Typographical conventions in mathematical formulae
As a set of formulas is defined in typography the sentence especially in mathematics, but also chemical formulas, for print or on-screen viewing. The inclusion of formulas is more difficult than that of pure text, as this character must not only side by side but also on top of each set ( eg, fractions ). Some characters must extend over subsequent characters (eg root sign ); also may be subject to subsequent characters ( such as parentheses ) or by its function (indices and exponents ) their size. A further difficulty lies in the variety of mathematical signs. Symbols must be clearly distinguishable.
- 2.1 Formula Editors
- 2.2 MathML
- 3.1 Chemical formula set
Traditions of the formula set
In several countries in the course of history several conventions for mathematical typesetting have developed. They differ mainly in the font sizes used and the letter spacing, ie the distribution of voids. Apart from these small differences, there are also some big differences, which are presented below.
Text formulas and remote formulas
In typesetting, a distinction between standing in the text as text formulas and remote formulas that are indented on a separate line, as
The two examples conform to the conventions of the U.S. formula set. They show up especially on the way upper and lower limits are set in -sum, product and integral sign. Here, the upper and lower limits are located on the right of the icon.
In other traditions, such as the Russian and the German, the boundaries are always written above and below the symbol to which they relate, as in the text or in a separate line
Formulas, which are components of a sentence are in accordance with DIN 1338 be even if they are free, treated in terms of punctuation, such as a phrase. It should be set between formula and punctuation exclusion. Tabulated list of formulas, however, are not accompanied by punctuation.
Font sizes
In the formula set different font sizes are used not only for indices and exponents, but also the limits of sums, products and integrals. In the American set of formulas also numerators and denominators of fractions are set in smaller type, depending on how big the surrounding text, and in what stage of the fracture is, if it is a multiple fracture:
In German and Russian formula set against it though limits and indices and exponents are set smaller, but the numerator and denominator in fractions always have the same size as the rest of the formula.
Appearance of mathematical symbols
Some differences in the set of formulas relate the shape of individual characters, such as the root, comparison and integral character that can look different in Anglo-American, German and Russian tradition.
Contrary to this representation is specified in DIN 1302 ( " General mathematical symbols and concepts " ) " equal less " set for the one character that is listed here as an American form. The German form of the square root sign is even in the DIN standards are not uniform, but typically as stated here.
Standing straight, inclined and italics
Whether it's Formula letters standing straight (eg, the small Latin a), tilted ( a) or italic ( ) can be written, also depends on the underlying tradition. The differential operator d is written to DIN upright and italic to AMS. Allen traditions it is common that defined mathematical function such as sin and exp signs are written upright.
In the Anglo - American world capital Greek letters and constants, such as the Euler number e, the imaginary unit i ( with ) or the Golden Section, upright written, variables, and small Greek letters, however, italic, for example. In French and Russian set of formulas, however all Greek letters are also written all uppercase upright in French as well as traditional. The formula according to the American tradition
Is written in the Russian formula set with upright θ:
In the French formula set next to the θ and the R would be written upright:
In the German formula set the table applies. Thus, the German version of the American differs only in:
In DIN 1304 ( " Symbols " ) and DIN 1338 ( " formula notation and formula set " ) is defined:
These rules also apply to Symbol indices (DIN 1338, section 3.4.2).
Greek letters
Most typesetting traditions using today's standard Greek capital letters. Some use it in an upright form, others in italics. Several small letters exist in different spellings and are shown in the adjacent picture. Furthermore, two forms are at Epsilon before: in addition. Greek letters that look like Latin, are not suitable as symbols.
Both forms of Pi were since the Middle Ages next to each other in use, is even in some Greek script fonts in the 19th century to find, but since then it has gone from the normal Greek script. As symbols in importance as a county number will be written today, as lemniscate constant. This form, however, occurs outside the astronomy quite rare. The two variants of the theta, and are identical in meaning in printed Greek texts, as symbols, however, they can refer to two different situations. The same applies to Phi and. The first form comes from the manuscript, while the second is the usual form in Greek printing. The sigma shape that occurs in printing at the beginning and in the middle of a word, and the form are to be distinguished, that occurs in pressure at the end of texts of words.
What forms of Greek letters are used depends on the formula set tradition. The Anglo-American set of formulas used all given forms parallel. According to ISO and are attracted to. According to the recommendations of the IUPAP up on and as well as to view the nonexchangeable variants and the variant forms as equivalent and should therefore have no independent significance.
The Greek small Ny used in mathematics and physics, ν written ( just standing) or ν ( inclined ) or ( in italics), can be derived from the Latin small Vau, written v ( standing straight ), v ( inclined ) or ( in italics), by suitable choice of font differ. Particularly difficult to distinguish is the Greek small Ypsilon, υ written, or υ.
Line breaks in formulas
Long formulas may be so large that they no longer fit on one line and must be split across multiple lines. The most important standard rules are:
- An equation, which continues with another equal sign is divided before the character. The more equal sign at the beginning of the new line below the previous equal sign.
- Long terms are divided in front of a plus or minus sign, but possibly not within brackets. The plus or minus sign at the beginning of the new line further to the right than the previous equal sign.
- Must be shared product, 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 separate line and repeated in the new row, the American record it appears, however, as in German, only in the second line.
Fonts for typesetting
In mathematical formulas in addition to letters and numbers and mathematical symbols are used, which are rarely available in text fonts. These include, for a special mathematical characters such as, on the other hand, the above mentioned Variant forms of Greek letters. To remedy this deficiency, it does have some fonts that have the Greek letters and the variants of Theta Pi and Sigma in addition to many other mathematical symbols.
