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The present Arithmetic page claims exponentiation as an arithmetic operation. I may be getting it wrong, but I class arithmetic as the art of manipulating numerals - the symbols - in manners isomorphic to the algebraic ops. eg/ie one does not multiply the platonic six hundred seventeen by the platonic nine hundred two, but writes down "617" and "902" and after a bunch of ciphering (the exact word) ends up with another symbol, "556534", representing the number which would come out of the field op on the two inputs. (I'm coming to the point.)
I was taught the four basics plus the extraction of roots (square and cube; I know how it extends to higher degrees but hope never to have to do it). But I've never encountered exponentiation in arithmetic (other than by repeated multiplication, obviously). Is there an actual symbol-manipulation procedure specialised for exponentiation? Please add it if there is one ('cause I'm dying to know what it is). Please change Arithmetic if there is not. 142.177.23.79 23:01, 14 May 2004 (UTC)
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)
Our heroes at Encyclopedia Britannica use the balanced terms "involution" & "evolution" as the standard to refer to the 3rd binary operation and its inverse. ---- OmegaMan
Every mathematician I know (and I am one and know many) calls this operation "exponentiation". Not one of them ever said "involution" in my hearing, or in writing that I read. Zaslav 23:13, 3 February 2006 (UTC)
"Extracting a root used to be called evolution." This is as uncommon as calling exponentiation for Involution. The article on Evolution does not mention that meaning of the word. I will delete the sentence. Bo Jacoby 10:57, 24 May 2006 (UTC)
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)
1
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)
This article doesn't say anthing about powers with non-integer exponents (i.e. decimals, fractions, etc.) I don't know much about it, so could somebody put it up? →[evin290]
It has been suggested that this article or section be merged with exponential function.
I am a high schooler, with not so good math skills. This article is very helpful for studying and basic learning about the math that I have trouble with. Making this merge with another would only make it more complictaed for me. Please keep them seperate so I can get the basics down. -Thanks Bigfoot
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)
Hello Arnold. § Fractional exponents depend on § Powers of e. Please repair your reorganisation. Bo Jacoby 15:06, 1 February 2006 (UTC)
Thank you. Its better, but still I am not happy. Fractional , real, and complex exponents are more advanced than integer exponents and these §s stops the uninitiated reader from continuing reading. The logical sequence is: integer exponents, powers of e, complex exponents. Then fractional and real exponents are special cases of complex exponents. You might have a § on exponent one. Why not move the advanced §s below the heading 'advanced' ? Bo Jacoby 07:20, 2 February 2006 (UTC)
The n'th root is multivalued, and that cannot be helped. The contradiction of terms, multivalued function, paralyses the uninitiated reader who is sensible to contradictions. The interest in powers with fractional exponents faded when it became clear that they cannot be used for solving polynomial equations after all. It was the wrong path to walk. The reader must unlearn fractional exponents on positive reals and learn about finding complex roots in polynomials. Do you intend not to talk about ex at all? If yes, do that. If no, utilize it to the limit. I strongly believe in the path: xn --> ex --> ez . Bo Jacoby 16:32, 2 February 2006 (UTC)
The power function f(x)=x=x1=eln x is not multivalued, even if ln x is multivalued. Bo Jacoby 16:45, 2 February 2006 (UTC)
Surely, any integer is a special case of a complex number, but when there are special cases where your conclusion is wrong, then your argument is wrong too, and that is confusing for your reader. It is not true that the fractional exponent approach does not require limits, because you cannot even compute the square root of two without taking limits. The fractional exponent approach requires the reader to solve a polynomial equation, which is advanced stuff compared to the original level of this article. It is true, however, that the fractional exponent approach does not require calculus, but the definition of ex given in the § powers of e does not require calculus either. Your § on fractional exponents rely on the concept of nth root, which has not yet been defined. Your example, 51.732, requires the reader to first solve the equation x1000=5 and then compute x1732. Some people might not find it easy to grasp. But in practice you use the exponential function, which you do not define although it is easier to define than the fractional exponent. Your information that e is transcendental is confusing and useless, because a reader of the elementary article exponentiation cannot be expected to know the more advanced theory of transcendental numbers. So the fractional exponent approach is no good in practice, you admit, and it is no good in theory either, because when restricted to positive real radicands it is insufficient, and when extended to complex radicands it is multivalued. The picture from logarithm belong there and not here because this article is about exponentiation. Articles on related subjects should be referred to for more advanced reading but not for foundation. This article should be for beginners. Bo Jacoby 10:10, 3 February 2006 (UTC)
This article on exponentiation need not depend on knowledge on square roots. To understand exponentiation with integer exponents you only need to know about multiplication and division. To understand ex you also need to know a little about limits, but not much. (As ((n+1)/n)n is almost constant for big values of n, just take some big value as an informal definition of e). On this very modest prerequisite the theory of exponentiation with complex exponents is erected. Things do not necessarily get more technical and should not get technical just because they got technical in school. Going from no knowledge to complex exponentiation in a single reading should of course be done if possible, and it is possible, and it was done. The fractional exponent approach has been used for generations, true, but that is no excuse to torture the generations to come. You tried to make it more approachable for beginners, but you failed completely. Your text is a disaster, illogical and incomprehensive. I didn't suggest that you add more about using ex. I repeat: Do you intend not to talk about ex at all? If yes, do that. If no, utilize it to the limit. You cannot avoid ex, and it is sufficient for defining exponentiation. If one of two methods is sufficient, then the other method is unnecessary. Radicals and square roots are unnecessary concepts in an article on exponentiation, irrespective of what you learned in school. Get rid of it. Bo Jacoby 14:29, 3 February 2006 (UTC)
I agree that there are many ways. Now I made a restructuring of the article. I used one of the possible ways, but included the other computation of fractional exponents as a possibility. I don't use calculus or continuity, but one limit. I look forwards to your comments. Bo Jacoby 12:50, 4 February 2006 (UTC)
In the present version the ln is used without being defined. Bo Jacoby 12:13, 22 February 2006 (UTC)
When you write any base is seems as if you mean any positive real base, although it is not obvious what you mean. That case is treated in a § on complex powers of positive real numbers. The problem of multivalued logarithm is not taken care of any more. You just refer to the inverse as if it was well defined. I think that an elementary article on exponentiation should not talk about logaritms at all, but that is too late now. I have a minimalistic taste, and you don't. It's OK. Bo Jacoby 14:09, 22 February 2006 (UTC)
Yes, the definition is ex = limn→oo(1+x/n)n where n is integer, so this point is not sticky at all. I'd like to emphasize at this stage that x might be a complex number or even a matrix, because neither of the two other methods for fractional exponents, (the one assuming knowledge of continuity of real functions, the other assuming knowledge of integration along the real axis), work for non-real exponents. When x is an integer it should be shown that the two definitions ex = limn→oo(1+x/n)n and ex = (limn→oo(1+1/n)n)x coincide. This is easy using n = mx . When x is not an integer, the definition ex = limn→oo(1+x/n)n is a generalization of the old definition of exponentiation with integer exponent. It cannot be proved, because it is a definition, and so the 'proof' is sticky. Note that the following real powers of one: 1x = ei·2π·x , are fundamental for the description of circular motion, and oscillation, and waves. Complex exponents are very useful and should not be considered advanced. Also I took care not to mention ln(a) but just: let b be any solution to the equation eb=a . Bo Jacoby 13:34, 23 February 2006 (UTC)
Well done. Bo Jacoby 16:50, 27 February 2006 (UTC)
"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)
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)
These things we who write in technical topics need to keep in mind. Best wishes all Fra nkB 19:10, 31 March 2006 (UTC)
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)
I have removed this section and related discussion, apparently inserted by User:Bo Jacoby, as a totally misleading invented notation.
This was already discussed ad nauseam in Talk:Root of unity, where Bo tried and failed to get the article to include his own invented notation.
