I gave up on my dream of being a mathematician around 1970, but I still enjoy occasionally taking a gander at the field from afar, even if I can no longer follow the details. In recent years I’ve run into something called p-adic numbers that were so unintuitive I cracked my brains trying to understand them without result (the Wikipedia page, like all their math pages, was singularly unhelpful); now, via this MetaFilter post, I have come as close to real understanding as I am likely to thanks to Derek Muller’s Veritasium video (33 min.). I normally prefer to absorb information by reading, but even a well-written piece like this one by Kelsey Houston-Edwards only made sense to me after watching the video.

But this isn’t MathHat, and I’m bringing it here because of the odd term “p-adic.” The “p” stands for prime, but why “-adic”? It was apparently first used in James Pierpont’s Lectures on the Theory of Functions of Real Variables (1905), p. 92: “When m is used as base, the numbers a are said to be expressed in an m-adic system.” But he’s just said “When m=10, we have the decimal system”; why would you go from “decimal” to “-adic”? Anybody know the history of this terminology?


  1. Simplicissimus says

    In older mathematical literature, the Greek-derived “dyadic” was not infrequently used instead of the Latin-derived “binary.” In fact, a rational number whose denominator is a power of 2 is still typically referred to as a “dyadic rational,” even if “binary” is now preferred in virtually all other contexts (e.g., “binary number system” in preference to “dyadic number system”). In any event, as a working mathematician, I have no doubt that “p-adic” is modelled on “dyadic.” And in case anyone’s wondering, “binary” and “ternary” do get generalised to “n-ary” in mathematical jargon.

  2. Trond Engen says

    Greek instead of Latin? Decadic -> m-adic.

    Edit: Or what Simplicissimus said.

  3. In any event, as a working mathematician, I have no doubt that “p-adic” is modelled on “dyadic.”

    Thanks very much, that makes perfect sense!

  4. Stu Clayton says

    Of course “tribadic” is not quite the same as “triadic”, although both are non-binary.

  5. ‘dyadic’, ‘n-adic’ and even ‘variadic’ are used commonly in programming, to talk of functions written prefix taking two, n or a variable number of arguments. OTOH operators written infix (like +, -, ×, ÷) are called ‘binary’; negation prefix – or factorial suffix ! called ‘unary’. There are ‘ternary’ operators.

    If you’re wanting to talk in general about functions/operators that take two arguments, prefer ‘dyadic’; if you’re talking about numbers-of-arguments in general ‘adicity’.

    (Not to be confused with ‘acidity’.)

  6. Yes, OED has this for -adic:

    ad suffix1 + –ic suffix, after monadic adj., dyadic adj., polyadic adj., etc.

    But that –ad suffix1 is the same as we see in more general use:

    1. Forming collective numerals, as monad n., myriad n., pentad n.; also (with reference to festivals or games occurring at a regular interval) Olympiad n., Pythiad n.
    2. … dryad n., naiad n., Pleiad n.
    3. … Iliad n., Columbiad n. 1. …
    4. Forming nouns denoting plants belonging to one of the natural orders of John Lindley (1799–1865), or (in later use) one of the families that superseded these orders (generally corresponding to scientific Latin names ending in –aceae), as magnoliad n. (family Magnoliaceae), musad n., protead n., etc. …

    The etymology:

    Of multiple origins. Partly a borrowing from Latin. Partly a borrowing from Greek.
    Etymons: Latin -ad-, -as; Greek -αδ-, -άς.

    classical Latin -ad-, -as (in e.g. hebdomas hebdomad n.) and its etymon ancient Greek -αδ-, -άς (in e.g. ἑβδομάς), suffix forming nouns. Some early borrowings are partly via French forms in -ade (in the case of decade n., this has given the usual modern form).
    Ancient Greek
    -άς is used to form nouns from both verbs and nouns; in particular it is used to form collective numerals, after δεκάς decad n. (of Indo-European origin), and feminine patronymics, as Δρυάς dryad n. (compare -id suffix3).

    And so on, in perhaps over-soigné detail.

  7. There is another noun meaning of dyadic (or older dyad) in mathematics, referring to a kind of notation that is particularly useful in mathematical physics dealing with vectors. The OED has a definition, but it looks to be quite dated and/or compiled by lexicographers who did not really understand what made dyadics useful—meaning that it emphasizes the wrong aspects of the mathematical formulation.

    The main idea of a dyadic is expressed in the first sentence of the Wikipedia article about the topic: “In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.” The point is to express a second-rank tensor as something that can be combined with other vectors and tensors via dot and cross products. The normal symbol for a dyadic has a double-headed arrow above it (where a vector would have just a right-pointing arrow), indicating that it can form separate and distinct vector products from the right and the left.

  8. jack morava says

    The secret of p-adic numbers is that if p is a prime number, \eg 3, then its powers, \eg 9 = 3^2, 27 = 3^3, 81 = 3^4 … are understood to decrease to zero as the exponent gets large. It’s like decimal numbers with infinitely many digits to the left of the period, and only finitely many (other than zero) to its right. [I’m not making this up.]

