Talk:Exponentiation/Archive 2011

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$0^{0}$ = x

x meaning any number. For those who don’t know what logarithms are, log(a) b is the same as saying "With $a^{x}$, what does x have to be to get b?". Lets look at log(2) 256 as an example, with $2^{x}$, what does x have to be to get 256? The answer is 8, because $2^{8}$ = 256. Now lets look at log(1) 1, with $1^{x}$, what does x have to be to get 1? x can of course be any number, so I will just keep the answer as x. Because log(a) b = log b/log a (with log being log(10), it still works the same with any other positive logarithm) we get

x = log(1) 1 = log 1/log 1 = 0/0 = $0^{1}$/$0^{1}$ = $0^{1-1}$ = $0^{0}$

Well okay maybe 'x' isn't a suitable answer, but you have to at least accept that the answer can be any number, and that it deserves at least a small mention in this article. Robo37 (talk) 15:40, 12 August 2009 (UTC)

the subject of 0⁰ has been discussed, argued and done to death in this article in the section Exponentiation#Zero to the zero power. Now I worry about whether I should have put a comma before the 'and' in that list, I think that also has people raising the cudgels. Dmcq (talk) 18:11, 12 August 2009 (UTC)

Yeah but I've just proved that $0^{0}$ is x when $1^{x}$ = 1, I just thought that that was worthy of a mention. Robo37 (talk) 21:59, 12 August 2009 (UTC)

Please see original research. You will need to get it published in some reputable journal first. Also your discovery must be notable, that your discovery is right and wkipedia is wrong does not matter as far as wikipedia is concerned, it is mainly concerned with whether other people have found your discovery notable. You need to convince some other people first. Dmcq (talk) 22:24, 12 August 2009 (UTC)

When an expression can equal any value (which is what you've just proven for 0^0) we call it an indeterminate form. That 0^0 is indeterminate is well known. --Tango (talk) 23:19, 16 August 2009 (UTC)

And from 0/0 = x we can infer that x ≠ x. Or perhaps x = x. Who knows? Well both James Anderson (computer scientist) and IEEE arithmetic come to different conclusions so who am I to judge? Dmcq (talk) 06:47, 17 August 2009 (UTC)

Ah, yes, that Dr. Anderson didn't really solve anything, as when he defined Φ he also managed to make sure it wasn't really a number anymore. --Professor Fiendish (talk) 02:18, 21 August 2009 (UTC)

Not exactly "original research" since this concept is at the core of understanding that division by zero is evil. It would be appropriate to use terms like "indeterminate" or "innumerate" which imply that the operation is nonsensical by definition. This goes along with the paraphrased quote: "A theory that predicts everything all at once, predicts nothing in the moment. " or simply "If all answers are right, then none are." Saying that 0^0 can be any value simply tells you that your equation just a had a great big hole shot in it due to the previous line's/lines' manipulation! Otherwise, you could prove that 2=1.  :) JWhiteheadcc (talk) 20:54, 21 December 2009 (UTC)

Just for the record, since nobody explicitly mentioned this, this has nothing to do with the "indeterminate status" of 0^0 (since that involves limits and the argument does not) but with deduction from undefined expressions (base 1 logarithms and 0/0), which never gives a reliable conclusion. The fallacy is in fact based on manipulations that are not allowed, compare with the following "proof" that 0 squared is undefined:

$\textstyle 0^{2}=0^{3-1}={\frac {0^{3}}{0}}={\frac {0}{0}},$ which is undefined (of course 0³ is in fact also undefined by a similar argument). Marc van Leeuwen (talk) 13:20, 18 March 2010 (UTC)

Since 0/0 is indeterminate as well, I think it having to do with 0/0 is the same as it having to do with its indeterminate status. --Tango (talk) 13:30, 18 March 2010 (UTC)

For the Mathematica support of 0^0 I added some links to WolframAlpha, which is a free website based on Mathematica and developed essentially by Wolfram which makes Mathematica. Then I discovered any external links should be in the citations section. Is someone willing to correct that? Should each link be a separate citation? 3 citations in the same sentence or paragraph seems excessive. Nickalh50 (talk) 22:47, 6 February 2011 (UTC)

log or ln?

There is inconsistent use of log(x) and ln(x) in the article. Unless it's being discussed in the context of some base other than e, shouldn't the preferred form be log(x)? And even then, doesn't log(x) always imply log_e(x)? — Loadmaster (talk) 03:55, 4 September 2009 (UTC)

I seem to recall that some sources use log for the base-10 logarithm (it could even be that this is the older usage). Because of this, I would tend to suggest ln, which is unambiguous in that it never means anything other than the natural logarithm to my knowledge. — Steven G. Johnson (talk) 05:45, 4 September 2009 (UTC)

There's a cultural issue here. Mathematicians use log to mean natural log, and may or may not also use ln. Mathematicians very rarely have a desire to talk about the base-10 log at all.