The problem is this: the expression by convention always denotes the principal value, i.e. it always equals 1 in standard notation.
Moreover, the section explaining that multi-valuedness arises for complex exponents of complex numbers is also misleading, for two reasons. First, the same ambiguity arises in general for any non-integer exponent...it is false to imply it only happens for complex exponents or powers of unity when it wasn't mentioned earlier. Second, the ambiguity is resolved by the standard convention for the principal value.
I've changed the article to mention principal values at the beginning of the section on real exponents, giving the well-known example of square roots, and linked to the appropriate articles. I changed the section on complex exponents to explain that their multivaluedness is not really any different than that for real exponents, and gave the classic i^i example. I removed the sectioon on powers of unity entirely, as this is misleading for the reason noted above and is no different in any case than powers of any other real number. The Root of unity article is already linked.
—Steven G. Johnson 17:08, 17 May 2006 (UTC)
You use the fact that i=eiπ/2, but you have deleted the explanation, so you made the article completely incomprehensible to the uninitiated reader for whom it was intended. (What is the word for doing that? I think the word is 'vandalism'). Whether or not you use the notation 1x, the article must explain the fractional powers of unity and define π before it is used. Bo Jacoby 11:08, 23 May 2006 (UTC)
Steven. In Eulers formula, eix=cos(x)+i sin(x), the left hand side is already defined as lim(1+ix/n)n, (where the limit is for large values of n). That has been explained to the reader of the article. However you use Euler's formula the other way round, now supposing the reader to know trigonometry and not exponentials. If you want to define trigonometry, then use cos(x)=(eix+e−ix)/2 and sin(x)=(eix−e−ix)/(2i). Why not start reading the article in the version it had before you began vandalizing it, then as step 2: understand the flow of logic, which by now you obviously do not understand at all, then, as step 3, do some serious thinking. Then, as step 4, write your suggested improvement on this discussion page. Then, as step 5, read the comments of your fellow wikipedians. That approach will do you honor and no shame. Bo Jacoby 08:43, 24 May 2006 (UTC)
Summarizing does not mean neglecting logic. The intelligent reader deserves better than that. Use my comments as an opportunity to improve on your writing. When you consider yourself perfect then you prevent yourself from improving the article. I do not suggest that sin and cos should be defined nor used in this article, because I completely agree that "this article is not going to develop all of mathematics".
It is perfectly reasonable to explain exponentiation without referring to trigonometry. If z is a nonzero complex number, and n is a positive integer, then by now we know that 00=z0=1 ; 0n=0 ; 0−n remains undefined ; zn=z·zn−1 ; z−n=1/zn ; e=lim(1+1/n)n and ez=lim(1+z/n)n. Everything is well-defined and single-valued, and the rules ez+w=ezew and zn+m=znzm and (zn)m=znm apply. You may continue like this:
Bo Jacoby 14:26, 26 May 2006 (UTC)
Thanks for asking my opinion. The goal is not to satisfy me, but to make a useful article. The reader must have some previous knowledge to understand any article. The more knowledge required by the reader, the fewer readers understand the article, and the less useful is the article. Perhaps you and I have different readers in mind, you're thinking of "The rest of the mathematical world" and I'm thinking of an intelligent young student? That is why I have this minimalistic point of view regarding assumptions on the part of the reader.
The algebraic definitions in the Trigonometric functions article depend on power series, which this article does not (and should not) rely on. That is why I would like this exponentiation article not to depend on trigonometry. My reader does not understand power series, and so talking about power series does not help him understanding exponentiation.
Understanding the formula abc=(ab)c requires the multivalued interpretation of the exponentiation: 14(1/4) = 1 while (14)1/4 = 11/4 = {+1,+i,-1,-i}.
On 00 you added the comment:
This should be omitted, as it increases confusion and decreases clarity. The link to empty product explains:
Also the remark that e can be also defined in other ways, goes without saying. Some readers gets confused by the remark, having plenty to do understanding a single definition. Some readers don't get confused, but nobody gets happier. Bo Jacoby 22:33, 28 May 2006 (UTC)
Does anyone know how you could when you know both x and z? Example: 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 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 = , the probabilty of it not happening = , or , so the number of times you must roll a die to make the odds of rolling a one in a round of rolls when
...but how do you find x?
Well yeah, I know that... let me make it clearer (I'm bad at stating questions) what does "log" exactly do? Heh... I annoyed my teachers the same way asking how trig. ratios were found.
How about this: (or some other number)
can be solved by the Lambert's W function.
I think its a dumb redirect for my entry of POW (prisoner of war) to be redirected here.
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)
In the "Fractional exponent" section of the article, the following...
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)
If the expression for =10 x 10 x 10, then what is the expression for , or how do we calculate it without using the button on a calculator?
As a consequence, do non-integer exponents also deserve a slightly more detailed sub-section?
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)
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)
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)
I would like some clarification on terminology, especially the word "power".
In the expression:
;
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:
; ; ;
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) revised for clarification Drj1943 01:46, 27 November 2006 (UTC)
I was recently presented with this interesting 'proof':
The poser of this problem told me the invalid part was that 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)
The actual page on exponentiation claims:
A non-zero integer power of e is
- .
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: which confuses me profoundly?
I especially can't see how this motivates using even for for x=0 as hardly will hold in that case. -- Qha 00:44, 12 December 2006 (UTC)
Trovatore, your own reference [1] concludes: "Consensus has recently been built around setting the value of 0^0 = 1". So you cannot use it for arguing otherwise. Bo Jacoby 23:52, 15 December 2006 (UTC)
Sorry for doing you injustice, sir. If you insist that "0^0 undefined" be included, you must provide modern references saying so. Bo Jacoby 00:08, 16 December 2006 (UTC) Don't fight an edit war. You claim that "others consider it undefined" but both references say that 0^0=1. Please understand that you must provide proof of your claim. Who are the authors that consider 0^0 undefined? Bo Jacoby 00:16, 16 December 2006 (UTC)
Trovatore, who are the authors that consider 0^0 undefined? Bo Jacoby 00:29, 16 December 2006 (UTC)
When do we want to be indeterminate? What is the point of writing something meaning nothing? What are the "situations in which 0^0 is best left undefined"? Yes, there is a place for integer exponents and another for complex exponents. The reader should consider integer exponents in peace before being bothered by advanced stuff. See Talk:Empty product. Bo Jacoby 09:11, 16 December 2006 (UTC)
By the way, there's a passage in the "Ask Dr. Math" ref that no one seems to have commented on:
This could actually be a reference for the "0.00.0 is undefined" formulation, depending on where it comes from, which is unclear. It's indented the same as the Knuth passage, making it seem a quote, but I don't know who's being quoted. Possibly it's Alex Lopez-Ortiz, the maintainer of the sci.math FAQ. Kahan is probably William Kahan. -- Trovatore 20:09, 16 December 2006 (UTC)
The article now says this:
“ | There are two differing conventions about whether the value of 00 should be defined to be 1 or left as an indeterminate form. | ” |
I'm really not comfortable with this phrasing. That this is an indeterminate form when construed in the way that is necessary in analyis is a demonstrable fact, not a convention. I think I could argue for the same conclusion in the contexts where it makes sense to consider it an empty product, but not in a brief comment like this. I'm going to think about ways of rephrasing this. Michael Hardy 02:30, 17 December 2006 (UTC)
Here is the requested comment on the "Ask Dr. Math" passage above. The premise: "when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0)" , does not imply the conclusion that "0.0^(0.0) is undefined". That (lim xy) is undefined implies only that no definition, or lack of definition, of (lim x)lim y can make xy a continuous function. So the logical implication of the passage is not valid, and it doesn't really matter that the premise itself is not quite valid either: When approaching from any nonvertical direction we have y=ax, and lim xy =lim xax = 1. Only when approaching from a vertical direction do we have lim xy = lim 0y = 0. In order to obtain a limit k≠0 and k≠1 the non-analytical curve, y = log k/log x, having vertical tangent at (0,0) must eventually be followed. WP deserves better logic than this. Bo Jacoby 14:31, 17 December 2006 (UTC).
