Polite Numbers.

March 17, 2023 by 37 Comments

Wikipedia says:

In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.

But it says nothing about why such numbers have the odd name “polite,” and that usage is not in the OED entry (updated September 2006). Anybody know?

Comments

  1. J.W. Brewer says
    March 17, 2023 at 11:59 am

    I can’t immediately find this usage in an English-language text in the google books corpus older than 1999. (James J. Tattersall’s “Elementary number theory in nine chapters”) But perhaps some hattic with specialized knowledge or more time to devote to googling will trace it to a deeper origin. I thought I had found a 1970 Dutch book where this English terminology was calquing Dutch terminology, but that was a bad-metadata situation and the book was in fact from the present century. But I wonder if some calque/loanword origin is possible. Number theory seems like the sort of field in which important new work was perhaps being published in languages other than English within living memory.

  2. Jen in Edinburgh says
    March 17, 2023 at 12:50 pm

    There are also friendly numbers and amicable numbers and sociable numbers (although don’t ask me to explain them), so maybe the polite numbers are less friendly than those, or maybe they were just running out of suitable words.

  3. Y says
    March 17, 2023 at 12:56 pm

    Earliest lead I can find is from a 1991 paper: “In an edition of the Open University newspaper, Sesame (no. 124, December 1988), there appeared the following problem. A positive integer is said to be polite if it can be represented as the sum of two or more consecutive positive integers.” I haven’t found that issue online.

  4. Stu Clayton says
    March 17, 2023 at 12:59 pm

    Also amenable numbers.

    It’s not polite to hide your backgound, as “polite number” is doing. I doubt it was disovered in a handbag.

  5. David L says
    March 17, 2023 at 1:10 pm

    Because the numbers that make them up are politely standing in line?

    I’m just guessing — I find lots of references to them but no explanation.

    I also found ‘polite vicar number’ on google but it turned out to be the phone number of a pub in England called “The Polite Vicar.”

  6. languagehat says
    March 17, 2023 at 1:10 pm

    There are also friendly numbers and amicable numbers and sociable numbers

    Ah, so it’s part of a series (like streets named after composers). That moves the problem to a higher level.

  7. rosie says
    March 17, 2023 at 1:25 pm

    It quite often happens in a mathematical argument that you need to refer to a concept more than once, so it would be convenient to refer to it by name. If it doesn’t already have a name, then you have to make one up. If you’re writing problems for students to solve, it helps them, too. It looks as if that’s what happened with the problem cited by Y.

    In the case of the term “polite numbers”, it’s managed to catch on in a small way — it has a Wikipedia page. I find even this to be surprising: it is easy to characterise which integers are polite and which are not. And in a mathematical context it’s better to refer to a property using terms mathematicians already well know than using one with little currency.

  8. C Baker says
    March 17, 2023 at 1:37 pm

    With no evidence, I still assume it’s for the same reason that some quarks are called “charm” or “strange”. The people who named them had that sense of humor.

  9. languagehat says
    March 17, 2023 at 1:46 pm

    Yes, that’s a plausible comparison.

  10. Martin says
    March 17, 2023 at 2:15 pm

    “Polite” is derived from Latin politus, meaning “refined” or “elegant.” So, some mathematician decided that numbers were refined or elegant if they could be expressed as the sum of two or more consecutive positive integers. And indeed that does seem like an elegant property. Perhaps it is based on the little staircase model that illustrates the property on the Wikipedia page.

  11. Frans says
    March 17, 2023 at 3:10 pm

    The first reference on Wikipedia is to https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/772-how-polite-is-x/BE0B086FA725A3BE26315DA67B094CF1 which mentions the Open University newspaper Sesame (No. 124, Dec 1988) which I can’t immediately locate.

  12. Brett says
    March 17, 2023 at 3:50 pm

    xkcd on an unnecessary generalization of the friendly numbers.

  13. J.W. Brewer says
    March 17, 2023 at 4:43 pm

    JenInEd’s examples apparently did not exhaust the genre. See, e.g., https://en.wikipedia.org/wiki/Betrothed_numbers (a/k/a “quasi-amicable”).

  14. mollymooly says
    March 17, 2023 at 4:43 pm

    As is often the case, the talk page of the Wikipedia article is informative, although in this case it does not answer Hat’s question. It seems the “polite number” concept is used in school-level mathematics education rather than in higher-level number theory.