On the other hand, engineers usually use log to mean base-10 log. Physicists and chemists, sometimes.

Personally, as a mathematician, I just don't much care for ln, because obviously log means natural. I think this is an issue we're not going to solve and shouldn't try. The important thing is that we make ourselves clear to the reader, not that we be always consistent in notation. --Trovatore

(talk) 06:51, 4 September 2009 (UTC)

This is an issue which has come up repeatedly at community.wolframAlpha.com; apparently the staff there get questions daily on the subject. Considering there seems to be differences of opinion in the math/science/engineering community in general as to the meaning of log, we should shift to log_e (subscript) and log_10. Inconsistency in notation within a single article will CREATE confusion. Part of the complication arises from an attempt to reach widely varying audiences with the same article. Grade school students from what I have seen and taught universally use log for log base 10 and ln for ln base e. In upperdivision college (past Calculus) and higher the use of log for log base e becomes more prevalent. A sentence somewhere mentioning the need to include the base when absolute clarity seems appropriate. In my opinion, an article desiring to be as quality as Wikipedia looks for the use of log without a subscript should completely be replaced by a subscript version, unless the base is not important.

B.S. MathNickalh50 (talk) 23:06, 6 February 2011 (UTC)

But in a single article, don't you think it makes sense to use a consistent notation? Although it's true that many mathematicians don't use ln, I'd be

surprised to encounter a mathematician who wouldn't recognize ln (I am in a math dept). Since ln is unambiguous in every context, whereas log is ambiguous in some contexts (not to mention that there are far more engineers out there than mathematicians, and far more people with only high-school math than either), that seems to make ln a better choice for the article. — Steven G. Johnson (talk) 15:24, 4 September 2009 (UTC)

I might have missed it, but I don't see where we use "log" to refer to the natural logarithm. I see two key sections where logarithms are discussed. In the section "Real powers" we use ln to refer to the natural logarithm. In the section "Powers of complex numbers" we use log to refer to the complex logarithm (and repeatedly point out this is the complex logarithm). It would be very strange to me to see ln used to denote the complex logarithm. We also say "Θ(log n)" in a lower section, but the context there seems clear, and we are not referring to the natural logarithm anyway. So I don't see that there is really an inconsistency in the article. Like Stevenj, I prefer ln(x) over log(x) for the natural logarithm, because grade-school students are often misinformed about what log(x) should mean. — Carl (CBM · talk) 15:42, 4 September 2009 (UTC)

log versus ln

I reverted a change which would have put ln everywhere in the complex numbers section instead of log. However we still have ln in some sections before that. Using ln is simply not standard in mathematics or computing though it appears on calculators, it seems to be more of a thing for schools to me. However exponentiation is a fairly introductory article for the first half. ANy feelings about this problem of ln versus log? It seems funny having ln in one part and log in another but I'm not sure how to resolve the problem. Dmcq (talk) 12:11, 24 September 2009 (UTC)

I think that the sections about real logarithms should use ln for natural logarithms, because that is the notation usually adopted by calculus textbooks. On the other hand, sections about complex logarithms should use log for the complex logarithm, which is of course not the same thing as the natural logarithm. Books on the complex numbers usually use log for the complex logarithm. So (1) there is no contradiction in using ln in one place and log in another, and (2) our article presently uses the notation commonly encountered in textbooks. — Carl (CBM · talk) 12:45, 24 September 2009 (UTC)

Perhaps it would be more helpful for most readers to add a "Notation" section which explains that natural log (log_e) is also commonly written as "ln", and that most mathematicians simply use "log", while engineers generally use "log" to represent log₁₀. And that for the purposes of this article "log" means "natural log" unless otherwise stated. In other words, transfer most of the discussions here on the talk page to an appropriate section in the article itself. — Loadmaster (talk) 22:58, 24 September 2009 (UTC)

That is in logarithm already; we could tell readers to see that section for details, rather than repeating it here. The difficulty is always finding references about notation. — Carl (CBM · talk) 23:52, 24 September 2009 (UTC)

Have we got actual evidence that engineers commonly use log base 10 nowadays? I know it used to be common when they used slide rules but nowadays is it actually common at all and is that what they mean? Dmcq (talk) 01:16, 7 February 2011 (UTC)

What is "a"?

In the section on zero to the zeroth, there is this : " A commentator who signed his name simply as "S" provided the counterexample of (e−1/t)t (which can be obtained in one example above by taking a = 1)...." I can't find "a" any-where in a formula. Kdammers (talk) 08:55, 26 January 2011 (UTC)

It's in the previous section. Given the difficulty of labeling equations, and the fact that the back reference is so far back, I just removed the parenthetical comment, which isn't really necessary to understand the text. — Carl (CBM · talk) 12:09, 26 January 2011 (UTC)

"No natural extension"

"The power aⁿ can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists..."