The three points of view regarding the values of 00 and 00.0 seems to be these.
Position 1: 00=1 and 00.0=1 (Knuth, Euler, Laplace, Kahan)
Position 2: 00=1 and 00.0 is undefined (Trovatore)
Position 3: 00 and 00.0 are both undefined (Cauchy, many textbooks)
As xy is discontinuous for (x,y)=(0,0), the (lack of) limit give no definition of 00.0.
The subsection Exponentiation#Powers_of_zero says on 00: "If the exponent is zero, some authors consider that 00=1, whereas others consider it undefined or indeterminate, as discussed below".
The article says on 00.0: "The zeroth power of zero is usually left undefined in complex analysis; this is discussed below". This seems to be position 3.
The subsection Exponentiation#Zero_to_the_zero_power contains many arguments for position 1, one, perhaps, for position 2: ("In discrete mathematics, the convention is often adopted that 00 = 1. In continuous mathematics such as calculus and complex analysis, the indeterminate form 00 is often left undefined"), and none for position 3.
We don't know any authors besides Trovatore in favour of position 2, and we don't know any authors since Cauchy in favour of position 3. (The words "some", "others", "usually" and "often" do not qualify as references).
Bo Jacoby 11:16, 18 December 2006 (UTC).
Surely we need far better references that "some authors" and the like. The author of a textbook is not necessarily a reseach mathematician. He takes most of the material for his textbook from other textbooks or from articles. There is nothing wrong in that. A textbook which does not define 00 does not necessarily argue that 00 is not or should not be defined, but only indicates that no definition is important in the context of the present textbook. We need the argumentation af the authors of the textbooks. So far we have found absolutely no argumentation in favour of leaving 00 undefined, only that "it is written in the scriptures" - a religious style of argumentation. You should not call the claim 00 = 1 unqualified. I repeat from Concrete Mathematics: "Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1, for all x, if the binomial theorem is to be valid when x=0, y=0, and/or x=−y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant". This means that the claim 00 = 1 is qualified, while the claim "00 is undefined" is unqualified. If we want to qualify that claim we need a mathematician saying the opposite, that "00 = 1 is a mistake". How could he possibly continue? "00 must be left undefined because the functions x0 and 0x have different limiting values when x decreases to 0" ?. That is simply not a valid reason. Leaving 00 undefined does not help at all. I do look forward to see some mathematical argumentation rather that religious argumentation in this matter. Bo Jacoby 14:21, 18 December 2006 (UTC).
What support are you talking about? I am requesting references, so far in vain. "a large group" is not a proper reference. These WP-articles were supposed to make sense to young people, but we are doing a bad job. Why does Knuth feel strong statements necessary? Obviously because "Some textbooks leave the quantity 00 undefined". Why doesn't he make similar statements about why 2+2 = 4 ? Obviously because no serious textbook leaves 2+2 undefined. Why do you feel strongly about undefining what has been successfully defined? Bo Jacoby 14:48, 18 December 2006 (UTC).
Quote: "The convention that 00 is 1 is not necessary here, because the series can be rewritten so that the first term is explicitly 1 rather than ". Of course no evaluation of any expression is ever necessary if you just replace the expression with the proper value. Bo Jacoby 15:02, 18 December 2006 (UTC).
No, WP doesn't define or undefine things, but obsolete ideas are in articles on history, not on mathematics. I look forward to seeing your references, though. Please quote the argumentation from the books. Bo Jacoby 15:06, 18 December 2006 (UTC).
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).
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).
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).
Hi CMummert. Please comment on your edits and reverts to Exponentiation#Zero_to_the_zero_power. The mathematical term for assigning meaning is: " definition", not " convention". The reference to " empty product" need not be repeated. The statement "Other power series identities are similar in this respect" is sloppy: either we tell the story or we don't. The message "Knuth in particular has used this to justify putting 00=1" is contained in the reference. I appreciate the new references to programming languages. Bo Jacoby 20:33, 19 December 2006 (UTC).
Thanks. The WP article on convention does not seem to explain your use of the word. I feel that the word 'convention' is implicating that this is not a matter of mathematical necessity but that anyone may pick the choice of his liking. Perhaps this is where we disagree. Note that only if . If we don't adopt the 'convention', then the left hand side is well defined for all values of x but the right hand side is undefined for x=0. If we do adopt the 'convention' then the equality holds everywhere. Psychologically I understand the opposition against the 'convention'. The different limits of x0 and 0x seems to pinpoint the troublespot: 00. So stay out of trouble by avoiding 00. But the discontinuity remains whether or not 00 is defined. Bo Jacoby 21:39, 19 December 2006 (UTC).
Yes, convention implies that "some alternative convention could be equally good but we accept this one for social convenience". That is not the case here. Therefore the word convention is misleading. It is far more than "a convenience to make it simpler to state certain theorems". It relieves computer programs and hand computations from special cases and provides a huge simplification. If the computer system had 00 undefined every programmer would have to declare a subroutine power(x,y) by "if x=0 and y=0 then 1 else x^y" and use this subroutine rather than the standard routine x^y. No alternative definition does the job. There is no freedom of choice here. It is really not a convention. In Exponentiation#Powers_of_e we have a similar example: A formula is proved true for all integers, and the right hand side is defined for all complex numbers. Then this formula becomes the unique definition of the left hand side for complex numbers. There is no freedom of choice, and we would not call it a convention. The discussion on 00 is even simpler, because 00 is defined for integer 0, and the discussion is about whether this definition should also apply to real and complex 0. No actual generalization is called for. Bo Jacoby 22:22, 19 December 2006 (UTC).
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). PS. See this. Remember to comment on this.
Do not make changes without consensus. We all tire of this. VectorPosse 08:47, 20 December 2006 (
To Bo Jacoby: Look, I'm trying really hard not to resort to knee-jerk reverts. As CMummert correctly pointed out earlier, many changes you make do improve the articles. But then you do some weird stuff too. You last edit summary was, "The examples from 'polar form' are redistributed". It appears that the whole section on polar forms was just deleted. So I don't know what you mean by "redistributed". I think that section was crap, so I don't mind that it was deleted, but at least state what it is you are doing. And better yet, why not improve that section instead of just deleting it? There was useful information there, even if it wasn't presented very well. Correct me if I'm wrong, or let me know to where you intend to "redistribute". VectorPosse 09:22, 20 December 2006 (UTC)
"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).
CMummert prefers "is often left undefined" rather than "some textbooks". The word "often" is unprecise and subjective. So far we have two textbooks in the reference list, and no quotations from them to support your claim. You too should keep NPOV. Bo Jacoby 15:13, 20 December 2006 (UTC).
Hi CMummert. Thanks a lot for the quotes.
When we, due to this controversy, cannot provide a clear article on exponentiation, we should at least provide clarity on the logic of both points of view, and elaborate on the consequences.
I see a danger in not defining, because the formulas must then be supplemented by an exception for zero. Alone in the short article on power series there are 14 formulas to correct, and we must expand the article with about the same number of lines, decreasing the clarity and quality of the article. It is hard work. You do it - not me. There must be an easier way. Why not add a note saying: "in this article 00 means 1". That must be done in some hundred mathematical articles in wikipedia. It could be centralized, though, saying in one article: "Note that in wikipedia 00 means 1". Which article? Exponentiation of course. Then we could add a note to the wikipedia articles where this is not the case, that "in this article 00 is undefined". That is not many articles because undefined expressions are not used. It doesn't make sense to say something that means nothing.
