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# ALPHABETS

#### Contents

This section is based only partly on lexicographical research. See Remarks about usage.

## The Greek alphabet

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.

• Many Greek letters are used as proper names of mathe­ma­tical objects, for example $\pi$. Here, I provide some usages that might be known to undergraduate math majors.  Many other usages are given in MathWorld and in Wikipedia. In both those sources, each letter has an individual entry.
• But any mathematician will feel free to use any Greek letter with a meaning different from common usage. This includes $\pi$, which for example is often used to denote a projection.
• Greek letters are widely used in other sciences, but I have not attempted to cover those uses here.

### The letters

• English-speaking mathematicians pronounce these letters in various ways.  There is a substantial difference between the way American mathe­maticians pronounce them and the way they are pronounced by English-speaking mathe­maticians whose background is from British Commonwealth countries. (This is indicated below by (Br).)
• Mathematicians speaking languages other than English may pronounce these letters differently. In particular, in modern Greek, most Greek letters are pro­nounced differ­ently from the way we pronounce them; β for example is pro­nounced vēta (last vowel as in "father"). The modern Greek pronunciation is not noted in this article.
• Newcomers to abstract math often don’t know the names of some of the letters, or mispronounce them if they do.  I have heard young mathe­maticians pronounce $\phi$ and $\psi$ in exactly the same way, and since they were writing it on the board I doubt that anyone except language geeks like me noticed that they were doing it.  Another one pronounced $\phi$ as  “fee” and $\psi$ as “fie”.

#### Pronunciation key

• ăt, āte, ɘgo (ago), bĕt, ēve, pĭt, rīde, cŏt, gō, ŭp, mūte.
• Stress is indicated by an apostrophe after the stressed syllable, for example ū'nit, ɘgō'.
• The pronunciations given below are what mathematicians usually use. In some cases this includes pronunciations not found in dictionaries.

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 Space-Time: 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 sum-of-divisors 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.

## Fraktur

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.

 $A,a$: $\mathfrak{A},\mathfrak{a}$ $H,h$: $\mathfrak{H},\mathfrak{h}$ $O,o$: $\mathfrak{O},\mathfrak{o}$ $V,v$: $\mathfrak{V},\mathfrak{v}$ $B,b$: $\mathfrak{B},\mathfrak{b}$ $I,i$: $\mathfrak{I},\mathfrak{i}$ $P,p$: $\mathfrak{P},\mathfrak{p}$ $W,w$: $\mathfrak{W},\mathfrak{w}$ $C,c$: $\mathfrak{C},\mathfrak{c}$ $J,j$: $\mathfrak{J},\mathfrak{j}$ $Q,q$: $\mathfrak{Q},\mathfrak{q}$ $X,x$: $\mathfrak{X},\mathfrak{x}$ $D,d$: $\mathfrak{D},\mathfrak{d}$ $K,k$: $\mathfrak{K},\mathfrak{k}$ $R,r$: $\mathfrak{R},\mathfrak{r}$ $Y,y$: $\mathfrak{Y},\mathfrak{y}$ $E,e$: $\mathfrak{E},\mathfrak{e}$ $L,l$: $\mathfrak{L},\mathfrak{l}$ $S,s$: $\mathfrak{S},\mathfrak{s}$ $Z,z$: $\mathfrak{Z},\mathfrak{z}$ $F,f$: $\mathfrak{F},\mathfrak{f}$ $M,m$: $\mathfrak{M},\mathfrak{m}$ $T,t$: $\mathfrak{T},\mathfrak{t}$ $G,g$: $\mathfrak{G},\mathfrak{g}$ $N,n$: $\mathfrak{N},\mathfrak{n}$ $U,u$: $\mathfrak{U},\mathfrak{u}$