Is this true? Why isn't the natural extension aⁿ = exp(nlog(a))? I know this involves multiple values, the choice of principal value, and excursions into complex numbers, but I see no reason why those things can't be a part of the "natural extension". In fact, the text continues ("but when the base a is a positive real number...") exactly as if it is allowing such possibilities as "extension". 86.160.209.202 (talk) 13:33, 26 May 2011 (UTC)

Exponentiation of a positive real number to a complex power is a single valued function. The real logarithm log(a) is single valued and only defined for a > 0 whereas the complex logarithm log(a) has multiple possible values. Dmcq (talk) 14:08, 26 May 2011 (UTC)

Yes, thanks, I understand that multiple values arise, as I mentioned. I still do not understand why this is not a "natural extension". Even something fairly benign like 2^(1/2) potentially has two values via the exp(nlog(a)) route. I think it would be better if the article said that the concept can be extended to all real and complex a and n, but flagged up that complexities can arise, as dealt with later in the article. 86.179.0.162 (talk) 17:49, 26 May 2011 (UTC)

A multivalued "extension" is not even a function, so it's not an extension of the exponentiation function to a larger function. — Carl (CBM · talk) 18:52, 26 May 2011 (UTC)

There is no mention that exponentation is a (single-valued) function. It is described as a "mathematical operation". Although in the first instance we expect a single value, the extension to multiple values and to all real and complex numbers is clearly a "natural extension" of that operation, albeit with additional complexities arising. Note also, later in the article, "If exponentiation is considered as a multivalued function...". 86.179.0.162 (talk) 19:48, 26 May 2011 (UTC)

2^1/2 only has the positive value 1.414..., not the negative value if 2 is a real. The square root of 2 can be positive or negative. Conventionally the 1/2 is sometimes used for either root or if we're dealing with complex numbers. Dmcq (talk) 22:21, 26 May 2011 (UTC)

Yes, I agree, by convention 2^1/2 is understood as the positive value. My argument is not really about such conventions, or about the specific notation or terminology used (so my comment above about multi-valued functions and such is probably a bit of red herring on reflection). My point is simply that, contrary to what the article claims, and irrespective of what we call it or how we write it, there is a natural extension of the concept/process/operation of exponentation to all real and complex numbers, and that natural extension is embodied by the formula exp(nlog(a)), and all its ramifications. 86.179.0.162 (talk) 00:56, 27 May 2011 (UTC)

The defined value is the same as the positive square root of 2. By convention it sometimes can also mean the negative square root but that is not the defined value of the exponentiation operation applied to 2 and 0.5. That convention only comes from us looking at the power and seeing it written as the particular rational fraction 1/2 and as the article points out there are problems in general with doing that sort of thing if the result is allowed to be negative. I don't think many people share your idea that an infinite set of different values is a natural extension to a single value, and especially when just doing something like (z^v)^w then goes and multiplies up the number of results. Dmcq (talk) 05:12, 27 May 2011 (UTC)

Instead of saying conventionally it might be better to say that x^1/2 is often used as a convenient notational device to mean a square root of x. Dmcq (talk) 05:22, 27 May 2011 (UTC)

0^0, etc.

The section in the article, and the discussions in the talk page, are extremely cagey about the idea of "0^0". I think this is just because people are confused about the philosophy of mathematics. (Even mathematicians like Knuth can be confused about the philosophy of mathematics; being good at X doesn't mean that you are good at understanding the philosophy of X!) The erroneous idea stuck in people's heads is that there is some "true fact hovering out there in maths-space" about what 0^0 "really is equal to" that we can "discover". This is an extremely problematic position, and doesn't make any sense if we interpret mathematical statements to be theorems in some set theory. See: Platonism, Formalism.

If 0^0 is not defined the same by all authors, then it is not the case that "we don't know" what it "really" is. It is just what it says on the tin: it is not defined the same by all authors! Nothing more, nothing less.

This issue comes up again and again, with "0.9... = 1", division by zero, and all that. It's all symptoms of the same underlying misconception.

Dissimul (talk) 02:34, 12 June 2011 (UTC)

Sorry, I take it back about Knuth. In the larger context of his quote "it has to be 1!", we can see that he is actually saying something like "the binomial theorem is extremely important, and if we want to make the binomial theorem pretty, 0^0 has to be 1!" So he understood that it is simply a decision, and he made the decision based on which theorems he wanted to make pretty and which theorems he wanted to make ugly. But the quote as it stands could mislead somebody who hasn't really thought about what "mathematical truths" are before. It makes Knuth sound like he has "discovered" that it "has to be 1!", as if he was in communion with the Maths Gods. Dissimul (talk) 03:15, 12 June 2011 (UTC)

Do rational exponents warrant a top level section?

