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Typographical conventions in mathematical formulae

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Title: Typographical conventions in mathematical formulae  
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Subject: Mathematical notation, Metasyntactic variable, Summation, Letter case, Subscript and superscript
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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, sub- 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.

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

AX = \Omega_{\mathrm{e}^x} + \begin{matrix}\frac{a}{b+\frac{c}{d}}\end{matrix}

might be rewritten as

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 \mathbb{N}. Similarly, the symbols \mathbb{Z}, \mathbb{Q}, and \mathbb{R}, may be used to denote the integers, rationals, and reals, respectively. But, as its name suggests, blackboard bold simulates the practice used on chalkboards to indicate boldface. So many non-mathematical publications, having boldface available, use it. Thus, for instance, the integers would be denoted by \bold{Z}. (In context of math, font variations such as bold/non-bold may encode an arbitrary relation between symbols; using specialized symbols for \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.

See also

References

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