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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• Comment or view
• Article history

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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)

Proposed merge of Base (exponentiation) into Exponentiation

stub article with limited potential for growth; it could be covered under Exponentiation#Terminology and Exponentiation#Powers via logarithms. fgnievinski (talk) 15:14, 23 January 2025 (UTC)

I agree to redirect of Exponentiation, but not to a section, as "base" is defined in the first sentence of this article. This is so evident that I'll do this immediately. D.Lazard (talk) 15:52, 23 January 2025 (UTC)basis

I don't think there's any benefit in making the word "base" bold here. Italic is completely fine for this. –jacobolus (t) 18:29, 23 January 2025 (UTC)

MOS:BOLDREDIRECT suggests that "base" and "exponent" must be bolded as targets of redirects. D.Lazard (talk) 21:04, 23 January 2025 (UTC)

"Terms which redirect to an article or section are commonly bolded" is not the same at all as a hypothetical "every term which redirects must be bolded". In my opinion it is worth using bold for (a limited number of) synonyms of the main title, or on words in later sections of an article which could plausibly be their own article. Bolding every glossary-type term related to a topic is overdoing it: distracting but not really helpful to anyone. (MOS even explicitly says: "Avoid using boldface for introducing new terms.") –jacobolus (t) 23:03, 23 January 2025 (UTC)

MOS:TWOSTYLES clarifies an exception: "Combined styles are also valid in articles about a term or when significant terms redirect to an article". In the present case, "base" is both a term and a redirect, so it deserves both italics and bold:

... bⁿ, is an operation involving two numbers: the base, b, and the exponent or power, n.

Basically, anytime a sentence could be rewritten to use the word "called", a word-as-word is involved:

... bⁿ, is an operation involving two numbers: b, called base, and n, called the exponent or power.

The boldface would not be needed only if the redirects were considered "insignificant or minor" (WP:RPLA). However, "base" and "exponent" are key concepts which are likely to be sought after by readers unfamiliar with the subject. "Base" would deserve italics with no bold if it remained as a separate article.

[Caveat: I wrote the above clarification in the MOS a couple of years ago. Since it went undisputed for many years, it is a "generally accepted standard that editors should attempt to follow". Any proposals for further clarifications can be discussed at Wikipedia talk:Manual of Style/Text formatting.] fgnievinski (talk) 03:16, 24 January 2025 (UTC)

In my opinion adding bold here makes the article worse, and is by no means either a consensus convention on Wikipedia or a requirement of the manual of style. It is distracting but not particularly helpful to readers. "TWOSTYLES" does not seem applicable to this particular situation. Whether a term is redirected here is largely arbitrary and historically contingent – we could easily imagine making many other things redirect to this page, e.g. Repeated multiplication, Fractional power, Multiplication rule (exponentiation), Raise to a power, ..., and these are not "insignificant or minor" topics. Those current red links, just like Base (exponentiation) could easily be turned into redirects, and that would be completely fine. However, bolding the appearance of all of those phrases here would be completely ridiculous and distracting. Similarly, when we just bolded exponentiation, also bolding exponent immediately following is not necessary. Anyone here looking for the word "exponent" is going to trivially find it. –jacobolus (t) 17:45, 24 January 2025 (UTC)

As for least surprise: I went through the links to base (exponentiation). Most of them were poorly targeted and instead should point at radix. Of the rest, most of those were about the base of a logarithm, which I agree could cause confusion if people end up here instead, since this article doesn't say anything about the base of a logarithm. It would probably be better to put "base" in some explicit glossary page and redirect base (exponentiation) to there instead of redirecting it to this article. –jacobolus (t) 18:46, 24 January 2025 (UTC)

There's base (mathematics). fgnievinski (talk) 03:31, 28 January 2025 (UTC)

Article wikilinks are not supposed to point at disambiguation pages.

