Talk:Exponentiation/Archive 2019

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Failure of associativity

@Stemdude: If you know logical quantifiers, then you should also know well that a mathematical identity must hold for every possible collection of values for the variables (that is permissible under the given assumptions). The negation of such a statement can therefore be proven using only one counterexample. See All Cretans are liars. I've added "in general" to clarify, though.--Jasper Deng (talk) 00:49, 20 January 2019 (UTC)

The fact that identities *must* hold over the specified ranges of variables is *exactly my point*.

The original article made the false claim that b^(q^p) != b^(qp) for *all* q, p integers in a section entitled "Identities and Properties". I simply corrected this inaccurate statement by pointing out the exceptions in N and R. You objected to this due to my (very elementary) use of logical quantifiers, which you seem to think are too advanced to use in this article. In that case it would have been a trivial matter to simply restate the equation in english, *which I already did in the R section and which you removed anyway for no apparent reason*.

Adding "in general" to a supposed identity is worthless, because you're not specifying the actual exceptions and mathematical identities do not hold "in general" they either hold for every value or they are not identities. Again, the section is entitled "*Identities* and Properties" and the equation is listed as if it were an identity. Telling a novice reader that an identity holds "in general" is a contradictory and confusing statement.

I don't think this is actually an "identity" at all, just a useful rule that holds over most values. I didn't simply remove it though in the interests of compromise, but I'm beginning to think I should have done exactly that.

Again, I'm willing to compromise my having the facts stated in English rather than using logical quantifiers. You seem unwilling to compromise whatsoever and are taking a "Scorched Earth" approach to all of my changes. It seems personal. Stemdude (talk) 01:25, 20 January 2019 (UTC)

Ok, I've done it myself. It took literally a second to fix. I'm astonished you spent so much time arguing about this when it's so easy to change. Stemdude (talk) 01:34, 20 January 2019 (UTC)

@Stemdude: Wrong. You will certainly agree with the statement that $\sin(x)\neq \cos(x)$ even though the two functions do take the same values at certain point; negation of equality does not have to hold for all possible instantiations of the variable(s) but only one, so for example, $1\neq (1\ {\text{if}}\ x\neq 0,0\ {\text{otherwise}})$. The phrase "generally not the same as" is very common in mathematics education and that is what we should be using here. Examples like these show why it is too strong to require an inequality to hold universally. It's not personal. On the Internet, nobody knows you're a dog, so it can't be based on your personality when I have said nothing specifically about your person.--Jasper Deng (talk) 01:35, 20 January 2019 (UTC)

Wrong. Logical negation of an equality is NOT the same thing as an inequality. Inequalities are expected to hold universally for all specified values. This is really basic stuff. Do you have any formal mathematical training?

 Stemdude (talk) 01:40, 20 January 2019 (UTC)

As for sinx and cosx, they may be intensionally different but extensionally speaking there exist values of x for which they are equal. Hence a universal inequality quantified over all values of x is incorrect. You're confusing intensional and extensional definitions. Please read Extensional_and_intensional_definitions. Stemdude (talk) 01:44, 20 January 2019 (UTC)

Yes, I am completing a degree in it, and in any case, ad hominem arguments are fallicious. You seem to have an issue with a statement like, "In general, $3^{x}\neq 2^{x}$" or "$3^{x}$ is generally not equal to $2^{x}$". The former is usually taken to be an abbreviation of the latter, even though we can agree that there is equality between the left- and right-hand expressions at $x=0$. The burden is on you to show why we should deviate from this established convention that is used by literally every mathematics textbook I have used. As for intensionality and extensionality, where in the article is there a universal quantifier over all possible values for the exponent? This is besides the point. Convention takes the expressions to be intensionally defined.

You won't get anywhere by just saying otherwise, because I have no a priori reason to believe what you say. You can bring this up at WT:MATH, but I really do not expect this to be treated as more than a case of bikeshedding. --Jasper Deng (talk) 01:49, 20 January 2019 (UTC)

Asking for your qualifications isn't ad hominen. Particular when you have no idea what you are talking about. "Convention takes the expressions to be intensionally defined." is a laughably wrong statement.

I've compromised by fixing my changes according to exactly what you originally wanted - I removed the logical quantification. You're not even specifying what is wrong with my edits now, just reverting them out of spite. Stemdude (talk) 02:08, 20 January 2019 (UTC)

"laughably wrong" – then why do some of the best textbooks in the world on mathematical analysis use statements in that form? The "textbook" statement of this form is path-dependence of line integrals, a case where there will be many cases where line integrals along different paths happen to be equal. The statement is usually given as, "In general, $\int _{C_{1}}\mathbf {F} (\mathbf {x} )\cdot d\mathbf {x} \neq \int _{C_{2}}\mathbf {F} (\mathbf {x} )\cdot d\mathbf {x}$"). My issue is that you are needlessly bikeshedding over something that matters to a small minority of readers of this article, and that your presentation is needlessly convoluted in that respect. MOS:MATH explicitly prescribes a conversational presentation of the content.

