Formula set

from Wikipedia, the free encyclopedia

In typography, a formula set is the set of primarily mathematical , but also chemical formulas for printing or display on the screen . The setting of formulas is more difficult than of pure text , since characters not only have to be placed next to each other, but also on top of each other (e.g. in the case of fractions ). Some characters must extend over subsequent characters (e.g. root characters ); their size can also depend on the following characters (e.g. brackets ) or on their function ( indices and exponents ). Another difficulty lies in the variety of mathematical symbols . Similar signs must be clearly distinguishable.

Logo of the German Institute for Standardization DIN 1338
Area Correspondence
title Formula notation and formula set
Brief description: regulates the spelling of formulas in typesetting
Latest edition 2011-03
ISO -

Traditions of the formula set

Different conventions for the set of mathematical formulas have developed in different countries throughout history . They differ mainly in the font sizes used and the spacing , i.e. the distribution of empty spaces. In addition to these small differences, there are also some major differences that are presented below.

Text formulas and separate formulas

In the formula set, a distinction is made between text formulas like in the running text and separate formulas that are indented in a separate line like

.

The two examples conform to the conventions of the US formula set. They are particularly evident in the way in which upper and lower limits are set for sum , product and integral symbols. Here the upper and lower limits are arranged to the right of the symbol.

In other traditions, such as the Russian and also the German, the borders are always written above and below the symbol to which they refer, as in the running text or on a separate line

.

At Wikipedia, the formula set is based on TeX , an American system for text and formula set, which also supports German spelling. American typesetting saves space and eliminates large spaces between lines, as in the following case. These empty spaces are unfavorable for the publisher because they leave more paper unprinted; the uneven line spacing can impair readability. For the same reasons, a break can be written in body text instead of in the manner . In Germany according to DIN 1302 and internationally according to ISO 80000-1 , both spellings are equally valid. As an alternative to this changed notation, TeX also supports a smaller form  .

According to DIN 1338, formulas that are part of a sentence should be treated like part of a sentence with regard to punctuation marks, even if they are free. An exclusion should be placed between the formula and punctuation marks . On the other hand, formulas listed in tabular form are not supplemented with punctuation marks.

Font sizes

In general, smaller font sizes are selected for indices and exponents than for the actual formula letters. How much smaller these formula elements are compared to the rest depends on the language area. Different measuring systems alone play a role here: American printers use the so-called pica point (also: printer's point), which measures around 0.351 mm. In contrast, the French point system is traditionally used in Europe, which goes back to François Ambroise Didot . This Didot point is about 0.376 mm, which is larger than the American pica point. With the introduction of desktop publishing (DTP), another dimension was introduced, the DTP point with a length of 0.353 mm. It is currently the only reliable measure in most application programs, such as Microsoft Word , Photoshop and printer communication, while CorelDRAW was programmed as metric. In LaTeX, and therefore also in Wikipedia, exponents and indices are set 3 pt smaller than the surrounding formula text. In Russia, however, exponents and indices are set 2 dd smaller.

In the formula set, different font sizes are used not only for indices and exponents, but also for the limits of sums, products and integrals. In the American formula set, the numerator and denominator of fractions are also set in smaller font, depending on how large the surrounding text is and what level the fraction is in, if it is a multiple fraction:

In the German and Russian formulas, on the other hand, limits and indices as well as exponents are set smaller, but the denominators and numerators in fractions always have the same size as in the rest of the formula.

Appearance of math symbols

Some differences in the formula set concern the shape of individual characters, such as the root , comparison and integral characters , which can look different in the Anglo-American, German and Russian traditions.

American form German form Russian form
American radical.svg Root.svg Russian radical.svg
American integral.gif German integral.gif Russian integral.gif

Contrary to this representation, DIN 1302 (“General mathematical symbols and terms”) defines the symbol for “less than or equal to” that is listed here as the American form. The German form of the root sign is not uniform even in the DIN standards , but typically as stated here.

