread and butter—to others it might be quite understandably confusing. But even to me, it seems to obscure the point that this is unity we're talking about. I would tentatively suggest stating the cases to begin with in terms of 1^x (the cases being integer x, rational x, real x, and—heaven help us—complex x). Then recall to the reader that e^{i2\pi} = 1, and explore these results further using powers of e.--Perey 16:07, 12 December 2006 (UTC)

text: I agree that the expression e2·π·i·x is spooky, involving transcendental numbers e and π. It could simply be called 1x with a caveat that it is the primitive value rather than the principal value. But I am overwhelmingly opposed. (See Talk:Function_%28mathematics%29#warning_to_editors). Bo Jacoby 23:03, 12 December 2006 (UTC)
thread_title:  An [[invalid proof]] 

message: The actual page on exponentiation claims:
A non-zero integer power of e is
:e^x 
= \left( \lim_{m \rightarrow \pm\infty} \left(1+\frac{1}{m} \right) ^m\right) ^x 
= \lim_{m \rightarrow \pm\infty} \left(\left(1+\frac{1}{m} \right) ^m\right) ^x 
= \lim_{m \rightarrow \pm\infty} \left(1+\frac{1}{m} \right) ^{mx} 
= \lim_{mx \rightarrow \pm\infty} \left(1+\frac{x}{mx} \right) ^{mx} 
= \lim_{n \rightarrow \pm\infty} \left(1+\frac{x}{n} \right) ^n 
 .
The right hand side generalizes the meaning of ex so that x does not have to be a non-zero integer but can be zero, a fraction, a real number, a complex number, or a square matrix. 
Would someone care to explain this particular step: \lim_{m \rightarrow \pm\infty} \left(1+\frac{1}{m} \right) ^{mx} 
= \lim_{mx \rightarrow \pm\infty} \left(1+\frac{x}{mx} \right) ^{mx}
 which confuses me profoundly?
I especially can't see how this motivates using e^x = \lim_{n \rightarrow \pm\infty} \left(1+\frac{x}{n} \right) ^n even for for x=0 as mx \rightarrow \pm\infty hardly will hold in that case. -- Qha 00:44, 12 December 2006 (UTC)
replies: text: Thanks for asking. The fraction 1/m is multiplied in numerator and denominator by x, which is not zero. When m goes towards plus or minus infinity, the so does mx. Then mx is renamed to n. Bo Jacoby 08:15, 12 December 2006 (UTC)
thread_title:  Powers of e 

message: The quote: "The function zz, viewed as a function of a complex number variable z and defined as ea ln z, has a logarithmic branch point at z = 0" is neither correct nor an argument for leaving 0^0 undefined. Please improve. Bo Jacoby 00:04, 19 December 2006 (UTC)
replies: text: That error was introducted (inadvertenly) by M Hardy when cleaning up the math notation    . Thanks for pointing it out; I fixed it. CMummert 00:44, 19 December 2006 (UTC)
replies: text: Thanks. The logarithm log(z) is discontinuous for z=0 as lim log(z)=∞ while zz is continuous for z=0 as lim zz=1, because lim z·log(z)=0. So z=0 is not a logarithmic branch point of zz. And still it is not an argument for leaving 00 undefined. Bo Jacoby 08:07, 19 December 2006 (UTC)
thread_title: logarithmic branch point

message: Trovatore: "While edits made in collaborative spirit involve considerably more time and thought than reflexive reverts, they are far more likely to ensure both mutually satisfactory and more objective articles." Wikipedia: edit war. Bo Jacoby 00:14, 19 December 2006 (UTC)
replies: text: The sentence after that is: "In the case of less experienced contributors, who have unknowingly made poor edits, reversion by two or more people often demonstrates that such reversions are probably not fundamentalistic or in bad faith, but instead closer to an objective consensus."  This is more applicable to you than your quote applying to Trovatore.  --C S (Talk) 00:33, 19 December 2006 (UTC)
replies: text: The edit in question is Trovatore's insisting in moving the important Knuth quote from the subsection (where in my opinion it belongs) to the reference subsection (which in my opinion is for references, not for quotes), without discussing the matter. I am not a less experienced contributor. Bo Jacoby 08:07, 19 December 2006 (UTC)
thread_title: quote in the subsection or in the reference section