It might be best to make "base" back into its own article again, direct both base (exponentiation) and base of a logarithm to the same place, and maybe also merge with radix, discussing the "base" of an exponential, a logarithm, a number system, etc. all in one place, since they are still more or less the same idea. –jacobolus (t) 07:12, 28 January 2025 (UTC)

A middle ground would be creating Base (algebra) as a set index article, which accepts incoming wikilinks, and retargeting the redirects Base of a logarithm and Base of exponentiation (with a see-also link to Base of computation). fgnievinski (talk) 21:37, 28 January 2025 (UTC)

Base (algebra) seems like a terrible title to me, but this also should not be a "Set index article" (whatever that is). All three are essentially the same idea, not different ideas with the same name. Edit: I think Wikipedia:Broad-concept article is the appropriate guideline page. –jacobolus (t) 22:39, 28 January 2025 (UTC)

What would be a better title for such a BCA? Base (logarithm and exponentiation) is rather long. Base (mathematics) has other non-related concepts. fgnievinski (talk) 02:30, 29 January 2025 (UTC)

I strongly oppose to such a BCA, and, by the way, to any new article about bases. Indeed the base of an exponentiation is simply the name given to the first operand of a specific operation, exactly similar to addend, numerator, dividend, ..., none of them having a specific article. The base of a logarithm is the base (common meaning) on which logarithms are built, and the radix of a numeral system is the base on which the system is built. So, a BCA could only be consist on vague considerations on the general meaning of the word "base". The fact that logarithm of base ⁠$b$⁠ and exponentiation of base ⁠$b$⁠ are each the inverse function of the other is well documented in both Logarithm and Exponentiation Moreover, this is more a property of logarithm and exponentiation rather than a property of bases. So, no new article is needed, a such a new article could be confusing. D.Lazard (talk) 10:02, 29 January 2025 (UTC)

Fair enough. I went through all of the inbound links pointed at Base (exponentiation) and retargeted those about the base of a logarithm to Base of a logarithm (redirects to Logarithm) instead and those about the base of a positional number system to Radix, so hopefully someone who clicks a link on the word "base" won't wind up at exponentiation and get confused. –jacobolus (t) 10:41, 29 January 2025 (UTC)

Terminology of "power"

The introduction to this page reads:

In mathematics, exponentiation, denoted bⁿ, is an operation involving two numbers: the base, b, and the exponent or power, n.

[...] This binary operation is often read as "b to the power n"; it may also be called "b raised to the nth power", "the nth power of b", or most briefly "b to the n".

Now, correct me if I'm wrong, but strictly speaking, isn't the "power" of an exponential expression its outpot? That is, p in bⁿ = p. My understanding is that refering to such an expression as "b to the power of n" is technically incorrect, and that the "proper" form is those listed after it, better reflecting the logic of "the nth power of b is p" >> "p is the nth power of b" >> "p is a power". "b to the power of n" is still very common, of course, but, assuming I haven't misunderstood something, perhaps this warrants a mention?

Also, not to split hairs, but shouldn't the final example read "b to the nth"? I suppose it's a difficult task to manage linguistic descriptivism and technical prescriptivism in articles of this nature. Kisequé (talk) 06:50, 27 February 2025 (UTC)

First of all, "prescriptivism" is forbidden in Wikipedia, which has only to report the common usage. I agree that "b to the power of n" is incorrect, but it is not in the quotation of the article. Also, "p is a power" is incorect, while "p is a power of b" is correct. Also, the ordinal (adjective) form is correct before the noun "power", but not for the name of the exponent (after "power" or alone). Finally, "nth" implies that the exponent is a natural number, which is not the general case. So, I do not see anything to change in the article. D.Lazard (talk) 08:05, 27 February 2025 (UTC)

Powers of one

Using polar form 1 can be exponentiated to become other numbers such as 2, 137, e³, there is not really a limit, so why does this page not talk about the fact that polar form allows such things? 24.83.129.58 (talk) 16:54, 27 May 2025 (UTC)

Probably, there is a mistake in your reasoning/computation. This is the reason for not talking about in Wikipedia. D.Lazard (talk) 17:07, 27 May 2025 (UTC)

