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The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power. G. B. Halsted QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. ZERO: THE IDEA AND THE NUMBER ERIN FITZGERALD-HADDAD Dr. Elena Marchisotto Math 331, Section #15946 17 May 2006 Fitzgerald-Haddad 2 Zero is both invention and discovery. The numeral zero was invented, for the symbol could have taken any shape and the meaning would not have been altered. While zero as a number was discovered, as the value had always existed, although not recognized. For example early man could have zero sheep, should all be slaughtered or sold, but the quantity of sheep, zero, would no longer be of concern. Thus, it was not until the 7th century that the number was discovered by Indian mathematicians. However, zero would not be fully accepted by the western world until the 17th century. This paper will explore the history of zero’s development as a number as well as the invention of the symbol. Charting the progress of zero from placeholder, to the inclusion of zero in the system of real numbers we will then explore modern mathematical operations acting on zero. Finally the paper will discuss elementary classroom lessons offering students greater conceptual understanding of the impossibility of division by zero. The earliest evidence of the use of zero as a placeholder has been found in Babylonian cuneiform. The Babylonian number system of base 60 is referred to as a sexagesimal system. Numbers greater than 59 were represented as repeated groupings of 60, as we use groupings of 10 today. Unlike our decimal number system, the Babylonians used only two symbols, one mark having the value of 10 and the other having the value of one. The Babylonian system was a positional number system written horizontally, with the symbols of greatest value to the left and lesser values to the right. The arrangement of the symbols was crucial to accurate numerical representation. For example the value 3,661, where we have, in base ten (3103) + (6102) + (6101) + (1100), or 3000+600+60+1=3661,the Babylonians would, in base sixty, have (1602) + (1601) + (1600), or 3600+60+1=3661. The Babylonians would represent this value with three marks of the symbol for one, as there is one unit of each of the positional values Fitzgerald-Haddad 3 comprising the number 3661. However, as you can see, because a single symbol could represent 1, 60, 3600, or even 216,000 (603) there was much confusion in this system. The written representation was based on the use of the abacus. Numerical value on the abacus was clear in the arrangement of positions of values, but the written representation was not as clear, because a numerical position having no value was represented by a space. Often the space would not be recognized or forgotten by a hurried scribe. To refer back to the previous example of the number 3661 written with three marks, if a space was to be placed between two of the marks, the number would then represent a very different value, one greater than 216,000. Around 300 BCE the mark of two slanted wedges was used to represent the empty place (Seife 15). Thus the numbers 61 and 3601, which were each represented by two single wedge marks, were distinguished from one another. 3601 was written with the two slanted wedges separating the single wedge marks. While the Babylonians developed a symbol as a placeholder for zero, they did not recognize the symbol as a value. As a result, zero was not considered a number. Instead it was a form of numerical punctuation. In fact, this symbol is also seen in ancient Babylonian texts used as a period or as a symbol to connect a word to its translation (Kaplan 12). Interestingly, the zero marker has not been found to have been used at the far right position, but only within a number. Explanation has been offered as that of current verbal use of numbers. For example, one may refer to the cost of an item as three-fifty. Understanding the context of discussion and value of the item, one can distinguish that a magazine would cost three dollars and fifty cents, or a kitchen appliance three hundred fifty dollars. Obviously the earliest use of a symbol for zero was quite imprecise and incomplete. The Mayan civilization in Central and South America, between the 4th and 6th centuries CE (Ifrah 495), had a more fully developed number system using zero as a placeholder. They Fitzgerald-Haddad 4 used their number system extensively in creating a complex calendar system and for astronomical calculations. The Mayan number system was a base-20 system, written vertically with three symbols; a dot represented one, a horizontal line represented five and the zero symbol was an oval shape resembling a clamshell. The system was not a true base twenty system, but place values were such that the lowest position of the vertical number was units up to 20, the next position was 20, the third position was 1820, the forth 18202, and so on (Bunt 230). For example the value 3661 in this system is shown as 