Dealing with non-English alphabets in mathematics.

SUMMARY

Technical writers and especially editors need to be familiar with mathematical notation, particularly the non-English characters. This article illustrates the proper mathematical use of the Greek alphabet, elements of the Germanic black letter alphabet, the Hebrew alphabet, and some nonstandard English characters, concluding with several hints for the practicing technical communicator.

The technical communicator's effectiveness in providing information to those who need it can be hampered by imperfect knowledge of the subject. We all have weak spots, but no single subject causes us as much frustration as mathematics. Much of that frustration stems from the myriad symbols found in mathematical texts, often incomprehensible to anyone but an expert. Many symbols--those that tell us what operations to perform or how various quantities relate to one another--clearly possess only a mathematical interpretation, having little or no meaning in other contexts. For example, the symbols [Mathematical Expression Omitted] and [Mathematical Expression Omitted] mean plenty to a mathematician but little to the rest of us.

Other symbols are letters from our own alphabet. These letters--the variables and constants of mathematics--represent numbers. They help us work out mathematical problems in a general way without forcing us to use the actual values dictated by a specific case. Thus, we derive formulas and perform calculations first, then substitute the numerical values for the letters afterward. Variables and constants taken from the English alphabet are typeset in italics so they can be distinguished from letters not related to mathematics.

Still other symbols are letters taken from non-English alphabets. Mathematicians and scientists use them not only as variables and constants but also as abbreviations of the names of mathematical objects: functions, spaces, sets, almost anything. The most common non-English letters are Greek, but we also see German black letters, and even a Hebrew letter or two. These characters may be unfamiliar to some of us, prompting all the usual questions: What is it? What does it mean? Is it written and aligned correctly?

By reviewing the uses of these non-English characters in mathematics and science, this article attempts to clarify their meanings and variations in form. It also provides some simple guidelines on how to use these letters to form plurals and prefixes and how to avoid embarrassing discrepancies between text and illustrations. This knowledge should help us improve our skill in writing, editing, illustrating, or processing technical documents.

THE GREEK ALPHABET

The Greeks developed philosophy, science, logic, and mathematics to an extent unmatched anywhere until the rise of western Europe as a scientific power in modern times. Led by such luminaries as Pythagoras, Zeno, Eudoxus, and Archimedes, the great minds of Greece laid the foundation of all mathematics to follow [1], and the Greek alphabet remains a major influence on mathematical symbolism. Table 1 lists the upper and lower case Greek letters and their equivalents in our modern English alphabet. Where two variants are shown for a letter, the second is a cursive form that often appears in technical documents.

The use of symbols is mathematical texts is a relatively new invention, originating only about 400 years ago. Early mathematical formulations had always been written out as word problems, much like the ones you and I despised in school, only worse. Some early symbolism existed, of course. For example, the third-century algebraist Diophantus always stated his problems in words but nonetheless invented a special symbol to denote the unknown quantity in equations, similar to our variable [chi] [2].

Translations of Greek mathematical texts began appearing in Europe in the twelfth century followed in the sixteenth century by reproductions of the Greek originals themselves. Thus, although Church-educated Renaissance scholars wrote in Latin, most were also fluent in Greek. As new scientific achievements demanded new words to describe them, these new words were first borrowed from the Greek language, then invented on the basis of Greek roots.

The seventeenth century saw the widespread shortening of mathematical words to abbreviations and finally to just symbols [3; 4]. Among these symbols were letter, often chosen as mnemonic clues to the names of the quantities or objects they represented; that is, the sound of the letter suggested the first letter or sound of the word. The texts were written in Latin, but many of the symbols were--and remain--letters of the Greek alphabet.

One familiar mnemonic is [Pi]. As most of us already recognize, the most common use of [Pi] is to represent the ratio of the circumference to a circle to its diameter. Mathematicians have known about this ratio for 400 years, but it was not until 1706 that William Jones suggested the abbreviation [Pi], for the periphery of a circle having unit diameter. Apparently Jones carried little weight with the mathematical community because no one paid any attention to him. In 1737, Leonhard Euler used [Pi] according to Jone's suggestion. Euler was one of the most respected mathematicians of his (and our) time, so everyone paid attention to him [5]. We still do.(*) Table 2 shows some other commonly used a Greek letters that are mnemonics in modern English. [Tabular Data 2 Omitted]

I do not propose to catalog all the uses of Greek letters in mathematics, but the following considerations can help us cope with some trickly situations.