But you are the one to see a danger of defining. Tell me. What can possibly go wrong by defining what is otherwise undefined? There is no contradiction, as if we were aiming at defining 1/0. The definition 00=1 gives an expressive power which is otherwise lost. If you at least admit that 00=1 in discrete mathematics, then please stop supporting VectorPosse. It will save you years of tedious editing work.
Bo Jacoby 19:18, 20 December 2006 (UTC).
Thanks for the nice words. I am not aware of any change in style. Of course I don't intend to offend anybody and I apologize if I did. I agree that dividing by zero is a disaster. The expression x=a/b means exactly the same thing as the equation a=bx. If a=0 and b=0, then the equation is true for all values of x. If a≠0 and b=0, then the equation is false for all values of x. In neither case is the equation suitable as a definition. But that x0=1, even for x=0, is the de facto standard in power series and in many more places. The motivation is explained: empty product, empty function, even continuity is valid when the exponents are integers only. Now 0=0.0 is also de facto standard: nobody cares whether 0 is integer or real. But while undefined 00 is a disaster, an undefined 00.0 would pass virtually unnoticed, because the function 0x is quite unimportant and almost never used, while x0 is used all the time. Nevertheless, no harm is done by setting 00.0 = 00 . It is the right thing to do. You set 2+2=2+2.0=2.0+2.0 without hesitation.
An editor of a WP mathematics article should be a mathematician. It is neither sufficient nor necessary to copy a textbook from the shelf. The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion. Perhaps this is where CMummert disagrees. Trovatore's original objection against defining 00.0 was not the textbook from the shelf, but the lack of interpretation of the expression 00.0 . That is a philosophical argument, not a mathematical argument. The same kind of argument has been made against x4, because space has only 3 dimensions and so x4 has no interpretation; and against 1/2, because you cannot count to 1/2, and against negative numbers, and against complex numbers. It is hard for me to explain why, but the argument is completely invalid mathematically. Mathematics does not depend on interpretation, but on logical consistency. This is why I disagree with Trovatore. I am completely convinced that the undefining of 00.0 and 00 is a bad mistake. It is OK to leave an expression undefined as long as there is no clue to a definition, but when a constructive definition has emerged, then there is no point in going backwards to the expression being undefined. Who are we to prevent people from the benefit of a good definition? We are free not to use it ourselves if we don't want to.
Bo Jacoby 23:22, 20 December 2006 (UTC).
CMummert wrote "Let me preface this by pointing out that these are non-editorial discussions we're having now; strictly speaking, they're off-topic". To that I answered that "The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion". I am not implying that we are not all mathematicians, but that our mathematical discussion is important in our roles as WP editors and not off-topic. I am sorry if that was misunderstood. The authors of polynomial and power series and binomial theorem and many others assume the definition 00=1. Where do CMummert, Trovatore and VectorPosse expect our readers to find this definition if not in exponentiation ? Bo Jacoby 12:27, 21 December 2006 (UTC).
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).
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).
The original discussion of 00 in the "empty product" article [5], written back in 2003 mainly by Michael Hardy and Toby Bartels, was perfect. I want it back!
A particularly silly claim of the present article is that we should define 00 = 1 because that's what programming languages do. Programming languages are based on math, not the other way around.
Besides, the list is not comprehensive. Mathematica, for example, takes 00 to be an indeterminate form. Curiously, Mathematica simplifies to a0 to 1 even if you don't put any restraints on a, but does not attempt to simplify 0a. It presumably takes 00 to be indeterminate for the purpose of symbolic limit evaluations.
The conclusion in the 2003 edition of the "empty product" article got it right:
I read the "empty product" article a few years ago when I first needed to know the deal with 00, and found it extremely helpful. The article we have here is not helpful the way it is presently written. It seems that certain editors (no names) are more interested in coming up with "more nuanced approaches" for their own pleasure than to actually serve the readers. Fredrik Johansson 13:56, 21 December 2006 (UTC)
An expression that "may be replaced by 1 for the purpose of calculation" has the value 1. A consistent no-nonsense point of view is that
Bo Jacoby 21:01, 21 December 2006 (UTC).
The viewpoint 00 = 5 is consistent but nonsense: When restricting the exponent to integer values, the definition 00 = 1 is in harmony with interpretation as well as with continuity: x0 = 1 for x≠0, and this function is continuous for x=0. When the variables are generalized to nonnegative reals, continuity does not survive. When generalized to complex numbers, singlevaluedness does not survive. This is not a reason why the definition of xn should not survive. There is no improvement in removing a definition, unless you want to use the now undefined expression for something else, which you might do if the expression is not already used. But you don't want to redefine 00. You want to undefine it. And for no reason but that there are some books on your shelf. Look at Derivative#Rules_for_finding_the_derivative.
That is continuous mathematics, well within the realm where you insist that 00 be undefined. Now insert r=1 and get the result that the derivative of x is the discontinuous function x0, which equals 1 everywhere except for x=0. That is an incorrect result. The 'consistent' CMummert-definition 00 = 5 gives another incorrect result. The definition 00 = 1 gives the correct result. You are doing everyone a disservice by you crusade against the definition 00 = 1. Please do some serious thinking before you answer by knee-jerk reaction. You claim to be a mathematician. Show me. Bo Jacoby 14:52, 22 December 2006 (UTC).
Retaining the definition 00 = 1 is not a mathematical necessity, only a mathematical convenience, as it avoids complications. Undefining it leaves a lot of formulas - in discrete as well as continuous mathematics - to be rewritten, showing that the authors of these formulas did assume the definition. While x=0/0 says nothing about x, (because 0x=0 for all values of x, so x might even be equal to 5), the statement x=00 says that x is an empty product. So there is a profound difference between 0/0 and 00. Your 'counterexample' is nevertheless to the point: One doesn't prove that 00 = 1. It's a definition. Bo Jacoby 17:10, 29 December 2006 (UTC).
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).
The dichotomy is correct, but the links were misleading. Did you ever click on those links? Now I removed them from the article. Bo Jacoby 14:06, 22 December 2006 (UTC).
Isn't it true that fractional exponents are only defined for ?
There is plenty of discussion in the section "Zero to the zero power" to support both points of view. Bo Jacoby needs to stop making changes to this section and others without consensus. People have worked extra hard to make sure the wording is neutral. VectorPosse 20:16, 28 December 2006 (UTC)
The very replacement of with actually assumes the definition x0 = 1. Whether necessary or not, the culture of mathematics is that for all x, and so that x0 = 1 for all x, even for x=0. The replacements are not actually made, not in wikipedia, not anywhere. The use of x0 = 1 is not limited to discrete mathematics. In calculus the formula is valid, and x=x1 is valid too. Now the general formula leads to . For consistency, calculus has to accept the definition x0 = 1 for all values of x. Even if "few books will even bother to mention it", it is assumed by all the mathematicians. So the claim of the article that "In continuous mathematics such as calculus and complex analysis, the indeterminate form 00 is often left undefined" is simply not correct. Even in calculus the definition 00 = 1 is assumed. Bo Jacoby 07:46, 29 December 2006 (UTC).
There is no difference between saying that 0^0 is a shorthand for 1 and saying that 0^0 equals 1. The empty product applies to integer exponents, and you say you agree that an empty product is one. Why not accept it at least in the subsection on integer exponents? I don't need consensus, nor do you, but we both need support. I count Fredrik Johansson and Hardy and EdC amoung my supporters, and you count VectorPosse and Trovatore amongst yours. You have lots of editing to do, in this article and in others, to prevail in your pointless crusade against a commonly used definition. Bo Jacoby 17:40, 29 December 2006 (UTC).
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).
Nor is there any reason why an obsolete word shall be the first one to meet the uninitiated reader. I'll move the sentence down to 'advanced topics'. Bo Jacoby 13:21, 31 December 2006 (UTC).
CMummert, please note the following surviving piece of heresy from Exponentiation#Exponentiation_in_abstract_algebra:
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).