10(1820) + 3(20) + 1(1), or 3600+60+1=3661. The Babylonians would represent this number with the dot symbol for one in the lowest position, three dots in the next position for the three groups of twenty, and two bars representing ten groups of 360. Unlike the Babylonians the Maya used the zero marker in the lowest, final numeral position. The number twenty was represented by a dot in the upper position and the zero symbol below to indicate the absence of additional units. This number system “enabled them to express numbers going beyond 100,000,000” (Ifrah 401). There is no evidence that the Maya used zero in calculations, but they did begin the calendar with zero, as the beginning of time as well as the first day of each month. Ideas about numbers evolved and as the Maya civilization fell into decline the Indian civilization began to flourish. By 500 CE the Hindus had developed a positional number system composed of nine digits. While it is not clear exactly how the current base-ten number system composed of ten digits evolved, there are a number of copperplate deeds, which provide evidence for the evolving number system. The oldest example of a copperplate property deed using a nine-digit place value number system dates 595 CE (Ifrah 437-38). Another example dated 876 CE, of a stone inscription, indicates usage of a ten-digit base ten number system (Ifrah 438). The deed specifies the numbers 270 and 50, the form of the zero is the circle as it is known Fitzgerald-Haddad 5 today, differing only by the fact that it is written in a smaller size than the other digits, suspended at a higher level than the rest of the script. From these artifacts we find examples of the current numeration system at a given time in history. The symbol for zero is said to have evolved from a dot to the circle form we use today. Kaplan describes the origin of the circle stemming from the use of sand on the counting boards of India (49). The sand provided a record of calculations as removed markers left an indentation in the sand. This indentation was a circular shape made by the round markers. The written record of calculations on the counting board included the mark holding the empty place, thus the circle came to be the symbol for zero. From the use of the digit zero to the study of the mathematical operations of zero, recognition of zero as a number followed. In early times, numbers represented concrete objects as a means of naming a collection. From the start zero operated differently from all other numbers, as it does not name a collection. Even today, one does not state that they have zero coins, but instead, “I have no change” is the common expression. To have something indicates the presence of that which one counts. Linguistically we do not represent by number that which does not exist. Zero enabled the abstraction of mathematics. This abstraction in which calculations could be made, apart from concrete objects of count, necessitated the exploration of numbers for the understanding of the relationships of operations and the result of such operations on the given number of zero. This abstraction necessitated rules of operation for such an unusual number. Zero could operate with other numbers, but it was observed that zero did not operate in the same manner as other numbers. For example, any two numbers added together results in a sum larger than either of the two original numbers, unless one of the numbers is zero. It was also Fitzgerald-Haddad 6 observed that when one number is subtracted from another number the resulting value is less than the original number, again unless the number subtracted is zero, then the value is unchanged. Conversely in multiplication zero causes great change, for any number multiplied by zero becomes zero. Division by zero, as we will later discuss, is another matter, as the concept causes much confusion in the 7th century and for centuries to come. “The Hindu mathematician, astronomer and poet, Brahmagupta (588 – 660 CE), is credited with both the introduction of negative numbers and zero into arithmetic” (Pogliani et al. 732). In 638CE Brahmagupta offered the first writings on arithmetic operations of zero in Brahmasphutasiddhanta (The Opening of the Universe), the first written text on zero (St. Andrews). Brahmasphutasiddhanta was a lengthy volume written in verse on astronomy and mathematics. In the text Brahmagupta defined zero as the result of subtracting a number from itself. He then addressed addition and subtraction of positive numbers, negative numbers and zero, and the resulting sum in terms of positive or negative in a manner that continues to hold true today. Regarding multiplication involving zero, Brahmagupta discerned the product to always be zero, but in division by zero, he simply described a fraction with a denominator of zero, leaving many questions about zero and the operation of division. Additionally, he wrongly stated that division of zero by zero is zero (Seife 70-71). The slow evolution of an understanding of zero is evidenced by the passage