Matematiksel Semboller-3


Although the great practical invention of zero has often been attributed to the Hindus, partial or limited developments of the zero concept are clearly evident in a variety of other numeration systems that are at least as early as the Hindu system, if not earlier. The actual effect of any one of these earlier steps in the full development of the zero concept - or, indeed, whether there was any actual effect - is by no means clear, however.

The Babylonian sexagesimal system used in the mathematical and astronomical texts was essentially a positional system, even though the zero concept was not fully developed. Many of the Babylonian tablets indicate only a space between groups of symbols if a particular power of sixty was not needed, so the exact powers of sixty that were involved must be determined partly by context. In the later Babylonian tablets (those of the last three centuries B.C.) a symbol was used to indicate a missing power, but this was used only inside a numerical grouping and not at the end. (NCTMp49)

Not to be overlooked is the fact that in the sexagesimal notation of integers the "principle of position" was employed. Thus, in 1.4 (=64), the 1 is made to stand for 60, the unit of the second order, by virtue of its position with respect to the 4. The introduction of this principle at so early a date is the more remarkable, because in the decimal notation it was not regurlarly introduced until about the ninth century after Christ. The principle of position, in its general and systemic application, requires a symbol for zero. We ask, Did the Babylonians possess one? Had they already taken the gigantic step of representing by a symbol the absence of units? Babylonian records of many centuries later -of about 200 B.C.-give a symbol for zero which denoted the absence of a figure, but apparently it was not used in calculation. It consisted of two angular marks as illustrated above on the right, one above the other, roughly resembling two dots, hastily written. About 130 A.D. Ptolemy in Alexandria used in his Almagest the Babylonian sexagesimal fractions, and also the omicron o to represent blanks in the sexagesimal numbers. This o was not used as a regular zero. It appears therefore that the Babylonians had the principle of local value, and also a symbol for zero, to indicate the absence of a figure, but did not use this zero in computation.Their sexagesimal fractions were introduced into India and with these fractions probably passed the principle of local value and the restricted use of the zero. (Cajori p5)

When the Greeks continued the development of astronomical tables, they explicitly chose the Babylonian sexagesimal system to express their fractions, rather than the unit-fraction system of the Egyptians. The repeated subdivision of a part into 60 smaller parts necessitated that sometimes “no parts” of a given unit were involved, so Ptolemy’s tables in the Almagest (c. A.D. 150) included both of these symbols for such a designation.

Considerably later, in approximately 500, Greek texts used this symbol, the omicron, the first letter of the Greek word ouden (“nothing”). Earlier usage would have restricted the omicron to symbolizing 70, its value in the regular alphabetic arrangement.

Perhaps the earliest systematic use of a symbol for zero in a place-value system is found in the mathematics of the Mayas of Central and South America. The Mayan zero symbol was used to indicate the absence of any units of the various orders of the modified base-twenty system. This system was probably used much more for recording calendar times than for computational purposes. (NCTM p49)

The Maya counted essentially on a scale of 20, using for their basal numerals two elements, a dot representing one and a horizontal dash representing five. The most important feature of their system was their zero, this character as illustrated, which also had numerous variants. (Smith p44)

It is possible that the earliest Hindu symbol for zero was the heavy dot that appears in the Bakhshali manuscript, whose contents may date back to the third or fourth century A.D., although some historians place it as late as the twelfth. Any association of the more common small circle of the Hindus with the symbol used by the Greeks would be only a matter of conjecture. (NCTM p50)

There is no probability that the origin will ever be known, and there is no particular reason why it should be. We simply know that the world felt the need of a better number system, and that the zero appeared in India as early as the 9th century, and probably some time before that, and was very likely a Hindu invention. In the various forms of numerals used in India, and in later European and Oriental forms, the zero is represented by a small circle or by a dot. Variations include these, as illustrated. (Smith p70)

Since the earliest form of the Hindu symbol was commonly used in inscriptions and manuscripts in order to mark a blank, it was called sunya, meaning “void” or “empty.” This word passed over into the Arabic as sifr, meaning “vacant.” This was transliterated in about 1200 into Latin with the sound but not the sense being kept, resulting in zephirum or zephyrum. Various progressive changes of these forms, including zeuero, zepiro, zero, cifra, and cifre, led to the development of our words “zero” and “cipher.” The double meaning of the word “cipher” today - referring either to the zero symbol or to any of the digits - was not in the original Hindu. In early English and American schools the term “ciphering” referred to doing sums or other computations in arithmetic. (NCTM p50)

The traditional Chinese numeration system is a base-ten system employing nine numerals and additional symbols for the place-value components of powers of ten. Before the eighth century A.D. the place where a zero would be required was always left absent. A circular symbol for zero is first found in a document dating from 1247, but it may have been in use a hundred years earlier. (NCTM p43)

Interestingly enough, the forms of the modern Arabic numerals are not the same as the Hindu-Arabic forms of the western world. For example, their numerical representation for five is 0 and their zero is representated by a dot. (NCTM p49)
This can be illustrated as shown; (Smith p70)

The various forms of the numerals used in India after the zero appeared may be judged from this table. (Smith p70)

This table illustrates some later European and Oriental forms. (Smith p71)

The name for zero is not settled even yet. Older names and variations include naught, tziphra, sipos, tsiphron, rota, circulus, galgal, theca, null, and figura nihili.(Smith p71)

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969

CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958


The ancient writers commonly wrote the word for root or side, as they wrote other words of similar kind when mathematics was still in the rhetorical stage. The symbol most commonly used by late medieval Latin writers to indicate a root was R , a contraction of radix, and this, with numerous variations, was continued in the printed books for more than a century. Thus it appears as such in the works of Boncompagni (1464), Chuquet (1484), Pacioli (1494), de la Roche (1520), Cardan (1539), Tartaglia (1556), Ghaligai (1521), and Bombelli (1572.) The symbol was also used for other purposes, including response, res, ratio, rex and the familiar recipe in a physician's prescription.

Meanwhile, the Arab writers had used various symbols for expressing a root, including this sign on the right, but none of them seem to have influenced European writers.

This symbol first appeared in print in Rudolff's Coss in 1525, but without our modern indices. It is frequently said that Rudolff used this sign because it resembled a small "r", for radix (root), but there is no direct evidence that this is true. The symbol may quite have been an arbitrary invention. It is a fact , however, that in and after the 14th century we find in manuscripts such forms as the following for the letter "r."

It was a long time after these writers that a simple method was developed for indicating any root, and then only as a result of many experiments. French, English, and Italian writers of the 16th century were slow in accepting the German symbol, and indeed the German writers themselves were not wholly favourable to it. The letter l (for latus, side; that is, the side of a square) was often used. In the 17th century our common square-root sign was generally adopted, of course with many variants. The different variants of the root sign are too numerous to mention in detail in this work, particularly as they have little significance. By the close of the 17th century the symbolism was, therefore, becoming fairly well standardised, although there still remained some work to be done. The 18th century saw this accomplished, and it also saw the negative and fractional exponent come more generally into use.

Some variations on the radical sign are as follows. The illustrations are the work of many different writers, including Stifel (1553), Gosselin (1557), Ramus-Schoner (1592), Rahn (1659), Stevin (1585), Vlacq, Biondini (1689) and Newton (1707).

(Smith p407 - p410)

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958

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