Straight, inclined and italic font

Whether formula letters are written straight (e.g. the small Latin a), inclined ( a ) or italic ( ) also depends on the tradition behind them. The differential operator d is written upright according to DIN and italic according to AMS . It is common to all traditions that defined mathematical function symbols such as sin and exp are written upright.

In the Anglo-American language area, capital Greek letters and constants, such as Euler's number e, the imaginary unit i (with ) or the golden ratio , are written upright, while variables and small Greek letters are in italics, for example . In the French and Russian formula sets, on the other hand, all Greek letters and all capital letters in French are traditionally written upright. The formula according to American tradition

is written in the Russian formula set with an upright θ:

θ

In the French set of equations, in addition to the θ, the R would be written upright:

θ

The following table applies to the German formula set. The German version differs from the American version only in :

In DIN 1304 ( "Symbol") and DIN 1338 ( "formula notation and formula rate"), now also internationally in ISO 80000 -1 (German version as EN ISO 80000-1: 2013 "Quantities and units - Part 1: General") , is set:

object Writing Examples
Numbers written in digits upright 1; 1234; −0.734
Numbers, represented by letters,
with free meaning
italic / inclined n times; 2 n ; a ik
Numbers, represented by letters,
with conventional meaning
upright e = 2.71828 ... (Euler's number)
i 2 = j 2 = −1 (imaginary unit)
Formula symbols for physical quantities,
including physical constants
italic / inclined i (electrical current); m (mass)
e ≈ 1.602 10 −19 As (elementary charge )
μ 0 ≈ 4 π 10 −7 H / m (magnetic field constant)
Characters for functions and operators,
if their meaning can be freely selected
italic / inclined y = f ( x )
u ( t ) = U 0 sin ωt
Characters for functions and operators
when with conventional meaning
upright d; ∂; Δ; Σ
div; lim; Re (real part); lg; sin
Characters for units upright m (meters); μF (microfarad)
Chemistry and atomic physics symbols upright Fe (iron); e - (electron)
Abbreviations of words upright p abs (absolute); F min (minimal)

These rules for formula symbols also apply to indices (DIN 1338, Section 3.4.2).

Greek letters

Variant forms of Greek letters

Most of the formula set traditions use today's Greek block letters. Some use them upright, others italicize. Several lowercase letters exist in different spellings and are shown in the picture opposite. Furthermore, there are two forms of the epsilon : next to . Greek letters that look like Latin are not suitable as formula symbols.

Both forms of Pi have been in use side by side since the Middle Ages, and can even be found in some Greek scripts in the 19th century, but since then it has disappeared from the normal Greek script. As a symbol in the meaning of a circle number is written today as a lemniscatic constant . However, this form occurs very rarely outside of astronomy . The two variants of theta , and , have the same meaning in printed Greek texts, but as symbols they can designate two different facts. The same applies to Phi with and . The first form comes from manuscript, while the second represents the form common in Greek prints. In the case of the sigma , a distinction must be made between the form that appears in print at the beginning and in the middle of a word and the form that appears in print text at the end of words.

Which forms of the Greek letters are used depends on the formula set tradition. The Anglo-American formula set uses all of the specified forms in parallel. According to ISO , and are preferable. According to the recommendations of the IUPAP , except for and as well as the non- interchangeable variants and the variant forms are to be regarded as equivalent and should therefore not have an independent meaning.

The Greek small Ny used in mathematics and physics , written ν (straight) or ν (inclined) or (italic), can be derived from the Latin small Vau, written v (straight), v (inclined) or (italic), by choosing a suitable font distinguish. It is particularly difficult to distinguish the Greek small Ypsilon , written υ, υ or .

Line breaks in formulas

Long formulas can be so large that they no longer fit on one line and have to be split over several lines. The most important standardized rules are:

  • An equation that continues with another equal sign is divided before that sign. The additional equal sign is at the beginning of the new line under the preceding equal sign.
  • Long terms are split before a plus or minus sign, but preferably not within parentheses. The plus or minus sign is at the beginning of the new line further to the right than the preceding equal sign.
  • If a product has to be shared, the same applies.
  • Numerical value and unit as well as the factors of a derived unit are not separated.