of five hundred years time until another Indian mathematician, Bhaskara (1114 – 1185 CE), returned to the topic of division by zero. In his text, regarding division by zero, Bhaskara follows the thinking of Brahmagupta in stating that a number divided by zero becomes the fraction with a denominator of zero. He then follows with further explanation, “In this quantity consisting of that which has cipher for its divisor, there is no alteration, though many may be inserted or Fitzgerald-Haddad 7 extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth” (qtd. in Kaplan 73). According to Kaplan, Bhaskara is asserting in this passage that any number divided by zero is equal to infinity. However, Bhaskara did not define infinity. Of course Bhaskara like Brahmagupta before him erred in his understanding of division by zero. Bhaskara’s error may be due to the line of thinking followed by some young children learning about division by zero today. It concerns a scenario of a basket of apples and the question of how many times one can remove zero apples from the basket. From this thinking it is conceivable that one would arrive at the solution that one could reach into the basket an infinite number of times, removing zero apples each time. The problem with this scenario is that it equates division with subtraction. One can subtract zero apples and the quantity remains, in fact there can be an infinite number of zero apples subtracted and the original quantity will remain unchanged, but division is another matter. Division is defined by multiplication, as it is the inverse operation. In order for division by zero to equal infinity, then it must also be true that zero multiplied by infinity is equal to any number, and from that then all numbers are equal. We know that this cannot be true because each number is unique. Bhaskara did make advancements in contemplating arithmetic operations with zero. He correctly stated that 02 = 0, which likely came of the understanding that any number, including zero, multiplied by zero is equal to zero, thus 0 0 = 0. Because we can say that 0 0 = 02, it then follows that 02 = 0. He also asserted that 0 = 0 (Kaplan 119), which comes of the fact that 0 0 = 0. While the Indian mathematicians were contemplating zero and the result of division by zero, the Arabs, through the conquest of India, were embracing the current knowledge of Indian Fitzgerald-Haddad 8 mathematics and their system of numeration. The Arabs readily absorbed the knowledge of their conquered peoples with scholars translating texts. These translated texts may explain why, although Indian mathematicians are responsible for our system of numeration and the discovery of zero, it was once disputed that the Arabs were responsible for such advances. Thus the term Arabic numerals, is more correct today as Hindu-Arabic numerals. The Arabs were instrumental in spreading the concept of zero, just as they were instrumental in the development of the English words zero and cipher. The Arab word for zero, sifr, gives the English cipher. Sifr is from the Hindu sunya, meaning void. Sunya was used around 400 CE to indicate the empty column of the abacus. The Latin zephirum, used by Fibonacci in the 12th century, is from the Arabic. And finally zero, from zephirum. Philippi Calandri, in his work De Arithmetica Opusculum, printed in Florence in 1491 is credited with the first print use of the word zero (Ifrah 481). In addition to the Arab influence, trade and commerce was a driving force in the spread of these new ideas about numbers. In the last part of the 11th century Leonardo of Pisa, or Fibonacci as he came to be known in mathematics, had been traveling throughout what had become the Arab world. Spending time in Northern Africa he learned mathematics from the Muslims (Seife 78). Fibonacci demonstrated characteristics of a mathematician himself. With this knowledge, upon returning to Italy, he wrote Liber Abaci (The Book of the Abacus) in 1202. Fibonacci described this numeration system as “the best of the calculating systems he came on [in his travels]” (Kaplan 107). At last, the Hindu-Arabic numeration system had been effectively introduced to Europe. The system was much more effective for calculations than the Roman numeral system. The new system enabled numerical calculations, unlike the Roman system in which calculations were made on an abacus and then the numerals were used solely for recording Fitzgerald-Haddad 9 purposes. This practice was due to the difficulty in combining the values of the symbols. For example, to add the quantities 1996, as the Roman numeral MCMXCVI and 14, as the Roman numeral XIV, the process requires combining values and then recombining until the simplest form is reached. Thus MCMXCVI + XIV = MCMXCXX = MCMCX = MMX = 2010. The new system enabled merchants and traders to quickly make calculations with numbers alone. While many in the business world embraced the new system for calculations, others viewed the numbers with suspicion. There was a fear of forgery, due to concern