ZERO: IN THE PAST AND TODAY

There are several opinions about the origin of the concept of zero¹. One view is that zero was borrowed from the Greeks. When there was no number, Ptolemy would write the letter ο (omicron)—probably from the word οὐδέν, meaning “nothing.” It is possible that the sexagesimal zero derives from this sign.

India is most often regarded as the birthplace of zero. Zero has been known in Indian mathematics since the time of “Brahmagupta” (about 600 AD). It was represented by a dot or a small circle. Later, zero reached Europe via the Arabs.

The Arabs rendered zero as as-sifr, which literally means emptiness. In some places, zero was initially called something similar (cf. German Ziffer, Russian цифра). Around the end of the 15th century, another word for this concept emerged in France and Italy. Later, many languages borrowed zero from Italian and therefore call it zero today.

The term nulla figura—“no sign”—appeared in handwritten Latin translations and adaptations of Arabic works in the 12th century. The term nulla (Latin nullus—no, not any) is found in N. Chuquet’s (N. Chuquet) manuscript (1484) and in the first printed arithmetic (1478). Even in the 15th century, zero was not considered a number. A. Girard (A. Girard) was probably the first to recognize zero as a number (1697). L. Euler (L. Euler) included zero among the natural numbers in his “Universal Arithmetic” (1768).

When zero entered the Lithuanian language is now difficult to say precisely. Apparently, this word was not needed in old religious writings. We do not find zero in the first Lithuanian dictionary—K. Sirvydas’s “Dictionarium trium linguarum” (c. 1620; the first edition of this dictionary and its later editions (III–V) are now considered separate dictionaries; the dictionaries were reissued in 1979 and 1997).

Most likely, zero appeared in Lithuanian lexicography only in the 18th century. We find it in P. Ruigys’s dictionary. As stated in “Brockhaus Enzyklopädie” (p. 39), in German, until the 15th century, zero was denoted by die

¹ Fakti pateikiami iš Александрова Н. В. Математические термины. Справочник. Москва, 1978. С. 85–86; Brockhaus Enzyklopädie. Mannheim, 1991. Bd. 16. S. 38–39.

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Ziffer. In the century mentioned, its meaning broadened, and die Ziffer acquired the meaning ‘digit’. P. Ruigys records both meanings of die Ziffer:

Ziffer, f. Zahl, Rokótinė, ės, f. Suskaitytinė, ės, f.

Ziffer, die Nulle, Nullė, lės, f. (Ruig 413²)

As can be seen, the feminine form nulė was formed by analogy with German. Half a century later, K. Milkus copied exactly the same entry from P. Ruigys’s dictionary into his own (Milk 557). F. Kuršaitis also includes the word under discussion in his dictionary. He gives the form niulis:

Null, die, niūlis, io. Da geht N. mit N. auf, czia wįkas lygiei išėit. er ist eine wahre N., jis čyštai per nieką; kaip niekas. das ist N. u. nichtig, tai wįjai nieks; od. per nieką (F. Kurš 96)

It is unclear whether F. Kuršaitis’s nulis means the number (digit), since the examples illustrate only the phraseological use of die Null. In this case, the meaning of niulis would most likely be ‘nothing’.

Later, other forms of nulio also appeared. Interesting facts are recorded in the enormous Lithuanian-German dictionary by Aleksandras Kuršaitis (Frydrichas Kuršaitis’s nephew). Giving nulio as the main form, A. Kuršaitis also lists others: apskritulys, núolis, zeras (A. Kurš 1576). Apskritulys was taken from A. Vilimavičius’s grammar:

apskritulys (nulis, zero) Null m. (Vilim 209).

Incidentally, the meaning ‘circle’ (‘zero’) of apskritulio (nulio) is found neither in LKŽ nor in its card index. This may be a term used only by A. Vilimavičius.

Taken from M. Miežinis’s dictionary is zeras (F. Kuršaitis did not include A. Vilimavičius’s indeclinable form zero):

zeras, v. nulís, m. nulle. zero. nulb (0) (Miež 288).

The source for Nuolio is J. Šlapelis’s dictionary. Here nuolis appears only as an illustration:

išsipūsti, išsipučiu, -pūčiau – раздуться. Kàs yra núolis? – Núolis... núolis?... Tai išsipūtęs taškas (Šlap 212).

Interestingly, the indeclinable form zero meaning ‘zero’ continued to be listed as an international word in Lithuanian dictionaries for a long time. Zero can-

² Here and below, the numbers indicate page numbers.

