Zero: The Exceptional Number

by: Stefanos Gialamas, Ph.D., President, ACS Athens
Miriam McCann

Consortium of Mathematics and its Applications
Number 36, Winter 1990

The history of zero has two basic branches: that meaning “empty space,” and that which represents a number and is used for computation. Today, the same symbol and the same words are used for both situations (in the United States, at least). The use of one symbol for both is tied to the adoption of our modem number system.

 

In spoken and written language, the use of a zero is virtually unnecessary. For instance, the number 3052 is “three thousand fifty-two.” The words indicate the value. It becomes necessary to use a zero when we wish to write symbols for the numbers and indicate value using a positional number system or when the number is zero. How did this come about?

 

The Babylonians provided us with some of the first recorded uses of symbols for numbers. Wedge shaped symbols produced by a stylus pressed into clay tablets indicated a sexagesimal (base 60), positional system. That is, the position of the symbol within the number indicated its value. No symbol for indicating an empty position in a number was found on these tablets, which date back to over 4000 years ago (2300-1600 B.C.).

 

In the sexagesimal system, there are few occasions when the use of a symbol to indicate an empty position is needed, and the early Babylonians relied on the context to make dear the value of the number system as written.

 

About the sixth century B.C., a specific symbol that indicated the absence of a symbol began appearing within numerals. This new addition to the symbolic notation did not appear at the end of a numeral, so some confusion about the value could still occur. Astronomical tables in later years began to show the “missing” symbol at the end of numerals [Menninger1969].

The Babylonians were not the only civilization to leave behind evidence of a positional system and some way to indicate a missing numeral. The Egyptians also used a symbolic system to indicate numerical values, but their system was not positional. It did not use or require the use of a zero to prevent misinterpreting the value. The Mayans of Central America and southern Mexico used a vigesimal (base20) system, which was also positional. They employed symbol for zero that looked roughly like a half-closed eye [Cajori, 1926]. This number system was being used during the first century A.D. and was found on early calendars [Kline 1953].

The Incas of Peru kept records of transactions by knots on cords, called quipu. This decimal (base 10) system used different types of knots as well as position of the knots to indicate value. A gap between sets of knots showed the absence of a value (position). The absence of a knot showed that the last (units) position was empty. Cords could be combined, and an additional cord was attached to indicate the sum. It was probably easy for the “reader” to tell when a missing number was indicated. Cords with no knots could also indicate zero [Ascher & Ascher 1981].

Native North American people used numeration systems of various bases (3, 4, 5, 8, 10, and 20), but there is no indication of place values or the use of a zero [Boyer 1944].

Ptolemy (A.D. 130) was using a missing numeral symbol in tables of cords [Boyer 1968] which resembled the Greek letter

omicron. Ptolemy apparently used the °0° symbol only with the sexagesimal fraction notation and not as part of the alphabetic numeral system of the Greeks [Heath 1981]. °0° was an abbreviation of the Greek word ouden or outhen meaning “nothing” [Menninger 1969].

The zero that represents a number and is used as such first appeared with the Hindu civilization in India. The earliest recorded evidence found so far is on the Gvalior inscription, found on the wall of a temple near Lashkar in Central India.

The inscription gives the date 933(our A.D.870) and lists gifts to a temple. The numbers “270” and “50” appear, using a small circle for zero [Menninger1969]. This zero was probably in use long before recorded evidence shows.

The mathematicians of the civilizations mentioned so far were aware of the concept of zero and references were made in their writings, but they did not have an established symbol representing the concept. Since today in mathematics we use a large number of symbols, it is hard to comprehend that the early mathematicians (actually until the Dark Ages) were limited in expressing their concepts in literature.

Sunya, in Sanscrit (sixth to eighth centuries A.D.) meant “empty” and became the Arabic as-sifr (“the empty”) as the Hindu digits began expanded Hindu numerals spreading westward in the ninth century. During the thirteenth century A.D., two Latin words, eifra and zefirum (or eephirum) were used to mean zero. They were adapted into Latin from Arabic as the numbers became known kept written material from the lower in Europe [Menninger 1969]. These Latin words later became cipher and zero. The confusion created by a single word (cipher) meaning a number and also a digit which had numerical value, or of two words (cipher and zero), which each meant the same thing, helps to explain why it seemed to take so long for the new number system to become adopted.