I'm not altogether sure about what the rational exponents section is all in aid of. I( don't see that it stands on its own before the real numbers section. I feel the version of 26 November [1] which simply had a small section in the real powers was far better. How do others feel? Do rational numbers deserve a big section to themselves rather than just being a subsection of real powers of positive numbers? They don't have particular interesting properties or anything that I can see in their own right. The only interesting thing about them is in the negative nth root section and I think that is in the right place at the end of the real powers. Dmcq (talk) 18:42, 2 December 2011 (UTC)

I must admit that I was not aware that the edit introducing rational exponents separately was so recent. My intention was mainly to improve the logical flow in the version of 1 December. Nevertheless, I think rational exponents may merit a separate section before the section on general real exponents, for two reasons. The first is the principle of going from simple to complicated, which suggests that since a rational exponent m/n is interesting in itself as it can be interpreted as the nth root of $a^{m}$, it makes sense to discuss this before mentioning how to treat irrational exponents based on logarithms and exponential functions. The other is that the article discusses how irrational exponents can be approximated by rational ones based on continuity. In the version of 26 November, it is not clear why the approximation with a rational exponent is treated after the subsection on the exponential function. Isheden (talk) 19:16, 2 December 2011 (UTC)

I think the article is already far too long for the topic, and the rational powers are not a necessary step in the process of generalization. Therefore discussing them is purely to illuminate its properties in relation to nth roots and negative bases. I'm not making any suggestion aside from that conciseness should be sought. Quondum^talk_contr 21:08, 2 December 2011 (UTC)

There's no need to extend the article. All that is needed is to collect the material that is related to rational exponents (including negative nth roots) and present it before 1) real (and later complex) powers of positive numbers and 2) powers of complex numbers. Isheden (talk) 21:53, 2 December 2011 (UTC)

I have made a suggestion what it might look like here: User:Isheden/sandbox2 Actually, I think the total length can be shortened with this order of presentation because some of the material on real exponents becomes superfluous. If you think it's good, we can incorporate it, otherwise feel free to roll back to the version of November 26. Isheden (talk) 23:07, 2 December 2011 (UTC)

Your rearrangement makes it clearer how the real powers may be arrived at via two independent logical routes, which makes it a sensible reorganization. There are some (IMO) superfluous examples and discussions; these can be addressed separately at the detail level within a sensible structure. The subheading "Real exponents" could be replaced with something like "Definition on terms of the natural logarithm". I assume your suggestion replaces the section Negative nth roots as well – that section was awkward on its own. What is still not brought out is that the extension to real exponents is not a true generalization of the rational case, inasmuch as it does not cover the negative base cases that rational powers can define; this can easily be added later as a detail. Quondum^talk_contr 06:29, 3 December 2011 (UTC)

I have carried out a reorganization based on the above. I hope the line of thought became a bit clearer. Feel free to edit further or roll back if necessary. Isheden (talk) 18:24, 5 December 2011 (UTC)

Base a then b

The article starts with base a and later goes on to base b, I think perhaps we should have b everywhere, any objections? Dmcq (talk) 13:07, 7 December 2011 (UTC)

Plus I think w^z would be better in the complex section than a^b when both are complex. Dmcq (talk) 13:12, 7 December 2011 (UTC)

Support from my side. Isheden (talk) 13:38, 7 December 2011 (UTC)

I think I'll just have b for the base even for the complex numbers, just looks a bit better than having w around the place I think. Dmcq (talk) 18:36, 7 December 2011 (UTC)

This might be worth a try, but I have a feeling that using b for a complex base may be problematic. With all the other cases it does not matter if they share a notation, but with a complex base and a non-integer exponent it must be treated separately, and use of a different variable name for the base may be the most obvious cue one can give. Quondum^talk_contr 19:07, 7 December 2011 (UTC)

Irrational exponents

The article claims that irrational exponents can be reached using continuity over the rationals and that a unique continuous extension exists. This needs to be developed further based on reliable sources. To my understanding, an irrational number can only be reached as the limit of a sequence of rational numbers, for example based on truncated continued fractions. Moreover, since the expansion as continued fraction is not unique, I don't understand what is meant. Isheden (talk) 08:25, 8 December 2011 (UTC)

Though I'm not a mathematician, it would seem to me that an irrational number can only be defined in terms of rationals as you describe – as a limit of a sequence, but that this would not prevent it being reached as a limit over rationals. It seems to make sense that the value of a function of an irrational number could be defined using the usual approach of a neighbourhood of that irrational number, but restricted to rational numbers. You are right about the need to develop the topic more.f the Quondum^talk_contr 09:07, 8 December 2011 (UTC)

