# Origins of Mathematical Symbols/Names

I'm not sure if this has been asked. I'll explain the question by an example.

Fields are often denoted by the letter k, which comes from the German word Körper, meaning body (like corpse, corporeal).

Most mathematical symbols relate directly or indirectly to the English names, so what other exceptions are there?

(Yes, this is inspired by the other post about languages in math)

• On a somewhat unrelated note, I only learned recently that the words manifold and variety are synonymous. The former is German, and the latter is French (I might be wrong). The French would call differential manifold "variété différentielle," while algebraic variety is just "variété algébrique." Dec 9, 2009 at 2:49
• Let me add to this question: what is the origin of the word "ring"? Dec 9, 2009 at 2:58
• Interesting! I had always assumed that K stood for something like "Kampen"... which, as I just found out, is not actually a German word. :P Dec 11, 2009 at 19:37
• In Spanish is the same as in French: we say "variedad" either for a manifold or an (algebraic, "algebraica") variety. Dec 12, 2009 at 0:13
• Some cognate of (English word) "variety" is "manifold" in all the Romance languages. "Manifold" comes from Germann, I believe actually from Riemann. Dec 18, 2009 at 19:43

$\mathbb{Z}$ comes from the German "Zahlen" which means "numbers".

Center of a group is denoted Z, from German word Zentrum

This one is pretty well-known: the notation $e$ for the identity of a group comes from the German word Einheit, meaning unit.

I'd be willing to bet that the notation $G$ for a group also comes from German... but we don't notice, because the German word for group is Gruppe!

Here's a fun one: the notation $Z$ for a topological quantum field theory comes indirectly from the notation $Z$ for a partition function in statistical mechanics, which comes from the German word Zustandssumme, meaning state sum. I said "indirectly" because partition function in quantum field theory isn't a statistical-mechanical partition function... it just looks like one after you Wick rotate! (Then again, maybe there's a deeper sense in which the QFT partition function really is a statistical-mechanical partition function. Does anybody know?)

• Who was the first to use $Z$ for a partition function? Jan 6, 2010 at 3:10
• Mar 19, 2018 at 3:58

As an undergraduate, I was told that $V$ is often used to denote a neighborhood because the French translation is voisinage. Anyone else hear this?

• Yes, I think this is true. And U is for 'Umgebung', the German word for neighborhood. Dec 9, 2009 at 10:21
• And U and V are consecutive letters, making them especially convenient for using to represent two neighborhoods that occur together. One often sees, say, a function f: U -> V where U and V are open subsets of A and B, respectively. Dec 9, 2009 at 13:47
• @Arminius: I hope I don't double post. Was having trouble posting earlier, but I did not know that. Thanks. Dec 10, 2009 at 0:22

I've been told that the notation $\mathcal{O}$ for the structure sheaf of a scheme/variety/whatever comes from the Italian word "olomorfo/olomorfa" for "holomorphic".

I should note that I don't have any evidence for this claim beyond "I heard it somewhere from somebody". It would be great if anybody could corroborate this.

• Great! I always wondered where this comes from. Dec 9, 2009 at 10:17
• This is new for me!! cool!! Dec 11, 2009 at 4:13
• finally I know why this letter is used :) Jan 22, 2010 at 21:40
• I used to believe that the O came as the geometrical approximation of a physical ring, from the local ring idea. May 16, 2010 at 8:54
• Um, can someone point to an actual reference justifying this? Why would a letter from an Italian word be used here when the topic was systematically developed by the French? It feels like an artificial etymology (and Kevin is conceding he has no source on this, but people's willingness to believe it surprises me). I once heard O was in honor of Oka. Hmm... Perhaps O is related to the very long (since 1870s) tradition of using O for rings in number theory, which was based on Dedekind's term for a ring: order, or rather Ordnung in German. He wrote fraktur o a lot in his work. Aug 13, 2010 at 4:59

The notation $\mathcal{F}$ for sheaves comes from the French word "faisceau" meaning "bundle".

Also "gerbe" means "sheaf" in French.

• It's in fact the same etymology for "fascism". Go look up your Roman history for why "bundles" have anything to do with government. Dec 9, 2009 at 6:45
• Indeed wikipedia has a brief explanation: en.wikipedia.org/wiki/Fascism#Etymology Dec 9, 2009 at 14:16
• Surely a native French speaker will read this? I'm not one, but I thought the French noun gerbe meant "spray", as in a spray (bouquet) of flowers. This also explains why gerber is slang for "to vomit". Apr 12, 2010 at 20:04
• In French, the word "gerbe" commonly refers to an arrangement of wheat like this upload.wikimedia.org/wikipedia/commons/thumb/0/06/… Apr 13, 2010 at 0:16
• I can't think of an English word that exactly matches the common usage of faisceau in French; I could use any of cone, beam, ray, spray, jet, stream, or sheaf (thanks to Tom) depending on context. The best description I can come up with is: things tied together in a directed way. It has lots of uses from light ray (faisceau lumineux) to muscle fibres (faisceau musculaire). Apr 13, 2010 at 1:08

You might like to take a look at this site:

Earliest Uses of Various Mathematical Symbols

I've heard that the "$K$" of $K$-theory comes from the German word "Klasse(n)" meaning "class(es)", but I don't have any concrete evidence for this.

