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 = (VE) 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, and everything else should be set in roman type. This applies equally to characters from the Latin/English alphabet (a, b, c, ...; A, B, C, ...) as to letters from any other alphabet, most notably Greek (α, β, γ, ...; Α, Β, Γ, ...). Any operator, such as cos (representing cosine) 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 initial height, while hi would represent one instance from a set of heights (h1, h2, h3, ...). 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 American mathematical typography

The rules of mathematical typography differ 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

${\displaystyle AX=\Omega _{e^{x}}+{\begin{matrix}{\frac {a}{b+{\frac {c}{d}}}}\end{matrix}}}$

might be rewritten as

${\displaystyle AX=\Omega _{\,\exp x}+{\begin{matrix}{\frac {a}{b+c/d}}\end{matrix}}}$

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

Incidentally, the above formula demonstrates the American 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 names of well-known functions, such as sin x (the trigonometric function sine) and exp x (the constant e raised to the power of x) are written in lowercase upright letters (and often, as shown here, without parentheses around the argument).

Certain important constructs are sometimes referred to by blackboard bold letters. For example, some authors denote the set of natural numbers by ${\displaystyle \mathbb {N} }$. 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 ${\displaystyle \mathbb {N} }$ etc. allows the author more freedom of expressing such relations.)

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

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 9 November 2012. 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.
Greek letters used in mathematics, science, and engineering

Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities. Those Greek letters which have the same form as Latin letters are rarely used: capital A, B, E, Z, H, I, K, M, N, O, P, T, Y, X. Small ι, ο and υ are also rarely used, since they closely resemble the Latin letters i, o and u. Sometimes font variants of Greek letters are used as distinct symbols in mathematics, in particular for ε/ϵ and π/ϖ. The archaic letter digamma (Ϝ/ϝ/ϛ) is sometimes used.

The Bayer designation naming scheme for stars typically uses the first Greek letter, α, for the brightest star in each constellation, and runs through the alphabet before switching to Latin letters.

In mathematical finance, the Greeks are the variables denoted by Greek letters used to describe the risk of certain investments.

Latin letters used in mathematics

Many letters of the Latin alphabet, both capital and small, are used in mathematics, science and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, physical entities. Certain letters, when combined with special formatting, take on special meaning.

Below is an alphabetical list of the letters of the alphabet with some of their uses. The field in which the convention applies is mathematics unless otherwise noted.

Letter case

Letter case (or just case) is the distinction between the letters that are in larger upper case (also uppercase, capital letters, capitals, caps, large letters, or more formally majuscule) and smaller lower case (also lowercase, small letters, or more formally minuscule) in the written representation of certain languages. The writing systems that distinguish between the upper and lower case have two parallel sets of letters, with each letter in one set usually having an equivalent in the other set. The two case variants are alternative representations of the same letter: they have the same name and pronunciation and are treated identically when sorting in alphabetical order.

Letter case is generally applied in a mixed-case fashion, with both upper- and lower-case letters appearing in a given piece of text. The choice of case is often prescribed by the grammar of a language or by the conventions of a particular discipline. In orthography, the upper case is primarily reserved for special purposes, such as the first letter of a sentence or of a proper noun, which makes the lower case the more common variant in regular text. In some contexts, it is conventional to use one case only. For example, engineering design drawings are typically labelled entirely in upper-case letters, which are easier to distinguish than the lower case, especially when space restrictions require that the lettering be small. In mathematics, on the other hand, letter case may indicate the relationship between objects, with upper-case letters often representing "superior" objects (e.g. X could be a set containing the generic member x).

List of mathematical symbols

This is a list of symbols used in all branches of mathematics to express a formula or to represent a constant.

A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations, a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. However, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.

Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image using TeX.

Mathematical Alphanumeric Symbols

Mathematical Alphanumeric Symbols are Unicode blocks of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The letters in various fonts often have specific, fixed meanings in particular areas of mathematics. By providing uniformity over numerous mathematical articles and books, these conventions help to read mathematical formulae.

Unicode now includes many such symbols (in the range U+1D400–U+1D7FF). The rationale behind this is that it enables design and usage of special mathematical characters (fonts) that include all necessary properties to differentiate from other alphanumerics, e.g. in mathematics an italic "A" can have a different meaning from a roman letter "A". Unicode originally included a limited set of such letter forms in its Letterlike Symbols block before completing the set of Latin and Greek letter forms in this block beginning in version 3.1.

Unicode expressly recommends that these characters not be used in general text as a substitute for presentational markup; the letters are specifically designed to be semantically different from each other. Unicode does not include a set of normal serif letters in the set (thus it assumes a given font is a serif by default; a sans-serif font that supports the range would thus display the standard letters and the "sans-serif" symbols identically but could not display normal serif symbols of the same).

All these letter shapes may be manipulated with MathML's attribute mathvariant.

The introduction date of some of the more commonly used symbols can be found in the Table of mathematical symbols by introduction date.

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; function symbols such as sin; operator symbols such as "+"; conceptual symbols such as lim and dy/dx; equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams.

Notation in probability and statistics

Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

Subscript and superscript

A subscript or superscript is a character (number, letter or symbol) that is (respectively) set slightly below or above the normal line of type. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, while superscripts are above. Subscripts and superscripts are perhaps most often used in formulas, mathematical expressions, and specifications of chemical compounds and isotopes, but have many other uses as well.

In professional typography, subscript and superscript characters are not simply ordinary characters reduced in size; to keep them visually consistent with the rest of the font, typeface designers make them slightly heavier (i.e. medium or bold typography) than a reduced-size character would be. The vertical distance that sub- or superscripted text is moved from the original baseline varies by typeface and by use.

In typesetting, such types are traditionally called "superior" and "inferior" letters, figures, etc., or just "superiors" and "inferiors". In English, most nontechnical use of superiors is archaic. Superior and inferior figures on the baseline are used for fractions and most other purposes, while lowered inferior figures are needed for chemical and mathematical subscripts.

Summation

In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Besides numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any types of mathematical objects on which an operation denoted "+" is defined.

Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.

The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result does not depends on the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with zero element) results, by convention, in 0.

Very often, the elements of a sequence are defined, through regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written 1 + 2 + 3 + 4 + ⋅⋅⋅ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where ${\displaystyle \textstyle \sum }$ is an enlarged capital Greek letter sigma. For example, the sum of the first n natural integers is denoted ${\displaystyle \textstyle \sum _{i=1}^{n}i.}$

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,

${\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.}$

Although such formulas do not always exist, many summation formulas have been discovered. Some of the most common and elementary ones are listed in this article.

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