  15. languagehat says
    March 17, 2023 at 5:23 pm

    Ah, that too is an interesting fact.

  16. jack morava says
    March 17, 2023 at 7:51 pm

    As David L says

    [IICC? I too am just guessing, have no prior knowledge] an integral number is polite if, like \eg

    0 + 2 + 7 + 11 + 17 + 38 = 75

    it can be presented as the sum of a sequence of strictly increasing integers. A lecture in hierarchy for us all.

  17. jack morava says
    March 17, 2023 at 8:08 pm

    I forgot to mention the numbers of

    https://en.wikipedia.org/wiki/Reo_Fortune

    which (I am told?) he discovered while his wife M Mead was knocking boots with G Bateson in a tent in PNG.

  18. J.W. Brewer says
    March 17, 2023 at 8:28 pm

    jack morava is missing the “consecutive” constraint. 75 can be expressed as 37 + 38 (or 24 + 25 + 26, or 13 + 14 + 15 + 16 + 17). But any odd number >1 (1 is impolite, but that’s because it’s 2 to the zeroth power), can be expressed as 2n+1, so the shortest “polite” sequence is always the two-integer combo of n + (n+1). Even numbers (that are not integral powers of 2) will take necessarily more than two consecutive integers. E.g., 30 is 9 + 10 + 11. Or 6 + 7 + 8 + 9. Or 4 + 5 + 6 + 7 + 8. But it’s probably easiest to stop at the fewest-necessary-integers option. Even numbers not divisible by 3 will typically require at least four consecutive integers, e.g. 38 is 8 + 9 + 10 + 11, while 76 may not have a shorter workable sequence than 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13. I *think* the general trick is to figure out how to express a given even number as Tn* + xn for the lowest available value of n, which will let you work out the n consecutive positive integers necessary, i.e. (1 + x) and the (n-1) following positive integers.

    *Imagine n as a subscript to the T (for “triangular”).

  19. jack moravaja says
    March 17, 2023 at 8:35 pm

    I stand corrected, thanks JW.

  20. Y says
    March 17, 2023 at 8:37 pm

    A timid number is one that may be expressed as the sum of some available numbers.
    An overweening number is one that any other number can be expressed as the sum of.

  21. John Cowan says
    March 18, 2023 at 12:41 am

    The WP article on Fortunate numbers starts by saying that they should not be confused with lucky numbers. The Lucky numbers are “one”, “four”, and “six hundred and something”.

  22. Hans says
    March 18, 2023 at 3:44 am

    The Lucky numbers are “one”, “four”, and “six hundred and something”.
    This lady is only in partial agreement on this matter.

  23. PlasticPaddy says
    March 18, 2023 at 5:40 am

    @jwb
    Your examples show at least the worst case: An even number M requires an n of at most the smallest power of 2 for which 2M divided by 2 raised to that power is odd. So 38 requires an n of at most 4, and 76 requires an n of at most 8.

  24. languagehat says
    March 18, 2023 at 9:06 am

    The Lucky numbers are “one”, “four”, and “six hundred and something”.

    You’ve omitted nine and eighteen. I’m sending your paper back for revision.

  25. Stu Clayton says
    March 18, 2023 at 10:52 am

    @JWB: that Lene Lovich lady sings Lucky Number in a manner that reminds me strongly of Nina Hagen (voice briefly breaking an octave upwards and then back down, for example). According to the WiPe, they even sang together in the 80s. Was either of them first (to popular knowledge) with that wigged-out delivery, or were there predecessoresses ?

  26. Craig says
    March 18, 2023 at 12:38 pm

    This seems like an appropriate place to mention The On-Line Encyclopedia of Integer Sequences®. https://oeis.org/

  27. David W says
    March 18, 2023 at 12:41 pm

    Predecessrices?

  28. Stu Clayton says
    March 18, 2023 at 3:18 pm

    Predecessrices?

    I concede defeat.

  29. Brett says
    March 19, 2023 at 1:59 am

    There are obviously a whole slew of descriptors for various numbers and types of numbers. The ancient Greeks knew of the perfect numbers and amicable pairs, and those numbers sometimes showed up in classical numerology. Some properties are more interesting that others. For example, whether a number is “narcissistic” depends on the base in which it is represented. (The smallest nontrivial narcissistic number in base 10 is 153, which is also the number of large fish caught by Simon Peter in John 21:11. This might or might not have been a number that was already deemed significant around the ancient Mediterranean by that time. It was used by Archimedes in his rational approximation for the area of the vesica piscis* lens; however, I am unclear on whether that association with something fish-related is older or newer than the gospel story.)