• Many of the forms are confusing and are commonly mispronounced by younger mathematicians.  (Ancient mathematicians like me have taken German classes in college that required learning fraktur.)  In particular the uppercase $\mathfrak{A}$ looks like $U$ but in fact is an $A$, and the uppercase $\mathfrak{I}$ looks like $T$ but is actually $I$.
• When writing on the board, some mathematicians use a cursive form when writing objects with names that are printed in fraktur.
• Unicode regards fraktur as a typeface (font family) rather than as a different alphabet. However, unicode does provide codes (range 120068 through 120223) for the fraktur letters that are used in math (no umlauted letters or ß). In TeX you type "\mathfrak{a}" to get $\mathfrak{a}$.
• In my (limited) experience, native German speakers usually call this alphabet “Altschrift” instead of “Fraktur”.  It has also been called “Gothic”, but that word is also used to mean several other quite different typefaces (black­letter, sans serif and (gasp) the alphabet actually used by the Goths.
• I have been doing mathematical research for around fifty years. It is clear to me that mathematicians' use of and familiarity with fraktur has declined a lot during that time. But it is not extinct. I have made a hasty and limited search of Jstor and found recent websites and research articles that use it in a variety of fields. There are also a few citations in the Handbook (search for "fraktur").
• It is used in ring theory and algebraic number theory, mostly to denote ideals.
• It is used in Lie algebra. In particular, the Lie algebra of a Lie group $G$ is commonly denoted by $\mathfrak{g}$.
• The cardinality of the continuum is often denoted by $\mathfrak{c}$.
• It is used occasionally in logic to denote models and other objects.
• I remember that in the sixties and seventies fraktur was used in algebraic geometry, but I haven't found it in recent papers.

#### Acknowledgements

Thanks to Fernando Gouvêa for suggestions.

## Hebrew alphabet

Aleph, $\aleph$ is the only Hebrew letter that is widely used in math.

• $\aleph$ is the cardinality of the set of integers. A set with cardinality $\aleph$ is countably infinite. More generally, $\aleph$ is the first of the aleph numbers $\aleph_1$, $\aleph_2$, $\aleph_3$, and so on.
• Cardinality theorists also write about the beth ($\beth$) numbers, and the gimel ($\gimel$) function. Daleth $\daleth$ and zayin ז have been used in cardinality theory as well. I am not aware of other uses of the Hebrew alphabet.
• LaTeX has codes for aleph, beth, gimel and daleth but not for any other Hebrew letters.
• Unicode has code for the Hebrew alphabet to be used in writing Hebrew (64288 through 64296). If you type several letters using these codes they print out right to left.
• It also has separate codes (64288 through 64296) intended to display them without changing directionality. (Thanks to Toby Bartels).
• There is more information about the Hebrew alphabet in Robin Knight's note The symbol $\aleph$.

## Cyrillic alphabet

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 Tate-Shafarevich 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:

• Ж. Use it for a ornate construction, like the Hopf fibration or a wreath product.
• Щ. This would be mean because it is hard to pronounce.
• Ъ. Guaranteed to drive people crazy, since it is silent. (It does have a name, though: "Yehr".)
• Э. Its pronunciation indicates you are unimpressed (think Fonz).
• ю. Pronounced "you". "ю may provide a counterexample". "I do?"

A typeface is a particular design of letters.  The typeface you are reading is Arial.  This is Times New Roman. This is Cambria.

## Boldface and italics

Typefaces typically come in several styles, such as bold (or boldface) and italic.

#### Examples

 Arial Normal Arial italic Arial bold Times Normal Times italic Times bold Cambria Normal Cambria italic Cambria bold

Boldface and italics are used with special meanings (conventions) in mathematics. Some of them are listed below. Not every author follows these conventions.

Styles (bold, italic, etc.) of a particular typeface are supposed to be called fonts.  In fact, these days “font” almost always means the same thing as “typeface”, so I  use “style” instead of “font”.

### Vectors

A letter denoting a vector is put in boldface by many authors.

