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).[1][2][3]

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 ∑ (representing summation), should therefore be set roman. Note that each element must be set depending upon its own merits, including subscripts and superscripts. Thus, hi would be suitable for the interior height of a dome (the subscript "i" being an abbreviation of the word interior), while hi 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, ui would be suitable for the initial velocity, while ui 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 super- 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

might be rewritten as

(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, as shown here, without parentheses around the argument).

Certain important constructs, mainly the basic number system, are sometimes referred to by blackboard bold letters. For example, some authors denote the set of natural numbers by . Other authors prefer to use bold Latin for these symbols.[4] (In context of math, font variations such as bold/non-bold may encode an arbitrary relation between symbols; using specialized symbols for 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

References
  1. Mills, I. M.; Metanomski, W. V. (December 1999), On the use of italic and roman fonts for symbols in scientific text (PDF), IUPAC Interdivisional Committee on Nomenclature and Symbols, retrieved 14 August 2019. This document was slightly revised in 2007* and full text included in the Guidelines For Drafting IUPAC Technical Reports And Recommendations and also in the 3rd edition of the IUPAC Green Book. *Refer to Chemistry International. Volume 36, Issue 5, Pages 23–24, ISSN (Online) 1365-2192, ISSN (Print) 0193-6484, DOI: 10.1515/ci-2014-0529, September 2014
  2. See also Typefaces for Symbols in Scientific Manuscripts, NIST, January 1998. This cites the family of ISO standards 31-0:1992 to 31-13:1992.
  3. "More on Printing and Using Symbols and Numbers in Scientific and Technical Documents". Chapter 10 of NIST Special Publication 811 (SP 811): Guide for the Use of the International System of Units (SI). 2008 Edition, by Ambler Thompson and Barry N. Taylor. National Institute of Standards and Technology, Gaithersburg, MD, US. March 2008. 76 pages. This cites the ISO standards 31-0:1992 and 31-11:1992, but notes "Currently ISO 31 is being revised [...]. The revised joint standards ISO/IEC 80000-1—ISO/IEC 80000-15 will supersede ISO 31-0:1992—ISO 31-13.".
  4. Krantz, S., Handbook of Typography for the Mathematical Sciences, Chapman & Hall/CRC, Boca Raton, Florida, 2001, p. 35.
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