See Construction of the real numbers for the various ways real numbers are constructed as a completion of the rational numbers. The usual wasy is by Dedekind cuts which is one of the ways described there. The reason exponentiation for reals is normally via the exponential function rather than by completing the rationals is because even though filling in the rationals is the more obvious method it is more work and messier to do rigorously. It's the same reason the exponential function is normally defined by the series rather than as the limit of the compound interest like Bernoulli and Euler did. The obvious and more easily explained reasons aren't always the most tractable mathematically. Dmcq (talk) 09:32, 8 December 2011 (UTC)

This would suggest a very simple definition in terms of Dedekind cuts without the need for limits, made easier by the monoticity of exponentiation. Given the cut r=[A,B], and b>1, b^r=[ℚ_≤0∪b^A,b^B], and the other way around for 0<b<1. Maybe not suitable, but then the question should be asked whether the limit over the rationals is not also a little esoteric. There must be innumerable ways of defining real powers, so perhaps it makes sense to remove the limit-over-rationals version and just to have a blanket statement that other ways of extending it to reals exist; the definition via logs seems most straightforward. Quondum^talk_contr 11:02, 8 December 2011 (UTC)

Just because something is monotonic doesn't mean it doesn't have jumps in it. You really need the continuity in it otherwise you might have the limit different in different directions. Dmcq (talk) 11:07, 8 December 2011 (UTC)

As to just removing it, the method just assuming the rationals can be filled in is how it is introduced at an elementary level and people normally assume it works. This isn't a textbook and besides it is introduced that way in some textbooks. The other business bringing an an integral of 1/t or something like that to introduce exponentiation of reals, that is a very inobvious way of getting there, rather like suddenly producing a Gudermannian when solving a calculus problem. Dmcq (talk) 11:27, 8 December 2011 (UTC)

OK, so now I understand what is meant by a unique continuous extension. The Dedekind cut is a partitioning of the rationals where the cut defines the irrational number. Still, in my opinion this section does not need to discuss how an irrational number is defined, but rather how to construct an infinite sequence of rationals that tends to the irrational number. One way would be to add a digit in the decimal representation in each step, but this method assumes that you know the decimal representation with infinite precision. Another way is to use an infinite continued fraction representation. There may be a way to construct a sequence of rationals from the Dedekind cut also. If the section should not treat how to construct such a sequence, then I guess it suffices to mention some ways to fill in the rationals. Isheden (talk) 17:32, 8 December 2011 (UTC)

Constructing one particular sequence does not prove anything. For instance one could choose to always use an odd power and negative values and so get a limit for negative bases, however there are even powers as well mixed in densely with them. It may be one gets a limit with a particular sequence but another sequence gives a different limit. All the limits must be the same. Dmcq (talk) 18:43, 8 December 2011 (UTC)

I agree that constructing a particular sequence that happens to converge to a limit does not prove anything. However, the continued fraction representation has certain properties that makes it useful for any irrational exponent, provided that the base is positive (which is what this subsection is about anyway): 1) Every irrational number has a unique representation as an infinite, regular continued fraction. 2) Each convergent of the continued fraction representation is the best rational approximation given the size of the denominator. 3) Even convergents are always smaller, and odd convergents always larger, than the irrational number. Thus, this method provides the smallest possible interval the irrational number given the size of the denominators of the rational bounds, and a method for shrinking the interval to any given precision. I do not see why these properties would not merit a paragraph. Isheden (talk) 11:36, 10 December 2011 (UTC)

Nice as the properties of continued fractions might be, they are largely irrelevant. Point 1 only ensures that there is a single method that constructs a reproducable sequence of rationals that has as its limit the irrational value. There are innumerable easy ways of doing this. Point 2 makes an assertion that only relates to using the smallest numerator/denominator for a given accuracy. Speed of convergence is completely irrelevant. Point 3 simply ensures that successive values in the sequence always bracket the irrational number. This provides no assurance that the corresponding rational powers bracket the corresponding. The proof of continuity of the result remains lacking. As a way of calculating approximate powers through integer powers and roots it might be nice (but this point is more relevant to continued fractions than to exponentiation) – what is needed is a definition and proof. Quondum^talk_contr 12:10, 10 December 2011 (UTC)

Just to emphasis that, note that the justification for continued fractions here did not mention exponent or power once. The article is supposed to be about exponentiation. Yes a proof that the values between are actually between and that the two values can be made arbitrarily close to each other is missing but that's what the textbook is for. We could put in a sentence or two more outlining what the proof in that book does but a full blown proof would be outside of Wikipedia's remit, it is long and simply isn't interesting enough. Dmcq (talk) 12:25, 10 December 2011 (UTC)