Cant you just raise 1 to the power of -iln(y)/2npi where n is a non-zero integer, and then exploit 1=e^i2npi, so you get e^(i2npi*(-iln(y)/2npi)), which turns to e^(i×-iln(y)), then you get e^(ln(y)), making it just simplify to y A slice of 8.5397342 (talk) 21:39, 27 May 2025 (UTC)

The answer is almost certainly no, but it's very hard to read exactly what you are trying to write if you don't use LaTeX for mathematical notation. You might want to look at Mathematical fallacy § Complex exponents. In any event, this page is for discussing improvements to the article, and is not a good venue for miscellaneous discussion about the topic. This kind of question belongs on some other online forum. –jacobolus (t) 23:20, 27 May 2025 (UTC)

edit request

section: matrices and linear operators "and if A is invertible, then A^{-n} = (A^{-1})^n = (A^n)^{-1}" please add source Username7382287 (talk) 00:58, 4 July 2025 (UTC)

A big bunch of changes

Hi @Sollyucko. I reverted your edit here which crammed in a lot of unrelated changes into one, most of which seem a bit out of place (e.g. we don't need to add any of these wikilinks to "the number of grains of sand that can be contained in the universe"). But this article could use a lot of help, so I don't want to discourage you from getting involved.

In particular, the lead section is both overstuffed with concrete details and also not a good summary of the article, the history section probably belongs closer to the bottom than the top, and we could do a much better job providing basic context and motivation for the concept and explaining some of the elementary mathematical details. Some of the more technical parts could be streamlined and don't have great narrative flow; they read like a mishmash of reference material written by different authors was collected and then smashed together in arbitrary order (which is probably more or less what did happen). We could do a much better job linking exponentiation to exponential growth and decay. Etc.

If you have concrete ideas, maybe we can discuss here. One place to start might be looking up nice accessible overviews aimed at an audience of high school students or teachers. –jacobolus (t) 07:21, 30 August 2026 (UTC)

I've made a new edit, refocused partly based on your feedback. What do you think?

Solomon Ucko (talk) 16:55, 30 August 2026 (UTC)

Ack, sorry. I was just in the process of re-applying more than half of your previous changes, and various miscellaneous copyedits of my own, resulting in a massive edit conflict that will now be quite annoying to resolve; I might take a break first. I think a few examples gave me an unfairly bad impression leading to the wholesale revert. –jacobolus (t) 17:16, 30 August 2026 (UTC)

A couple points about the most recent one:

• Putting more about rare notations like the up arrow notation in the lead is unnecessary. We don't need to belabor the point about code syntax here either (but can discuss at length later in the article, as appropriate).
• I think cramming the content of the previous "terminology" section into the lead is more distracting than helpful. We should instead be trying to take things out of the lead.
• I don't think it's necessary to extend $$x^{\frac {p}{q}}=(x^{p})^{\frac {1}{q}}={\bigl (}x^{\frac {1}{q}}{\bigr )}^{p}$$ to $$x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}={\sqrt[{q}]{x^{p}}}=(x^{\frac {1}{q}})^{p}=\left({\sqrt[{q}]{x}}\right)^{p}$$ when $x^{1/n}$ and ${\sqrt[{n}]{x}}$ are equated immediately above. Putting the same thing in two different notations at once distracts from the point. If we were in any place other than this article, I'd replace the fractions in exponents by horizontal ones $$x^{p/q}={\bigl (}x^{p}{\bigr )}^{1/q}={\bigl (}x^{1/q}{\bigr )}^{p}$$ though in this particular context it seems plausible that using a vertical fraction makes the fractions more explicit; I'm still tempted though.
• I don't think "Raising 0 to a negative exponent results in division by zero" seems quite accurate, since it doesn't "result" in anything in the context of real or complex numbers. But if you want we could say something along the lines of "As with division by 0, Raising 0 to a negative exponent is undefined..."

–jacobolus (t) 17:29, 30 August 2026 (UTC)

I decided to dissolve the "terminology" section into the lead because:

• It seemed like a needlessly-separate digression into certain aspects of phrasing.
• Roughly half of it wasn't even about terminology, but instead illustrating some integer exponents.
• Most of the discussion about terminology was in the lead, the "Etymology" section, and the "History" section, not there.