Also, from the point of view of equality of expressions and functions, two expressions are deemed "unequal" or "inequal" precisely if they differ in value for some, not all, instantiations of variables. When functions are considered as subsets of the Cartesian product of the domain with the codomain, we would most certainly write the inequality sign in such situations.

We are not going to explicitly write the existential quantifier for every statement of this form. Nor do respected authors like Walter Rudin. You can say I'm wrong however much you'd like, but I see no reason why we should assume the failure of an identity to be universally quantified.--Jasper Deng (talk) 02:30, 20 January 2019 (UTC)

You are using this talk page to argue about issues that are irrelevant to improving this article, and are refusing to discuss the reasons you continue to revert. Stemdude (talk) 08:59, 20 January 2019 (UTC)

With respect to both the heavy use of quantifiers and the use of the ambiguous symbol ≠, it would be better to use English prose to make statements that are clear, correct, and unambiguous. Best would be to have citations to reliable sources to support a particular formulation. --JBL (talk) 02:58, 20 January 2019 (UTC)

Hi, JBL. I absolutely agree and using clear, correct and unambiguous English prose is *exactly* what I've done. I've clearly stated the exceptions to the "identity" in question. But Jasper Deng prefers to use the ambiguous term "in general" which gives the reader less information and besides, mathematical identities are never said to hold "in general"; it's a contradiction of the term "identity". Stemdude (talk) 08:56, 20 January 2019 (UTC)

Stemdude's edits are definitively wrong, as calling "identity" a succession of equalities and inequalities. Such a succession is generally a bad idea. In this case, this is specially confusing as the succession is aimed to present several different ideas. Thus I have split the formula for clarifying it. D.Lazard (talk) 10:39, 20 January 2019 (UTC)

(edit conflict) "In general" is not at all ambiguous. An inequality is not an identity. Rather, we are stating the negation of an identity, in which context "In general" means "there exist ... such that $f(...)\neq g(...)$". This is considered fully rigorous.--Jasper Deng (talk) 10:39, 20 January 2019 (UTC)

I interpret the target of the edited paragraphs as dealing with preference rules for evaluating expressions involving exponentiation. It is not about identities or equations that hold for certain domains. I think that for this purpose the original formulation, especially when adapted with "in general", is preferable to mathematically correct identities that veil the target. Personally, I would keep the use of "≠" and would not introduce quantifiers. However, suggesting better English prose is beyond my abilities. Purgy (talk) 11:10, 20 January 2019 (UTC)

The current version (after edits from D.Lazard and Dmcq) is much better than the versions over which the edit war was being fought. --JBL (talk) 15:39, 20 January 2019 (UTC)

Thanks; that prose is much clearer. Would someone like to look at Exponentiation#Identities and properties? Certes (talk) 16:05, 20 January 2019 (UTC)

Thanks for more helpful edits but that link doesn't lead where I hoped it would. I meant "Real exponents → Identities and properties", 5.5 in the TOC a subsection which has since been deleted. Certes (talk) 18:02, 20 January 2019 (UTC) updated 12:04, 21 January 2019 (UTC)

@Certes: I don't think anything from that section was salvageable as valuable encyclopedic content, and it was completely unsourced, so I've deleted it. --JBL (talk) 19:39, 20 January 2019 (UTC)

I agree. This section was added by Stemdude. His last edit was in conflict with the edit where I split the formula in the other section of the same name. So I did not notice that this section was added again. Sorry for this omission. D.Lazard (talk) 20:23, 20 January 2019 (UTC)

Thanks, yes that shouldn't have been in there at all. Possibly part of the integer section should just be general properties - but real exponentiation is an extension of the integer case. Dmcq (talk) 20:26, 20 January 2019 (UTC)

Powers via logarithms

After briefly introducing the logarithm, that section introduces $b^{x}$:

$b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\cdot \ln b}$

That definition sounds circular to someone jumping directly to that section; that is, without having read the previous section. Since $x\cdot \ln b$ is not an integer or rational number, raising e to that power doesn't seem to be easier than raising b to the xth. It doesn't help that the notation $exp()$ is never used. ale (talk) 17:00, 20 February 2019 (UTC)