* Many upper case Greek letters so closely resemble their modern English equivalents (Table 1) that distinguishing between them can be almost impossible. For that reason, upper case Greek letters are rarely used as mathematical symbols.

* Several lower case Greek letters so closely resemble English letters that the two should not be used together. For example, both nu ([Nu]) and upsilon ([Upsilon]) are easily confused with the italic English letter "v," [Upsilon]. If [Nu] or [Upsilon] appears in a manuscript along with [Nu], it might be best to replace one of them with some other symbol. Note also that the cursive lower case alpha ([Alpha]) is preferred over the noncursive form (a), which looks too much like our modern letter "a" to be distinctive. The author should be able to choose acceptable alternatives when these problems arise.

* Sometimes we see the cursive forms curly theta, [Theta], and curly phi, [Phi], instead of the usual [Theta] and [Phi], respectively. In some cases, these forms are the traditional symbols introduced centuries ago. In other cases, modern authors choose these characters because [Theta] and [Phi] are already in use or because the authors want to avoid confusion with common meanings of [Theta] and [Phi].

These letters do, however, possess traditional mathematical meanings of their own. Theta-functions, for instance, are often identified by [Theta] rather than by [Theta]. Recall from school that complex numbers are written in the form a + bi, where a is a real number (the common numbers we use every day) and bi is a so-called imaginary number (in which i = [square root of -1]. In the study of topological Reimann surfaces, the four [Theta] -functions, generally denoted by [[Theta].sub.1] through [[Theta].sub.4], describe the relationship between certain complex numbers connected with these surfaces. Theta-functions are usually written with a variable representing the complex number in question. For example, if z represents the complex number a + bi, then the first [Theta]-function can be given in standard function notation as [[Theta].sub.1] (z).

The character [Theta] has several uses. In the traditional notation of spherical coordinates, (r, [Theta], [Theta]), the [Theta] denotes the longitude [9] whereas in geocentric coordinates, (r, [Theta], [Lambda], the [Theta] denotes the latitude [10]. In number theory, Euler's [Theta]-function provides a way to calculate how many integers (the positive and negative whole numbers plus zero) are relatively prime to a specified integer. In standard function notation, this function is [Phi] ([Chi]).

This list by no means exhausts all the uses of [Theta] and [Phi], but these examples should be sufficient to show that the choice of these characters over [Theta] and [Phi] is not always arbitrary. Note, however, that the current trend in mathematics is to replace the traditional [Theta] and [Phi] with [Theta] and [Phi].

* Authors occasionally use Greek characters to represent vector-valued quantities. Recall that a vector can be visualized as an arrow whose length and direction are both important. Normally vectors are typeset as nonitalic boldface letters, but it may be impossible to get boldface Greek letters with a typewriter or word processor (or with many typesetting systems, for that matter). If Greek letters must be used to denote vectors, they should be embellished in the same way as other symbols used for vectors: [Alpha] [right arrow], [Alpha] [bar], [Alpha] [tilde], and so forth.

* Beware of inconsistencies between next and illustrations. Often, illustrations are drawn separately, then pasted into the text later with the result that the style of Greek callouts in the illustration may not match that of their counterparts in the text. This sort of inconsistency is especially noticeable when the callouts are referred to in the caption, highlighting, the difference in style. Sometimes the match is wrong because the illustrator uses letters from rub-on sheets or Leroy templates and the compositor uses computer fonts. At other times, the Greek letters provided in graphics software differ from those provided in the text software. Whatever the reason, too often we see style differences such as [Phi] versus [Phi] or even [Phi] versus [Phi]. One easy way out of this dilemma is to typeset the callouts along with the text, then paste them onto the illustrations as needed.

* The delta has a special form, written [Delta], called curly delta or curly dee (round dee to printers), which is used almost exclusively to indicate partial derivatives, as [Delta] x/ [Delta] y.

