Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word Zahl(en) 'number(s)'. What are some more examples?
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4$\begingroup$ i'm not sure if this is a relevant example, but this TED talk explains (more or less accurately, apparently) why $x$ prevailed for the unknown in equations ted.com/talks/terry_moore_why_is_x_the_unknown.html $\endgroup$– citedcorpseAug 13, 2013 at 13:23
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2$\begingroup$ @MJD that claim is made in a footnote (typeset humorously in fraktur) in paolo aluffi's algebra book $\endgroup$– citedcorpseAug 13, 2013 at 13:47
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2$\begingroup$ Why does $p$ stand for momentum? $\endgroup$– Spine FeastAug 13, 2013 at 13:49
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3$\begingroup$ a latin scholar will explain better, but the $p$ is from "impetus", a modified "petere" $\endgroup$– citedcorpseAug 13, 2013 at 13:52
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1$\begingroup$ @MJD : math.duke.edu//education/webfeats/Slope/Slopederiv.html $\endgroup$– John GowersAug 13, 2013 at 15:14
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17 Answers
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In topology the letter $F$ is commonly used to denote a closed set, from French fermé 'closed [set]'. The common use of $K$ to denote a compact set is probably from German kompakt, as in kompakte Menge 'compact set' and kompakter Raum 'compact space'. The common use of $k$ to denote an arbitrary field is probably from German Körper 'field'. The common use of $G$ for an open set is probably from German Gebiet 'region', though as a mathematical term it now means 'non-empty, connected, open set'. The notation $G_\delta$-set for the intersection of countably many open sets combines this $G$ with $\delta$ for German Durchschnitt 'intersection'. Presumably $F_\sigma$-set for the union of countably many closed sets is from the $F$ above and $\sigma$ for French somme 'sum'. The $T$ in the names of the separation axioms $T_1,T_2$, etc. is from German Trennungsaxiom 'separation axiom'.
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$\begingroup$ And in algebra $K$ is often used to denote a field from the German körper if I'm not mistaken. $\endgroup$ Aug 13, 2013 at 13:12
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1$\begingroup$ And $G$ as in $G_\delta$ sets is probably from the German "Gebiet" (domain). I've also heard the $\delta$ being short for "Durchsnitt". I've always found it funny that $F_\sigma$ and $G_\delta$ are from French and German respectively. $\endgroup$– mrfAug 13, 2013 at 13:28
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2$\begingroup$ @MTurgeon: Very unlikely: one speaks of eine offene Menge, not eine geöffnete Menge. $\endgroup$ Aug 13, 2013 at 13:41
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1$\begingroup$ @SamiBenRomdhane: I suspect that most English speakers who use $O$ for an open set do so on account of open and aren't even aware of ouvert. (I prefer not to use $O$ at all, for anything, owing to its resemblance to $0$.) $\endgroup$ Aug 13, 2013 at 13:43
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2$\begingroup$ The use of $U$ for open sets probably comes from the German Umgebung "neighbourhood". $\endgroup$ Aug 13, 2013 at 14:53
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Eigen (as in the eigen vectors of a matrix) is Dutch/German for "own".
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A function is often called càdlàg if it is right-continuous and admits left limits. This term is from the french continue à droite, limite à gauche.
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3$\begingroup$ See also càglàd (obvious) and càllàl (continue à l'une, limite à l'autre, meaning continuous on one side, limit on the other side, and which side is continuous can depend on the point). $\endgroup$– jwgAug 13, 2013 at 14:49
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The Klein $V$-group is the four-element group with generators $a$ and $b$ and $a^2 = b^2 = (ab)^2 = 1$. The $V$ is for vierergruppe = "four-group".
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$\ln()$ for "logarithmus naturalis"?
My advisor also told me that the "socle of a ring" makes a little more sense when you know that "socle" is an architecture term for the support underneath a column or pedastal, and so the socle of a ring acts as a kind of "support for the ring." In some languages, the word for "pedestal" is something like "socle," so the meaning is less hidden there.
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When I put "socle" into google translate, it autodetects it as "plinth" which is a relatively better-known word in English. It turns into "zócalo" in Spanish, sòcol in Catalan, Sockel in German, zoccolo in Italian, cokół in Polish, soco in Romanian, and 虹晶 in Mandarin.