My question then is why the decimal representation is relevant? After all, similar theoretical issues apply to this representation and it is a completely arbitrary way of constructing the intervals. Also, you will need to know x very accurately to get an accurate result in the exponentiation. Is it just for illustrational purposes? Isheden (talk) 12:47, 10 December 2011 (UTC)

I've been debating with myself when I should raise that same point... I personally think that even as an illustration it has little relevance; it merely demonstrates the interpretation of the nthe root for power 1/n which is a point that belongs under rational exponents, not reals. I don't think it contributes much. Quondum^talk_contr 12:56, 10 December 2011 (UTC)

Yes it was an illustration. That's why it says 'for example'. The arbitrary nature of the limits was deliberate and hopefully a reader would easily see how they related to the desired real power and how to get more accurate fractions. I think a different power would be better as the sqrt(3) might be confusing. Dmcq (talk) 13:12, 10 December 2011 (UTC)

Aside from the labelling as an example (which as I said I feel serves little purpose; I guess it is meant to illustrate the extension through continuity over rationals, but really it has the feel of illustrating progressively accurate approximation, and continuity is merely a prerequisite), this seems to be largely subject to the same criticisms as was the continued fraction example. The question is simply whether any example is warranted. It's already pretty obvious that if there is any such thing as continuity over rationals, then extension in this manner will work; no example needed for this. The example most certainly does not illustrate whether the continuous extension exists. The example also refers to a level of accuracy, which I already indicated was spurious. So my feeling would be to remove it for much the same reasons as the continued fraction was removed. Quondum^talk_contr 15:03, 10 December 2011 (UTC)

I have checked in various books on elementary mathematics and reworked the example to reflect how the topic is typically covered. There is probably no need for a discussion on the accuracy of the rational approximations; it suffices to show the principle by means of an example. Isheden (talk) 21:02, 10 December 2011 (UTC)

I changed the original example which was like that because it seemed to me that there was a misunderstanding about what a limit was in what was said earlier in this discussion. Having a particular sequence converge to a limit is described in Limit of a sequence. The limit meant here is the (ε, δ)-definition of limit which is a bit more complex. Unfortunately what has been put back looks like the limit of a sequence again. At least the idea of the special sequence has been removed for which I'm thankful for but the example no longer illustrates what the limit described here is about. Dmcq (talk) 21:46, 10 December 2011 (UTC)

If the intention is to illustrate the (ε, δ) limiting process in action, I guess one could re-insert the mention of (strict) monotonicity and replace the sequence of values with a sequence of δ intervals based on the decimal expansion, saying that each maps onto a minimum ε interval, which goes to zero as the δ interval containing the exponent goes to zero. This may satisfy us all. Quondum^talk_contr 22:29, 10 December 2011 (UTC)

In the original example [2], some irrational exponent was rounded to 1.732 without any further comment. Regarding the limit definitions, I think we are safe since with each new digit, r gets closer to x (within the rationals) and for any given ε (maximum difference between the limit and $b^{r}$) you need to add enough digits so that the difference between x and r is at most δ. However, making the connection clear in a line or two might be beneficial. Isheden (talk) 22:59, 10 December 2011 (UTC)

That looks good to me. It's quite hard getting the right level of detail so people get the basic ideas without putting in too much unnecessary detail for an encyclopaedia. Dmcq (talk) 00:03, 11 December 2011 (UTC)

While the narrowing of intervals in the example works, I'm not sure if it is needed. Isn't it sufficient to say that for any given accuracy ε, you need to add enough digits to make |x-r| smaller than some δ? Isheden (talk) 10:14, 11 December 2011 (UTC)

By the way it is possible to construct other extensions from the rationals to the reals which don't satisfy continuity, they are pretty nasty and require the Axiom of Choice. One just chooses another real not yet in ones set and gives it a value for the power and then constructs all linear combinations with the ones there already until one exhausts all the reals. (The 'just' is rather a biggie!) but you can see one could have a power of pi for instance equal to the 100th power and there would be no contradiction if one did not require monotonicity or continuity. There is just one continuous extension though. (square root of 2 doesn't work as (x^sqrt(2))^sqrt(2) = x^2 Dmcq (talk) 18:50, 8 December 2011 (UTC)

Circularity

A removed a number of changes to the real powers section because they seem to have degenerated to defining e^x as that because it is defined that way with little reference to what it would mean if x was an integer for instance. I think improvements to this section should really be based on sources with citations which is what the sections are really missing. Dmcq (talk) 12:38, 8 December 2011 (UTC)