    This leads to the idea that there are different (p-adic) number systems (fields), all containing the field of rational numbers \eg 7/19… The real number system is the only one of these that’s `connected’ in a certain (both formal and intuitive) sense, the (countably many) rest of them being `totally disconnected’.

    To quote the arithmetic geometer Kazuya Kato: the real numbers, like the sun, fill our field of vision in the day; but at night the primes, like the stars, come out.

  9. This video is also an extremely good introduction to p-adics:

  10. Thanks!

  11. Yes, definitely from dyadic. Dyads and triads are also commonly used terms in network theory, including in the social sciences, for the same reason – a sociologist would talk about a dyad and not a pair or a binary relationship.

  12. I was a mathematics major at university and was introduced to p-adic numbers as part of the very first course I took, but they were really unintuitive concepts to me for a long time. Rather than having a mental picture of what they were, I had to figure out their properties and solve problems involving them by using the definitions we were given. It probably took me a whole year to get at a basic level of understanding that the Veritasium video explains in about 30 minutes (this wasn’t the fault of the instructor at university was a fantastic communicator; there simply wasn’t time to devote a whole lecture to p-adics as there were so many concepts to introduce to us). I wish this video had been available back then.

  13. John Cowan says

    Other terms: polyadic ‘accepting at least two arguments’, polyvariadic ‘accepting at least one argument’, niladic or nullary ‘accepting exactly zero arguments’.

  14. Gavin Wraith says

    I think you have to be careful in extending etymology to mathematical terminology. Beware of the whimsy. Because the terrain of mathematics constantly expands, neonymy is a permanent necessity. Once bad neonymy has taken root it is hard to change.

  15. jack morava says

    In good conscience I can’t leave this

    It’s like decimal numbers with infinitely many digits to the left of the period, and only finitely many (other than zero) to its right…

    as it stands, without emphasizing that

    \vdash : [I am not making this up.] :

    Whether you’re adding or multiplying, you carry on carrying to the left and sort of quit when you get tired, \cf also John H Conway, on Numbers and Games \qv.

    This, in connection with the Chinese Remainder theorem, can be extremely useful in quite gritty applied computational problems but IANANTheorist. The Ramans tell me they use 3 for routine work and 691 for serious stuff.

  16. David Marjanović says

    What does \vdash mean?

  17. It’s doubtless part of his LaTeX dialect.

  18. It must actually be a custom macro (or from a more obscure package), since the usual thing is “\Vdash”—and LaTeX commands are case sensitive.

    However, if you want to know what an unknown LaTeX command produces, it’s almost always findable by searching Google for “\[whatever] LaTeX” (not case sensitive).

  19. Stu Clayton says

    Beware of the whimsy.

    Yes, it’s -adictive.

  20. David Marjanović says

    So it’s this? I didn’t know that symbol at all…

  21. jack morava says

    @ DM,

    \vdash prints as something called a turnstile, currently used mostly to indicate adjoint functors, but it has an honorable etymology going back to the Principia Mathematica of Whitehead and Russell, where it means `it is asserted that’ or something similar; it’s usually prefixed to a proposition.

    [? Recall the mythopoetic battle between Whitehead and Wittgenstein the poker-holder, over the definition of a logical proposition, resolved according to story when it was realized that for Whitehead a proposition was, well, a proposition, but for Wittgenstein it was something written on a little piece of paper.]

  22. David Marjanović says

    Recall the mythopoetic battle between Whitehead and Wittgenstein

    Sorry, I’m completely unfamiliar with all of this. Your comment doubled my knowledge of Whitehead.

  23. jack morava says

    I realized post-sending that my mythopoetic memory had conflated Whitehead with Russell and then after checking


    that never mind it was Popper if it was anyone at all.

  24. Stu Clayton says

    Here’s a little more: Whitehead wrote a conundrous book called Process and Reality. All that many people understood in it, including yours truly, was:

    The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato.

    That’s a fine example of thoughtful exaggeration.

  25. jack morava says

    I’ve seen arguments that all of rock and roll is the legacy of Chuck Berry…

  26. it was Popper if it was anyone at all.

    The at least equally known confrontation between poker-wielding Wittgenstein and another ultra-intellectual Mathematician is with Alan Turing. The way I heard it they had a blazing row about something inconsequential, and never spoke again. (I’m failing to find any reference to support that memory.)

    I hope that Oxbridge colleges (and indeed the LSE) have withdrawn all pokers.

  27. something called a turnstile, … has an honorable etymology going back to the Principia Mathematica of Whitehead and Russell, where it means `it is asserted that’ or something similar; it’s usually prefixed to a proposition.

    Not it doesn’t mean ‘it is asserted that’. It means: it is derivable from given axioms using only rules of inference. Sometimes you see a formula written to the left of the turnstile, meaning the r.h. formula can be derived from the l.h. Or a comma-list of formulas to the left meaning derive from those taken together. Or a meta-symbol to the left denoting ‘all axioms and all formulae derivable from them so far’, plus this extra formula to give the derivation.