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$\begingroup$ I always wondered what socle was supposed to mean! $\endgroup$– mdpAug 13, 2013 at 13:11
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2$\begingroup$ Not quite true: socle has that meaning in architecture, but the Latin base is actually socculus, a diminutive of soccus 'a slipper, a sock'. $\endgroup$ Aug 13, 2013 at 13:16
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1$\begingroup$ I made changes to pacify those with a deep knowledge of Latin roots. Apparently Latin roots are completely useless for understanding words if you go back far enough. $\endgroup$– rschwiebAug 13, 2013 at 19:50
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1$\begingroup$ @JyrkiLahtonen Yeah it does mean that in English, but English speakers hardly ever use it unless they know about architecture. $\endgroup$– rschwiebAug 14, 2013 at 19:28
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The reason the "Klein bottle" is called a bottle has its origin in something of a German pun on Fläche/Flasche; see here
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$\begingroup$ this is also why when one wants to talk about two surfaces, one uses $S$ and $F$ $\endgroup$ Aug 13, 2013 at 13:48
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$\begingroup$ Similarly, the Witch of Agnesi takes its English name from a pun in Italian, at least according to Wikipedia. $\endgroup$ Aug 14, 2013 at 1:39
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$\begingroup$ These answers should be posted as a separate answers to give them proper visibility. Not everybody is aware of these facts. $\endgroup$ Aug 14, 2013 at 7:39
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$\begingroup$ Well, I thought about it, but it isn't really a piece of notation or a mnemonic as such, more an interesting tidbit (although I don't want to imply disapproval of your choice to post yours as an answer). $\endgroup$ Aug 14, 2013 at 21:49
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1$\begingroup$ To a German speaker, Fläche and Flasche are quite different words. According to your link, it may be that 'Fläche' was mistranslated into English (by an English speaker) as 'bottle', and then back-translated into German as 'Flasche' -- which is undeniably a snappier name for it. $\endgroup$– TonyKAug 1, 2014 at 20:28
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Łukasiewicz notation for logic represents $\land \lor \leftrightarrow$ with the letters $K A E$ respectively, so that for example $r\lor(p\land q)$ is $ArKpq$. $K A E$ are the initials of the Polish words koniunkcja, alternatywa, ekwiwalencja.
I don't know why Łukasiewicz used $C$ to represent material implication.
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$\begingroup$ Are you sure this is the reason? I am not a Polish speaker, but the examples I have found of czyli do not seem very much like material implication. $\endgroup$– MJDAug 13, 2013 at 15:09
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$\begingroup$ It's synonymous with 'therefore' and definitely can be used in this context. Whether this was his reason for $C$ I can't say, but it certainly is a (perhaps somewhat archaic) way of expressing implication. $\endgroup$ Aug 13, 2013 at 15:58
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$\begingroup$ I am a Polish speaker, and it certainly seems reasonable. Here's a link to the relevant Wiktionary article: en.wiktionary.org/wiki/czyli $\endgroup$ Aug 13, 2013 at 18:21
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3$\begingroup$ I had read that when I posted my comment above. I also consulted with Maciej Cegłowski, who though not a linguist is a native Polish speaker, and who agreed with me that it seems unlikely. My own suspicion is that $C$ is short for "conditional". $\endgroup$– MJDAug 14, 2013 at 19:08
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Gabriel introduced the notation $\text{Sex}(\mathcal A,\mathcal B)$ to denote the category of left exact functors from $\mathcal A$ to $\mathcal B$. This because the Latin word for left (which is sinister) starts with an S.
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2$\begingroup$ Is the category of right exact functors called $\text{Dex}(\mathcal A,\mathcal B)$ from dexter? $\endgroup$ Aug 13, 2013 at 18:42
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In homology one has a sequence of "differentials". Their images are usually denoted $B(X)$, apparently from the german word for "images", and their kernels $Z(X)$ from the german word for "cycles".