    Discussing the naming of possible interesting properties of numbers (or sets, or functions) is a commonplace occurrence on the Math Overflow site. People will post questions starting like: “Suppose we call an integer flibbertigibbety if it obeys….” Then one or more other mathematicians will answer, pointing out whether the proposed definition is a) pathological; b) equivalent to some other existing notion; c) new and potentially interesting; or d) some some more subtle combination of these.

    * I don’t know what the name was for this figure in Greek, or if it even had a specific one.

  30. Lars Mathiesen (he/him/his) says
    March 19, 2023 at 6:01 am

    What would actually happen to predeced- + -trix in Latin? The change -tt- > -ss- is expected between vowels in predecessus and predecessor, but in front of /r/?

    Let me adduce tonstrix to tondeo and suggest predecestrix.

  31. Craig says
    March 19, 2023 at 7:21 am

    Intercessor corresponds to intercestrix per the Wiktionary entry, so one might infer predecestrix(-ices).

    French substitutes -eur with -rice, so prédécessrice(s) is the logical form there.

  32. J.W. Brewer says
    March 19, 2023 at 8:57 am

    @Stu: The answer to your question is that I don’t know but am flattered that you thought I might know. Which is at least probably some evidence that there is not any standard consensus view among weird-rock historians affirming the existence or identity of a predecestrix of that vocal style. I note that Ms. Hagen did a German-lyrics adaptation of “Lucky Number” (retitled Wir Leben Immer … Noch) on the Nina Hagen Band’s _Unbehagen_ album, released in ’79 a year after Ms. Lovich’s original came out in the U.K., so awareness/influence in that direction is pretty obvious.

    Whether Ms. Lovich for her part had heard anything by Ms. Hagen when she did her recording is much less clear to me. She recorded it at least a few months prior to the release of the Nina Hagen Band’s first “Western” album at the end of ’78, and I’m not sure if even the hipster/weirdo music community in London was aware of Hagen’s pre-defection/explusion DDR-era recordings. (And I’m likewise unsure if the first BRD-era Hagen album attracted much initial attention in the Anglophone world, although it was released near-contemporaneously in the UK by the local affiliate of CBS Schallplatten GmbH.)

  33. Hans says
    March 19, 2023 at 11:26 am

    According to German WP, Hagen lived in the UK and was active in the British Punk scene for a time in 1976, after leaving the GDR and before settling in Western Germany. So Lovich and her may have met there.

  34. Athel Cornish-Bowden says
    March 19, 2023 at 4:03 pm

    As this is LanguageHat, not MathematicsHat, I want to ask a language question. In the Wikpedia article, ref. 5 includes

    Resolucion de Problemas en los Albores del Siglo XXI: Una vision Internacional desde Multiples Perspectivas y Niveles Educativos

    Is “Albores” a typo for “Arboles”, perhaps influenced by Italian, or is that a legitimate variant?

    No doubt the Dictionary of the RAE can tell me, but I’m too lazy to check that right now?

  35. Eduardo G says
    March 19, 2023 at 4:19 pm

    albor in Spanish means dawn, it has nothing to do with arbol; “en los Albores del Siglo XXI” can be translated as “at the beginning of the 21st century” or “early in the 21st century”

  36. languagehat says
    March 19, 2023 at 4:19 pm

    Is “Albores” a typo for “Arboles”

    No, albor is a fancy word for ‘dawn’ or ‘beginnings’: los albores de la civilización.

    (EDIT: Pipped by Eduardo.)

  37. Pancho says
    March 19, 2023 at 4:24 pm

    Is “Albores” a typo for “Arboles”, perhaps influenced by Italian, or is that a legitimate variant?

    It basically means at the dawn or the start of the 21st Century.

    https://dle.rae.es/albor

    albor
    Del lat. tardío albor, -ōris.
    1. m. albura (‖ blancura perfecta).

    2. m. Luz del alba. U. m. en pl. con el mismo significado que en sing.

    3. m. Comienzo o principio de algo. U. m. en pl. con el mismo significado que en sing.

    4. m. Infancia o juventud. U. m. en pl. con el mismo significado que en sing.

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