that a zero could easily be changed to a six or a nine. “At first local chambers of commerce resisted the adoption of the new Hindu-Arabic numerals” (Baumgart et al. 242). Outside the business world there was much resistance to the change, stemming from a fear of a symbol that represented nothing. The Catholic Church resisted converting to the new system and due to the great influence of the church many were skeptical. Further, zero did not operate in the same manner as other numbers, which lead to further resistance of a system with such a number. Europeans could not accept the logic that while in operations of addition or subtraction zero had no consequence, but in multiplication zero was of tremendous consequence. The idea that a number multiplied by zero resulted in a product of zero was inconceivable. In operations of division, from Bhaskara, the connection between zero and infinity was unacceptable. Matters of zero were further confused by the fact that zero alone signified nothing, but when added to the end of a series of numerals, the value was increased ten-fold. In 1299 Hindu-Arabic numerals were banned in Florence, requiring that all figures of accounts be written out (Seife 80). One explanation for the ban, offered by Bunt, was that the representatives responsible for verifying accounts were not knowledgeable about the system, thus the government discouraged the usage (229). However, usage of the numerals persisted and Fitzgerald-Haddad 10 in the end commercial interests brought about the reversal of the law. As a side note, the term cipher, as in the root of the word decipher is from the usage of the Hindu-Arabic numeration system during this period. The system with the cipher was used as a code to “send encrypted messages – which is how the word cipher came to mean secret code” (Seife 81). Competitions between abacists and algorists became popular, as the argument continued on the most efficient system of calculation. The competitions were documented in artwork. Ultimately the algorists defeated the abacists, as with training the algorists could perform calculations involving larger numbers at greater speeds. From the widespread adoption of the Hindu-Arabic numerals, mathematics was able to progress in a manner that would have been impossible without zero. Let us now consider some of the exclusive properties of zero within a field. In general terms a field is defined as a set of numbers operating under addition and multiplication in which the result of an operation is a unique solution and a member of the given field. Further there are given properties of the field, including closure, commutativity, associativity, distributivity, identity elements, and inverses (West, 175). Zero is the additive identity, as adding zero to any number does not change the number, the identity is retained. The multiplicative property of zero is unique, as the product of any number and zero, is zero. Zero as an exponent, causes any non-zero number to become one. To consider the implications of exponents we must recall the laws of exponents. 1. am = a a a … a, (for m factors), where m is a positive integer 2. a0 = 1, where a 0 3. a-m = 1/am, where a 0 4. am an = am+n Fitzgerald-Haddad 11 5. am/an = am-n, where a 0 6. (am)n = amn 7. (a/b)m = am/am, where b 0 8. (ab)m = am bm 9. (a/b)-m = (b/a)m, where a 0, b 0 (Billstein, 301) The result of zero acting as an exponent can be arrived at by considering the pattern: n1 = n, n2 = n n, n3 = n n n, n4 = n n n n, for each successive exponential value, the previous term is multiplied by n. We can reverse the sequence: n4, n3, n2, n1, n0, n-1, n-2. For each successive term we are now dividing by n. Thus from n1 = n, we divide by n to arrive at n0 = 1. Additionally we can consider the fraction n2/n2: we know this fraction to be n/n which is equal to 1, but in exploring implications of exponents n2/n2 = n2-2 = n0 and again we find that zero as an exponent acting on any number results in 1. But what about the exponent of zero acting on the base of zero? From the previous discussion on zero as an exponent, we found that in exploring the base of n with the exponent zero, n0 = 1. If n = 0, then can we say 00 = 1? Because this would necessitate a division by zero, which as we will soon see cannot be done we must say 00 1. We can say 04 = 0 0 0 0 = 0, 02 = 0 0 = 0, and 01 = 0. It should follow then that 00 = 0. But because of the discrepancy of two different values of 00 = 0 and 00 = 1, we find that zero acting as an exponent on zero is indeterminate. Many can recall from elementary school that one cannot divide by zero, but what is the rationale for such a rule? Let us consider division as the inverse of multiplication. By definition 3 ÷ 0 = n, if there is a unique number n, such that 0 n = 3. But the zero property of multiplication states that for n, 0 n = 0. Thus for any number represented by n, 0 n will never Fitzgerald-Haddad 12 have a product of 3. Therefore 3 ÷ 0, or any number divided by zero, is undefined because there is no number for n that