Talk:Exponentiation/Archive 2021

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'power' terminology

I think the intro of the article wrongly states that 'power' mean the same as 'exponent' and this should be corrected. Below, in the section 'Terminology', this is then reverted and their actual usage conventions are described a little bit. Please reply if you disagree: 'Power' can sometimes mean 'exponent', but often it means the same as square just for other exponents than 2. A power is thus (in the simplest case) a product with only one, repeatedly multiplied factor and it is not how often the factor appears in said product (which would be the exponent). So just like the exponent 2 is not the square and the exponent -1 is not the reciprocal, a power in general is not an exponent. If one wishes to avoid potential confusion, one could use the word exponential instead, as any power is also an exponential and vice versa (not however that a^b is a power of a and an exponential of b). If there are no objections, I will try to clarify the intro in this regard without complicating it. Ninjamin (talk) 20:45, 23 April 2020 (UTC)

In modern usage, "power" is a synonym for "exponent".

It's true that the phrase, "the nth power of x", preserves an older usage, where "power" means the result of exponentiating, the equivalent of a "sum" for addition. But there are many other expressions where "power" clearly means the exponent, e.g., "n raised to the power p".

So please don't change the phrasing in the article. --Macrakis (talk) 21:17, 23 April 2020 (UTC)

I have to disagree, the 'older usage' is actually the more frequent one in the article (I checked all occurences) and I found it in most other sources I looked up. The lack of other conventional terms for the result of exponentiation (exponential is not very common) probably makes for part of the wide usage of 'power' in this meaning. Sometimes, a single occurence of the word 'power' is even given both meanings, each in regard to a different part of the whole phrase, like in 'complex powers of complex numbers can be real numbers', where the power is complex as an exponent but it is real as result of the exponentiation. (The article also talks about monotonous and discontinuous powers, which I think are meant to be power functions, not powers…) I didn't want to say that either of the two meanings was more important, but just criticize that one is treated as irrelevant or minor while both are actually very common and widespread. So there is clearly something missing in the article. Ninjamin (talk) 16:53, 27 April 2020 (UTC)

Here are some prominent sources:

From the Bronstein (Handbook of Mathematics, 5th edition, Springer 2007), first two sentences of the 'POWERS' chapter 1.1.4.1:

The notation $a^{x}$ is used for the algebraic operation of raising to a power. The number $a$ is called the base, $x$ is called the exponent or power, and $a^{x}$ is called the power.

The 'power' entry in Wolfram's MathWorld uses the word 21 times and 5 of those are in the 'older' meaning, with only one of those five using the 'nth power of' syntax.

Even the 'Arithmetic operations' box to the right of the article has the fomula base^exponent=power.

This should be enough evidence that the'older' meaning is still very much in use. I will try again soon and improve the intro accordingly, while changing as little as possible. --Ninjamin (talk) 18:21, 13 June 2020 (UTC)

References

Some comments:

• This is a case for WP:Verifiability, not truth: The question is not whether the use of a word in the first sentence is correct. It is whether the word is commonly used with this meaning in the literature. It appears that both "exponent" and "power" are commonly used in the literature for similar sentences.
• The first sentence does not asserts that these two words are synonymous, which is wrong. It means that both can be used in similar sentences.
• The two words are not exactly synonymous, but the diference is too subtle for belonging to the lead.
• The difference between the two words is similar to the difference between "sum" and "addition". Generally (this may vary with authors), "exponent" refers to the second operand in an expression like $a^{b},$ and "power" refer to the second operand in the action of exponentiating. In other words, one term refers to the syntax, while the other refers to the semantics. I have not checked the whole article in details, but the use of the words in the table of content corresponds exactly to the above distinction (it is not me who has named the sections).
• In the first sentence, it is not clear whether one talks of syntax or semantics; thus both words are correct.

In summary, nothing needs to be changed in the terminology of the article. D.Lazard (talk) 20:22, 14 June 2020 (UTC)

″I have not checked the whole article in details″ – please do so instead of interpreting the section headers according to your prejudice. Most section headers contain the plural form "powers", and in all those cases, the meaning is the same: the results (1, 2, 4, 8, ...), not the operands (0, 1, 2, 3, ...). --Rainald62 (talk) 16:24, 1 September 2021 (UTC)

the caret(^) sign

It might be useful to include the history of the caret and its meaning as it is used fairly widely to suggest exponents or superscript, and this is the logical place to look for information on the symbol as the name is fairly unknown. a full explanation may not be needed but at least a reference to the page on the caret and its popular use starting from its meaning 'it lacks' in latin would be useful for people looking for information on it.81.103.233.161 (talk) 19:43, 21 October 2021 (UTC)elias

Nah, that's off topic for this article. Anyone who is searching for that can just put a '^' in the search bar, which directs them to the proper place. MrOllie (talk) 19:48, 21 October 2021 (UTC)

I don't think it's off-topic at all. I've added some info on this (with source). (By the way, the exponentiation usage has nothing to do with the Latin name caret 'it is missing'.) --Macrakis (talk) 16:49, 22 October 2021 (UTC)