Zero was also still used as a placeholder (the ancient notion of “empty” position) as well as a number in calculations. The dual role of zero, however, was not an easy concept to accept.

Early computations were done on various forms of counting boards, which existed in many civilizations. Some counting devices, such as the various forms of the abacus, are still being used today, while others are only known to us because they were illustrated on such items as vases, tombstones, or other artifacts [Menninger 1969J.

Sand tables were used prior to the ninth century for computations on a flat surface covered with a light coating of sand. Marks in the sand indicated number values; computations were completed and the sand smoothed to begin again. Other computational devices included counting boards divided into columns or rows for place value, with various types of markers used to indicate value. The markers were moved on the board as the computations were made [Dantzig 1959] and the results of the computations were recorded on something more permanent. A form of zero was used on these devices to indicate an empty cell, column, or row.

From India, the concept of zero spread to China and was incorporated into an existing rod system of numbers [Boyer 1968]. The rod numbers were positional and decimal, but there is no indication that they were used in computation. The Chinese also used a named place-value system using Chinese characters for written numbers, which did not require the use of a zero.

Hindu numerals spread to Europe as early as A.D. 1000 when Gerbert (who became Pope Sylvester II) used the digits learned from the Arabs on counters of a counting board. The digits were not accepted then because it was not understood that they could be used for computations without a counting board. Hence, their usefulness was not appreciated. The Hindu numbers, including zero, gradually gained acceptance in Europe through their use by merchants and tradesmen. Texts written by arithmeticians and calculators also contributed to the spread of the new system [Menninger 1969].

The invention of the printing press in the fifteenth century expanded Hindu numerals to more of the population as printed matter became more accessible and available. Some forces, however, helped to slow the spread, including the social structure, which kept written material from the lower classes. Counting devices, when used by those who were adept, were speedy and accurate, and therefore, made a written numeral system somewhat obsolete.

What one can recognize as a zero today did not come into common use until much later in history. Besides the Babylonian, Mayan, and Greek symbols, a few other symbols were used prior to the “0” of modern times.

The Hindu symbol was first a dot and later a “0” crossed by a horizontal or slanting line. In the translations of Al Khowarizmi’s astronomical tables, three signs for zero appear (j, O, and t-an abbreviation of teca or theca). These symbols were used periodically by various writers over the years [Cajori 1928]. Even today, zero is occasionally written with the slash through it to distinguish it from the letter 0 (particularly in computer applications).

The fact that computations with zero seemed to have some strange results did not help people to understand the number. The difficulties with understanding the results of some calculations with zero were more evident as the new numbers were being assimilated into Western Europe.

In the ninth century, Mahavira wrote that a number divided by zero remains unchanged [Kline 1972]. Ehaskara (A.D. 1114) felt that a fraction with a denominator of zero remained the same even when anything was added to or subtracted from it. He also stated that a number divided by zero is an infinite quantity [Kline 1972). These ideas were undoubtedly difficult to comprehend for those who were unfamiliar with computations using the 10 digits. If computational results could not be explained in physical terms, the results did not have meaning.

zero_table1Subtraction of a number from zero was a particularly difficult concept to grasp. The resulting negative number was incomprehensible to some. Pascal (sixteenth century A.D.) regarded the subtraction of four from zero as “utter nonsense” [Kline 1972]. As late as the nineteenth century, Augustus DeMorgan felt that a negative number as a solution to an equation was “inconsistent, absurd” and had no “real meaning” [Kline 1972].

One fifteenth century French writer felt that zero was the creator of confusion and difficulties [Menninger 1969]. He noted that zero in front of a number (e.g., 03) did not affect the value of the number but zero behind the number multiplies it by 10 (e.g., 30).

 

zero_table2zero_table3

It is the number zero and its unique computational “rules” that seem to cause the most problems. In order for students to use the number properly, they must have some understanding of more advanced mathematical concepts. The first exposure is at the elementary level with zero as “nothing,” which seems adequate for the young child, but which does not aid understanding when division using zero is encountered. In this case, it is important to understand the idea of inverse relationships in mathematics (specifically, that multiplication is the inverse operation for division). Without this understanding of mathematical structure, students will continue to have difficulty understanding division by zero.

