Produced by Charles Wells Revised 20140901 Introduction to this website website TOC website index blog back to languages head
The Handbook has 428 citation for usages in the mathematical research literature. After finishing that book, I started abstractmath.org and decided that I would quote the Handbook for usages when I could but would not spend any more time looking for citations myself, which is very time consuming. Instead, in abmath I have given only my opinion about usage. A systematic, well funded project for doing lexicographical research in the math literature would undoubtedly show that my remarks were sometimes incorrect and very often, perhaps even usually, incomplete.
Every letter of the Greek alphabet except omicron is used in math. All the other lowercase forms and all those uppercase forms that are not identical with the Latin alphabet are used.
Alpha: $\text{A},\, \alpha$: ă'lfɘ. Used occasionally as a variable, for example for angles or ordinals. Should be kept distinct from the proportionality sign "∝".
Beta: $\text{B},\, \beta $: bā'tɘ or (Br) bē'tɘ. The Euler Beta function is a function of two variables denoted by $B$. (The capital beta looks just like a "B" but they call it “beta” anyway.) The Dirichlet beta function is a function of one variable denoted by $\beta$.
Gamma: $\Gamma, \,\gamma$: gă'mɘ. Used for the names of variables and functions. One familiar one is the $\Gamma$ function. Don’t refer to lower case "$\gamma$" as “r”, or snooty cognoscenti may ridicule you.
Delta: $\Delta \text{,}\,\,\delta$: dĕl'tɘ. The Dirac delta function and the Kronecker delta are denoted by $\delta $. $\Delta x$ denotes the change or increment in x and $\Delta f$ denotes the Laplacian of a multivariable function. Lowercase $\delta$, along with $\epsilon$, is used as standard notation in the $\epsilon\text{}\delta$ definition of limit.
Epsilon: $\text{E},\,\epsilon$ or $\varepsilon$: ĕp'sĭlɘn, ĕp'sĭlŏn, sometimes ĕpsī'lɘn. I am not aware of anyone using both lowercase forms $\epsilon$ and $\varepsilon$ to mean different things. The letter $\epsilon $ is frequently used informally to denoted a positive real number that is thought of as being small. The symbol ∈ for elementhood is not an epsilon, but many mathematicians use an epsilon for it anyway.
Zeta: $\text{Z},\zeta$: zā'tɘ or (Br) zē'tɘ. There are many functions called “zeta functions” and they are mostly related to each other. The Riemann hypothesis concerns the Riemann $\zeta $function.
Eta: $\text{H},\,\eta$: ā'tɘ or (Br) ē'tɘ. Don't pronounce $\eta$ as "N" or you will reveal your newbieness.
Theta: $\Theta ,\,\theta$ or $\vartheta$: thā'tɘ or (Br) thē'tɘ. The letter $\theta $ is commonly used to denote an angle. There is also a Jacobi $\theta $function related to the Riemann $\zeta $function. See also Wikipedia.
Iota: $\text{I},\,\iota$: īō'tɘ. Occurs occasionally in math and in some computer languages, but it is not common. (See omicron).
Kappa: $\text{K},\, \kappa $: kă'pɘ. Commonly used for curvature.
Lambda: $\Lambda,\,\lambda$: lăm'dɘ. An eigenvalue of a matrix is typically denoted $\lambda $. The $\lambda $calculus is a language for expressing abstract programs, and that has stimulated the use of $\lambda$ to define anonymous functions. (But mathematicians usually use barred arrow notation for anonymous functions.)
Mu: $\text{M},\,\mu$: mū. Common uses: to denote the mean of a distribution or a set of numbers, a measure, and the Möbius function. Don’t call it “u”.
Nu: $\text{N},\,\nu$: nū. Used occasionally in pure math,more commonly in physics (frequency or a type of neutrino). The lowercase $\nu$ looks confusingly like the lowercase upsilon, $\upsilon$. Don't call it "v".
Xi: $\Xi,\,\xi$: zī, sī or ksē. Both the upper and the lower case are used occasionally in mathematics. I recommend the ksee pronunciation since it is unambiguous.
Omicron: $\text{O, o}$: ŏ'mĭcrŏn. Not used much since it looks just like the Roman letter. The only occurrence I know of is that Roger Penrose used iota and omicron to denote the basis vectors for spinor space (thanks, Matthew R. Francis). There is an example on page 58 of Spinors and SpaceTime: Volume 1, by Roger Penrose and Wolfgang Rindler, Cambridge University Press, 1987.
Pi: $\Pi \text{,}\,\pi$: pī. The upper case $\Pi $ is used for an indexed product. The lower case $\pi $ is used for the ratio of the circumference of a circle to its diameter, and also commonly to denote a projection function or the function that counts primes. See default.