Given how entangled they are, should I create a combined section, perhaps titled "Terminology and history", containing relevant content from the lead (mentioning the primary names and notation in both places, and moving the rarer ones here), as well as all of "Etymology" and "History"?

---

I added the roots into that equation for a couple of reasons:

• The previous lead stated, "$b^{n/m}={\sqrt[{m}]{b^{n}}}$", and I figured that might be worth keeping somewhere.
• Using the reciprocal exponent notation may be circular reasoning here, whereas the root notation is clearly distinct.

I meant to change "The equality on the right" to something like "The commutation equality", but lost track when copying my changes out of a pending edit that VE screwed with.

---

By "X results in Y", I meant that X causes Y, or X leads to Y, not that X has the same numerical result as Y. For example, 0^-123 = 1/(0^123) = 1/0, which is an instance of division by zero. Is there a clearer way to state that? To me, "as with" implies that the similarity is coincidental or due to parallel reasoning, not one causing the other.

---

Solomon Ucko (talk) 18:01, 30 August 2026 (UTC)

I don't think we should mix history with terminology. I'd move the history section down toward the bottom, but sections about notation/terminology belong near the top. Stuff about the second power being called the "square", etc. seem distractingly out of scope for the lead here. –jacobolus (t) 18:03, 30 August 2026 (UTC)

Ok, fair enough. How about organizing the article like this?

• lead
• Notation
  • Hatnote: "For the history of notation, see § History"
  • As per Wikipedia:Manual of Style/Mathematics § Article body, this could also include that digression about multiplication notation
• Terminology
  • Hatnote: "For the history of terminology, see § History"
• Integer exponents
• ...
• In programming languages
• History
• See also
• ...

Solomon Ucko (talk) Solomon Ucko (talk) 18:14, 30 August 2026 (UTC)

"Notation" and "terminology" can be safely combined if it seems simpler. We should try to keep any such section(s) that are very high up the page as concise as we can, while keeping information that seems essential for understanding the rest of the article. Anything especially tricky can be cut or moved to a footnote. –jacobolus (t) 18:17, 30 August 2026 (UTC)

I'm not sure how useful the material in § Etymology is to most readers. It probably doesn't need to be near the top. –jacobolus (t) 18:20, 30 August 2026 (UTC)

I've merged it into § History, which I've put at the bottom of the content sections.

Solomon Ucko (talk) 19:58, 30 August 2026 (UTC)

Changing "operation involving two numbers" into "binary operation" is the kind of thing we should avoid in the lead section of articles aimed at a wide audience. The latter is harder to make sense of, and the wikilink points at a less useful place for someone who doesn't already know what these various terms mean. –jacobolus (t) 20:45, 30 August 2026 (UTC)

There's a better (though more time-consuming) way to solve that :) Special:Diff/1372231551

What do you think of it now?

Solomon Ucko (talk) 23:30, 30 August 2026 (UTC)

There wasn't necessarily anything wrong with the article Binary operation (in fact, I find the beginning of the version before your change to be more accessible than the version after).

The issue is just that "operation involving two numbers" is more concrete, explicit, and comprehensible to an audience not familiar with mathematical jargon than "binary operation". To someone who doesn't already know what "binary operation" means, it's an opaque and potentially confusing phrase (someone might speculate that it has to do with binary numbers, or computer programs). If the jargon word were of very high importance to the topic, we could use an inline gloss, like "binary operation (rule for combining two numbers to produce a third)" or whatever, but the previous phrase was just fine: Someone who does already know what a binary operation will clearly recognize that that's what we're talking about, and Binary operation is prominently wikilinked from the beginning of Operation (mathematics) so anyone looking for that link per se should have little difficulty finding it.

I don't think our article here benefits from using this jargon in the lead. If we want to include it somewhere later down the page, we could, but we should try to have the lead section use the simplest and most accessible language we can while still conveying the message accurately. –jacobolus (t) 00:43, 31 August 2026 (UTC)

I see, yeah... What about keeping the old phrasing but changing the link?