Is the learning of both numbers and the concept of zero by children parallel to the development of the number system? Some aspects of the process seem so. Young children learn to count verbally starting with the number “one” just as our ancient ancestors did. Their early experiences with numbers are associated with the number of items in a set of objects, called numerosity. This first exposure does not include zero; however, the concept of an empty set is not a difficult idea and young children are quite aware when all of the items in question are gone. Pre-schoolers are also able to add and subtract, as long as the operations are performed on sets of objects. They are even able to do this in a verbally presented hypothetical situation, as long as the numbers are small (1,2, 3) and the numbers being used are associated with items [Hughes 1986].

For instance, if a child is asked, “What does 1 and 2 make?” he does not usually know. However, “What does 1 block and 2 blocks make?” elicits the correct answer. The pre-school child even accepts the removal (subtraction) of all of the items in the set resulting in a cardinality of zero.

These early childhood experiences remind us of the early civilizations who were counting and calculating using objects. What is surprising to many is that it took many centuries for people to adopt the ten-digit numerical system for computational purposes. A series of studies reported by Martin Hughes [1986] with pre-school-and early primary-aged children in England sheds some light on this difficulty.

The children in Hughes’ study were presented with 4 “tins” containing “bricks.” Each tin contained 0, 1, 2, or 3 bricks. After the child saw and counted the number in each, the tins were covered and moved and the child was asked to tell how many were in each tin. Since the tins were identical, correct answers were purely chance. Then, the children were told they could give themselves hints about the numerosity of the contents by marking the paper attached to the top of each tin. The type of mark was entirely up to the child [Hughes 1986].

Successful students, those whose marks allowed them to identify the contents of the tin immediately after making the marks as well as at a later time, used either the symbols 0, 1, 2, 3, or tally marks, or the same number of some created symbol or picture. In a similar study, the students were given magnetic numbers to use on the tins. Some students responded by placing the appropriate number of pieces on the tin rather than the appropriate number symbol [Hughes 1986]. When the tester showed the students how they could indicate the number in each tin, they caught on quickly and were successful in marking the tins and identifying the contents from their indications.

The same type of game was played where the number of bricks was increased or decreased, and the children were to indicate what had happened. Only the students with the greater mathematical ability used formal symbols to indicate the operation when initially introduced to the game [Hughes 1986].

Children in both studies indicated that the empty tin had nothing in it either by placing nothing on the tin or using the symbol 0. The students who used the number of pieces or tally to indicate the numerosity left the top of the tin blank, while those who used the number symbols to indicate the count used the zero to indicate the empty tin [Hughes 1986].

These studies with children indicated that they did not pick up the ideas that the counting, adding, and subtracting they were doing on paper in their math classes had any connection to the counting, adding, and subtracting that they could do with objects. With instruction, however, the students did see the connection and were successful in future games.

zero_table4      zero_table5

The same lack of understanding seemed to be part of the reason the Hindu system of numbers and calculations was so long in being accepted. The use of calculating devices was widespread, and the objects on the boards represented sets of things; the abstract computations required a whole new way of dealing with numbers.

Just as children need to be taught how to relate counting and the concept of zero as a number, so did the people in our history. Until printing enabled the skill of computation to spread quickly and people opened their minds to something new, the spread of the Hindu numbers and zero was bound to be slow.

zero_table6 zero_table7

References

Ascher, Marcia and Ascher, Robert. 1981. Code of the Quipu. Ann Arbor, MI: University of Michigan Press.

Boyer, Carl B. 1944. Zero: the Symbol, the Concept, the Number. National Mathematics Magazine. 18:323-330.

Boyer, Carl B. 1968. A History of Mathematics. New York: John Wiley & Sons, Inc.

Cajori, Florian. 1926. A History of Mathematics. New York: The Macmillan Co.

Cajori, Florian. 1928. A History of Mathematical Notations, Vol. I. LaSalle, IL: The Open Court Publishing Company.

Dantzig, Tobias. 1959. NUMBER: The Language of Science. 4th ed. New York: The Macmillan Co.

Hughes, Martin. 1986. Children and Numbers. New York: Basil Blackwell Inc.

Kline, Morris. 1953. Mathematics in Western Culture, New York: Oxford University Press.

Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.

Menninger, Karl. 1969. Number Words and Number Symbols: A Cultural History of Numbers. Cambridge, MA: M.I.T. Press.

The Figures In This Article Are From Karl Menninger’s Number Words And Number Symbols: A Cultural History Of Numbers. Cambridge, Ma: The M.Lt. Press, 1969.

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