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3$\begingroup$ I always thought $B$ stood for "boundary". $\endgroup$– Zhen LinAug 13, 2013 at 13:20
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1$\begingroup$ @ZhenLin i think that's more of an ad-hoc thing that we do when working in english, to try to make more sense of it. one could then say we use $Z$ to stand for "zeroes", which is actually kind of pleasant $\endgroup$ Aug 13, 2013 at 13:22
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The etymology of the $\sin$ function has a colorful history - it comes from sinus, the latin word for... well, bosom. This was due to a mistranslation from Arabic text in the 12th century: The word jaib means bosom, and since Arabic is written without short vowels, it was written essentially as jb. But jb was also the spelling of jiba, which was a transliteration of the Sanskrit word for chord (the mathematical chord, ie a line passing through a circle, half the length of which is the sine of the angle from the center of the cicle).
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8$\begingroup$ According to my Latin dictionary, sinus also means "curve". Which might be a simpler explanation than the mistranslation one. $\endgroup$– detlyAug 13, 2013 at 22:07
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$\begingroup$ And I doubt it's a mistranslation, the relation of meanings ("curved things") is apparent. See also this figure: 1.bp.blogspot.com/-DniqyzYnS0Y/Tf5Xx7mpoLI/AAAAAAAAANo/… $\endgroup$– leonbloyAug 16, 2013 at 14:02
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$\begingroup$ Interesting, I wasn't aware of this. I'd be interested to see if there's more reasoning behind scholars believing that this was a mistranslation, but I don't have access to the sources in the wikipedia article: en.wikipedia.org/wiki/Sine#Etymology $\endgroup$ Aug 16, 2013 at 15:40
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The logical-or symbol $\lor$ is a stylized letter ‘V’, the first letter of the Latin word vel.
(The $\land$ symbol arose later, derived by analogy from $\lor$.)
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$\begingroup$ I always remember "and" because it looks like an "A". And "or" because it's an upside-down "and". $\endgroup$– Joe Z.Aug 13, 2013 at 17:09
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QED comes from the latin quod erat demonstrandum
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$\begingroup$ In french, it's CQFD for "ce qu'il faut démontrer" == "Which had to be demonstrated". Just for your curiosity. $\endgroup$– FabinoutAug 14, 2013 at 8:03
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$\begingroup$ @Fabinout Being french, I know that, although I'd say "Ce qu'il fallait démontrer". ;) $\endgroup$ Aug 14, 2013 at 14:31
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The identity element of a group is $e$ for einheit, the German word for "identity".
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$\begingroup$ I wondered about this just yesterday. $\endgroup$ Jul 28, 2016 at 12:23
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In old Polish textbooks for secondary school complex numbers were denoted $\bf Z$, as zespolone and integers — $\bf C$, as całkowite.
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The center of a group $G$ is denoted $Z(G)$. The $Z$ is for “Zentrum”, which is the German word for ‘center’.
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Most of the good examples have been mentioned already.
- $+$ probably comes from Latin et, "and"
- $\tau$ is sometimes used for the golden ratio, from Greek tomos, "section" (same root as atom).
- $\sinh$ from Latin sinus hyperbolus (and so on)
- $\in$ from Latin est.
- $M$ for an arbitrary set from German Menge, crowd.
- $\mathrm E[x]$ for expectation is from German Erwartung or French éspérance.
- $\aleph$ (the cardinality of the countable numbers) is maybe from Hebrew einsof, infinity, or maybe just because it's the first letter of the Hebrew abjad.
- $m$ for slope may be from Latin modulus, "quantity", or French monter, "to climb".
- $e$ may be from German Einheit, "unity", because $\ln e = 1$.
- $c$ for the speed of light from Latin celeritas, "swiftness".
If you want to be pedantic, you could point out that there are a large number of mnemonics from other languages which also happen to work in English. For example
- $\mathbb{Q}$, $\mathbb{R}$ from German Quotient, Reel
- $\pi$ from Greek perimitron or from Latin peripheriam
- $I$ from French intensite de courant
- $\%$ comes from Italian per cento, "per hundred"
- $x$ is used for variables because it's uncommon in French
You might also count all symbols for the SI prefixes (atto-, yotta-, etc.) except the ones which have become productive in English (micro-, mega-, etc.) But that would be far too pedantic, even for me
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$\begingroup$ As long as we're here, we might as well include the actual reason that $x$ is used for variables. $\endgroup$ Mar 13, 2016 at 0:29
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In Polya's enumeration theorem the letter $Z(G)$ which is used for the cycle index of the permutation group $G$ originates with the German word Zyklenzeiger, I think.