message: The subsection is close to position 2, except that one distinguishes between discrete and continuous math rather than between integers and reals. In consequence of that peculiar point of view one should state 00=1 in the subsection on discrete mathematics, and allow 00 to be left undefined in the subsection of continuous mathematics. Please comment on that. Bo Jacoby 08:07, 19 December 2006 (UTC)
thread_title: discrete and continuous

message: 00=1 is supported by Euler, Laplace, Libri, Möbius and Knuth, while Cauchy leaves it undefined. These are mathematicians, not textbooks. If the authors of the textbooks following Cauchy shall count as authors we must quote their argumentation. Bo Jacoby 22:37, 19 December 2006 (UTC)
thread_title:  textbooks for one, textbooks for all 

message: Do not make changes without consensus.  We all tire of this.  VectorPosse 08:47, 20 December 2006 (
:There is consensus that in discrete mathematics 00=1. Even Trovatore wrote: "If n is a nonnegative integer, there is no problem. As I said, I find 0.00=1.0 to be completely convincing and unproblematic". (Talk:Empty_product#Remarks_on_.22binomial_theorem.22_argument). The controversy is for continuous mathematics. My change was for integer exponents. See also Exponentiation#Exponentiation_in_abstract_algebra. I requested your comment here and again here and you didn't respond. Bo Jacoby 09:44, 20 December 2006 (UTC)
replies: text: The examples from the polar form subsection are now found in Exponentiation#Multivalued_power and Exponentiation#Singlevalued_power. The rest belong in complex number but not in exponentiation and that's why I don't improve it. Bo Jacoby 09:44, 20 December 2006 (UTC)
replies: text:  Well, a few of them were relocated to the sections you mentioned.  I still think your edit summary was a bit disingenuous.  Nevertheless, I agree that the content could go elsewhere.  VectorPosse 09:52, 20 December 2006 (UTC)

text: Discuss first and revert later. See Exponentiation#Primitive_and_principal_roots_and_powers_of_unity for the concept that confused you. Bo Jacoby 09:48, 20 December 2006 (UTC)
replies: text:   Don't pretend like your opinions on 1x have not been discussed ad nauseum before.  It didn't confuse me.  It was "confusing" terminology and notation.  And I didn't revert; I removed one sentence that violated the parallel structure of the two sections in question and said nothing about single-valuedness.  VectorPosse 09:54, 20 December 2006 (UTC)

text: No I don't pretend. Here I stated standard interpretations of 1x : the complete set of values (e2πixn), a primitive value (e2πix),  and the principal value (1). My controversial observation, that the primitive value is the useful one, was not promoted here. Bo Jacoby 10:05, 20 December 2006 (UTC)
thread_title:  Bo Jacoby, you don't have consensus 

message: "The principal value has the advantage of being singlevalued, but the price to be paid is that it ceases to be continuous". CMummert continues: "A branch cut for the logarithm must be defined in order to make it an analytic function". No, that is not correct. A branch cut was needed in the first place in order to define the principal value, but the branch cut does not make the function neither analytical nor continuous on points on the branch cut. When walking around the singularity, you experience either entering another branch (multivalued) or a discontinuity (singlevalued). The better solution is to consider the function value to be a multiset. Then after walking around the singularity the value returns to the same multiset as before, even if each point in the multiset has moved continuously into another point in the multiset. Example. The square root of x for x=1 is the multiset {+1,−1}. After moving x once around the unit circle from 1 via i, −1, −i and back to 1, the square root moves half a turn and ends in {−1,+1}, which is the same multiset, even if it is not represented by the same ordered pair of numbers. Please improve. Bo Jacoby 15:04, 20 December 2006 (UTC)
thread_title:  branch cut

message: Not all power series have a nonzero constant term. Not log(1-x)=x+x2/2+x3/3+x4/4+...
I have fixed it. Bo Jacoby 15:17, 20 December 2006 (UTC)
thread_title:  Not all power series have a nonzero constant term.  