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The present Arithmetic page claims exponentiation as an arithmetic operation. I may be getting it wrong, but I class arithmetic as the art of manipulating numerals - the symbols - in manners isomorphic to the algebraic ops. eg/ie one does not multiply the platonic six hundred seventeen by the platonic nine hundred two, but writes down "617" and "902" and after a bunch of ciphering (the exact word) ends up with another symbol, "556534", representing the number which would come out of the field op on the two inputs. (I'm coming to the point.)
I was taught the four basics plus the extraction of roots (square and cube; I know how it extends to higher degrees but hope never to have to do it). But I've never encountered exponentiation in arithmetic (other than by repeated multiplication, obviously). Is there an actual symbol-manipulation procedure specialised for exponentiation? Please add it if there is one ('cause I'm dying to know what it is). Please change Arithmetic if there is not. 142.177.23.79 23:01, 14 May 2004 (UTC)
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)
Our heroes at Encyclopedia Britannica use the balanced terms "involution" & "evolution" as the standard to refer to the 3rd binary operation and its inverse. ---- OmegaMan
Every mathematician I know (and I am one and know many) calls this operation "exponentiation". Not one of them ever said "involution" in my hearing, or in writing that I read. Zaslav 23:13, 3 February 2006 (UTC)
"Extracting a root used to be called evolution." This is as uncommon as calling exponentiation for Involution. The article on Evolution does not mention that meaning of the word. I will delete the sentence. Bo Jacoby 10:57, 24 May 2006 (UTC)
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)
1
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)
This article doesn't say anthing about powers with non-integer exponents (i.e. decimals, fractions, etc.) I don't know much about it, so could somebody put it up? →[evin290]
It has been suggested that this article or section be merged with exponential function.
I am a high schooler, with not so good math skills. This article is very helpful for studying and basic learning about the math that I have trouble with. Making this merge with another would only make it more complictaed for me. Please keep them seperate so I can get the basics down. -Thanks Bigfoot
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)
Hello Arnold. § Fractional exponents depend on § Powers of e. Please repair your reorganisation. Bo Jacoby 15:06, 1 February 2006 (UTC)
Thank you. Its better, but still I am not happy. Fractional , real, and complex exponents are more advanced than integer exponents and these §s stops the uninitiated reader from continuing reading. The logical sequence is: integer exponents, powers of e, complex exponents. Then fractional and real exponents are special cases of complex exponents. You might have a § on exponent one. Why not move the advanced §s below the heading 'advanced' ? Bo Jacoby 07:20, 2 February 2006 (UTC)
The n'th root is multivalued, and that cannot be helped. The contradiction of terms, multivalued function, paralyses the uninitiated reader who is sensible to contradictions. The interest in powers with fractional exponents faded when it became clear that they cannot be used for solving polynomial equations after all. It was the wrong path to walk. The reader must unlearn fractional exponents on positive reals and learn about finding complex roots in polynomials. Do you intend not to talk about ex at all? If yes, do that. If no, utilize it to the limit. I strongly believe in the path: xn --> ex --> ez . Bo Jacoby 16:32, 2 February 2006 (UTC)
The power function f(x)=x=x1=eln x is not multivalued, even if ln x is multivalued. Bo Jacoby 16:45, 2 February 2006 (UTC)
Surely, any integer is a special case of a complex number, but when there are special cases where your conclusion is wrong, then your argument is wrong too, and that is confusing for your reader. It is not true that the fractional exponent approach does not require limits, because you cannot even compute the square root of two without taking limits. The fractional exponent approach requires the reader to solve a polynomial equation, which is advanced stuff compared to the original level of this article. It is true, however, that the fractional exponent approach does not require calculus, but the definition of ex given in the § powers of e does not require calculus either. Your § on fractional exponents rely on the concept of nth root, which has not yet been defined. Your example, 51.732, requires the reader to first solve the equation x1000=5 and then compute x1732. Some people might not find it easy to grasp. But in practice you use the exponential function, which you do not define although it is easier to define than the fractional exponent. Your information that e is transcendental is confusing and useless, because a reader of the elementary article exponentiation cannot be expected to know the more advanced theory of transcendental numbers. So the fractional exponent approach is no good in practice, you admit, and it is no good in theory either, because when restricted to positive real radicands it is insufficient, and when extended to complex radicands it is multivalued. The picture from logarithm belong there and not here because this article is about exponentiation. Articles on related subjects should be referred to for more advanced reading but not for foundation. This article should be for beginners. Bo Jacoby 10:10, 3 February 2006 (UTC)
This article on exponentiation need not depend on knowledge on square roots. To understand exponentiation with integer exponents you only need to know about multiplication and division. To understand ex you also need to know a little about limits, but not much. (As ((n+1)/n)n is almost constant for big values of n, just take some big value as an informal definition of e). On this very modest prerequisite the theory of exponentiation with complex exponents is erected. Things do not necessarily get more technical and should not get technical just because they got technical in school. Going from no knowledge to complex exponentiation in a single reading should of course be done if possible, and it is possible, and it was done. The fractional exponent approach has been used for generations, true, but that is no excuse to torture the generations to come. You tried to make it more approachable for beginners, but you failed completely. Your text is a disaster, illogical and incomprehensive. I didn't suggest that you add more about using ex. I repeat: Do you intend not to talk about ex at all? If yes, do that. If no, utilize it to the limit. You cannot avoid ex, and it is sufficient for defining exponentiation. If one of two methods is sufficient, then the other method is unnecessary. Radicals and square roots are unnecessary concepts in an article on exponentiation, irrespective of what you learned in school. Get rid of it. Bo Jacoby 14:29, 3 February 2006 (UTC)
I agree that there are many ways. Now I made a restructuring of the article. I used one of the possible ways, but included the other computation of fractional exponents as a possibility. I don't use calculus or continuity, but one limit. I look forwards to your comments. Bo Jacoby 12:50, 4 February 2006 (UTC)
In the present version the ln is used without being defined. Bo Jacoby 12:13, 22 February 2006 (UTC)
When you write any base is seems as if you mean any positive real base, although it is not obvious what you mean. That case is treated in a § on complex powers of positive real numbers. The problem of multivalued logarithm is not taken care of any more. You just refer to the inverse as if it was well defined. I think that an elementary article on exponentiation should not talk about logaritms at all, but that is too late now. I have a minimalistic taste, and you don't. It's OK. Bo Jacoby 14:09, 22 February 2006 (UTC)
Yes, the definition is ex = limn→oo(1+x/n)n where n is integer, so this point is not sticky at all. I'd like to emphasize at this stage that x might be a complex number or even a matrix, because neither of the two other methods for fractional exponents, (the one assuming knowledge of continuity of real functions, the other assuming knowledge of integration along the real axis), work for non-real exponents. When x is an integer it should be shown that the two definitions ex = limn→oo(1+x/n)n and ex = (limn→oo(1+1/n)n)x coincide. This is easy using n = mx . When x is not an integer, the definition ex = limn→oo(1+x/n)n is a generalization of the old definition of exponentiation with integer exponent. It cannot be proved, because it is a definition, and so the 'proof' is sticky. Note that the following real powers of one: 1x = ei·2π·x , are fundamental for the description of circular motion, and oscillation, and waves. Complex exponents are very useful and should not be considered advanced. Also I took care not to mention ln(a) but just: let b be any solution to the equation eb=a . Bo Jacoby 13:34, 23 February 2006 (UTC)
Well done. Bo Jacoby 16:50, 27 February 2006 (UTC)
"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)
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)
These things we who write in technical topics need to keep in mind. Best wishes all Fra nkB 19:10, 31 March 2006 (UTC)
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)
I have removed this section and related discussion, apparently inserted by User:Bo Jacoby, as a totally misleading invented notation.
This was already discussed ad nauseam in Talk:Root of unity, where Bo tried and failed to get the article to include his own invented notation.
