## The Evolution of a System

Our own number system, composed of the ten symbols {0,1,2,3,4,5,6,7,8,9} is called the *Hindu**-Arabic system*. Furthermore, this system is positional, which means that the position of a symbol has bearing on the value of that symbol within the number. For example, the position of the symbol 3 in the number 435,681 gives it a value much greater than the value of the symbol 8 in that same number. We’ll explore base systems more thoroughly later. The development of these ten symbols and their use in a positional system comes to us primarily from India.[i]

It was not until the 15^{th} century that the symbols that we are familiar with today first took form in Europe. However, the history of these numbers and their development goes back hundreds of years. One important source of information on this topic is the writer al-Biruni, whose picture is shown here.[ii] Al-Biruni, who was born in modern day Uzbekistan, had visited India on several occasions and made comments on the Indian number system. When we look at the origins of the numbers that al-Biruni encountered, we have to go back to the third century B.C.E. to explore their origins. It is then that the Brahmi numerals were being used.

The Brahmi numerals were more complicated than those used in our own modern system. They had separate symbols for the numbers 1 through 9, as well as distinct symbols for 10, 100, 1000,…, also for 20, 30, 40,…, and others for 200, 300, 400, …, 900. The Brahmi symbols for 1, 2, and 3 are shown below.[iii]

These numerals were used all the way up to the 4^{th} century C.E., with variations through time and geographic location. For example, in the first century C.E., one particular set of Brahmi numerals took on the following form[iv]:

From the 4^{th} century on, you can actually trace several different paths that the Brahmi numerals took to get to different points and incarnations. One of those paths led to our current numeral system, and went through what are called the Gupta numerals. The Gupta numerals were prominent during a time ruled by the Gupta dynasty and were spread throughout that empire as they conquered lands during the 4^{th} through 6^{th} centuries. They have the following form[v]:

How the numbers got to their Gupta form is open to considerable debate. Many possible hypotheses have been offered, most of which boil down to two basic types[vi]. The first type of hypothesis states that the numerals came from the initial letters of the names of the numbers. This is not uncommon…the Greek numerals developed in this manner. The second type of hypothesis states that they were derived from some earlier number system. However, there are other hypotheses that are offered, one of which is by the researcher Ifrah. His theory is that there were originally nine numerals, each represented by a corresponding number of vertical lines. One possibility is this:[vii]

Because these symbols would have taken a lot of time to write, they eventually evolved into cursive symbols that could be written more quickly. If we compare these to the Gupta numerals above, we can try to see how that evolutionary process might have taken place, but our imagination would be just about all we would have to depend upon since we do not know exactly how the process unfolded.

The Gupta numerals eventually evolved into another form of numerals called the Nagari numerals, and these continued to evolve until the 11^{th} century, at which time they looked like this:[viii]

Note that by this time, the symbol for 0 has appeared! The Mayans in the Americas had a symbol for zero long before this, however, as we shall see later in the chapter.

These numerals were adopted by the Arabs, most likely in the eighth century during Islamic incursions into the northern part of India.[ix] It is believed that the Arabs were instrumental in spreading them to other parts of the world, including Spain (see below).

Other examples of variations up to the eleventh century include:

Devangari, eighth century[x]:

West Arab Gobar, tenth century[xi]:

Spain, 976 B.C.E.[xii]:

Finally, one more graphic[xiii] shows various forms of these numerals as they developed and eventually converged to the 15^{th} century in Europe.

## The Positional System

More important than the form of the number symbols is the development of the place value system. Although it is in slight dispute, the earliest known document in which the Indian system displays a positional system dates back to 346 C.E. However, some evidence suggests that they may have actually developed a positional system as far back as the first century C.E.

The Indians were not the first to use a positional system. The Babylonians (as we will see in Chapter 3) used a positional system with 60 as their base. However, there is not much evidence that the Babylonian system had much impact on later numeral systems, except with the Greeks. Also, the Chinese had a base-10 system, probably derived from the use of a counting board[xiv]. Some believe that the positional system used in India was derived from the Chinese system.

Wherever it may have originated, it appears that around 600 C.E., the Indians abandoned the use of symbols for numbers higher than nine and began to use our familiar system where the position of the symbol determines its overall value.[xv] Numerous documents from the seventh century demonstrate the use of this positional system.

Interestingly, the earliest dated inscriptions using the system with a symbol for zero come from Cambodia. In 683, the 605^{th} year of the Saka era is written with three digits and a dot in the middle. The 608^{th} year uses three digits with a modern 0 in the middle.[xvi] The dot as a symbol for zero also appears in a Chinese work (*Chiu**-chih** li*). The author of this document gives a strikingly clear description of how the Indian system works:

*Using the [Indian] numerals, multiplication and division are carried out. Each numeral is written in one stroke. When a number is counted to ten, it is advanced into the higher place. In each vacant place a dot is always put. Thus the numeral is always denoted in each place. Accordingly there can be no error in determining the place. With the numerals, calculations is easy…”***[xvii]**

## Transmission to Europe

It is not completely known how the system got transmitted to Europe. Traders and travelers of the Mediterranean coast may have carried it there. It is found in a tenth-century Spanish manuscript and may have been introduced to Spain by the Arabs, who invaded the region in 711 C.E. and were there until 1492.

In many societies, a division formed between those who used numbers and calculation for practical, every day business and those who used them for ritualistic purposes or for state business.[xviii] The former might often use older systems while the latter were inclined to use the newer, more elite written numbers. Competition between the two groups arose and continued for quite some time.

