Typographical conventions in mathematical formulae

Typographical conventions in mathematical formulae provide uniformity across mathematical texts and help the readers of those texts to grasp new concepts quickly.

Mathematical notation includes letters from various alphabets, as well as special mathematical symbols. Letters in various fonts often have specific, fixed meanings in particular areas of mathematics. A mathematical article or a theorem typically starts from the definitions of the introduced symbols, such as: "Let G = (V, E) be a graph with the vertex set V and edge set E...". Theoretically it is admissible to write "Let X = (a, q) be a graph with the vertex set a and edge set q..."; however, this would decrease readability, since the reader has to consciously memorize these unusual notations in a limited context.

Usage of subscripts and superscripts is also an important convention. In the early days of computers with limited graphical capabilities for text, subscripts and superscripts were represented with the help of additional notation. In particular, n² could be written as n^2 or n**2 (the latter borrowed from FORTRAN) and n₂ could be written as n_2.

International recommendations

Various international authorities, including IUPAC, NIST and ISO have produced similar recommendations with regard to typesetting variables and other mathematical symbols (whether in equations or otherwise). However, these recommendations do not always follow the common practice of mathematicians as they are described in the AMS Style Guide of the American Mathematical Society, and reflected in the manual of style of Wikipedia.

In general, anything that represents a variable (for example, h for a patient's height) should be set in italic type, and everything else should be set in roman type. This applies equally to characters from the Latin/English alphabet (a, b, ..., z, A, B, ..., Z) as to letters from any other alphabet, most notably Greek (α, β, γ, ..., ω, Α, Β, Γ, ..., Ω). Any operator, such as cos (representing the cosine function) or Σ (representing summation), should therefore be set roman. Note that each element must be set depending upon its own merits, including subscripts and superscripts. Thus, h_i would be suitable for the interior height of a dome (the subscript "i" being an abbreviation of the word interior), while h_i would represent one instance (the i^th) from a set {h₁, h₂, h₃, ...} of heights. Notice that numbers (1, 2, 3, etc.) are not variables, and so are always set roman. Likewise, in some special cases symbols are used to represent general constants, such as π used to represent the ratio of a circle's circumference to its diameter, and such general constants can be set in roman. (This does not apply to parameters which are merely chosen to not vary.)

For vectors, matrices and tensors, it is recommended to set the variable itself in boldface (excluding any associated subscripts or superscripts). Hence, u_i would be suitable for the initial velocity, while u_i would represent one instance from a set of velocities {u₁, u₂, u₃, ...}. Italic is still used for variables, both for lowercase and for uppercase symbols (Latin, Greek, or otherwise). The only general situation where italic is not used for bolded symbols is for vector operators, such as ∇ (nabla), set bold and roman.

General rules in mathematical typography

The rules of mathematical typography differ slightly from country to country; thus, American mathematical journals and books will tend to use slightly different conventions from those of European journals.

One advantage of mathematical notation is its modularity—it is possible to write extremely complicated formulae involving multiple levels of superscripting or subscripting, and multiple levels of fraction bars. However, it is considered poor style to set up a formula in such a way as to leave more than a certain number of levels; for example, in non-math publications

[math]\displaystyle{ \textstyle AX = \Omega_{\,e^x} + \frac{a}{b+\frac{c}{d}} }[/math]

might be rewritten as

[math]\displaystyle{ AX = \textstyle\Omega_{\,\exp x} + \frac{a}{b+c/d} }[/math]

(Even in mathematical publications, where 3 or 4 levels of indices are frequent, avoiding multilevel fractions is productive.)

Incidentally, the above formula demonstrates the rule that italic type is used for all letters representing variables and parameters except uppercase Greek letters, which are in upright type. Upright type is also standard for digits and punctuation; currently, the ISO-mandated style of using upright for constants (such as e, i) is not widespread. Bold Latin capital letters usually represent matrices, and bold lowercase letters are often used for vectors. The symbols of functions that are not reduced to a single letter, such as sin(x) (the trigonometric function sine) and exp(x) (the exponential function) are written in lowercase upright letters (and often without parentheses around the argument).

Certain important constructs, mainly the basic number systems, are sometimes referred to by blackboard bold letters. For example, some authors denote the set of natural numbers by [math]\displaystyle{ \mathbb{N} }[/math]. Other authors prefer to use bold Latin for these symbols. (In the context of math, font variations such as bold/non-bold may encode an arbitrary relation between symbols; using specialized symbols for [math]\displaystyle{ \mathbb{N} }[/math] etc. allows the author more freedom of expressing such relations.)

Donald Knuth's TeX typesetting engine incorporates a large amount of additional knowledge about mathematical typography.

See also

• Mathematical notation
• Language of mathematics
• Greek letters used in mathematics
• Roman letters used in mathematics
• Mathematical alphanumeric symbols
• Table of mathematical symbols
• Blackletter

References

• Knuth, Donald E. (1979), "Mathematical typography", Bull. Amer. Math. Soc. (N.S.) 1 (2): 337–372, doi:10.1090/s0273-0979-1979-14598-1, http://projecteuclid.org/euclid.bams/1183544082, retrieved 2010-04-15