The problem is this: the expression by convention always denotes the principal value, i.e. it always equals 1 in standard notation.
Moreover, the section explaining that multi-valuedness arises for complex exponents of complex numbers is also misleading, for two reasons. First, the same ambiguity arises in general for any non-integer exponent...it is false to imply it only happens for complex exponents or powers of unity when it wasn't mentioned earlier. Second, the ambiguity is resolved by the standard convention for the principal value.
I've changed the article to mention principal values at the beginning of the section on real exponents, giving the well-known example of square roots, and linked to the appropriate articles. I changed the section on complex exponents to explain that their multivaluedness is not really any different than that for real exponents, and gave the classic i^i example. I removed the sectioon on powers of unity entirely, as this is misleading for the reason noted above and is no different in any case than powers of any other real number. The Root of unity article is already linked.
—Steven G. Johnson 17:08, 17 May 2006 (UTC)
You use the fact that i=eiπ/2, but you have deleted the explanation, so you made the article completely incomprehensible to the uninitiated reader for whom it was intended. (What is the word for doing that? I think the word is 'vandalism'). Whether or not you use the notation 1x, the article must explain the fractional powers of unity and define π before it is used. Bo Jacoby 11:08, 23 May 2006 (UTC)
Steven. In Eulers formula, eix=cos(x)+i sin(x), the left hand side is already defined as lim(1+ix/n)n, (where the limit is for large values of n). That has been explained to the reader of the article. However you use Euler's formula the other way round, now supposing the reader to know trigonometry and not exponentials. If you want to define trigonometry, then use cos(x)=(eix+e−ix)/2 and sin(x)=(eix−e−ix)/(2i). Why not start reading the article in the version it had before you began vandalizing it, then as step 2: understand the flow of logic, which by now you obviously do not understand at all, then, as step 3, do some serious thinking. Then, as step 4, write your suggested improvement on this discussion page. Then, as step 5, read the comments of your fellow wikipedians. That approach will do you honor and no shame. Bo Jacoby 08:43, 24 May 2006 (UTC)
Summarizing does not mean neglecting logic. The intelligent reader deserves better than that. Use my comments as an opportunity to improve on your writing. When you consider yourself perfect then you prevent yourself from improving the article. I do not suggest that sin and cos should be defined nor used in this article, because I completely agree that "this article is not going to develop all of mathematics".
It is perfectly reasonable to explain exponentiation without referring to trigonometry. If z is a nonzero complex number, and n is a positive integer, then by now we know that 00=z0=1 ; 0n=0 ; 0−n remains undefined ; zn=z·zn−1 ; z−n=1/zn ; e=lim(1+1/n)n and ez=lim(1+z/n)n. Everything is well-defined and single-valued, and the rules ez+w=ezew and zn+m=znzm and (zn)m=znm apply. You may continue like this:
Bo Jacoby 14:26, 26 May 2006 (UTC)
Thanks for asking my opinion. The goal is not to satisfy me, but to make a useful article. The reader must have some previous knowledge to understand any article. The more knowledge required by the reader, the fewer readers understand the article, and the less useful is the article. Perhaps you and I have different readers in mind, you're thinking of "The rest of the mathematical world" and I'm thinking of an intelligent young student? That is why I have this minimalistic point of view regarding assumptions on the part of the reader.
The algebraic definitions in the Trigonometric functions article depend on power series, which this article does not (and should not) rely on. That is why I would like this exponentiation article not to depend on trigonometry. My reader does not understand power series, and so talking about power series does not help him understanding exponentiation.
Understanding the formula abc=(ab)c requires the multivalued interpretation of the exponentiation: 14(1/4) = 1 while (14)1/4 = 11/4 = {+1,+i,-1,-i}.
On 00 you added the comment:
This should be omitted, as it increases confusion and decreases clarity. The link to empty product explains:
Also the remark that e can be also defined in other ways, goes without saying. Some readers gets confused by the remark, having plenty to do understanding a single definition. Some readers don't get confused, but nobody gets happier. Bo Jacoby 22:33, 28 May 2006 (UTC)
Does anyone know how you could when you know both x and z? Example: 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 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 = , the probabilty of it not happening = , or , so the number of times you must roll a die to make the odds of rolling a one in a round of rolls when
...but how do you find x?
Well yeah, I know that... let me make it clearer (I'm bad at stating questions) what does "log" exactly do? Heh... I annoyed my teachers the same way asking how trig. ratios were found.
How about this: (or some other number)
can be solved by the Lambert's W function.
I think its a dumb redirect for my entry of POW (prisoner of war) to be redirected here.
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)
In the "Fractional exponent" section of the article, the following...
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)
If the expression for =10 x 10 x 10, then what is the expression for , or how do we calculate it without using the button on a calculator?
As a consequence, do non-integer exponents also deserve a slightly more detailed sub-section?
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)
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)
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)
I would like some clarification on terminology, especially the word "power".
In the expression:
;
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:
; ; ;
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) revised for clarification Drj1943 01:46, 27 November 2006 (UTC)
I was recently presented with this interesting 'proof':
The poser of this problem told me the invalid part was that 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)
The actual page on exponentiation claims:
A non-zero integer power of e is
- .
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: which confuses me profoundly?
I especially can't see how this motivates using even for for x=0 as hardly will hold in that case. -- Qha 00:44, 12 December 2006 (UTC)
Trovatore, your own reference [1] concludes: "Consensus has recently been built around setting the value of 0^0 = 1". So you cannot use it for arguing otherwise. Bo Jacoby 23:52, 15 December 2006 (UTC)
Sorry for doing you injustice, sir. If you insist that "0^0 undefined" be included, you must provide modern references saying so. Bo Jacoby 00:08, 16 December 2006 (UTC) Don't fight an edit war. You claim that "others consider it undefined" but both references say that 0^0=1. Please understand that you must provide proof of your claim. Who are the authors that consider 0^0 undefined? Bo Jacoby 00:16, 16 December 2006 (UTC)
Trovatore, who are the authors that consider 0^0 undefined? Bo Jacoby 00:29, 16 December 2006 (UTC)
When do we want to be indeterminate? What is the point of writing something meaning nothing? What are the "situations in which 0^0 is best left undefined"? Yes, there is a place for integer exponents and another for complex exponents. The reader should consider integer exponents in peace before being bothered by advanced stuff. See Talk:Empty product. Bo Jacoby 09:11, 16 December 2006 (UTC)
By the way, there's a passage in the "Ask Dr. Math" ref that no one seems to have commented on:
This could actually be a reference for the "0.00.0 is undefined" formulation, depending on where it comes from, which is unclear. It's indented the same as the Knuth passage, making it seem a quote, but I don't know who's being quoted. Possibly it's Alex Lopez-Ortiz, the maintainer of the sci.math FAQ. Kahan is probably William Kahan. -- Trovatore 20:09, 16 December 2006 (UTC)
The article now says this:
“ | There are two differing conventions about whether the value of 00 should be defined to be 1 or left as an indeterminate form. | ” |
I'm really not comfortable with this phrasing. That this is an indeterminate form when construed in the way that is necessary in analyis is a demonstrable fact, not a convention. I think I could argue for the same conclusion in the contexts where it makes sense to consider it an empty product, but not in a brief comment like this. I'm going to think about ways of rephrasing this. Michael Hardy 02:30, 17 December 2006 (UTC)
Here is the requested comment on the "Ask Dr. Math" passage above. The premise: "when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0)" , does not imply the conclusion that "0.0^(0.0) is undefined". That (lim xy) is undefined implies only that no definition, or lack of definition, of (lim x)lim y can make xy a continuous function. So the logical implication of the passage is not valid, and it doesn't really matter that the premise itself is not quite valid either: When approaching from any nonvertical direction we have y=ax, and lim xy =lim xax = 1. Only when approaching from a vertical direction do we have lim xy = lim 0y = 0. In order to obtain a limit k≠0 and k≠1 the non-analytical curve, y = log k/log x, having vertical tangent at (0,0) must eventually be followed. WP deserves better logic than this. Bo Jacoby 14:31, 17 December 2006 (UTC).