* Another special symbol is the nabla [9], an inverted upper case delta: [delta]. Mathematics call this symbol a del and use it with partial derivatives as an operator: [Mathematical Expression Omitted] where i, j, and k are called unit vectors. With this symbol, we can denote the gradient of a function, [Delta] F (pronounced gradF); the divergence of a vector function, [Delta] * F (pronounced the divergence of F); or the curl of a vector function, [Delta] x F (pronounced the curl of F). A related symbol is the Laplacian operator or just Laplacian, [[Delta].sup.2]. The Laplacian is not del-squared; the superscript is part of the notation. The symbol is alternatively denoted by [Delta].

* Most compositors are sufficiently familiar with handwritten Greek letters that the common ones usually need not be specially marked for typesetting as long as they are written legibly [11]. However, the lower case epsilon ([Epsilon]), the upper case sigma ([Sigma]), and the upper case pi ([Pi]) should be clearly identified to distinguish them from the more common relational symbol [Epsilon] (in) and operational symbols [Sigma] (summation) and [Pi] (product) [12]. Keep in mind that [Epsilon] or even [Epsilon] is sometimes incorrectly used in place of [Epsilon]. The correct symbol can usually be obtained from rub-on sheets.

* Some people are thrown by the alignment of Greek letters relative to the baseline of the text [13], especially when the letters must be applied from rub-on sheets after the rest of the text has been typed. The correct alignment of Greek letters relative to the baseline is shown in Figure 1.

When letters that extend below the baseline are modified by a subscript, the subscript can appear on about the same level as the baseline character. For example, in [[Chi].sup.i], the i appears to be a multiplier rather than a subscript of [Chi]. We can clarify the meaning by lowering the subscript as in [[Chi.sub.i], reducing its point size as in [[Chi].sub.i], or moving the subscript into parentheses after the main character as in [[Chi].sup.(i)].

THE BLACK LETTER ALPHABET

As Europe emerged from the Middle Ages, the Church-sponsored Latin was progressively supplanted by national tongues. Advanced mathematics continued to be written in Latin by university scholars until the beginning of the nineteenth century, but textbooks were written in German, French, Italian, English, or Dutch [5]. As nationalism in Europe grew and Germany rose to scientific and mathematical prominence, German became an important technical language [14]. The black letter alphabet (sometimes incorrectly called Gothic, which is really a somewhat heavy sans serif typeface) was the German national hand until the end of World War II [15], so we should not be surprised that it has contributed a few symbols to mathematics.

Among the most common black letters are F and R. The first, F (I or nonitalic Im in English), is the black letter "I." This symbol commonly represents the imaginary part (imaginarteil in German) of a complex number, such as the bi in a + bi. The second common black letter, R (R or nonitalic Re in English), indicates the real part (realteil in German) of a complex number, the a in a + bi. These two symbols are usually written with a variable that represents the complex number in question. If, as before, the variable z represents the complex number a + bi, then the usual notation is F (z) and R (z).

Vectors can be multiplied by one another in two ways. The outer product, or cross product, just gives another vector. The inner product, or dot product, gives an ordinary number, called a scalar. Many readers will recall the modern [Chi] x y and [Chi] * y for the outer and inner products, respectively. An older notation that we still sometimes see incorporates the black letter "r" and "h" as variables to represent the two vectors being multiplied: [r, h] denotes the outer product and (r, h) denotes the inner product [7].

Abstract space is not a place; it is a mathematical system containing undefined geometric objects and axioms. Thus, a space is just a collection, or set, of objects. A Hilbert space is a set of vectors for which the inner product of one of the vectors times itself equals the square of the absolute value of the vector ([Chi] * [Chi] = ~). A Hilbert space is traditionally denoted by the back letter "H," h.

In algebra, a Farey sequence is the sequence of all fractions between 0 and 1 in which both the numerator and denominator are nonnegative and have no common divisors other than 1. The traditional designator of a Farey sequence is the black letter "F," F. A subscript indicates the order of the set, that is, the number of elements it contains: [F.sub.n].

The lower case black letter "c," c, commonly denotes the number of elements in certain infinitely large sets of objects [10], such as all the points that make up a line. The c stands for continuum, in reference to the continuum of the straight line [6].

Similar to the many fonts in the English alphabet, black letters are available in a variety of styles. Thus, the ones we see in another document may not match. exactly the ones seen here. For consistency, all the black letters used in a document should be from the same font. Regardless of the style used, eyes unaccustomed to these characters may find it difficult to tell the various black letters apart, especially the upper case letters. When dealing with these letters, keep at hand a listing of the font to verify that the letters used are the correct ones.