I was trying to break the circularity implied by the notation (perhaps not the one you're referring to) by arguing that e^x is defined in terms of exp(x). (e)ⁿ for integer n is not defined the same way as e^x for real x; the definition of the latter (and its notation) is merely chosen because it (mostly) generalizes the former. Perhaps you are thinking e^x is exp(x), not (e)^x? I realized that in my changes the mathematical motivation for the direction of the generalization is lost; the choice is merely retrospectively justified. In an encyclopedic context I feel one should not assume that the reader "knows" what the notation means; I have found context-specific notation problematic before in this way. However, I hear you on the matter of sourcing. Aside from the circularity (or confusion in the notation, take your pick) that I feel is present, these sections seem rather clumsy. Quondum^talk_contr 13:47, 8 December 2011 (UTC)

What is there shows that e^x is the same as exp(x) when x is an integer. There is no circularity in that. It is a major step to showing that using exp(x) is a reasonable way to extend exponentiation to the reals. What you did was define e^x as exp(x) and then show they were the same using that definition. I could substitute addition for exponentiation in what you said and it would still be equal. That's circularity. Dmcq (talk) 14:20, 8 December 2011 (UTC)

I have no objection to keeping a proof of correspondence for integer exponents; deletion thereof was only one part of my changes. I did not define e^x as exp(x), but as b^x with b=e, and then showed that e^x=exp(x). Everything was defined in terms of exp(x), even the definition of e. Nothing circular in that; my objective was to rigorously remove circularity. Quondum^talk_contr 14:41, 8 December 2011 (UTC)

But you defined b^x in terms of exp(x ln(b)). |That's where the circularity arose, or else totally vacuous proof whichever way one wants to look at it. Any connection between the integer and real cases was lost. Anyway I've tried to distinguish better between integer and real powers now. Dmcq (talk) 18:33, 8 December 2011 (UTC)

I think we are misunderstanding each other. I merely showed the equivalence of notation e^x=exp(x), not that it was equivalent to a definition using an integer or rational exponent. That would have required a separate demonstration. At the moment, under section Exponentiation#Powers via logarithms, b^x seems to be defined in terms of the real power of the specific base e. I do not see a definition for that other than this same line. This seems circular to me. Nowhere is e^x defined for real numbers. Quondum^talk_contr 19:17, 8 December 2011 (UTC)

The definition of $e^{x}$ for reals is in the exponential function section. One could certainly remove the definition of $e^{x}$ for reals from the exponential function section and simply leave the demonstration that $\exp(x)=e^{x}$ for integer values of x. The demonstration that if $b^{x}$ is defined as $\exp(x\ln(b))$ then $e^{x}$ is $\exp(x)$ because $\ln(e)$ is 1 is pretty much a waste of space though and looks like it is trying to prove something special when it does nothing of the sort. It definitely was not worth putting in when removing the demonstration of the equivalence of the real and integer definitions for integer powers. Dmcq (talk) 22:52, 8 December 2011 (UTC)

I have problem with just defining $e^{x}$ as a special case of $b^{x}$ rather than $b^{x}$ being defined in terms of $e^{x}$. $e^{x}$ is $\exp(x)$ in any case but $e^{x}$ is used in many cases like with matrix powers and having to have logarithm in the first place makes a bit of a mess of that notation and means one would always have to refer to $\exp$. Dmcq (talk) 23:56, 8 December 2011 (UTC)

This is a common notational issue; in principle e^M for a matrix M is not the same function as e^x when x is real. In the contenxt of real analysis, if b^x is defined simultaneously for all non-negative b and x by using continuity to extend from the case where x is rational, then e^x really is just a special case of b^x. But even if we define e^x as exp(x), at some level a proof is still needed that this function agrees with e^x in the other sense. Similarly, if we define arctan in terms of a power series and then define π in terms of some value of inverse arctan, we still have to prove at some level that this number π really is the ratio of circumference to diameter of a circle. — Carl (CBM · talk) 00:16, 9 December 2011 (UTC)

In an article such as this, I would feel that the meaning of the notation e^x must be clearly defined. Without this subtle distinction (and I do not think one could assert that one interpretation is obvious), the target audience that stands to gain most from what is being said will end up wrestling with exactly this notational ambiguity. It either denotes exp(x) or b^x with e=b, not both. The chain of definitions cannot be made to hang together without being clear what you are defining something in terms of. My own feelings are that the inconsistency in the notation demands that one consider the notation e^x as just a convenient shorthand for exp(x), but only where the ambiguity is unimportant. But at least we have located a contributary souce of a problem: what the notation e^x means is ambiguous. Quondum^talk_contr 09:21, 9 December 2011 (UTC)

This is ridiculous. Saying that there are multiple starting points for defining e^x is not the same thing as saying that the notation is "ambiguous". Ambiguous notation would indicate that different definitions give different results, which is not the case here. (Even exp can be defined in multiple ways, but they are just different starting points to obtain provably the same function.) — Steven G. Johnson (talk) 23:11, 10 December 2011 (UTC)