#### Examples

• “Suppose $\mathbf{v}$ be an vector in 3-space.”  Its coordinates typically would be denoted by $v_1$, $v_2$ and $v_3$.
• You could also define it this way:  “Let $\mathbf{v}=({{v}_{1}},{{v}_{2}},{{v}_{3}})$ be a vector in 3-space.”  (See parenthetic assertion.)

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.

### Definitions

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.

#### Example

“A group is Abelian if its multiplication is commutative,” or  “A group is Abelian if its multiplication is commutative.”

### Variables

It is standard practice in printed math to put single-letter variables in italics. Multiletter identifiers are usually upright.

#### Example

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 $f$.

#### Example

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

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 use is to represent certain sets of numbers:

#### Remarks

• Mathe­ma­tica uses some lower case blackboard bold letters.
• Many mathe­ma­tical writers disapprove of using blackboard bold in print.  I say the more different letter shapes that are available the better.  Also a letter in blackboard bold is easier to distinguish from ordinary upright letters than a letter in boldface is, particularly on computer screens.

## Script

Mathematicians use the word script to refer to two rather different styles. Both of them apply only to uppercase letters.

#### Script

 $A$: $\scr{A}$ $H$: $\scr{H}$ $O$: $\scr{O}$ $V$: $\scr{V}$ $B$: $\scr{B}$ $I$: $\scr{I}$ $P$: $\scr{P}$ $W$: $\scr{W}$ $C$: $\scr{C}$ $J$: $\scr{J}$ $Q$: $\scr{Q}$ $X$: $\scr{X}$ $D$: $\scr{D}$ $K$: $\scr{K}$ $R$: $\scr{R}$ $Y$: $\scr{Y}$ $E$: $\scr{E}$ $L$: $\scr{L}$ $S$: $\scr{S}$ $Z$: $\scr{Z}$ $F$: $\scr{F}$ $M$: $\scr{M}$ $T$: $\scr{T}$ $G$: $\scr{G}$ $N$: $\scr{N}$ $U$: $\scr{U}$

#### Calligraphic

 $A$: $\cal{A}$ $H$: $\cal{H}$ $O$: $\cal{O}$ $V$: $\cal{V}$ $B$: $\cal{B}$ $I$: $\cal{I}$ $P$: $\cal{P}$ $W$: $\cal{W}$ $C$: $\cal{C}$ $J$: $\cal{J}$ $Q$: $\cal{Q}$ $X$: $\cal{X}$ $D$: $\cal{D}$ $K$: $\cal{K}$ $R$: $\cal{R}$ $Y$: $\cal{Y}$ $E$: $\cal{E}$ $L$: $\cal{L}$ $S$: $\cal{S}$ $Z$: $\cal{Z}$ $F$: $\cal{F}$ $M$: $\cal{M}$ $T$: $\cal{T}$ $G$: $\cal{G}$ $N$: $\cal{N}$ $U$: $\cal{U}$

### Using script

• In LaTeX, script letters are obtained using "\scr" and calligraphic using "\cal". For example, "{\scr P}" gives ${\scr P}$.
• Both script and calligraphic are used to provide yet another type style for naming mathematical objects.
• One of the most common uses is to refer to the powerset of a set $S$: ${\scr P}(S)$, ${\scr P}S$, ${\cal P}(S)$, ${\cal P}S$.
• There may be some tendency to use script or cal to name objects that are in some way high in the hierarchy of objects or else a space that contains a lot of the stuff you are talking about. Lie algebra may be an exception. This requires lexicographic research.
• The names of categories are commonly denoted by script or calligraphic. Some authors have trouble because they want to put names of categories such as "Set" and "Grp" in cal or scr but don't have lower case letters in those styles. In Toposes, Triples and Theories the online version went through several changes over the years. Category Theory for Computing Science uses bold for category names.
• I have never run across a paper that used both script and calligraphic to mean two different things.

#### Acknowledgments

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.