I guess that hypothetical someone should have read the previous sections and not jumped straight into this one, then, huh? –Deacon Vorbis (carbon • videos) 18:02, 20 February 2019 (UTC)

I added a sentence that hopefully makes it clearer. Danstronger (talk) 04:27, 21 February 2019 (UTC)

division by zero is implied

Can somebody please elaborate a bit this sentence? "division by zero is implied" --Backinstadiums (talk) 12:29, 10 March 2019 (UTC)

... implied, because -n > 0. — Preceding unsigned comment added by Purgy Purgatorio (talk • contribs) 12:45, 10 March 2019 (UTC)

@Purgy Purgatorio: Since imply/implication are terms used in logic, I'd add a numerical example to clarify its use in this section --Backinstadiums (talk) 14:24, 10 March 2019 (UTC)

I won't, the terms are used ubiquitously in math, too. Purgy (talk) 20:33, 10 March 2019 (UTC)

"naturally defined"

I'm not sure of the meaning of "naturally defined" in "When n is a positive integer and b is not zero, b⁻ⁿ is naturally defined as ⁠1/bⁿ⁠".
Could someone please elaborate?
Is it used as synonym for "consequently", or is it natural in the sense that it's stemming from the fundamental properties of the operations?

I have added "for" between your quotation and the end of the sentence. I hope that this answer your question. D.Lazard (talk) 10:45, 28 March 2019 (UTC)

Thank you, the latest revision does make it clearer. — Preceding unsigned comment added by 212.5.158.95 (talk) 08:44, 29 March 2019 (UTC)

Many other programming languages ... provide library functions

It'd enhance the article to add a brief line about what a library function is and how it differs from syntax --Backinstadiums (talk) 09:25, 6 July 2019 (UTC)

What are "the power identities"?

The section on Rational exponents contains the sentence: "This sign ambiguity needs to be taken care of when applying the power identities." To what identities does this sentence refer?

The examples following this remark are not clearly explained in terms of previously defined concepts. For example, the term "surd" is introduced without a previous definition.

Tashiro~enwiki (talk) 02:29, 10 November 2019 (UTC)

See the section Exponentiation#Identities and properties.Dmcq (talk) 13:42, 10 November 2019 (UTC)

That section says its identities apply to integer exponents. So they are irrelevant to manipulating non-integer rational exponents. If the article wishes to state similar identities for rational exponents, it should do so explicitly.

Tashiro~enwiki (talk) 16:13, 10 November 2019 (UTC)

Real exponents

I am not surprised that my recent edits of the top sentences in this chapter were reverted as less clear, but I think that it deserves to be remarked that extending the validity of certain identities, shown to hold for integers and rationals, to the realm of reals is possible, but not straightforward on an elementary level. Purgy (talk) 10:58, 21 January 2019 (UTC)

You took a sentence with no commas and you converted it into a sentence with four commas. This creates a huge amount of work for the reader to determine what binds to what, with no benefit whatsoever. D.Lazard was 100% correct to revert.

Separately, in response to the question in your edit summary, Certes explained above what their true target was, and I responded above after removing the section that they were referring to. --JBL (talk) 13:22, 21 January 2019 (UTC)

One is supposed to read what I wrote about the scope of my surprise, and about my expectations what were remarkable: extending the validity of certain identities, shown to hold for integers and rationals, to the realm of reals is possible, but not straightforward on an elementary level (WARNING: 3 commata).

Thanks for settling Certes' concern! BTW, there are no questions intended to be answered in the summaries (even when Certes now did so), but only rhetorical questions, trying to give my stimuli for the edits. Perhaps you can imagine possible timestamps of when what information went which way, or was available at all, so that my actions are reasonable. I assume good faith in your explications. Purgy (talk) 14:21, 21 January 2019 (UTC)

Since I recall several public highbrow discussions and resulting papers about the steep difference between introducing exponentiation for integers/rationals and reals (I faintly recall some "Fundamental Theorem of Highschool Algebra", or similar, decreeing the validity of specific theorems, holding for rationals, as valid for reals, too), I dare to repeat my above question, whether it shouldn't be mentioned in the article –in appropriately styled fluency– that:

- even though it is possible to meaningfully extend the notion of exponentiation to the real numbers,

- this extension from the rationals to the reals is drastically less straightforward than the one from the integers to the rationals.

Thanks. Purgy (talk) 08:31, 25 January 2019 (UTC)

For positive real basis, this extension is straightforward, although not algebraic (it involves continuity). The difficulty is for a negative basis. I have rewritten the introduction of the section on real exponents for emphasizing this difficulty, which is not restricted, as it was suggested, to a single identity. D.Lazard (talk) 10:30, 25 January 2019 (UTC)

A minor point: The choice of language in "On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values" suggests that the thing defined exists and has properties before its definition is settled. To say "is much more difficult" suggests that such a definition had been accomplished, albeit with difficulty. The adjective "consistently" could be interpreted as 1) a definition that defines a unique mathematical object, 2) consistency among various authors or 3) A definition that defines something that obeys various identities we desire to hold.