The problem here is that $e^{k}$ uses the integer rules but $e^{x}$ uses the exp definition. And it causes further trouble later in Clausen's paradox in the section Exponentiation#Failure of power and logarithm identities where a use of e refers to the result of a prior exponentiation and so isn't referring to exp but to a power of a complex number which just happens to be the same as e. People have lived with the problem for a long time now and it ain't our job to fix the world but still it is a little troublesome. Dmcq (talk) 00:12, 11 December 2011 (UTC)

I completely agree that it is not our job to try to fix this. — Carl (CBM · talk) 02:35, 11 December 2011 (UTC)

It's true that the notation gives the same results for real exponents, but that doesn't help with the problem of proving things. When we want to prove e² = e·e we have to decide if this is by definition, or if it is a special property of exp(x).

The notation does become ambiguous when we move to complex powers. The complex exponential exp(z) is a single-valued analytic function. The notation e^z can mean either this, or the complex power function of e to the power z, which is only defined relative to a branch of the complex logarithm. — Carl (CBM · talk) 02:35, 11 December 2011 (UTC)

If you are worried about branch cuts, then you have a problem even for real x [e.g. e^1/2 could denote ±exp(1/2)]. However, in most of mathematics, when one writes a^b without qualification then one means the principal value, in which case e^x is consistent with exp(x). This is a non-issue in actual usage.

Regarding the problem of proving that different definitions are equivalent, we already have a whole article on that: Characterizations of the exponential function. Any additional discussion that is needed can go there. — Steven G. Johnson (talk) 03:09, 11 December 2011 (UTC)

I understand that we can allow the reader to wrestle with notation to some extent, but it helps in this context to clarify pertinent notational use where this may otherwise be confusing. It is therefore appropriate to state, at least in the definition, that for the purposes of definition and anywhere this might lead to different results, e^x with an explicit base e denotes exp(x). In particular, note the violation of this interpretation in "A short proof that e to a positive integer power k is the same as exp(k)", where in fact it crucially and specifically means the former and not the latter; this omission should be fixed. Quondum^talk_contr 05:14, 11 December 2011 (UTC)

I thought it was clearly marked as different there otherwise why would one need to try and prove them the same? I guess putting a bracket around the e might make it even more obvious. I'll do that and see what it looks like. Dmcq (talk) 10:35, 11 December 2011 (UTC)

I must admit I think of $e^{x}$ as a different type function depending on the type of x, and even e in the case of Clausen's paradox, and that $\exp(k)$ where k is an integer involves an implicit function mapping the integer to a real. Dmcq (talk) 10:41, 11 December 2011 (UTC)

Notation is overloaded so often in a mathematics that there is nothing strange in your approach. I've added a suitable statement, and I like the added parentheses. Now at least I can't claim it to be circular. As to "why would...", I do not like mathematical definitions that depend upon posing such questions.

It often occurs that the actual formal function is determined by the parameter type. Confusion of the actual type with a natural embedding is typically benign. Because it is rare that the results are nonequivalent, it is worth dealing with such cases more explicitly, as we have now done. Clausen's "paradox" is really just applying of a rule that simply does not apply, before we even consider what the function types are. There is no rule that (a^b)^c = a^bc with complex b and c. Quondum^talk_contr 16:50, 11 December 2011 (UTC)

The danger is that we run the risk of just confusing people if we don't state up-front (not buried several paragraphs down) that exp(x) is exactly the same as "ordinary" exponentiation in the usual case (taking the positive real result for real x, and taking the principal value more generally). — Steven G. Johnson (talk) 16:40, 11 December 2011 (UTC)

I am for up-front clarity of this type. Have I improved the position with my last edit where I make a specific interpretation of e^x take precedence? Is more still to be done/reordered? Quondum^talk_contr 16:50, 11 December 2011 (UTC)

@Stevenj I think the text is better now okay and it is true the integer power rule follows trivially because the exponential rule follows the exponentiation law. However I think it is worthwhile providing the proof that they are equal for integers as the equivalence is simply stated in general rather than proven and is not straightforwardly obvious and the proof is quite short. Dmcq (talk) 17:45, 11 December 2011 (UTC)

It seems a bit redundant to me; if you are going to include any additional proof, it seems like the most pertinent thing to prove would be the exp(x+y) identity. — Steven G. Johnson (talk) 17:59, 11 December 2011 (UTC)

Well that follows immediately because integer exponentiation satisfies it, so using that proof shows it for the sum of two positive integers. Just saying that makes it longer though and I'd prefer any proofs to be very short unless they are reasonably notable. Dmcq (talk) 21:26, 11 December 2011 (UTC)