Talk:Exponentiation

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: rejected by Theleekycauldron (talk) 10:57, 25 December 2022 (UTC)

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• ... that exponentiation is basically repeated multiplication? Source: https://mathinsight.org/exponentiation_basic_rules
  • Reviewed:
  • Comment: My first nomination submission.

Created by Slaythe (talk). Self-nominated at 01:05, 24 December 2022 (UTC).

• Sorry, Slaythe, but it's not eligible, because it hasn't been created, 5x expanded, or designated a Good Article within the last week. Please see WP:DYKCRIT. It you get it promoted to Good Article status, you may resubmit a DYK nomination. MANdARAX • XAЯAbИAM 01:34, 24 December 2022 (UTC)

Three notations for multiplication

This article uses 3 different notations for multiplication. IMO, either $\times$ must be replaced with $\cdot ,$ or $\cdot$ must be replaced by $\times ;$ in any case, some occurrences of $\cdot$ should be removed, especially in exponents. As I have no clear opinion on the best choice, I wait for a consensus here. For clarification (see the preceding thread), I have added an explanatory footnote. D.Lazard (talk) 10:28, 9 January 2023 (UTC)

As you wrote in the first thread, the primary issue with using $\cdot$ is that it becomes ambiguous whether this represents ellipses or multiplication (to be fair, it's not really ambiguous, because one could figure out from context that one dot means multiplication and three means ellipses). So, I think that defaulting to $\times$ in this article is probably best, even though this notation feels quite elementary-school-y. Duckmather (talk) 15:20, 9 January 2023 (UTC)

Now that I know what is meant, I would suspect that most uninitiated readers (in the UK at least) would recognize a $\times$ sign as a multiplication sign, but that the $\cdot$ sign for multiplication would be much less familiar, and might well be a source of confusion. Additionally, I have come to this article in following-up work on the biographical WP article on William Oughtred, and I find that the introduction of the $\times$ sign, in W.O.'s Clavis Arithmeticae (1631), followed on very soon after, and in the context of, the description of logarithms (in the English edition of John Napier's Description of the Admirable Table of Logarithmes (S. Waterson, London 1618), Appendix, at p. 4 (Google)). (An explanation of the sign is given in William Forster's Forster's Arithmetick (1673), at pp. 43-44 and pp. 113-14 (Google).) Hence there is an historical association between Exponentiation and this $\times$ usage which some readers may want to understand. I find some explanations in F. Cajori's History of Mathematics (Macmillan, New York/London 1919), at pp. 157-58 (Internet Archive). Perhaps the pedagogic example is the better for being elementary? Eebahgum (talk) 18:38, 9 January 2023 (UTC)

I adjusted the intro to be consistent with /times. I left the rest of the article as is. Emschorsch (talk) 04:49, 24 July 2023 (UTC)

Definition of principal value of log(z)

In the section:

Principal value

[...]

and the imaginary part of z satisfies

-π < Im (z) < π [this does not make sense to me: Isn't it a condition on the Arg(z) or equivalently on the Im(log(z))? Since log(z)=log(|z|)+i(Arg(z)+2nπ), n in Z and the principal value of log(z) can be defined as Log(z) when chosing -π < Im(log((z))=Arg((z)) < π, i.e. n=0]

[...] 217.10.52.10 (talk) 09:52, 20 April 2023 (UTC)

 Fixed. D.Lazard (talk) 16:12, 21 April 2023 (UTC)

incomplete proof in explanation of rational exponents

In the top of the page, the article demonstrates how exponents of rational numbers correspond to nth-roots by proving that b^(1/2) == sqrt(b). Part of this proof relies upon the property that (b^M)*(b^N) == b^(M+N) (see excerpt pasted below). However, the article only proved this property based on the definition that natural-number exponents are equivalent to repeated multiplication. This proof does not apply when M or N are rational because rational exponents are not defined as a repeated multiplication.

Therefore, the article needs to have a separate proof that (b^M)*(b^N) == b^(M+N) when M and N are rational numbers.

Proving this property holds true for rational numbers is a fair bit more complicated than proving it holds true for natural numbers, but it's not so complicated as to be out of the scope of a wiki article. One such proof can be found on this stack overflow page[1]. unfortunately i do not know of any proofs that meet wikipedia's credibility requirements.

this is the specific excerpt from the wiki article that i take issue with: "Using the fact that multiplying makes exponents add gives b^(r+r) == b". It is located at the top of the page. Snickerbockers (talk) 16:14, 24 November 2023 (UTC)

The general case is an easy generalization of the given case: if M and N are rational numbers, one may reduce them to the same denominator, that is $M={\frac {m}{q}}$ and $N={\frac {n}{q}},$ with $m,n,q$ integers. Setting $y=x^{1/q},$ one has

$x^{M}x^{N}=y^{m}y^{n}=y^{m+n}=x^{\frac {m+n}{q}}=x^{M+N}.$

D.Lazard (talk) 17:08, 24 November 2023 (UTC)

I want to make the following claim, but where should I write it?

$\left(a\cdot b\right)^{r}=a^{r}\cdot b^{r}\cdot e^{-i2\pi nr}$
$\left(a^{r}\right)^{s}=a^{r\cdot s}\cdot e^{-i2\pi ns}$
取らぬタヌキ (talk) 21:41, 4 January 2024 (UTC)

$x^{-a}={\frac {1}{x^{a}}}$

There are no branches.

$x^{a+b}=x^{a}\cdot x^{b}$

There are no branches.