Rho: $\text{P},\,\rho$: rō. The lower case $\rho$ is used in spherical coordinate systems. Do not call it pee.
Sigma: $\Sigma,\,\sigma$ or ς: sĭg'mɘ. The upper case $\Sigma $ is used for indexed sums. The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sumofdivisors function. The ς form for the lower case has not as far as I know been used in math writing, but I understood that someone is writing a paper that will use it.
Tau: $\text{T},\,\tau$ or τ: tăoo (rhymes with "cow"). The lowercase $\tau$ is used to indicate torsion, although the torsion tensor seems usually to be denoted by $T$. There are several other functions named $\tau$ as well.
Upsilon: $\Upsilon ,\,\upsilon$ ŭp'sĭlŏn; (Br) ĭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.
Phi: $\Phi ,\,\phi$ or $\varphi$: fē or fī. Used for the totient function, for the “golden ratio” $\frac{1+\sqrt{5}}{2}$ (see default) and also commonly used to denote an angle. Historically, $\phi$ is not the same as the notation $\varnothing$ for the empty set, but many mathematicians use it that way anyway, sometimes even calling the empty set “fee” or “fie”.
Chi: $\text{X},\,\chi$: kī. (Note that capital chi looks like $\text{X}$ and capital xi looks like $\Xi$.) Used for the ${{\chi }^{2}}$distribution in statistics, and for various math objects whose name start with “ch” (the usual transliteration of $\chi$) such as “characteristic” and “chromatic”.
Psi: $\Psi, \,\psi$; sē or sī. A few of us pronounce it as psē or psī to distinguish it from $\xi$. $\psi$, like $\phi$, is often used to denote an angle.
Omega: $\Omega ,\,\omega$: ōmā'gɘ, ō'māgɘ; (Br) ōmē'gɘ, ō'mēgɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.
In some math subjects, a font tamily (typeface) called fraktur, formerly used for writing German, Norwegian, and some other languages, is used to name math objects. The table below shows the upper and lower case fraktur letters.
$A,a$: $\mathfrak{A},\mathfrak{a}$  $H,h$: $\mathfrak{H},\mathfrak{h}$  $O,o$: $\mathfrak{O},\mathfrak{o}$  $V,v$: $\mathfrak{V},\mathfrak{v}$ 
$B,b$: $\mathfrak{B},\mathfrak{b}$  $I,i$: $\mathfrak{I},\mathfrak{i}$  $P,p$: $\mathfrak{P},\mathfrak{p}$  $W,w$: $\mathfrak{W},\mathfrak{w}$ 
$C,c$: $\mathfrak{C},\mathfrak{c}$  $J,j$: $\mathfrak{J},\mathfrak{j}$  $Q,q$: $\mathfrak{Q},\mathfrak{q}$  $X,x$: $\mathfrak{X},\mathfrak{x}$ 
$D,d$: $\mathfrak{D},\mathfrak{d}$  $K,k$: $\mathfrak{K},\mathfrak{k}$  $R,r$: $\mathfrak{R},\mathfrak{r}$  $Y,y$: $\mathfrak{Y},\mathfrak{y}$ 
$E,e$: $\mathfrak{E},\mathfrak{e}$  $L,l$: $\mathfrak{L},\mathfrak{l}$  $S,s$: $\mathfrak{S},\mathfrak{s}$  $Z,z$: $\mathfrak{Z},\mathfrak{z}$ 
$F,f$: $\mathfrak{F},\mathfrak{f}$  $M,m$: $\mathfrak{M},\mathfrak{m}$  $T,t$: $\mathfrak{T},\mathfrak{t}$  
$G,g$: $\mathfrak{G},\mathfrak{g}$  $N,n$: $\mathfrak{N},\mathfrak{n}$  $U,u$: $\mathfrak{U},\mathfrak{u}$ 
Thanks to Fernando Gouvêa for suggestions.
Aleph, $\aleph$ is the only Hebrew letter that is widely used in math.
The Cyrillic alphabet is used to write Russian and many other languages in that area of the world. Wikipedia says that the letter Ш, pronounced "sha", is the only Cyrillic letter used in math. I have not investigated further.
The letter is used in several different fields, to denote the TateShafarevich group, the Dirac comb and the shuffle product.
It seems to me that there are a whole world of possibillities for brash young mathematicians to name mathematical objects with other Cyrillic letters. Examples:
A typeface is a particular design of letters. The typeface you are reading is Arial. This is Times New Roman. This is Goudy. (Goudy may not render correctly on your screen if you don't have it installed.)