If the current state of mathematical literature about the real number system is that some authors declined to define negative numbers raised to real powers, and others define it in different ways, that could be stated explicitly.

Tashiro~enwiki (talk) 16:42, 10 November 2019 (UTC)

Raising a negative number to a real power cannot be defined in the literature about the real number system, as the result is, in general, not a real number. The "difficulty of defining" means that a consistent definition requires to be non-elementary. The simplest definition is to define $a^{b}=e^{b\ln a}.$ If $b<0,$ this involves a complex logarithm, which is defined up to the addition of an integer multiple of $2i\pi .$ This shows that $a^{b}$ must be defined either as a multi-valued function or as the principal value of a multi-valued function. These non-elementary concepts are not avoidable.D.Lazard (talk) 17:25, 10 November 2019 (UTC)

Explain the peculiarity of notation for rational exponents

For real numbers, if $a=b$ and $x^{a}$ exists, many readers will expect that $x^{b}$ will also exist and that $x^{a}=x^{b}$. In the notation for rational exponents we can have $a=2/3,b=4/6,x=-8$ and, as I interpret the definitions in the article, $x^{a}$ exists but $x^{b}$ does not. This peculiar property of the notation for rational exponents should be explained.

The simplest explanation is that $x^{m/n}$ denotes the result of an algorithm that has 3 inputs $x,m,n$. Exponential notation such as $x^{a}$ denotes the result of an algorithm that has two inputs.

Tashiro~enwiki (talk) 02:29, 10 November 2019 (UTC)

Note the bit "in lowest terms" in "Taking a negative real number b to a rational power u/v, where u/v is in lowest terms". You have to reduce the power to lowest terms first. I wuldn't worry about this much, the power laws in general fail for this sort of stuff. It's the sort of thing people worried about a century ago but is ignored as pretty much irrelevant nowadays. Dmcq (talk) 13:50, 10 November 2019 (UTC)

The bit about "in lowest terms" is not included in the article's definition of $b^{m/n}$. It appears in subsequent remarks that describe a rule for the sign of $b^{m/n}$ in various cases. To say "where $m/n$ is in lowest terms" in that rule does not convey that $m/n$ must be reduced to lowest terms in the previous definition.

I agree that defining how a negative number is raised to a rational exponent is not treated consistently in texts. (Some wisely leave it undefined.) However, high school students in the USA might encounter exercises and exam questions that ask for the evaluation of $(-8)^{2/3}$ or even $(-8)^{4/6}$. Do rigourous texts that treat the development of real number systems define negative numbers raised to rational exponents? Or is the definition only advanced by writers of secondary school material?

Tashiro~enwiki (talk) 16:13, 10 November 2019 (UTC)

$(-8)^{2/3}$ has an unambiguous meaning as a real number. $(-8)^{4/6}$ is not an expression that anyone with a competent math teacher would ever come across in high school, in the US or otherwise. If you have reliable sources that discuss this question (or the others you have raised above) then you may propose adding material based on them to the article (or just do it), but this is not a forum for discussion of what some hypothetical source somewhere might say, or for your original research. (Probably you would like the ref desk.) --JBL (talk) 18:11, 10 November 2019 (UTC)

Most of my remarks concern defects in the language of the article. If the article intends to define $b^{m/n}$ as a procedure involving reducing $m/n$ to lowest terms, it should say so. If article gives examples purporting to explain fallacies, it should make it clear how those fallacies relate to previous material. If article intends have "the power identities" for rational exponents it should not do by referring to identities only stated for integer exponents

As to whether a competent high school teacher would ever discuss the notation $(-8)^{4/6}$ with students - why not? What is the position of the current article about this notation? Are we to understand that the notation does not define a unique number? - or is the intent of the current article that, by definition, $(-8)^{4/6}$ is to be evaluated as $(-8)^{2/3}$?

Insofar as these remarks are discussion, they are discussion about the current article, not discussion about the general topic of exponentiation or personal research.

Tashiro~enwiki (talk) 18:55, 10 November 2019 (UTC)

If you have reliable sources that discuss this question (or the others you have raised above) then you may propose adding material based on them to the article (or just do it), but this is not a forum for discussion of what some hypothetical source somewhere might say, or for your original research about what is or is not correct. --JBL (talk) 19:02, 10 November 2019 (UTC)