In the Russian formula set the operator is placed at the end of the separated line and repeated in the new line, in the American sentence, however, as in German, it only appears in the second line.

Fonts for the formula set

A serif font is recommended for the formula set in order to make it easier to distinguish . For details and exceptions see under formula symbols . (Since this font is only accessible in the formula editor on Wikipedia, the recommendation is not consistently implemented here.)

In addition to letters and numbers, mathematical formulas also use mathematical symbols that are rarely found in text fonts. These include, on the one hand, special mathematical symbols such as , on the other hand, the variant forms of Greek letters mentioned above. To remedy this deficiency, there are various fonts that contain the Greek letters and the variants of theta, pi and sigma in addition to many other special mathematical characters.

Programs for formula set in electronic media

Formula editors

A frequently used format for formulas on the computer is TeX or LaTeX . TeX is particularly widespread in the natural sciences and is increasingly becoming the standard for scientific publications. The reasons for this are not only the clean layout, but also the excellent formula set.

With LaTeX, we recommend using the AMS-LaTeX packages, which enable further specialties of the formula set or make them more accessible. For example, while the makeshift LaTeX environment creates eqnarraythe serious typographical error of too large a space around the equal sign when setting multi-line formulas and quickly reaches its limits with more complex requirements for the alignment of terms, AMS-LaTeX offers numerous specialized environments for this purpose on (e.g. align).

Many WYSIWYG - word processors provide for entering formulas so-called equation editors. Entering the formulas with a formula editor is intuitive, but at the same time somewhat cumbersome. The output quality often leaves a lot to be desired, unless the formula editors use TeX as the backend. An example of a formula editor that uses TeX as a backend is the commercial plug-in MathType for MS Word .

In the OpenOffice.org and LibreOffice packages, the Math component is responsible for creating formula sets. Formula sets created in Math can be integrated into text documents ( OOo Writer ), presentations ( OOo Impress ) or spreadsheets ( OOo Calc ). Formulas written in Math can also be saved directly in MathML format.

A Windows-based formula editor that allows MathML to create and integrate formulas as a picture in any application is the MathCast tool (for Windows) , which is available under Open Source .

MathML

MathML is a markup language for mathematical formulas, which is primarily intended for use on web pages, and makes extensive use of the mathematical characters in Unicode . The formula set in HTML documents is currently still relatively difficult, as few web browsers support MathML. The currently most common method of displaying more complex formulas on websites is to create the formulas with TeX and integrate them as graphics. LaTeX2html does this automatically and at the same time creates <ALT> attributes that contain the source code of the formula.

Special forms of the formula set

Chemical formula set

Structural formula of benzoic acid

The set of chemical formulas also falls under the term formula set. In this case, the typesetter has to deal with the comparatively simple reaction equations on the one hand, and the significantly more complex structural formulas on the other . Reaction equations are similar to mathematical equations in that they also combine the elements with symbols placed in superscript, subscript and on top of each other. The charge of an ion or, in general, of an elementary particle is expressed by a superscript number that appears to the right of the element or particle symbol. The number of atoms of a certain element in a molecule is represented by a subscript to the right of the element symbol. The oxidation number of an atom is represented by a superimposed Roman or, more rarely, Arabic number. The atomic number, i.e. the number of protons in the atomic nucleus, which determines the type of atom, is written at the bottom left, the mass number, i.e. the number of all particles in the atomic nucleus, is written at the top left of the element symbol. These elements of chemical formulas can be realized with the usual mechanisms of the mathematical formula set, be it with computer programs or with lead letters.