The three points of view regarding the values of 00 and 00.0 seems to be these.
Position 1: 00=1 and 00.0=1 (Knuth, Euler, Laplace, Kahan)
Position 2: 00=1 and 00.0 is undefined (Trovatore)
Position 3: 00 and 00.0 are both undefined (Cauchy, many textbooks)
As xy is discontinuous for (x,y)=(0,0), the (lack of) limit give no definition of 00.0.
The subsection Exponentiation#Powers_of_zero says on 00: "If the exponent is zero, some authors consider that 00=1, whereas others consider it undefined or indeterminate, as discussed below".
The article says on 00.0: "The zeroth power of zero is usually left undefined in complex analysis; this is discussed below". This seems to be position 3.
The subsection Exponentiation#Zero_to_the_zero_power contains many arguments for position 1, one, perhaps, for position 2: ("In discrete mathematics, the convention is often adopted that 00 = 1. In continuous mathematics such as calculus and complex analysis, the indeterminate form 00 is often left undefined"), and none for position 3.
We don't know any authors besides Trovatore in favour of position 2, and we don't know any authors since Cauchy in favour of position 3. (The words "some", "others", "usually" and "often" do not qualify as references).
Bo Jacoby 11:16, 18 December 2006 (UTC).
Surely we need far better references that "some authors" and the like. The author of a textbook is not necessarily a reseach mathematician. He takes most of the material for his textbook from other textbooks or from articles. There is nothing wrong in that. A textbook which does not define 00 does not necessarily argue that 00 is not or should not be defined, but only indicates that no definition is important in the context of the present textbook. We need the argumentation af the authors of the textbooks. So far we have found absolutely no argumentation in favour of leaving 00 undefined, only that "it is written in the scriptures" - a religious style of argumentation. You should not call the claim 00 = 1 unqualified. I repeat from Concrete Mathematics: "Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1, for all x, if the binomial theorem is to be valid when x=0, y=0, and/or x=−y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant". This means that the claim 00 = 1 is qualified, while the claim "00 is undefined" is unqualified. If we want to qualify that claim we need a mathematician saying the opposite, that "00 = 1 is a mistake". How could he possibly continue? "00 must be left undefined because the functions x0 and 0x have different limiting values when x decreases to 0" ?. That is simply not a valid reason. Leaving 00 undefined does not help at all. I do look forward to see some mathematical argumentation rather that religious argumentation in this matter. Bo Jacoby 14:21, 18 December 2006 (UTC).
What support are you talking about? I am requesting references, so far in vain. "a large group" is not a proper reference. These WP-articles were supposed to make sense to young people, but we are doing a bad job. Why does Knuth feel strong statements necessary? Obviously because "Some textbooks leave the quantity 00 undefined". Why doesn't he make similar statements about why 2+2 = 4 ? Obviously because no serious textbook leaves 2+2 undefined. Why do you feel strongly about undefining what has been successfully defined? Bo Jacoby 14:48, 18 December 2006 (UTC).
Quote: "The convention that 00 is 1 is not necessary here, because the series can be rewritten so that the first term is explicitly 1 rather than ". Of course no evaluation of any expression is ever necessary if you just replace the expression with the proper value. Bo Jacoby 15:02, 18 December 2006 (UTC).
No, WP doesn't define or undefine things, but obsolete ideas are in articles on history, not on mathematics. I look forward to seeing your references, though. Please quote the argumentation from the books. Bo Jacoby 15:06, 18 December 2006 (UTC).
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).
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).
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).
Hi CMummert. Please comment on your edits and reverts to Exponentiation#Zero_to_the_zero_power. The mathematical term for assigning meaning is: " definition", not " convention". The reference to " empty product" need not be repeated. The statement "Other power series identities are similar in this respect" is sloppy: either we tell the story or we don't. The message "Knuth in particular has used this to justify putting 00=1" is contained in the reference. I appreciate the new references to programming languages. Bo Jacoby 20:33, 19 December 2006 (UTC).
Thanks. The WP article on convention does not seem to explain your use of the word. I feel that the word 'convention' is implicating that this is not a matter of mathematical necessity but that anyone may pick the choice of his liking. Perhaps this is where we disagree. Note that only if . If we don't adopt the 'convention', then the left hand side is well defined for all values of x but the right hand side is undefined for x=0. If we do adopt the 'convention' then the equality holds everywhere. Psychologically I understand the opposition against the 'convention'. The different limits of x0 and 0x seems to pinpoint the troublespot: 00. So stay out of trouble by avoiding 00. But the discontinuity remains whether or not 00 is defined. Bo Jacoby 21:39, 19 December 2006 (UTC).
Yes, convention implies that "some alternative convention could be equally good but we accept this one for social convenience". That is not the case here. Therefore the word convention is misleading. It is far more than "a convenience to make it simpler to state certain theorems". It relieves computer programs and hand computations from special cases and provides a huge simplification. If the computer system had 00 undefined every programmer would have to declare a subroutine power(x,y) by "if x=0 and y=0 then 1 else x^y" and use this subroutine rather than the standard routine x^y. No alternative definition does the job. There is no freedom of choice here. It is really not a convention. In Exponentiation#Powers_of_e we have a similar example: A formula is proved true for all integers, and the right hand side is defined for all complex numbers. Then this formula becomes the unique definition of the left hand side for complex numbers. There is no freedom of choice, and we would not call it a convention. The discussion on 00 is even simpler, because 00 is defined for integer 0, and the discussion is about whether this definition should also apply to real and complex 0. No actual generalization is called for. Bo Jacoby 22:22, 19 December 2006 (UTC).
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). PS. See this. Remember to comment on this.
Do not make changes without consensus. We all tire of this. VectorPosse 08:47, 20 December 2006 (
To Bo Jacoby: Look, I'm trying really hard not to resort to knee-jerk reverts. As CMummert correctly pointed out earlier, many changes you make do improve the articles. But then you do some weird stuff too. You last edit summary was, "The examples from 'polar form' are redistributed". It appears that the whole section on polar forms was just deleted. So I don't know what you mean by "redistributed". I think that section was crap, so I don't mind that it was deleted, but at least state what it is you are doing. And better yet, why not improve that section instead of just deleting it? There was useful information there, even if it wasn't presented very well. Correct me if I'm wrong, or let me know to where you intend to "redistribute". VectorPosse 09:22, 20 December 2006 (UTC)
"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).
CMummert prefers "is often left undefined" rather than "some textbooks". The word "often" is unprecise and subjective. So far we have two textbooks in the reference list, and no quotations from them to support your claim. You too should keep NPOV. Bo Jacoby 15:13, 20 December 2006 (UTC).
Hi CMummert. Thanks a lot for the quotes.
When we, due to this controversy, cannot provide a clear article on exponentiation, we should at least provide clarity on the logic of both points of view, and elaborate on the consequences.
I see a danger in not defining, because the formulas must then be supplemented by an exception for zero. Alone in the short article on power series there are 14 formulas to correct, and we must expand the article with about the same number of lines, decreasing the clarity and quality of the article. It is hard work. You do it - not me. There must be an easier way. Why not add a note saying: "in this article 00 means 1". That must be done in some hundred mathematical articles in wikipedia. It could be centralized, though, saying in one article: "Note that in wikipedia 00 means 1". Which article? Exponentiation of course. Then we could add a note to the wikipedia articles where this is not the case, that "in this article 00 is undefined". That is not many articles because undefined expressions are not used. It doesn't make sense to say something that means nothing.