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can be found in several “Dictionaries of International Words” (1936, compiled by K. Boruta et al.; 1951, translated by O. Katinskaitė et al.; 1969, translated by O. Doveikienė et al.). It is absent only from the 1985 dictionary (ed. V. Kvietkauskas) zero. One wonders whether zero was used in Lithuanian for so long. It most likely made its way into dictionaries simply because they were translations.

The first meaning of zero is stated very similarly in encyclopedias and dictionaries of various languages. The following should be regarded as a typical encyclopedic definition: the number 0, which, when added to any number, yields that same number (LTE VIII 254; TŽŽ 345).

In the main Lithuanian dictionaries, the meanings of zero are presented a little differently. Chronologically, the changes look as follows:

zero (2) 1. The numeral sign “0,” denoting the absence of magnitude and quantity; the boundary between positive and negative numbers. 2. figuratively nothing, no quantity (DŽ₁ 506)

zero noun (2) TrpŽ 1. math. The numeral sign 0, denoting the absence of quantity, and, when written to the right of a number, increasing it tenfold. 2. A conventional magnitude from which counting begins: As the temperature rises, the surface tension of liquids decreases and reaches zero at the critical temperature. 3. figuratively something very small, insignificant; nothing: How can that be: the state educated a person, teachers gave him a piece of their heart, and the result is zero (LKŽ VIII 892)

zero (2) 1. The numeral sign “0,” denoting the absence of magnitude or quantity; when written to the right of a number, increasing it tenfold; the boundary between positive and negative numbers: The thermometer fell, rose to ~zero. 2. figuratively nothing, no quantity. △ Absolute zero (temperature −273.16° C) (DŽ₂ 444)

zero (2) 1. A number denoting the absence of magnitude or quantity; the digit that denotes it: The thermometer fell, rose to ~zero.

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2. figuratively nothing, no quantity: △ Absolute zero (temperature – 273.16° C) (DŽ₃ 439, DŽ₄ 439)

Although at first glance the meaning of zero in the dictionaries seems essentially the same, closer inspection reveals differences. The meanings can be divided into three types. The most obviously similar presentations are those in DŽ₁ and DŽ₂. Here the emphasis is first on the fact that zero is the numeral sign “0,” that is, zero is understood as a digit or graphic unit. It is also added that it is the boundary between positive and negative numbers (though it is not specified whether that boundary is itself a number).

Another type of presentation is found in DŽ₃ and DŽ₄. Here it seems that the earlier meaning has been reversed. First, zero is said to be a certain category of number, and then the graphic expression of that number. In the definitions of both types, the logical category of number and its material expression—the graphic sign—are treated as one (first) meaning of zero.

Things are somewhat different in LKŽ VIII (the third type). Here the first meaning of zero is only the numeral sign, while the number (magnitude) is separated out and presented as the second meaning. This separation of meanings seems quite justified, though one could argue about their hierarchy. After all, logically, a certain magnitude comes first, not its graphic expression (although historically the concept of zero developed later than the sign 0 appeared). In this respect, the choice in DŽ₃ (DŽ₄) to define zero first as a number is welcome. The question of whether there is one meaning or two remains open. Apparently, in all DŽ dictionaries, number and sign are presented as a single meaning, following the analogous treatment of numerals in DŽ and LKŽ. In this respect, the distinction of two meanings of zero in LKŽ seems somewhat irregular. The reason was most likely the conditional nature of the magnitude of zero.

Incidentally, it is interesting that in the 1936 “Dictionary of International Words,” the first meaning of zero is not mathematical at all:

Zero (Latin) 1. nothing; 2. in mathematics—no quantity; the boundary between positive and negative quantities, 0; 3. figuratively—a worthless person, an insignificant person, someone who means nothing. (p. 386)

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What part of speech is zero³? It is not easy to answer this question in just one way. In the dictionaries and grammars of many languages (including Lithuanian), zero is treated as a noun (we infer that grammars classify zero as a noun from the fact that it is not even mentioned in their discussion of numerals). Why is zero not a numeral? It is difficult to say, since the morphological treatment of the numeral itself is problematic. LKM states that “a numeral is a nominative part of speech consisting of words that denote units of quantity—numbers, quantities of objects expressed by numbers, and their order” (p. 79). The definitions of zero show that it is a number, which means that, at least formally, it should belong among the units of quantity mentioned in the definition of a numeral. This is all the more so because LKG I states that numerals are “words that express an abstract number” (p. 604). Is zero not an abstract number?