Black letters are not available in all font sets, but the most common ones (F and R in particular) can be found on rub-on sheets. If not, acceptable substitutes such as Im and Re or italic English letters can be used, though they must be carefully defined to prevent any misunderstanding of their meanings.

THE HEBREW ALPHABET

Hebrew letters are rare in mathematics, but one appears often enough to warrant our attention. It is aleph, X, the first letter of the Hebrew alphabet. In modern set theory, some sets of objects are infinitely large. Among them are the set containing all the natural numbers (positive whole numbers, sometimes called the counting numbers) and the set containing all the points that make up a line (the set of the continuum). Georg Cantor, the founder of modern set theory, demonstrated in 1874 that the number of elements in most infinitely large sets can be counted [16, 17]. The number of elements in one of these denumerable sets is called the cardinal number, or cardinality. Cantor named this number [X.sub.o], pronounced aleph-null or aleph-nought. The number of elements in some other sets, however, such as the continuum, cannot be counted. Although the cardinality of an indenumerable set is usually denoted by c, in set theoretical notation we often see [Mathematical Expression Omitted], suggesting the role of power sets in its definition.

The letter X is not an English "N." Moreover, lacking sophisticated typesetting equipment, many of us resort to rub-on sheets. If you have to rub one of these symbols on, take care not to apply it upside-down: Inversion of this symbol has been the primary complaint about its use [12].

The X seems to be healthy and in common use (common in discussions of infinite sets, at any rate). Only occasionally is it replaced by some nontraditional, usually d (for denumerable infinity) or A, its English equivalent.

NONSTANDARD ENGLISH ALPHABETS

Upper case script letters from the English alphabet have their own meanings in mathematics. For example, script "R," R, sometimes indicates regions in space: [R.sup.2] represents two-dimensional space (the plane), [R.sup.3] represents three-dimensional space, and so on. Script "P," P, indicates a power set, as in P(A) = [2.sup.A], the set of all subsets of set A. If the appropriate script characters are unavailable, italic English letters can be substituted if they are defined clearly.

In the study of differential equations, script "L," L, denotes the Laplace transform [7], which usually looks something like L{f(t)} = F(s). Mathematicians use the Laplace transform to convert a difficult function into a simpler one so it can be evaluated more easily, then transform it back into the original. Often the symbol for British pounds Sterling, [pounds], is incorrectly used instead.

As with non-English letters, English script letters are progressively being replaced by more easily obtainable regular forms. A first choice may be any fancy-looking font available, perhaps a calligraphic font: A, B, C, D, and so forth. Script letters should be used whenever possible, but if not, make sure that the letters used are sufficiently different from other special characters that no confusion can result. Further, take care that they are clearly defined. The modern trend is to use italic English letters, giving [R.sup.2], P(A) L{f(t)}. Occasionally someone substitutes black letters for script, but it is better to avoid mixing alphabets, a tendency that can only lead to confusion.

Elliptic functions are so called because of their role in calculating the arc of an ellipse. The Weierstrassian elliptic function is traditionally denoted by the Icelandic "p," p, suggesting that this function is doubly periodic (periodisch in German). Sometimes the black letter "p," p, is used instead, but more often we see the italic English P or p. This symbol is usually written in standard function notation as p([Chi]).

The English alphabet has provided us with one symbol that no longer exists outside the pages of mathematical texts. The medieval long "s," written [Mathematical Expression Omitted] was used in all positions except at the end of a word and as a capital letter. Designed to save space in precious parchment manuscripts, it served in normal text from its invention in the eighth century until it was abandoned in the late eighteenth century [18]. Euler introduced the [Mathematical Expression Omitted] as an abbreviation for the Latin word summa to denote integration, a mathematical process of summing infinitesimally small elements. The correct size of the [Mathematical Expression Omitted] seems to mystify some people. It should be as tall as, but not substantially taller than, the function being integrated: ~ x dx in text but [Mathematical Expression Omitted] in display form. Moreover, the symbol ideally should stand as near to vertical as possible. When the [Mathematical Expression Omitted] is properly oriented, its limits are aligned vertically. Unfortunately, some word processors and typesetting systems furnish integral signs that are tilted to varying degrees, forcing us to live with what we can get.

