14 Science, mathematics, and computing
14.6 Mathematics
14.6.1 Introduction
The de facto standard for mathematical typesetting is T_{E}X (usually reproduced as TeX), a mark-up language available for many computing platforms, which consists of a coded, plain-text file that can be output to a device-independent file for printing or to PDF to produce high-quality results in terms of the complex layout, formatting, and special characters often demanded in mathematical work. LaTeX is the most popular TeX-based system. Some mathematical societies and publishers have their own plain TeX or LaTeX style files to help authors prepare their submissions according to the required specifications. Mathematical mark-up languages such as MathML and TeXML can be used to embed formulas in webpages, ebooks, and apps.
As a result, former constraints in setting ‘difficult’ material such as embellishments, fractions, fences, and overbars have been overcome, and digital product has removed the necessity to minimize displayed matter to reduce page extents and production costs.
When mathematical notation is required in text that is not created in a TeX-based system, authors should prepare their files using a word-processor equation editor or a stand-alone WYSIWYG (graphical) application for creating mathematical notation based on MathML.
In a mathematical equation each letter represents a quantity or operation, but the style (bold, italic, script, sans serif) and the relative positioning and sizes of the symbols (subscript, superscript) also convey important information.
Some typefaces are not suited to setting mathematics and should be avoided: sans-serif faces often suffer from indistinguishable characters (l, I, 1 are common examples). Subscripts and sub-subscripts must be legible and the distinction between roman, italic, bold, and sans serif clear even in one isolated character. Plain serifed faces such as Times, Computer Modern, and Lucida Math are generally to be favoured for mathematical work. The choice of typeface should be carried through to headings and labelling of figures where maths may occur. The typography of equations should not be altered to conform with design specifications for headings and labels.
14.6.2 Notation
The established style is to use roman type for operators and certain numerical constants, and to label points in a diagram. Italic is used for letters expressing a variable, and bold is used for vectors and matrices. Each description should be represented by a single symbol: avoid using abbreviated words. Here are some specific points:
• It is common to set the headword of definitions, theorems, propositions, corollaries, or lemmas in capitals and small capitals, with the remaining text of the heading in italic. The full text of the proof itself is set in roman.
• Avoid starting a sentence with a letter denoting an expression, so that there is no ambiguity about whether a capital or lower-case letter is intended.
• Most standard abbreviations, for example ‘log(arithm)’, ‘max(imum)’, ‘exp(onential function)’, ‘tan(gent)’, ‘cos(ine)’, lim(it)’, and ‘cov(ariance)’, are set in roman, with no full points.
• Distinguish between a roman ‘d’ used in a differential equation (dx/dy), the symbol ∂ for a partial differential, and the Greek lower-case delta (δ). Authors should preclude further potential for confusion by avoiding an italic d for a variable. US authors may use italic for differential d
• The exponential ‘e’ remains in roman (but may be italic in US texts); sometimes it may be preferable to use the abbreviation ‘exp’ to avoid confusion. The letters i and j are roman when symbolizing imaginary numbers, and italic when symbolizing variables.
• In displayed equations, the integral, summation, and product symbols (∫ ∑ ∏) may have limits set directly above them (upper limit) or below them (lower limit). In running text, the convention is to place these after the symbol wherever possible (as in the first example)
• Missing terms can be represented by three dots which are horizontal (⋯), vertical (⋮), or diagonal (⋱), as appropriate. Include a comma after the three ellipsis dots when the final term follows, for example x_{1}, x_{2}, …, x_{n}.
14.6.3 Symbols
Operational signs are of two types: those representing a verb (e.g. = ≈ ≢ ≥) and those representing a conjunction (e.g. + − ⊃ ×). All operational signs take a space on either side, consistently either a thin space or a word space. (If setting in TeX, such spaces are built in, so there is no need to add any.) They are not closed up unless they are being used to modify a value, for example, a difference of ±2; magnification ×200.
Set a multiplication point as a medial point (⋅) (U+22C5 dot operator) but it should be used only to avoid ambiguity and is not needed between letters, unless a vector product is intended (e.g. a∙b), in which case U+2219 bullet operator should be used; there are no spaces either side of a medial point.
In print product where space-saving is desirable, any symbol that involves setting a separate line of type should be avoided when an alternative form is available. So, for angle ABC, prefer ∠ ABC to and for vector r, prefer r (in bold) to
The space around a colon used as a ratio sign (mixed in the proportion 1: 2) is narrower than that used around an operational sign, so when using either word or thin spaces around operators, use a thin or hair space respectively on either side of the colon; some styles set the colon closed up. A colon in a set or function such as {x: x > 0} is closed up to the preceding letter with a non-breaking space after.
A solidus is set closed up to the characters either side (12/28, a/b) and there is no space before a factorial (12!). In TeX-based systems there is no constraint in using the vinculum or overbar—the horizontal rule above the square-root sign (
14.6.4 Superscripts and subscripts
Reserve superscript letters for variable quantities (set in italic); reserve subscript letters for descriptive notation (set in roman). Asterisks and primes are not strictly superscripts and so should always follow immediately after the term to which they are attached, in the normal way.
When first a subscript and then a superscript are attached to the same symbol or number, mark the subscript to align with the superscript in a ‘stack’.
If it is necessary to have multiple levels of superscripts or subscripts, the relationships must be made clear for the typesetter if rekeying is necessary.
Wherever possible, it is customary—and a kindness to the reader’s eyesight—to represent each superscript or subscript description by a symbol rather than an abbreviated word. Those subscript descriptions that are standardly made up of one or more initial letters of the word they represent are set in roman type.
