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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

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                          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 (3103) + (6102) +

(6101) + (1100), or 3000+600+60+1=3661,the Babylonians would, in base sixty, have (1602)

+ (1601) + (1600), 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
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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
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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 1820, the forth 18202, and so on (Bunt 230). For

example the value 3661 in this system is shown as 10(1820) + 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


       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
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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


        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


       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 ÷ 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


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                                                                       Fitzgerald-Haddad 15

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Fitzgerald-Haddad 16

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