• Now I'm wondering why class field theory was not called K-field theory... Dec 11, 2009 at 4:21
• It is stated in the first paragraph of this introduction by Karoubi: arxiv.org/ftp/math/papers/0602/0602082.pdf Dec 12, 2009 at 0:17
• I think this is also mentioned at the beginning of Rosenberg's book on K-theory. May 3, 2010 at 2:07

In homological algebra, one sometimes uses Z and B to denote cycles (or closed form) and boundaries (or exact forms), respectively. Z must be for Zycle.

• Z is for "Zykel" which means cycles, and B is for "Bilder" which means "images". (I'm german and that is what I learned in my german algebraic topology class) Dec 9, 2009 at 8:40
• Probably one reason for using the first letters of the German words here is that in English 'cycle' and 'chains' have the same first letter. Dec 9, 2009 at 9:06
• Also, boundary is Rand in german, with boundaries translating to Ränder. Dec 10, 2009 at 20:48
• Mar 19, 2018 at 4:03

$\mathbb{N}$ comes from the German "Natürliche Zahlen"=natural number
$\mathbb{Z}$ comes from the German "ganZe Zahl"=integer numbers
$\mathbb{Q}$ comes from the Latin "Quotient"= result of a division
$\mathbb{R}$ comes from the German "Reelle Zahl"=real numbers
$\mathbb{C}$ comes from the French "nombre Complexe"=complex numbers

• maybe \mathbb R comes from the french "nombres reelles" or the english "real numbers" instead? how do you know? For complex numbers, I would even argue for a german/latin origin, since germans used "complex" instead of "komplex" at the time of Gauss. Dec 11, 2009 at 20:47
• I was under the impression that Q, R, C are all from French (Quotient, Réel, Complexe). I doubt the Romans needed a symbol for Q... Apr 13, 2010 at 0:21
• Apparently Z, Q, R are all eventually due to Bourbaki and stand for the German Zahlen, Quotient, Reelle. However, they were all randomly used by someone else before... jeff560.tripod.com/nth.html Apr 13, 2010 at 1:34
• The word Quotient is actually a Latin word, inherited by many modern languages. Apr 27, 2010 at 10:41

The letter $T$ in the names for the separation axioms $T_1$, $T_2$, etc in point set topology comes from "Trennungsaxiom" in German. http://de.wikipedia.org/wiki/Trennungsaxiom

There is a "classic" book about the history of mathematical notations by Florian Cajori though there has been some "revision" of his work by more recent scholars.

http://en.wikipedia.org/wiki/Florian_Cajori

• Why did you put the word revision in quotes? Apr 12, 2010 at 23:50
• Some but not all have taken issue with some of Cajori's scholarship. Apr 20, 2010 at 13:11

I asked a while ago about the etymology of the name conductor. Often the conductor of an order in a number field is denoted by $\mathfrak f$. This comes from the original German name Führer given by Dedekind.

Oh, but of course $\emptyset$ comes from Bourbaki. Interestingly, so does $\Rightarrow$ to denote implication, and $\in$ instead of $\varepsilon$. The "Dangerous bend" comes from Bourbaki as well.

However, my all time favorite is the set of associated primes of a module M. $Ass(M)$ is in fact called the assassinator of $M$, and its elements are called assassins.

• In particular $\emptyset$ is due to André Weil, yet in comparison with other marks he has left on mathematics, it is the most vacuous. Apr 12, 2010 at 22:35
• I enjoyed the wordplay, Zavosh. Cheers! Apr 12, 2010 at 22:41
• Oh, I had always assumed that "dangerous bend" was due to Knuth! But you are correct: en.wikipedia.org/wiki/Bourbaki_dangerous_bend_symbol Apr 12, 2010 at 23:54
• $Ass$ doesn't sound polite in modern English though... Apr 15, 2010 at 8:25

$x,y,z$, and in particular that $x$ is the independent variable and $y$ the dependent variable, are due to Descartes, if I'm not mistaken.