"[...] is an operation involving two numbers: [...]"

Solomon Ucko (talk) 00:58, 31 August 2026 (UTC)

Why though? The wikilink Operation (mathematics) seems entirely fine. –jacobolus (t) 02:07, 31 August 2026 (UTC)

Also, the use of exponents in units of measurement is not really an example of exponentiation per se, and is substantially off topic for the lead. It can be mentioned somewhere else in the article if you like. –jacobolus (t) 21:21, 30 August 2026 (UTC)

That usage is not exponentiation of numbers, but it is exponentiation of something, namely units. Most units generally act like variables representing positive real numbers, and as such, follow the relevant identities and properties. Therefore, I think they are sufficiently connected.

If you'd rather not have it in the lead, I guess it could technically go in § In a group?

Alternatively, how about removing the example, keeping just the first sentence, and tacking it onto the end of the preceding paragraph about the contexts in which exponentiation is used?

Solomon Ucko (talk) 21:52, 30 August 2026 (UTC)

I think it's a non sequitur in the context of the lead, since it's about a different subject than what the article is focused on, which is exponentiation of numbers (and secondarily exponentiation of more general number-like quantities).

I don't think it quite belongs in a section about abstract algebra either; even though units are abstract, they aren't really conceived of as an algebraic structure, and are more relevant to science and engineering than to mathematics per se. Whether this should even be considered "exponentiation" at all is a question probably most at home in the philosophy of science. It could perhaps be discussed somewhat in the history section, in a discussion of the way a variable to the 2nd or 3rd power was historically considered to represent an area or volume, and this conception may have limited more generic understanding. But that might not really achieve whatever your purpose was in adding this note. It could also plausibly be mentioned in a notation section, or in a separate section about generalized uses of the exponential concept. But the most appropriate place for discussion of this topic at a non-trivial depth is probably some other article, e.g. Dimensional analysis.

To figure out what belongs in the lead here, we should try to put ourselves in the shoes of a middle or high school student encountering this topic for the first time, or perhaps a curious layperson with limited mathematical experience. We should try to answer the basic questions we anticipating them wondering about, and keep the narrative flowing as best we can. –jacobolus (t) 22:15, 30 August 2026 (UTC)

We could more explicitly say that this is related to the convention for division by zero, or that the two conventions are consistent with each-other, but I don't think it's inherently "caused" by it; maybe this point seems like a bit of an arbitrary semantic quibble, but I think the "results" or "causes" phrasing can potentially lead to some confusion. –jacobolus (t) 18:07, 30 August 2026 (UTC)

Hi @Sollyucko, sorry, I had to run an errand the other day, and later my browser crashed taking my edit-conflicted re-revert with it. I've tried to re-apply some of the changes I reverted before. If there are some that you still think are important that remained reverted, feel free to put them back or discuss. I think we should still further discuss some of the other higher-level recent changes, and the structure of the article in general.

I think the new versions of the notation / terminology sections at the top seem to be too long and a bit unfocused. I don't think we need to mention up arrow notation at all, and a discussion about the use of exponent notation for iterated functions seems out of place as the first thing to talk about.

While previously the list of basic properties was kind of cumbersome to be crammed directly in the lead section, we probably still want to preserve it quite near the top of the page. Maybe we can make up some kind of post-lead introductory section (called something like "basic properties" or "concept" or "overview" or ...?) that includes this material right near at the top. Specifically, we want to make clearer right away that almost everything about exponentiation flows from the property that multiplications/divisions of the base are turned into addition/subtraction of exponents. We should mention explicitly that anything to the 0th power is 1 and anything to the 1st power is the thing itself, and we should talk about how this basic property sets up a definition of negative and fractional exponents.