But you are the one to see a danger of defining. Tell me. What can possibly go wrong by defining what is otherwise undefined? There is no contradiction, as if we were aiming at defining 1/0. The definition 00=1 gives an expressive power which is otherwise lost. If you at least admit that 00=1 in discrete mathematics, then please stop supporting VectorPosse. It will save you years of tedious editing work.
Bo Jacoby 19:18, 20 December 2006 (UTC).
Thanks for the nice words. I am not aware of any change in style. Of course I don't intend to offend anybody and I apologize if I did. I agree that dividing by zero is a disaster. The expression x=a/b means exactly the same thing as the equation a=bx. If a=0 and b=0, then the equation is true for all values of x. If a≠0 and b=0, then the equation is false for all values of x. In neither case is the equation suitable as a definition. But that x0=1, even for x=0, is the de facto standard in power series and in many more places. The motivation is explained: empty product, empty function, even continuity is valid when the exponents are integers only. Now 0=0.0 is also de facto standard: nobody cares whether 0 is integer or real. But while undefined 00 is a disaster, an undefined 00.0 would pass virtually unnoticed, because the function 0x is quite unimportant and almost never used, while x0 is used all the time. Nevertheless, no harm is done by setting 00.0 = 00 . It is the right thing to do. You set 2+2=2+2.0=2.0+2.0 without hesitation.
An editor of a WP mathematics article should be a mathematician. It is neither sufficient nor necessary to copy a textbook from the shelf. The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion. Perhaps this is where CMummert disagrees. Trovatore's original objection against defining 00.0 was not the textbook from the shelf, but the lack of interpretation of the expression 00.0 . That is a philosophical argument, not a mathematical argument. The same kind of argument has been made against x4, because space has only 3 dimensions and so x4 has no interpretation; and against 1/2, because you cannot count to 1/2, and against negative numbers, and against complex numbers. It is hard for me to explain why, but the argument is completely invalid mathematically. Mathematics does not depend on interpretation, but on logical consistency. This is why I disagree with Trovatore. I am completely convinced that the undefining of 00.0 and 00 is a bad mistake. It is OK to leave an expression undefined as long as there is no clue to a definition, but when a constructive definition has emerged, then there is no point in going backwards to the expression being undefined. Who are we to prevent people from the benefit of a good definition? We are free not to use it ourselves if we don't want to.
Bo Jacoby 23:22, 20 December 2006 (UTC).
CMummert wrote "Let me preface this by pointing out that these are non-editorial discussions we're having now; strictly speaking, they're off-topic". To that I answered that "The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion". I am not implying that we are not all mathematicians, but that our mathematical discussion is important in our roles as WP editors and not off-topic. I am sorry if that was misunderstood. The authors of polynomial and power series and binomial theorem and many others assume the definition 00=1. Where do CMummert, Trovatore and VectorPosse expect our readers to find this definition if not in exponentiation ? Bo Jacoby 12:27, 21 December 2006 (UTC).
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).
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).
The original discussion of 00 in the "empty product" article [5], written back in 2003 mainly by Michael Hardy and Toby Bartels, was perfect. I want it back!
A particularly silly claim of the present article is that we should define 00 = 1 because that's what programming languages do. Programming languages are based on math, not the other way around.
Besides, the list is not comprehensive. Mathematica, for example, takes 00 to be an indeterminate form. Curiously, Mathematica simplifies to a0 to 1 even if you don't put any restraints on a, but does not attempt to simplify 0a. It presumably takes 00 to be indeterminate for the purpose of symbolic limit evaluations.
The conclusion in the 2003 edition of the "empty product" article got it right:
I read the "empty product" article a few years ago when I first needed to know the deal with 00, and found it extremely helpful. The article we have here is not helpful the way it is presently written. It seems that certain editors (no names) are more interested in coming up with "more nuanced approaches" for their own pleasure than to actually serve the readers. Fredrik Johansson 13:56, 21 December 2006 (UTC)
An expression that "may be replaced by 1 for the purpose of calculation" has the value 1. A consistent no-nonsense point of view is that
Bo Jacoby 21:01, 21 December 2006 (UTC).
The viewpoint 00 = 5 is consistent but nonsense: When restricting the exponent to integer values, the definition 00 = 1 is in harmony with interpretation as well as with continuity: x0 = 1 for x≠0, and this function is continuous for x=0. When the variables are generalized to nonnegative reals, continuity does not survive. When generalized to complex numbers, singlevaluedness does not survive. This is not a reason why the definition of xn should not survive. There is no improvement in removing a definition, unless you want to use the now undefined expression for something else, which you might do if the expression is not already used. But you don't want to redefine 00. You want to undefine it. And for no reason but that there are some books on your shelf. Look at Derivative#Rules_for_finding_the_derivative.
That is continuous mathematics, well within the realm where you insist that 00 be undefined. Now insert r=1 and get the result that the derivative of x is the discontinuous function x0, which equals 1 everywhere except for x=0. That is an incorrect result. The 'consistent' CMummert-definition 00 = 5 gives another incorrect result. The definition 00 = 1 gives the correct result. You are doing everyone a disservice by you crusade against the definition 00 = 1. Please do some serious thinking before you answer by knee-jerk reaction. You claim to be a mathematician. Show me. Bo Jacoby 14:52, 22 December 2006 (UTC).
Retaining the definition 00 = 1 is not a mathematical necessity, only a mathematical convenience, as it avoids complications. Undefining it leaves a lot of formulas - in discrete as well as continuous mathematics - to be rewritten, showing that the authors of these formulas did assume the definition. While x=0/0 says nothing about x, (because 0x=0 for all values of x, so x might even be equal to 5), the statement x=00 says that x is an empty product. So there is a profound difference between 0/0 and 00. Your 'counterexample' is nevertheless to the point: One doesn't prove that 00 = 1. It's a definition. Bo Jacoby 17:10, 29 December 2006 (UTC).
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).
The dichotomy is correct, but the links were misleading. Did you ever click on those links? Now I removed them from the article. Bo Jacoby 14:06, 22 December 2006 (UTC).
Isn't it true that fractional exponents are only defined for ?
There is plenty of discussion in the section "Zero to the zero power" to support both points of view. Bo Jacoby needs to stop making changes to this section and others without consensus. People have worked extra hard to make sure the wording is neutral. VectorPosse 20:16, 28 December 2006 (UTC)
The very replacement of with actually assumes the definition x0 = 1. Whether necessary or not, the culture of mathematics is that for all x, and so that x0 = 1 for all x, even for x=0. The replacements are not actually made, not in wikipedia, not anywhere. The use of x0 = 1 is not limited to discrete mathematics. In calculus the formula is valid, and x=x1 is valid too. Now the general formula leads to . For consistency, calculus has to accept the definition x0 = 1 for all values of x. Even if "few books will even bother to mention it", it is assumed by all the mathematicians. So the claim of the article that "In continuous mathematics such as calculus and complex analysis, the indeterminate form 00 is often left undefined" is simply not correct. Even in calculus the definition 00 = 1 is assumed. Bo Jacoby 07:46, 29 December 2006 (UTC).
There is no difference between saying that 0^0 is a shorthand for 1 and saying that 0^0 equals 1. The empty product applies to integer exponents, and you say you agree that an empty product is one. Why not accept it at least in the subsection on integer exponents? I don't need consensus, nor do you, but we both need support. I count Fredrik Johansson and Hardy and EdC amoung my supporters, and you count VectorPosse and Trovatore amongst yours. You have lots of editing to do, in this article and in others, to prevail in your pointless crusade against a commonly used definition. Bo Jacoby 17:40, 29 December 2006 (UTC).
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).
Nor is there any reason why an obsolete word shall be the first one to meet the uninitiated reader. I'll move the sentence down to 'advanced topics'. Bo Jacoby 13:21, 31 December 2006 (UTC).
CMummert, please note the following surviving piece of heresy from Exponentiation#Exponentiation_in_abstract_algebra:
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).