• Did he choose these letters more or less at random, or did he have some particular reason for the choice? Dec 9, 2009 at 17:35
• Well, now I'm not sure. Wikipedia en.wikipedia.org/wiki/Ren%C3%A9_Descartes credits Descartes with suggesting $x^2$ for "$x$ times $x$". Leibniz invented the words "coordinate, abscissa, ordinate". Dec 9, 2009 at 23:14
• I once heard that the typesetter of Descartes' book on geometry asked him whether any particular choice of letters is important and when Descartes replied that it is not, he suggested to use x and y because they are rarely used in French and so he will not run out of them when he typesets the book. Dec 10, 2009 at 15:26
• Dmitri: Interesting! I wonder if anybody can corroborate this story. May 4, 2010 at 12:34
• @KevinH.Lin Apparently not, cf. Explanation why $x, y, z$ are always variables. Feb 25, 2020 at 13:25

Utile erit scribit ∫ pro omnia. (It is useful to write ∫ instead of omnia) – Leibniz (1675-10-29)

(Source for this quotiation: Eriksson, Estep, Hansbo, Johnson: Computational differential equations, end of Ch. 3)

In response to some comments: omnis means “all”. Compare omnivore. Here endeth the Latin lesson.

• What is "omnia"? Dec 9, 2009 at 14:03
• Also, I think the $\int$ symbol is supposed to be an elongated "S", which is supposed to stand for "summa"(?) in Latin(?), meaning "sum". Dec 9, 2009 at 14:14
• I guess integrating is stretching the notion of sum :) Dec 9, 2009 at 15:07
• It's meant to be Greek to Latin: $\Sigma\to\int$ $\Delta\to d$ Dec 9, 2009 at 15:47
• I'm assuming omnia (literally all, I think) is used to mean something along the lines of "sum". Dec 9, 2009 at 16:28

F for a closed set comes from the French ferme (=firm, cf. fermer=to close).

What about G for an open set? Is this also an example of the next-letter phenomenon? (as in Michael's comment to this answer to the question.)

• According to Wikipedia (en.wikipedia.org/wiki/G%CE%B4_set) it comes from the German word Gebiet. Dec 11, 2009 at 21:01
• You're confusing fermé (closed) with ferme (firm, rigid); the accent makes a big difference! Apr 15, 2010 at 17:29
• Touche ;) Apr 27, 2010 at 18:53
• You mean touché . Touche without accent is not an adjective but the word for a piano key, the equivalent of the english "touch" as in "the painter's delicate touch" or a hit on a target. May 16, 2010 at 8:45
• No, I don't; note the emoticon. (On the other hand, ferme was a mistake.) May 17, 2010 at 11:30

Pat Ballew's blog Math Words has interesting stuff.

$E$ is sometimes used for vector spaces, from the French word "espace"="space".

• Could also be Euclid. Apr 15, 2010 at 14:24
• @François: that's more likely for Euclidean space $\operatorname{E}^n$. Apr 15, 2010 at 17:52

Wolfram has nice a little paragraph on the history of the term "Ring" right after the list of ring axioms.

Ring (from Wolfram Mathworld)

• Yeah, I had seen that. That does not explain why 'ring' was chosen, though. Dec 9, 2009 at 4:37
• I think it mentions on there that "ring" comes from the cyclic structure you get in many rings, e.g. successive powers of the same element in $\mathbb{Z}/k\mathbb{Z}$. Dec 13, 2009 at 19:23
• Certainly thanks to the chosen word (ring, anneau in french) I have always thought of a ring as a torus, like the product of one operation (+) by the other (x) and closed in these two dimensions. I guess that I would also like to see sub-rings and modules as small rings with the initial ring like a thread passing in their holes. May 16, 2010 at 8:51
• This is a false etymology. Everyone thinks it's from cycling in modular arithmetic (I once did), but please look at mathoverflow.net/questions/35286/… for the correct history of the term ring. Aug 13, 2010 at 4:47

In design theory we talk of a t-(v,k,λ). I think v originally meant "varieties", but I don't know if any of the other symbols meant anything; it would be nice to find out that they did. λ seems an odd choice for an integer... in many other contexts it gets used as a real number.

I'm not sure how relevant this is outside of Ireland, but while doing basic mechanics, if you ever see acceleration denoted as $f$, as it is in the "log tables" here, as in $v=u+ft$, the $f$ in this case stands for the Latin for acceleration, festinatio (with festino meaning "I hurry", so festinatio would very roughly and more literally translate as "hurriedness"), which is funny because adcelero is the Latin for "I speed up" which looks a lot more like acceleration.

Similarly, displacement denoted by $s$ as in $s=ut+\frac12 at^2$ is from the Latin for displacement, summoveo (with moveo meaning "I move [something]").

And, of course, velocitas, the Latin for speed. I can imagine u being used for velocity as well since the Romans actually pronounced "v" as "u", so the two are pretty much interchangeable.