multiplied by zero will result in a number other than zero. We can however find the solution when zero is the dividend. For example: 0 ÷ 3 = 0 because we can state that 0 3 = 0, thus zero is the unique number that satisfies both the division and multiplication equations. Let us now consider zero as both dividend and divisor, 0 ÷ 0 = n. In this case we find the solution, n, to be undefined or indeterminate because for the equation n 0 = 0, there is no unique number for n. Again, the zero property for multiplication states that the solution to an equation of equality must be unique. In this example n could be any number, thus 0 ÷ 0 cannot be solved, as there is no one unique solution. The idea of the undefined solution for division by zero can be difficult for children to grasp. One reason for such difficulties may be that for the first time students are faced with a mathematical problem that does not have a solitary solution. It does however provide an opportunity for students to wrestle with an abstract mathematical idea. Further, as an exercise in exploring division by zero, the mathematics department at Harvey Mudd College offers the following: 1. Consider two non-zero numbers x and y such that x = y 2. We can the say: x2 = xy 3. Subtract y2 from both sides: x2 – y2 = xy – y2 factoring to: (x+y)(x-y)=y(x-y) 4. Dividing by (x-y), obtain: x+y=y 5. Since x = y, we see that: 2y = y 6. Thus 2 = 1, since we started with y nonzero. Subtracting 1 from both sides: 1 = 0. Fitzgerald-Haddad 13 This demonstrates, to students, the chaos of division by zero. If x = y, then x-y = 0, and in step #4, we divided by zero, leading us to the inaccurate conclusion that 2 = 1. Another exploration for students to better understand the implications of division by zero involves the consideration of a number of examples of division equations revealing a pattern, such that division by zero could never equal zero. Let’s begin with the following equations, having 5 as the dividend and the divisor of decreasing values from 10 to .05. 5 ÷ 50 = 0.1 5÷5=1 5 ÷ 0.5 = 10 5 ÷ 0.05 = 100 5 ÷ 0.005 = 1000 From these examples we find that as the divisor approaches zero, the quotient becomes increasingly larger in value, moving further from the value of zero. Advancements in mathematics and science would not have been possible without the Hindu-Arabic system of numeration. The use of ten symbols simplified arithmetic calculations, enabling one to perform complex calculations. The discovery of zero enabled the development of the Cartesian coordinates. Zero provided the bridge to negative numbers on the number line. These advancements may have taken ten-centuries to develop, but in the last four-hundred years science has more than made up for the lost time. One need only consider calculations with Roman numerals to recognize the significance of the simple system developed in India over 1300 years ago and specifically the implications of the number zero. Fitzgerald-Haddad 14 Bibliography Baumgart, John K. et al. eds. Historical Topics for the Mathematics Classroom. Reston: National Council of Teachers of Mathematics, 1989. Billstein, Rick, Shlomo Libeskind, and Johnny W. Lott. A Problem Solving Approach to Mathematics for Elementary School Teachers. 8th ed. Boston: Pearson, 2004. Bunt, Lucas N. H., Phillip S. Jones and Jack D. Bedient. The Historical Roots of Elementary Mathematics. Edgewood Cliffs: Prentice, 1976. Filliozat, Pierre-Sylvain. “Making something out of nothing.” UNESCO Courier Nov. 1993: 30- . Academic Search Elite. CSUN Lib. 9 Apr 2006 <http://search.epnet.com.libproxy. csun.edu:2048/login.aspx?direct=true&db=afh&an=9312107674> Ifrah, Georges. From One to Zero: A Universal History of Numbers. Trans. Lowell Bair. New York: Viking, 1985. Kaplan, Robert. The Nothing That Is. New York: Oxford UP, 2000. Kak, Subhash C. “The Sign for Zero.” Mankind Quarterly Spr 90: 199- . Academic Search Elite. CSUN Lib. 9 Apr 2006 <http://search.epnet.com.libproxy.csun.edu:2048/login. aspx?direct =true &db=afh&an=9608261836> Pogliani, Lionello and Milan Randic. “Much Ado About Nothing – An Introductive Inquiry About Zero.” International Journal of Mathematical Education in Science and Technology. Sept/Oct 98: 729- . Academic Search Elite. CSUN Lib. 1 May 2006 < http://search.epnet.com.libproxy.csun.edu:2048/login.aspx?direct=true&db=afh&an=124 5075> School of Mathematics and Statistics University of St. Andrews, Scotland. 28 Apr. 2006< http://www-history.mcs.st-andrews.ac.uk/Biographies/Brahmagupta.html> Fitzgerald-Haddad 15 Seife, Charles. Zero: The Biography of a Dangerous Idea. New York: Viking, 2000. Su, Francis E., et al. "One Equals Zero!." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>. Teresi, D. “Zero.” Atlantic Monthly. July 1992: 88- . Academic Search Elite. CSUN Lib. 18 Apr 2006 <http://search.epnet.com.libproxy.csun.edu:2048/login.aspx?direct=true&db =afh&an=9707310687> West, Beverly Henderson et al. eds. The Prentice-Hall Encyclopedia of Mathematics. Edgewood Cliffs: Prentice, 1982. Fitzgerald-Haddad 16