Typographical conventions in mathematical formulae
Encyclopedia
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 font
s 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, 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.
s 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 one level; for example,
might productively be rewritten as
Incidentally, the above formula demonstrates the American rule that italic type
is used for all letters representing variables except uppercase Greek letters, which are in upright type. Upright type is also standard for digits and punctuation. 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 number
s by
. Similarly, the symbols 
, 
, and 
, 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 most publications, having boldface available, use it. Thus, for instance, the integers are more typically denoted by
.
Donald Knuth
's TeX
typesetting engine incorporates a large amount of additional knowledge about American-style mathematical typography.
Mathematical notation
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...
includes letters from various alphabets, as well as special mathematical symbols. Letters in various font
Font
In typography, a font is traditionally defined as a quantity of sorts composing a complete character set of a single size and style of a particular typeface...
s often have specific, fixed meanings in particular areas of mathematics. A mathematical article or a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
typically starts from the definitions of the introduced symbols, such as: "Let G = (V, E) be a graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
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
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
) 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 journalMathematical journal
A mathematics journal is a scientific journal which publishes exclusively mathematics papers. A practical definition of the current state of mathematics, as a research field, is that it consists of theorems with proofs published in a reputable mathematics journal, and which usually have passed...
s 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 one level; for example,
might productively be rewritten as
Incidentally, the above formula demonstrates the American rule that italic type
Italic type
In typography, italic type is a cursive typeface based on a stylized form of calligraphic handwriting. Owing to the influence from calligraphy, such typefaces often slant slightly to the right. Different glyph shapes from roman type are also usually used—another influence from calligraphy...
is used for all letters representing variables except uppercase Greek letters, which are in upright type. Upright type is also standard for digits and punctuation. Bold Latin capital letters usually represent matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, and bold lowercase letters are often used for vectors. The names of well-known functions, such as sin x (the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
sine) and exp x (the constant e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
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 number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s by
. Similarly, the symbols , , and , may be used to denote the integers, rationalsRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
, and reals
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, respectively. But, as its name suggests, blackboard bold simulates the practice used on chalkboards to indicate boldface. So most publications, having boldface available, use it. Thus, for instance, the integers are more typically denoted by
.Donald Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
's TeX
TeX
TeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....
typesetting engine incorporates a large amount of additional knowledge about American-style mathematical typography.
See also
- Mathematical notationMathematical notationMathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
- Greek letters used in mathematicsGreek letters used in mathematicsGreek 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...
- Roman letters used in mathematicsRoman letters used in mathematicsMany 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...
- Mathematical alphanumeric symbolsMathematical alphanumeric symbolsMathematical Alphanumeric Symbols is a Unicode block of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles .Unicode now includes many such symbols Mathematical Alphanumeric Symbols is a Unicode block of Latin and Greek...
- Table of mathematical symbolsTable of mathematical symbolsThis is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
- BlackletterBlackletterBlackletter, also known as Gothic script, Gothic minuscule, or Textura, was a script used throughout Western Europe from approximately 1150 to well into the 17th century. It continued to be used for the German language until the 20th century. Fraktur is a notable script of this type, and sometimes...