The numeral is considered the most limited part of speech, that is, numerals contain only a small number of roots (although the treatment of some words, such as both, -i, half, one and a half, and so on, varies). It is rightly thought that the numeral is a highly heterogeneous part of speech: grammatically very different words are assigned to a single class. If we treated the facts morphologically, the class of numerals would disappear. DLKG acknowledges that some are close to adjectives (e.g., one, feminine one), others to nouns (ten, cf. eye), and still others to adverbs (ten, cf. much) (p. 238). But for now, we are not breaking with tradition.

Thus, if we look at zero purely morphologically, it is a noun, but semantically (and in our linguistics, numerals are distinguished semantically) it tends toward the numerals. It does so also because numerals are treated rather mathematically in our tradition: we have fractional numerals (used almost exclusively in mathematical language; after all, in ordinary speech we say half, not one second; and what should we call the verbal expression of 0.5?).

³As far as I have been able to ask, Lithuanian linguists (even scholars) are taken aback when they hear this question. Most often they start considering the numeral, but quite frequently the final answer is: “I don’t know.”

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Since numerals have no common morphological distinguishing criteria, their most characteristic feature as a part of speech is considered to be the categorical meaning of abstract quantity, an exact number (LKM 80; LKMA 221). It is also said that, because their meaning is definite, words with a descriptive meaning are not used with numerals—they do not take modifiers (LKMA 221).

The precision of zero as a number is defined as follows: it is a number which, when added to another number, yields that same number. The matter of having modifiers is another question. In DŽ₂, DŽ₃ (DŽ₄), we have already seen absolute zero. But do we understand zero in this phrase in the same way as in the phrase zero degrees (cf. one hundred degrees)? Obviously not. Or compare he gave three apples and the teacher wrote a large three. In the latter example, the substantivized numeral takes a modifier. In the case of zero, we can discern the reverse process. Zero (initially denoting the symbol 0—cf. a grade of three (3)) has now, in some cases, become numeral-like (e.g., another zero hours, match score—two:zero [goals]). The conversion of zero into a numeral is also supported by the reversal of the meaning of zero in DŽ₃ (DŽ₄), where zero is first and foremost a number, not a symbol. Thus, the morphological treatment of zero could depend on context—in one case a noun, in another a numeral (just as one, -a can be a numeral, adjective, or pronoun).

Approximately this kind of treatment of zero can be found in German dictionaries and grammars. There, cardinal numerals (Grundzahlen, or Kardinalzahlen) are listed beginning with zero (null) (e.g., Jud 174; Mikal 159). In dictionaries, null (numeral) and die Null (noun) are given separate entries, since their spelling also differs (VLŽ 95). Incidentally, VDR (p. 399) marks null as an adverb, but that would be a discussion at the level of German.

In French, numerals are sometimes not distinguished at all (a purely morphological approach); instead, they are discussed alongside adjectives as cardinal determiners (déterminants numéraux) (e.g., PPKG 78). Here, too, we find zero (zéro).

LKM states: “the numeral is a relatively small part of speech, hardly ever augmented with new words (emphasis mine—A. U.)” (p. 80). That “hardly ever” makes it possible, in certain cases, to draw zero toward the numerals.

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Zero is not a word that is often needed in spoken language. Apparently, it is more a word of written language. As the studies by L. Grumadienė and V. Žilinskienė show (DDRLKŽ 29), zero is among the 2,200 most frequently used words in written language (if these data reflect reality). In 28 samples, 76 occurrences of zero were found (a total of 1,200 samples of 300,000 word occurrences each were examined). The dictionary alone does not show where and how, specifically, zero is used.

Numeral use of zero is likely characteristic of nonspecialized spoken and written language, whereas in the exact sciences zero is quite thoroughly terminologized; there are numerous specialized terms, which clearly indicate its use as a noun. Here, for example, are several terms in which zero has a concordant modifier: right (MTŽ) / binary AKTŽ) / terminal AKŽ, AKTŽ) / left (MTŽ) / repeating (MTŽ) / computer (AKŽ, AKTŽ) / insignificant (AKŽ, AKTŽ) / normative (AKŽ, AKTŽ) / simple (MTŽ) / front (AKŽ, KTŽ) / conditional (MTŽ) / identical (MTŽ) zero. Two concordant modifiers are also possible: negative computer zero (AKTŽ), positive computer zero (AKTŽ). Less often, zero takes a nonconcordant modifier: zero of a function (MTŽ), zero of the time axis (AKTŽ), zero with a minus sign (MTŽ), zero with a plus sign (MTŽ).

