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ALPHABETS

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 , .

).

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 .

).   Don’t refer to  as “r”, or snooty cognoscenti may ridicule you.  The Gamma function, denoted by , has the property that

).   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.

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 (Br.)  ).  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 (Br.)  ).

or (Br.)  ).  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.

).

).

).  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.

).   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.

)  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.

) .

## 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.

 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.

 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.

### 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:

Complex numbers

Quaternions

Integers.  Also the unit interval.

Natural numbers, either including or excluding 0.

Real numbers

Integers.

#### 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.