14.6.5 Brackets
The preferred order for brackets is {[( )]}. When a single pair of brackets has a specific meaning, such as [n] to denote the integral part of n, they can, of course, be used out of sequence. The vertical bars used to signify a modulus— │x│ —should not be used as brackets.
Two further sorts of mathematical brackets may be used: double square brackets 〚 〛 and ‘narrow’ angle brackets 〈 〉. Angle brackets are used singly in Dirac bra and ket notation and in pairs they may be used to indicate the value of a quantity over a period of time, but they can also be used generally. Note that in mathematical work, the inequality signs < and > are not used as brackets. Double square brackets can be placed outside, and narrow angle brackets inside, the bracket sequence, and are handy for avoiding the re-arrangement of brackets throughout a formula or, especially, a series of formulas. Thus for comparison’s sake the formula
14.6.6 Fractions, formulas, and equations
Definitions
The terms equation, formula, and expression are sometimes confused by non-mathematicians; the following explanations may help to clarify usage.
An equation is a statement in which two mathematical expressions are linked by an equals sign:
y = 12a + b or 10 ÷ 5 = 2
A mathematical expression does not have an equals sign—
v^{2} + f or 3 − 7i
A formula is a mathematical rule expressed in words or symbols:
(in words) the circumference of a circle is equal to twice the radius multiplied by pi
(in symbols) c = 2πr
Layout
As discussed in
and simple fractions such as
But just because the equations can have complex formatting, it does not necessarily mean that they should: work can be reduced and appearance improved by writing such a formula as
In printed product, displayed formulas three or four lines deep can thus be reduced to a space-saving, neater, and more manageable two-line form in almost all instances.
Displayed formulas are usually centred on the page. If there are many long ones, or a wide discrepancy in their length, it may be better to range them all left with a 1- or 2-em indent.
If it is necessary to break a formula—whether displayed or run-in—at the end of a line, it should be done at an operational sign, with the sign carried over to the next line. If an equation takes up two or more lines it should be displayed, with turnover lines aligned on the operational sign (preferably =):
Any equation referred to at another point in the text should be numbered; any numbered equation should be displayed. It is usually better to include the chapter number in front of the sequence number, such as 2.1 for the first equation in
14.6.7 Mathematical symbols
Table 14.4 Some mathematical symbols and corresponding Unicode code points
Equivalent character and numerical entity codes can be sourced online.
Unicode U+ | Unicode character name | Alternative common name (where different) | |
= | 003D | equals sign | |
≠ | 2260 | not equal to | |
≡ | 2261 | identical to | identically equal to |
≢ | 2262 | not identical to | not identically equal to |
≈ | 2248 | almost equal to | approximately equal to |
≉ | 2249 | not almost equal to | not approximately equal to |
≃ | 2243 | asymptotically equal to | |
≄ | 2244 | not asymptotically equal to | |
∼ | 223C | tilde operator | equivalent to, of the order of |
≁ | 2241 | not tilde | not equivalent to, not of the order of |
∝ | 221D | proportional to | |
→ | 2192 | rightwards arrow | approaches |
⇒ | 21D2 | rightwards double arrow | implies |
⇔ | 21D4 | left right double arrow | two-way implication |
> | 003E | greater-than sign | |
≯ | 226F | not greater-than | |
< | 003C | less-than sign | |
≮ | 226E | not less-than | |
≫ | 226B | much greater-than | |
≪ | 226A | much less-than | |
≥ | 2265 | greater-than or equal to | |
≤ | 2264 | less-than or equal to | |
π | 03C0 | greek small letter pi | pi |
∞ | 221E | infinity | |
( ) | 0028 and 0029 | left parenthesis (and right) | |
[ ] | 005B and 005D | left square bracket (and right) | |
{ } | 007B and 007D | left curly bracket (and right) | braces |
〈 〉 | 27E8 and 27E9 | mathematical left angle bracket (and right) | narrow angle brackets |
〚 〛 | 27E6 and 27E7 | mathematical left white square bracket (and right) | double brackets |
˄ | 02C4 | modifier letter up arrowhead | vector product |
∅ | 2205 | empty set | |
+ | 002B | plus sign | |
− | 2212 | minus sign | |
⋅ | 22C5 | dot operator | medial or multiplication dot |
ab, a·b, a × b | a multiplied by b | ||
a/b, a ÷ b, ab ^{−1} | a divided by b | ||
a^{n} | a raised to the power of n | ||
|a| | the modulus (or magnitude) of a | ||
√a, a^{½} | square root of a | ||
p! | factorial p | ||
′ | 2032 | prime | minute |
″ | 2033 | double prime | second |
° | 00B0 | degree sign | |
∠ | 2220 | angle | |
∶ | 2236 | ratio | |
∷ | 2237 | proportion | |
∴ | 2234 | therefore | hence |
∵ | 2235 | because | |
exp x, e^{x} | exponential of x | ||
log_{a}x | logarithm to base a of x | ||
ln x, log_{e}x | natural logarithm of x | ||
lg x, log_{10}x | common logarithm of x | ||
sin x | sine of x | ||
cos x | cosine of x | ||
tan x | tangent of x | ||
sin^{−1} x, arcsin x | inverse sine of x | ||
cos^{−1} x, arccos x | inverse cosine of x | ||
tan^{−1} x, arctan x | inverse tangent of x | ||
∫ | 222B | integral | |
∑ | 2211 | n-ary summation | summation |
∆ | 2206 | increment | delta |
∏ | 220F | n-ary product | product |
∆xx | finite increase of x | ||
δx | variation of x | ||
dx | total variation of x | ||
f(x) | function of x |