I don't think our current organization is very effective, with top-level sections of "Integer exponents", "Rational exponents", "Real exponents", "Complex exponents". This structure ends up cramming too much trivia about minor details related to integer exponents at the top, including stuff that is more advanced than we should expect for a lot of the audience, while deferring basic information that readers likely care about and forcing them to skim down the page. I think we might be better off making top-level sections about (1) exponentiation of rational/real numbers; then perhaps (2) a section about exponentiation with variables (representing numbers), including discussion of polynomials (positive integer exponents and a variable base), power functions (constant exponent and variable base), exponential functions (constant base and variable exponent), and possibly topics like logarithms; then (3) exponentiation of complex numbers; (4) exponentiation in abstract algebra; (5) exponentiation of square matrices; (6) other uses of index/exponent notation, including stuff like function composition or dimensional analysis. Material about history and practical computation can fit down toward the bottom somewhere. –jacobolus (t) 20:30, 4 September 2026 (UTC)

Hi @Jacobolus:

Gotcha, I understand. I might compare my edit to the current state of the page when I get a chance.

I think way the Notation and Terminology sections are organized is fine:

• I don't think everything needs to have some kind of overall narrative; in fact, since Wikipedia is a reference source rather than a novel, I think it's better for it to focus on listing facts, grouping them together in a way that helps readers find answers to their questions, rather than trying to craft a story.
• What's wrong with linking to Knuth's up-arrow notation here? I'd even consider having something like the following in the lead: "Just like multiplication is iterated addition, exponentiation is iterated multiplication, and Knuth's up-arrow notation can generalize this further."
• Should we merge the discussion of function exponentiation into "Iterated functions", and perhaps rename that section to something more general? Or perhaps keep just a short summary in Notation, and leave the details for iterated function?

If we're listing definitions and properties in the lead, I think we should just state the facts, and leave the proofs for the body. Perhaps something like this?

• When the exponent is a positive integer, exponentiation means multiplying that many copies of the base: $b\in \mathbb {Z} ^{+}\implies a^{b}=\underbrace {a\times \cdots \times a} _{b{\text{ times}}}$
• When the exponent is 0, the result of exponentiation is generally 1: $a\in \mathrm {numbers} \land a\neq 0\implies a^{0}=1$
• Negating the exponent results in the reciprocal result: $a^{-b}={\frac {1}{a^{b}}}$
• Addition of exponents is equivalent to multiplication of results, with the same base: $a^{b+c}=a^{b}a^{c}$
• etc.

There is indeed a lot of random stuff in the "Integer exponents" section. Complicating this, a lot of properties depend on each other, and it would be best to explain prerequisites first. How about something like this?

• Notation
• Terminology
• Definitions and fundamental propertiesq
  • Positive integer exponents
  • Combining addition, multiplication, and exponentiation
  • Zero and negative exponents; subtraction and division
  • Non-integer exponents on positive bases
  • Complex numbers
• Numeric examples
• Exponential function
• Inverses
• Polynomials
• Limits
• Abstract algebra
• Linear algebra
• Set theory
• Computation
• History

Solomon Ucko (talk) 00:52, 5 September 2026 (UTC)

The biggest audience for an article about such a broadly relevant topic as exponentiation is going to be laypeople and students, and our article needs to be as accessible as possible, especially in the first few sections until we have addressed elementary topics. See Wikipedia:Make technical articles understandable for some elaboration about this goal.

So I'd like us to end up with an article which can be looked at by, say, a 15 year old high school student which will nearly immediately answer typical questions they might have about what exponentiation is, how it works for ordinary numbers, and what its elementary properties are. Down toward the middle or bottom of the article we can go on whichever technical tangents we like, and include as much detail as we think is meaningfully relevant to an article-length summary of the topic. –jacobolus (t) 02:23, 5 September 2026 (UTC)

As for up-arrow notation, I would leave it to the section about § Repeated exponentiation. It would be fine to include a bit more explicit detail in that section showing how up-arrow notation works and how it can be applied to exponentiation per se. But in practice nobody uses up-arrow notation for exponentiation outside of that narrow context, so discussing it near the top puts, in my opinion, "undue weight" on what is kind of a niche side topic. (On the other hand, the use of a caret character ^ to represent exponentiation is pervasive in programming and computer markup languages, including notably (La)TeX, so is reasonable to mention near the top of the article.) –jacobolus (t) 02:28, 5 September 2026 (UTC)