$\left(a\cdot b\right)^{x}=a^{x}\cdot b^{x}\cdot e^{-i2\pi nx}$

There are branches.

$\left(x^{a}\right)^{b}=x^{a\cdot b}\cdot e^{-i2\pi nb}$

There are branches.

These will help you understand the following:

${\sqrt {x^{2}}}=x\cdot e^{-i\pi n}=\left\{x,\quad -x\right\}$

If it is a real number domain, the square root cannot take a negative value, so ${\sqrt {x^{2}}}=|x|$.

Also,

$(e^{i2\pi n})^{i2\pi m}=e^{-4\pi ^{2}nm}\cdot e^{4\pi ^{2}nm}=1$

取らぬタヌキ (talk) 19:37, 5 January 2024 (UTC)

We must understand correctly that a exponentiation is a multivalued function. The following formula transformation is important.

$x=x\cdot e^{-i2\pi n}$

$x^{a}=x^{a}\cdot e^{-i2\pi na}$

$\left(x^{a}\right)^{b}=x^{a\cdot b}\cdot e^{-i2\pi \left(n_{1}a+n_{2}\right)b}$

Furthermore,

$x=x\cdot e^{-i2\pi n}$

$\ln x=\ln x-i2\pi n$

$\ln x^{a}=a\ln x-i2\pi \left(n_{1}a+n_{2}\right)$

取らぬタヌキ (talk) 23:36, 10 January 2024 (UTC)

table of powers of 2, e, and 10

Moved from user talk:jacobolus § exponentiation table

with reference to https://en.wikipedia.org/wiki/Exponentiation#Particular_bases i see that my table has been removed, perhaps in part because it is a large exhibit that does contrast to the rest of the article - but i argue that something like it would make it easy for, say, a high school student, to clearly see

1. the distinction between BASE and EXPONENT,
2. a clear difference between how the 3 bases behave, and
3. what kind of magnitudes result when those bases and 21 exponents are combined in a tabular way.

so is there some way we could do this?

Lee De Cola (talk) 20:49, 21 November 2024 (UTC)

For reference, the table, removed in special:diff/1258811465:

                                      BASE                                   base-10
EXPONENT      2                     e*                      10                prefix
------------------------------------------------------------------------------------
   10     1,024                22,026.465 795   10,000,000,000
    9       512                 8,103.083 928    1,000,000,000                 giga-
    8       256                 2,980.957 987      100,000,000
    7       128                 1,096.633 158       10,000,000
    6        64                   403.428 793        1,000,000                 mega-
    5        32                   148.413 159          100,000
    4        16                    54.598 150           10,000
    3         8                    20.085 537            1,000                 kilo-
    2         4                     7.389 056              100                hecto-
    1         2                     2.718 282               10                 deca-
    0         1                     1                        1                 unit
   -1         0.5                   0.367 879                0.1               deci-
   -2         0.25                  0.135 335                0.01             centi-
   -3         0.125                 0.049 787                0.001            milli-
   -4         0.062 5               0.018 316                0.000 1
   -5         0.031 25              0.006 738                0.000 01
   -6         0.015 625             0.002 479                0.000 001        micro-
   -7         0.007 812 5           0.000 912                0.000 000 1
   -8         0.003 906 25          0.000 335                0.000 000 01
   -9         0.001 953 125         0.000 123                0.000 000 001     nano-
  -10         0.000 976 562 5       0.000 045                0.000 000 000 1
                               *values rounded to
                                6 decimal places

I took this table out because it seems unnecessary, distracting, and unhelpful to readers. While constructing such a table might be instructive for a high school student, it seems out of scope and somewhat off topic here. Literally nobody ever cares what the 10th power of e is, and we already have dedicated articles power of two and powers of 10 which describe the other columns here at length, including tables of values. A couple sentences of prose can just as effectively convey the message that powers of a larger number grow faster than powers of a smaller number. –jacobolus (t) 21:36, 21 November 2024 (UTC)

I understand your perspective, and must, I suppose, defer to your judgement; but still assert that there are many routes to understanding, including words, formulas, images, and interesting patterns. Wikipedia visitors aren't always 'readers' but often - myself for example - people who scan an article for the answers they seek or merely insights, as the table attempts to provide.

For decades I have taken the works of Edward Tufte - himself a vigorous challenger of rigid rules! - as my guide in these matters. Somebody might profit from comparing the 10th powers of 2, e, and 10; and folks keep saying that (bit for bit?) pictures are more powerful than words!

Oh, and the exhibit can be reverse engineered to see how to do a simple Wikipedia table.

Anyhow, I appreciate your perspicacity, guardianship, and knowledge of this deep subject; would that all important articles were overseen by such as you... Lee De Cola (talk) 22:42, 21 November 2024 (UTC)

You don't have to defer to me – I'm just one among many editors here: you can try to build consensus for including a table like this if you want.

I do think this article could use significant work to make it more accessible, but I don't plan to do that in the foreseeable future (I have been working on drafts of exponential function, which is somewhat related). More and better illustrations would certainly be welcome here. And, for example, I don't think § Etymology and § History sections need to be at the top. Much of the text of later section assumes too much of readers. But if you run an article with vs. without this table by a high school student or similar reader, I don't think they are going to find the table especially instructive. Helping such readers make useful inferences about a table like this is going to take a significant amount of additional description that isn't here, and just dropping it on people is I think going to mainly be distracting.

We already do have some (perhaps too much) similar tables here, e.g. in § Table of powers of decimal digits. –jacobolus (t) 23:15, 21 November 2024 (UTC)