    You can derive palpable untruths from untrue axioms. (Indeed that’s a useful test to make sure you’re not mis-deriving where the conclusion is true but the axioms turn out insufficient to derive it.) Then the turnstile doesn’t ‘assert’ anything more than the mechanical process of derivation.

    And it’s the weakness of those forms of derivation that give rise to Gödel’s theorem.

    The turnstile also gets used in programming languages meta-theory to mean this program ‘can’t go wrong’ — in a unhelpfully technical sense of ‘go wrong’.

  28. jack morava says

    As an indigenous LaTeX writer (it’s not a spoken language) I assumed the turnstile symbol was a capital T for True turned sideways. Humpty Dumpty and I use it rhetorically when needed to mean `I assert (something)’.

  29. John Cowan says

    Your comment doubled my knowledge of Whitehead.

    Allow me to triple it: “A Unitarian is someone who believes in at most one God.”

  30. David Marjanović says


  31. p-адические always impressed me by inconveinence of the term.
    The wanted a name for them and they wanted it to be an adjective, so they invented a suffix -adic.

    But speakers don’t recognise it, and it is attached to… an ideogram or what is p?
    Note, in Russian it is still Latin p.

  32. Stu Clayton says

    In math contexts, “p” is often used to represent a prime (number) variable. “i” is used to represent an integer (number) variable

    Curiously, ” ” is used to represent a blank (space) constant, not “b”. It’s all rather complex, where “i” does not represent an integer variable.

  33. John Cowan says

    However, “␢”, named BLANK SYMBOL, can be used to represent a blank space in a visible way. It dates back to the keypunch era, when the semantics of “␢” and “␢␢” were often distinct. A more modern synonym is “␣”, named OPEN BOX.

  34. Lars Mathiesen (he/him/his) says

    Mathematicians don’t deal in variables; a symbol means what it means, and in the next paragraph it’s another symbol with the same name and a different meaning. (In a programming language, they would be constants in different scopes).

    In this case, another p denoting a prime number that may or may not be the same as the previous p. (The conflation of symbols and the mathematical objects that they stand for is endemic, but usually not a problem).

  35. Yes, it is p in Russian mathematical texts as well. What I meant is that the term intends to use the outer (syntactic) interface of the suffix -адический /-adic, but horrible things are happening at its inner/left/earlier (morphonological) interface/surface:(

    (Something like “time-invariant” has something quite different from a suffix as its right part)

  36. PlasticPaddy says

    Not to mention that I thought your numbers were named for someone callled Radichevskii (or perhaps Paddy Chaevskii).

  37. P-addic.

  38. @PlasticPrime, rather the latter. In mathematical contexts I tend to read isolated letters that seem designate something as Latin: they seldom Cyrillic in formulas:)

    But much of what I said relates to orthography. I think Russians just don’t say peh ʔadi… as the spelling suggests. Unfortunately I keep reading it so each time I see it written:(

    But there is still no identifiable stem to which the suffix is attached… Unless it is paddy-cheskij.

    Also in Russian “pee” is the name of the Greek letter pi.
    п is “peh”, with a hard /p/.
    Neither “padi…” nor even “pe.adi…” sounds similar to diada and triada.

    I don’t know if this parallelism is maintained in English (/ˈdaɪ.æd/ but /piː/).

  39. It is interesting that even in mathematical equations, there can turn out to be quite a bit of linguistic-like redundancy. In at least one of my published papers, I used the letter e two mean two different things, even within the same equations. One was the basis of the natural logarithm, and the other was the charge of the electron. However, there was no realistic possibility for confusion between the two, based on where they appeared in the equations involved. I don’t think anybody even noticed the ambiguity, except for the copyeditor who edited the LaTeX to match the journal style, who changed one of the types of e to be printed upright, rather than in italic like most mathematical quantities. Of course, the editor did this entirely on their own, without needing to consult me, which just indicates that it was already transparent which type of e each instance was.

    On the other hand, I have seen multiple uses of i trip people up once or twice. Most of the time, one is unlikely to confuse the square root of –1 with an integer-valued index, particularly when the index i only appears in subscripts (or superscripts), like in bᵢ. However, sometimes the numerical value of the index i actually comes into things (for example, in the usual metric used to show that a countable product of metrizable spaces is itself metrizable). That same journal also tried to print the imaginary unit in upright type, to distinguish the two meanings. However, I just try to avoid the confusion completely; I follow J. J. Sakurai and start my integer indices from the letter j. (Of course, electrical engineers use j for the square root of –1, to avoid confusion with the current i. This tends to get mocked by physicists, who figured out that we could just use a majuscule letter for the current instead).

  40. Keith Ivey says

    Dyad and triad have /aɪ/, but that has nothing to do with the parallelism. After all tetrad, pentad, etc, don’t have it.

  41. Parallelism exists when native speakers can (does not matter how) recognise dyad, triad…. based on the pronunciation of p-adic. In Russian I think we can’t, I can’t. But maybe the association works in the language (I don’t know which one) where the term was coined, or in English.

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