The so-called blackboard bold [12] is not a separate alphabet. Rather, these letters are hand-drawn characters used by teachers during lectures to indicate boldface type. They usually incorporate double vertical strokes, as in A, B, and C. In a manuscript these letters should probably be marked for boldface, although sometimes they are meant to be typeset in script or black letter.

A FEW EXTRA HINTS

A few more points are worth noting.

* Plurals of non-English letters, as with English ones, are formed with an apostrophe: [Gamma]'s, [Alpha]'s, [Phi]'s.

* Non-English characters used as prefixes should be connected to their associated words with hyphens, just as we do with English letters, for example, T-value, H-bomb, [Mu]-meson, [Theta]-function, or [Alpha]-particle. Even when the letter is spelled out, as at the beginning of a sentence, the compound should remain hyphenated: Theta-function.

* When your output device is a dot matrix printer, non-English letters, if they are even available, probably will not print very well. An alternative is to use the English equivalents, but they must be carefully defined so the reader can figure out what they symbolize.

* If you are unsure of the exact letters or forms that appear in your manuscript, request from the author a list of special characters and ensure that they are clearly identified in the text.

SUMMARY

The appearance of non-English alphabets in mathematical documents often creates some confusion for technical communicators. That confusion sometimes stems from our inability to distinguish between the correct and incorrect letters and other symbols in context. Problems can arise from the similar appearance of different letters, the different appearance of the same letter in different fonts, the use of cursive variants, the author's or typist's substitution of incorrect characters for unavailable ones, or simply our unfamiliarity with the characters and their usage. The preceding review of non-English alphabets is not complete by any means, but it covers most of the situations we are likely to encounter in our daily work. I hope it clarifies some of the problems we see and equips us with the knowledge we need to deal with them effectively.

Acknowledgments

I wish to thank Charles Gregg, B. I. Literary Services, and Brian Thompson, Los Alamos National Laboratory, for their helpful and critical review of this manuscript. (*)In a business where an individual is lucky to have introduced one or at the most two new symbols, Euler invented a dizzying number, including [Mathematical Expression Omitted] and [Infinity].

REFERENCES

[1]E. T. Bell, Men of Mathematics (New York: Touchstore Books, Simon & Schuster, Inc., 1985). [2]B. L. van der Waerden, Geometry and Algebra in Ancient Civilizations (Berlin: Springer-Verlag, 1983). [3]G. Flegg, Numbers, Their History and Meaning (New York: Schoken Books, 1983). [4]J. E. Hofmann, The History of Mathematics (New York: Philosophical Library, 1957). [5]P. Beckman, A History of [Pi] (Pi) (New York: St. Martin's Press, 1971). [6]R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods (London: Oxford University Press, 1969). [7]S. B. Parker (Ed.), McGraw-Hill Dictionary of Scientific and Technical Terms, third ed. (New York: McGraw-Hill Book Company, 1984). [8]James & James Mathematical Dictionary, third ed. (New York: Van Nostrand Reinhold Company, 1968). [9]Van Nostrand's Scientific Encyclopedia, fourth ed. (Princeton, New Jersey: D. Van Nostrand Company, Inc., 1968). [10]W. H. Beyer (Ed.), CRC Handbook of Mathematical Sciences, sixth ed. (Boca Raton, Florida: CRC Press, 1987). [11]The Chicago Manual of Style, thirteenth ed. (Chicago: The University of Chicago Press, 1982). [12]E. Swanson, Mathematics into Type: Copyediting and Proofreading of Mathematics for Editorial Assistants and Authors, rev. ed. (Providence, Rhode Island: American Mathematical Society, 1979). [13]R. Schenkman, The Typing of Mathematics (Santa Monica, California: Repro Handbooks, 1978). [14]L. Hogben, Mathematics in the Making (Garden City, New York: Doubleday & Company, 1960). [15]Encyclopedia Americana, International Ed. (Danbury, Connecticut: Grolier, Inc., 1989). [16]P. J. Davis and R. Hersh, The Mathematical Experience (Boston: Houghton Mifflin Company, 1981). [17]C. B. Boyer, A History of Mathematics (Princeton, New Jersey: Princeton University Press, 1968). [18]R. Streckfuss, "The Rise and Fall of the Long S," Righting Words 4, no 3 (1990): 21-23.
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