Programs for the set of formulas in electronic media
Formula editors
A commonly used format for typesetting on the computer is TeX or LaTeX. TeX is widely used especially in the natural sciences and sits there as the standard for scientific publications by. The reasons are in addition to the clean layout especially at the excellent set of formulas.
With the use of the LaTeX AMS- LaTeX packages that allow other specialties of the formula set or make them more accessible is recommended. For example, while in the setting of multi-line formulas eqnarray the notbehelfsmäßige LaTeX environment creates the serious typographical error of a large empty space around the equal sign and for more complex requirements of standing on the alignment with one another Terme quickly reaches its limits, provides AMS- LaTeX for this purpose numerous specialized environments (eg, align) to.
Many WYSIWYG word processing systems provide so-called equation editors for entering formulas. The input of formulas with a formula editor, although intuitive, but at the same time a bit awkward. The output quality often leaves much to be desired, unless the formula editors use TeX as a backend. An example of a formula editor that uses TeX as a backend, is the commercial MathType plugin for MS Word.
In OpenOffice.org and the LibreOffice package the Math component is responsible for creating sets of formulas. In Math created sets of formulas can be included in text documents (OOo Writer), presentations (OOo Impress) or spreadsheets (OOo Calc). In Math -written formulas can be stored directly in the MathML format.
A Windows-based formula editor, which allows to generate MathML and integration of formulas as an image into any application that provided under open source tool MathCast is ( for Windows).
MathML
MathML is a markup language for mathematical formulas, which is primarily intended for use on Web pages, and it relies heavily on the mathematical symbols in Unicode.
The set of formulas in HTML documents designed currently still relatively difficult, since few Web Browser MathML support.
The currently most common method to represent complicated formulas on web pages is to create formulas with TeX and integrate as graphics.
Latex2html does this automatically and simultaneously created
Special forms of the formula set
Chemical formula set
Also typesetting chemical formulas falls under the concept set of formulas. In this case the setter on the one hand to do it with the comparatively simple reaction equations, on the other hand with the much more complex structural formulas. Reaction equations resemble mathematical equations insofar as the elements with high, low and on another to symbols also be combined with them. The charge of an ion or generally of an elementary particle is expressed by a superscript number, which is right next to the item or Teilchensymbol. The number of species in a molecule atoms of a particular element on the right side represented by a subscript number of the element symbol. The oxidation number of an atom is represented by an over asked Roman or Arabic numeral rare. The atomic number is the number of protons in the nucleus, which determines the type of atom, is the bottom left, the mass number, ie the number of all particles in the nucleus, written at the top left of the element symbol. These elements are chemical formulas by the usual mechanisms of the mathematical formula set, it is to be realized with computer programs or with lead letters.
But still occurs in the chemical rate on another type of sentence: the typesetting of chemical structural formulas. The inclusion of chemical structural formulas, an example is shown in the picture structural formula of benzoic acid, is set programs, but even more so lead translator remains a major challenge. In Theorem programs can usually for this purpose a graph are drawn in LaTeX about with the picture environment. In the lead sentence, however, the translator must first make an accurate picture of the outcome, which he wishes to place the horizontal, vertical and diagonal lines in various angles right along with the element symbols. Older books that are printed with lead and early phototypesetting, therefore you also sometimes see a small gap between the lines representing the chemical bonds, which take testimony from the problems of the complex task of the sentence structure of chemical formulas.
In computer programs you will be able to make do with a few structural formulas with graphics. But as soon emerge more than a few structures that are also probably very complex, it is usually necessary to use an additive for chemical formulas. This usually resort to ready-made structures into which the element symbols must be inserted only. For unusual structures but it is necessary to describe the formulas logical. An example of such a solution represents the packets XyMTeX and PPCH TeX
History
The set of formulas is almost as old as the printing press itself the first printed books like the Gutenberg Bible was soon followed by even mathematical works. Until the beginning of the 19th century, the quality of the formula rate was modest. From about 1850 he improved to 1900 to attain a climax.
The spread of the typewriter from about 1925 entered a throwback: many mathematical texts (especially theses and dissertations ) were no longer set professionally in print, but the author himself with a typewriter. Formulas were initially grudge by hand in gaps left open, since the 1960s, there were also special typewriters halbzeiligem distance for high and low positions, which were basically simple formulas can be placed. In this way, a whole series of books with the characteristic typewriter typeface and ragged created (example: Hans Opolka, Winfried Scharlau: From Fermat to Minkowski 1980. ).
This development Donald Knuth has provided since 1977 with the development of TeX stop. From about 1994 it is considered bad form not to put mathematical texts with TeX. On the other hand, there were parallel to the typewriter set in the 1960s, a second peak in the quality of the hand set (for example, Oskar Perron: Irrational numbers in 1960. ).
With the set of formulas dealt until the introduction of TeX essentially the professional group of the formula Coasters:
" Training in the set of mathematical and chemical formulas provided in the training documents for typesetter; but there is still often the opinion that this record type is only a few specialized translators reserved. Therefore is then placed mostly low value in the general training of translators, and so it is that, the trainees only a few companies can cope with the tasks posed by the [ ... ] substantial increases in the production of mathematical and chemical books to the typesetter land. "
Standardize
- DIN 1338 formula notation and set of formulas
- DIN 1302 General mathematical symbols and terms
- DIN 1304 symbols
- ISO / IEC 80000 Quantities and Units ( Quantities and Units )