There are also numerous terms in which zero has become a dependent component of the term. In such cases, it is most often in the genitive form: elimination of insignificant zeros (AKTŽ), zero environment (MTŽ), zero divisor (MTŽ, VLMŽ), zero-one law (MTŽ), zero property (AKTŽ), zero product (VLMŽ), number of zeros (VLMŽ), zero density (VLMŽ), zero theorem (MTŽ). There are terms in which zero appears as a dependent component in other case forms as well, for example: division by zero (AKTŽ), padding with zeros (AKTŽ). A few atypical compound terms (not in the form of a noun phrase) containing zero were also found, for example: without zero divisors (VLMŽ), homologous to zero (VLMŽ), homotopic to zero (VLMŽ).

As can be seen, zero has already made considerable headway in Lithuanian. It is no longer merely an elliptical numeral (and, at the same time, a conditional number)—new terminological meanings have emerged, most relevant in

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mathematics and computing. On the other hand, the use of zero as an ordinary number in nonspecialized spoken and written language prompts reflection on its morphological treatment as well, a matter that current Lithuanian linguistics consciously or unconsciously leaves unaddressed. Perhaps it is worth considering a dual treatment of zero, which would make it possible, in certain cases, to expand our semantically distinguished class of numerals.

Abbreviations

A. Kurš – Kurschat A. Littauisch–deutsches Wörterbuch. Thesaurus linguae Lithuanicae. Göttingen, 1970. Bd. II.

AKTŽ – Aiškinamasis anglų–lietuvių kalbų kompiuterijos terminų žodynas. Kaunas, 1997.

AKŽ – Paulauskas K. V., Jasinevičius R. Aiškinamasis kompiuterijos žodynas. Kaunas, 1995.

DDRLKŽ – Grumadienė L., Žilinskienė V. Dažninis dabartinės rašomosios lietuvių kalbos žodynas (mažėjančio dažnio tvarka). Vilnius, 1997.

DLKG – Dabartinės lietuvių kalbos gramatika. Vilnius, 1997.

DŽ – Dabartinės lietuvių kalbos žodynas: DŽ₁ – I leid., Vilnius, 1954. DŽ₂ – II leid., Vilnius, 1972. DŽ₃ – III leid., Vilnius, 1993. DŽ₄ – IV leid., Vilnius, 2000.

F. Kurš – Kurschat F. Deutsch–littauisches Wörterbuch. Halle, 1874. Bd. II.

Jud – Jude W. K. Deutsche Grammatik. Braunschweig, 1966.

LKG I – Lietuvių kalbos gramatika. Vilnius, 1965. T. 1.

LKM – Paulauskienė A. Lietuvių kalbos morfologija. Vilnius, 1994.

LKMA – Paulauskienė A. Lietuvių kalbos morfologijos apybraiža. Vilnius, 1983.

LKŽ VIII – Lietuvių kalbos žodynas. Vilnius, 1976. T. 8.

LTE VIII – Lietuviškoji tarybinė enciklopedija. Vilnius, 1981. T. 8.

Miež – Miežinis M. Lietuviszkai–Latviszkai–Lenkiszkai–Rusiszkas Žodynas. Tilžė, 1894.

Mikal – Mikalauskaitė A. Trukanaitė D. Deutsche Grammatik. Vokiečių kalbos gramatika. Kaunas, 1997.

Milk – Mielcke Ch. G. Littauisch-deutsches und deutsch–littauisches Wörter-Buch <...>. Königsberg, 1800.

MTŽ – Matematikos terminų žodynas. Vilnius, 1994.

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PPKG – Balaišienė J., Mickienė V. Praktinė prancūzų kalbos gramatika. Vilnius, 1996.

Ruig – Ruhig Ph. Littauisch–Deutsches und Deutsch–Littauisches Lexicon <...>. Königsberg, 1747.

Šlap – Šlapelis J. Lietuvių ir rusų kalbų žodynas. Vilnius, 1921.

TŽŽ – Tarptautinių žodžių žodynas. Vilnius, 1985.

VDR – Wörterbuch deutsch–russisch / von E. Daum und W. Schenk. Leipzig, 1990.

Vilim – Vilimavičius A. Mažas vokiškas kalbamokslis <...>. Heidelbergas, 1919.

VLMŽ – Mačienė V. Vokiečių–lietuvių kalbų matematinis žodynas. Vilnius, 1994.

VLŽ – Križinauskas J. Smagurauskas S. Vokiečių–lietuvių kalbų žodynas. Vilnius, 1992. T. 2.

Alvydas UMBRASAS

Alvydas UMBRASAS Lietuvių kalbos institutas Antakalnio g. 6 LT-2055 Vilnius El. paštas [email protected]

Received 20 October 2000

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