Typefaces typically come in several styles, such as bold (or boldface) and italic.
Arial Normal  Arial italic  Arial bold 
Times Normal  Times italic  Times bold 
Goudy Normal  Goudy italic  Goudy bold 
Boldface and italics are used with special meanings (conventions) in mathematics. Not every author follows these conventions.
Styles (bold, italic, etc.) of a particular typeface are supposedly called fonts. In fact, these days “font” almost always means the same thing as “typeface”, so I use “style” instead of “font”.

A letter denoting a vector is put in boldface by many authors.
It is hard to do boldface on a chalkboard, so lecturers may use $\vec{v}$ instead of $\mathbf{v}$. This is also seen in print.
The definiendum (word or phrase being defined) may be put in boldface or italics. Sometimes the boldface or italics is the only clue you have that the term is being defined. See Definitions.
“A group is Abelian if its multiplication is commutative,” or “A group is Abelian if its multiplication is commutative.”
Italics are used for emphasis, just as in general English prose. Rarely (in my experience) boldface may be used for emphasis.
It is standard practice in printed math to put singleletter variables in italics. Multiletter identifiers are usually upright.
Example: "$f(x)=a{{x}^{2}}+\sin x$". Note that mathematicians would typically refer to $a$ as a “constant” or “parameter”, but in the sense we use the word “variable” here, it is a variable, and so is $f$.
On the other hand, “e” is the proper name of a specific number, and so is “i”. Neither is a variable. Nevertheless in print they are usually given in italics, as in ${{e}^{ix}}=\cos x+i\sin x$. Some authors would write this as ${{\text{e}}^{\text{i}x}}=\cos x+\text{i}\,\sin x$. This practice is recommended by some stylebooks for scientific writing, but I don't think it is very common in math.
Blackboard bold letters are capital Roman letters written with double vertical strokes. They look like this:
\[\mathbb{A}\,\mathbb{B}\,\mathbb{C}\,\mathbb{D}\,\mathbb{E}\,\mathbb{F}\,\mathbb{G}\,\mathbb{H}\,\mathbb{I}\,\mathbb{J}\,\mathbb{K}\,\mathbb{L}\,\mathbb{M}\,\mathbb{N}\,\mathbb{O}\,\mathbb{P}\,\mathbb{Q}\,\mathbb{R}\,\mathbb{S}\,\mathbb{T}\,\mathbb{U}\,\mathbb{V}\,\mathbb{W}\,\mathbb{X}\,\mathbb{Y}\,\mathbb{Z}\]In lectures using chalkboards, they are used to imitate boldface.
In print, the most common uses is to represent certain sets of numbers:
Mathematicians use the word script to refer to two rather different styles. Both of them apply only to uppercase letters.
$A$: $\scr{A}$  $H$: $\scr{H}$  $O$: $\scr{O}$  $V$: $\scr{V}$ 
$B$: $\scr{B}$  $I$: $\scr{I}$  $P$: $\scr{P}$  $W$: $\scr{W}$ 
$C$: $\scr{C}$  $J$: $\scr{J}$  $Q$: $\scr{Q}$  $X$: $\scr{X}$ 
$D$: $\scr{D}$  $K$: $\scr{K}$  $R$: $\scr{R}$  $Y$: $\scr{Y}$ 
$E$: $\scr{E}$  $L$: $\scr{L}$  $S$: $\scr{S}$  $Z$: $\scr{Z}$ 
$F$: $\scr{F}$  $M$: $\scr{M}$  $T$: $\scr{T}$  
$G$: $\scr{G}$  $N$: $\scr{N}$  $U$: $\scr{U}$ 
$A$: $\cal{A}$  $H$: $\cal{H}$  $O$: $\cal{O}$  $V$: $\cal{V}$ 
$B$: $\cal{B}$  $I$: $\cal{I}$  $P$: $\cal{P}$  $W$: $\cal{W}$ 
$C$: $\cal{C}$  $J$: $\cal{J}$  $Q$: $\cal{Q}$  $X$: $\cal{X}$ 
$D$: $\cal{D}$  $K$: $\cal{K}$  $R$: $\cal{R}$  $Y$: $\cal{Y}$ 
$E$: $\cal{E}$  $L$: $\cal{L}$  $S$: $\cal{S}$  $Z$: $\cal{Z}$ 
$F$: $\cal{F}$  $M$: $\cal{M}$  $T$: $\cal{T}$  
$G$: $\cal{G}$  $N$: $\cal{N}$  $U$: $\cal{U}$ 
Thanks to JM Wilson for suggesting this topic and to the various people on Math Stack Exchange and Math Educators Stack Exchange who discussed script and cal.
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