message: There is no consensus, not even support, for VectorPosses point of view that 00 is undefined in discrete mathematics. Bo Jacoby 15:38, 20 December 2006 (UTC)
replies: text: True, but it doesn't matter. I don't believe there is a consensus in the literature that it is defined. We are not empowered to create our own consensus here. We are not slavish stenographers; certainly we're informed in our choices by our own experience and expertise, but we can't make up a consensus that doesn't exist. --Trovatore 20:31, 20 December 2006 (UTC)

text: I have already explained why I made that change.  I will say no more on the matter.  If and when you get consensus, Bo Jacoby, you are welcome to change it back.  VectorPosse 21:20, 20 December 2006 (UTC)
thread_title:  VectorPosse lacks support  

message: The links to discrete mathematics and to continuous mathematics are incorrect. We are talking about integer exponents rather than about finite mathematics, and about non-integer exponents rather than about numerical analysis. Somebody please correct it. Bo Jacoby 09:34, 22 December 2006 (UTC)
replies: text: I think the dichotomy is essentially correct, although I would say "discrete mathematics and power series" versus "continuous mathematics". The article used to say this. CMummert 12:57, 22 December 2006 (UTC)

text: I did click them.  There is an article entitled discrete mathematics.  The fact that continuous mathematics redirects to numerical analysis doesn't mean we shouldn't link to it; it means that the redirect will someday need to be fixed.  In the meantime, there is no harm in linking to the redirect from this article, because the redirect is supposed to point to the best article currently available on continuous mathematics. CMummert 22:23, 28 December 2006 (UTC)
thread_title:  confusion links in zero to the zero power 

message: Isn't it true that fractional exponents a^{\frac{m}{n}} are only defined for a>=0 ?
:That is quite at tricky question, actually. The answer depend on the book you are reading, because different books have different definitions. It is not entirely objective. 
:That x=am/n means that x is a solution to the equation xn=am. (Assume that m/n cannot be reduced).
:If a is a positive real number, then some positive real number solves the equation. So the fractional power is defined. However, if n is even, then there is also a negative solution. So the fractional power is not uniquely defined. People don't like that. So they arbitrarily discard the negative solution, and then the fractional power is defined for nonnegative reals.
:If a is a negative real number and m and n are odd numbers, then some negative real number x solves the equation. In that case the fractional power is defined for a<0. This is unimportant.
:Using complex numbers, the equation has n solutions. So the fractional power is a set consisting of n complex numbers. From this point of view the answer to your question is 'no'. Bo Jacoby 20:34, 22 December 2006 (UTC)
replies: text: I appreciate your answer. Don't you think there should be some reference to these remarks in the article itself? —The preceding unsigned comment was added by 83.130.76.97 (talk) 11:29, 23 December 2006 (UTC)

text: Yes. Bo Jacoby 12:23, 24 December 2006 (UTC)
thread_title: Fractional exponent

message: The sentence, (Exponentiation used to be called "involution".), has popped up once more. See above for the earlier discussion on the subject. The sentence adds more to confusion than to clarification, and the article on involution says something completely different. Bo Jacoby 13:08, 29 December 2006 (UTC)
replies: text: Both m-w.com and oed.com say that one meaning of the word "involution" is exponentiation.  The article on involution is describing a different concept with the same name. Ther eis no reason this article shouldn't include a sentence pointing out former terminology. CMummert 13:24, 29 December 2006 (UTC)
thread_title:  involution again 

message: CMummert, please note the following surviving piece of heresy from Exponentiation#Exponentiation_in_abstract_algebra: 
:suppose that the operation has an identity element 1. Then we can define x0 to be equal to 1 for any x. Now xn is defined for any natural number n, including 0.
Why didn't you purge it and placed it on your index librorum prohibitorum? My friend, you have plenty of work to do. Happy new year! Bo Jacoby 18:34, 31 December 2006 (UTC)
thread_title:  censorship