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ALPHABETS
Table of contents
Alphabets
The Greek alphabet
Every letter of the Greek alphabet except omicron is used in mathematics.
All the other lowercase forms and all those uppercase forms that are not identical with the Roman alphabet are used.
Mathematicians pronounce these letters in various ways. There is a substantial difference between the
way American mathematicians pronounce them and they way English-speaking
mathematicians whose background is British pronounce them. (This is indicated below by (Br).)
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 and
in exactly the same way, and since they were
writing it on the board I doubt that anyone except language nuts like me
noticed that they were doing it. Another
one pronounced
as “
” and
as “
”.
Many Greek letters are used as proper names of mathematical objects. I have indicated the most widely known ones here. They are all discussed in MathWorld and in Wikipedia. Greek letters are widely used in other sciences, but I have not attempted to cover those uses here.
Mathematicians freely use Greek letters,
including ,
to many anything they want,
without regard to their use as a particular proper name.
Pronunciation
key
.
Stress is indicated by an apostrophe after
the stredded syllable, for example ,
.
alpha (
).
beta (
or (Br)
). The
Euler Beta function is a function of two variables denoted by B. (The
capital beta looks just like a B but they write it that way and call it “beta”
anyway.) The Dirichlet beta function is
a function of one variable denoted by
.
gamma (
). Don’t refer to
as “r”, or snooty cognoscenti may ridicule
you. The Gamma function, denoted by
, has the property
that
delta (
). The
Dirac delta function and the Kronecker delta are denoted by
.
denotes the change or increment in x and
denotes the Laplacian of a multivariable
function.
epsilon (
or
;
is occasionally heard). The
letter
is frequently used informally to denoted a
positive real number that is thought of as being small. The symbol
for elementhood is strictly
speaking not an epsilon, but many mathematicians use an epsilon for it anyway.
zeta (
or (
).
There are many functions called “zeta functions” and they are mostly
related to each other. The unproved
Riemann hypothesis concerns the Riemann
-function.
eta (
or (
).
theta (
or (
).
Lower case theta may also be written
. The letter
is commonly used to denote an angle. There is also a Jacobi
-function related to the Riemann
-function.
iota (
).
kappa (
).
lambda (
). An
eigenvalue of a matrix is typically denoted
. The
-calculus is a language for expressing abstract
programs.
mu (
). Used
in statistics to denote the mean of a population. Don’t refer to
as “u”.
nu (
). Used
more in physics (frequency or a type of neutrino) than in pure math. The
lowercase
looks confusingly like the lowercase upsilon,
.
xi (
or
). I
recommend the ksee pronunciation since it is unambiguous.
omicron. Not used since it looks just like the Roman
letter.
pi (
). The
upper case
is used for an indexed product. The lower case
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 (
). The
lower case
is used in polar coordinate systems. Do not call it pee.
sigma (
). The
upper case
is used for indexed sums. The lower case
is used for the standard deviation and also
for the sum-of-divisors function.
upsilon (
)
Rarely used in math.
phi (
or
). Also
written
. Used for the Euler
-function (totient function) and for the
“golden ratio”
(see default). Widely used to denote an angle. Historically,
is not the same as the notation
for the empty set, but many mathematicians
use it that way anyway, sometimes even calling the empty set “fee” or
“fie”.
chi (
). Used
for the
distribution in statistics, and for various
math objects whose name start with “ch” (the usual transliteration of
) such as “characteristic” and “chromatic”.
psi (
or
). A
few of us pronounce it
to distinguish it from
.
is often used to denote an angle. These pronunciations have sociological
implications.
omega (
) .
Fraktur
In some subjects, especially ring theory and Lie algebra, an alphabet called fraktur, formerly used
for writing German, is used to name math objects. The table shows the upper and lower case
fraktur letters.
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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 used to mean several other typefaces (blackletter, sans serif and (gasp) the alphabet actually used by the Goths. |
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 looks like U
but in fact is an A, and the uppercase
looks like T
but is actually I.
Type styles
A typeface is a particular design of letters. The typeface you are reading is Arial. This is Times Roman.
Typefaces typically come in several styles, such as bold (or boldface) and italic. Some of these styles are used in special
ways in mathematics.
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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”. |
Boldface
Vectors
A letter denoting a vector is put in boldface by many
authors. You might write “Let v be an vector in 3-space.” Its coordinates typically would be denoted by v1, v2 and v3. This could also be written this way: “Let be an vector in 3-space.” (See parenthetic assertion).
It is hard to do boldface on a chalkboard, so lecturers may
use instead of v. This is also seen in
print.
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Definitions
The definiendum (word or phrase being defined) may be put in boldface, for example, “A group is Abelian if its multiplication is commutative.” Italics are also very commonly used for the definiendum. Sometimes the boldface or italics is the only clue you have that the term is being defined.
Italic
Italics are used for emphasis, just as in general English prose.
Italics may also be used to mark the word or phrase being defined. When a lecturer is writing at the blackboard they will typically underline a phrase that would be italicized in print.
In the symbolic language
It is standard practice in printed math to put single-letter variables in italics. Multiletter identifiers are usually upright.
Example
We write . Note that mathematicians would typically ref
er 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”. Nevertheless in
print they are usually given in italics, as in . Some authors would write this as
. This practice is recommended by some
stylebooks for scientific writing, but it is rarely done in math.
Blackboard bold
Blackboard bold letters are capital Roman letters written with double vertical strokes. They look like this:
In lectures using
chalkboards, they are used to imitate boldface. In print, the most common uses is to
represent certain sets of numbers:
Integers. Also the unit interval.
Natural numbers,
either including or excluding 0.
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.