Yet another type of theorem occurs in the chemical proposition: the setting of chemical structural formulas. The setting of chemical structural formulas, an example is shown in the picture Structural formula of benzoic acid , poses typesetting programs, but especially lead types, a great challenge. A graphic can usually be drawn in typesetting programs for this purpose, in LaTeX for example with the picture environment. In lead type, on the other hand, the typesetter must first get an exact picture of the result he wants in order to correctly arrange the horizontal, vertical and diagonal lines at various angles together with the element symbols. In older books printed with lead or early phototypesetting, therefore, one sometimes sees a small space between the lines that represent the chemical bonds that testify to the problems of the complex task of the set of chemical structural formulas.

In computer programs one will be able to make do with a few structural formulas with graphics. As soon as more than just a few structures appear, which may also be very complex, it is usually necessary to use an additive for chemical formulas. These mostly use ready-made structures into which the element symbols only have to be inserted. For unusual structures, however, it is necessary to describe the formulas logically. Examples of such a solution are the XyMTeX, ChemFig and PPCH-TeX packages.

history

The formula set is almost as old as the printing press itself. The first printed books such as the Gutenberg Bible were soon followed by mathematical works. Until the beginning of the 19th century, the quality of the formula set was modest. From around 1850 it improved, reaching a preliminary peak around 1900.

The widespread use of the typewriter from around 1925 onwards led to a step backwards: many mathematical texts (especially diploma theses and dissertations ) were no longer professionally set in print shops , but by the author himself with a typewriter. At first, formulas were entered by hand in the gaps that had been left blank, but from the 1960s there were also special typewriters with half-line spacing for superscripts and subscripts, which meant that simple formulas could in principle be set. In this way, a whole series of books with the characteristic typewriter typeface and fluttering edge were created (example: Hans Opolka, Winfried Scharlau : Von Fermat bis Minkowski. 1980).

Donald Knuth put a stop to this development with the development of TeX in 1977 . From around 1994 it was considered a bad style not to type mathematical texts with TeX. On the other hand, parallel to typewriter typesetting, there was a second high point in the quality of hand typesetting in the 1960s (example: Oskar Perron : irrational numbers. 1960).

Until the introduction of TeX, the formula set was mainly occupied by the professional group of formula setters:

“The training in typesetting mathematical and chemical formulas is provided for in the training documents for typesetters; but there is still often the opinion that this type of sentence is reserved for only a few special typesetters. Therefore, in the general training of typesetters, this is usually only given little value, and so it happens that the young professionals are only able to cope with the tasks that the [...] significantly increasing production of mathematical and chemical books places on typesetters. "

- P. Fritzsche and H. Wunderlich, 1952

Norms

  • DIN 1338 Formula notation and formula set
  • DIN 1302 General mathematical symbols and terms
  • DIN 1304 symbols
  • ISO / IEC 80000 quantities and units ( Quantities and Units )

literature

Formula sets are rarely dealt with in detail in the literature. Most works on typography leave out the formula set. Good information on the formula set can be found in the following works:

  • Friedrich Forssman , Ralf de Jong : Detailed typography - reference work for all questions about writing and typesetting. 2nd Edition. Hermann Schmidt, Mainz 2005, ISBN 3-87439-642-8 .
  • Paul Fritzsche, Herbert Wunderlich: The formula set in mathematics, chemistry and technology. With numerous exclusive examples . Fachbuchverlag Leipzig, 1952.
  • Jörg Knappen: Reach your goal quickly with LATEX 2e. 3rd, completely revised edition. Oldenbourg Munich 2009, ISBN 978-3-486-59015-9 .
  • Donald E. Knuth : The TeXbook (Computers and Typesetting Volume A). Addison-Wesley, Reading / Mass. 1984, ISBN 0-201-13448-9 .
  • Romeo Thieme: Theorem and meaning of mathematical formulas. Educational verb d. German book printer, Berlin 1934.
  • Herbert Voss : Mathematical theorem with LaTeX. Lehmanns Media, Berlin 2012, ISBN 978-3-86541-485-4 .

Web links

Commons : Mathematical symbols  - collection of images, videos, and audio files

Individual evidence

  1. XyMTeX in the English language Wikipedia