In a 14^{th} century manuscript of Boethius’ *The Consolations of Philosophy*, there appears a well-known drawing of two mathematicians. One is a merchant and is using an abacus (the “abacist”). The other is a Pythagorean philosopher (the “algorist”) using his “sacred” numbers. They are in a competition that is being judged by the goddess of number. By 1500 C.E., however, the newer symbols and system had won out and has persevered until today. The Seattle Times recently reported that the Hindu-Arabic numeral system has been included in the book *The Greatest Inventions of the Past 2000 Years*.[xix]

One question to answer is *why* the Indians would develop such a positional notation. Unfortunately, an answer to that question is not currently known. Some suggest that the system has its origins with the Chinese counting boards. These boards were portable and it is thought that Chinese travelers who passed through India took their boards with them and ignited an idea in Indian mathematics.[xx] Others, such as G. G. Joseph propose that it is the Indian fascination with very large numbers that drove them to develop a system whereby these kinds of big numbers could easily be written down. In this theory, the system developed entirely within the Indian mathematical framework without considerable influence from other civilizations.

[i] www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

[ii] www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html

[iii] www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

[iv] www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html

[v] Ibid

[vi] Ibid

[vii] Ibid

[viii] Ibid

[ix] Katz, page 230

[x] Burton, David M., *History of Mathematics, An Introduction*, p. 254-255

[xi] Ibid

[xii] Ibid

[xiii] Katz, page 231.

[xiv] Ibid, page 230

[xv] Ibid, page 231.

[xvi] Ibid, page 232.

[xvii] Ibid, page 232.

[xviii] McLeish, p. 18

[xix] seattletimes.nwsource.com/news/health-science/html98/invs_20000201.html, Seattle Times, Feb. 1, 2000

[xx] Ibid, page 232.

## Fibonacci's 'Numbers': The Man Behind The Math

The Latin phrase *filius bonacci,* in the first line of the *Liber Abaci* manuscript (above), gave rise to Leonardo da Pisa's modern nickname, Fibonacci. Click Here For A Closer Look **National Library of Florence** **hide caption**

Fibonacci's Arithmetic Revolution

### Buy Featured Book

Your purchase helps support NPR programming. How?

A page from the *Liber Abaci* manuscript. Leonardo da Pisa wrote symbolic calculations in the margin to illustrate the methods described in the text. Click Here For A Closer Look **Siena Public Library** **hide caption**

A page from the *Liber Abaci* manuscript. Leonardo da Pisa wrote symbolic calculations in the margin to illustrate the methods described in the text. Click Here For A Closer Look

Though generations of schoolchildren have cursed arithmetic, the world was a much more inconvenient place without it. Before the advent of modern arithmetic in the 13th century, basic calculations required a physical abacus.

But then came a young Italian mathematician named Leonardo da Pisa — no relation to da Vinci — who, in 1202, published a book titled *Liber Abaci*. That's Latin for "Book of Calculation."

And though it doesn't necessarily *sound* like an overnight best-seller, it was a smash hit. *Liber Abaci* introduced practical uses for the Arabic numerals 0 through 9 to Western Europe. The book revolutionized commerce, banking, science and technology and established the basis of modern arithmetic, algebra and other disciplines.

*Weekend Edition* "Math Guy" Keith Devlin tells the story of this arithmetic revolution in his new book, *The Man of Numbers.* Numerals 0 to 9 had been around in Hindu and Arabic cultures for centuries, but the problem was, Europeans didn't really do business with the numbers.

"They recorded everything in good old Roman numerals and if they wanted calculations, they went down the street to someone who was adept at using a physical abacus," Devlin tells NPR's Scott Simon. "It was actually a board with lines on it on which you moved pebbles around it was a crude and inefficient way of doing business."

The first edition of *Liber Abaci* was a dense, detailed book that was hard for the average person to grasp. So da Pisa released a simplified version to reach the traders and commercial people of Pisa — and the result spread around the world.

"Within a few decades of *Liber Abaci* appearing you've got what may have been 1,000 or more different people writing practical arithmetic textbooks," says Devlin. "Ordinary people who wanted to set up a business — and didn't have a lot of money to pay people to do the accounting for them — could do it for themselves."

The basics of accounting, banking, insurance and double entry bookkeeping all came out of 13th century Pisa, Devlin says. And that was thanks to the new ability to do arithmetic efficiently.

Sure, basic arithmetic may seem a simple thing today, but Devlin says its introduction to the world was comparable to the invention of the computer. Tedious and complicated tasks that required a specialist were suddenly faster and easier — and something you could do for yourself. "[Da Pisa] is Steve Jobs, Bill Gates. It's the computer revolution that we lived through in the 1980s, and the parallels are actually uncanny," says Devlin.

Despite his lasting impact on the modern world, da Pisa is not exactly a household name. But you might recognize him by his nickname: Fibonacci. In addition to writing *Liber Abaci,* da Pisa also introduced the famous Fibonacci sequence to Western Europe. (Remember that one from high school math? It starts with 0 and then 1, and then every subsequent number is the sum of the two numbers that precede it.) The name was given to him by a historian in the 19th century who read the phrase *filius Bonacci* — "son of Bonacci" — at the beginning of *Liber Abaci* and gave da Pisa his moniker.

Although Fibonacci can take credit for practical arithmetic in the Western world, Devlin says that even without him, it's unlikely that people would have had to rely on the abacus forever.

"One of the things about almost all of mathematics is that it will eventually surface and get used," Devlin says. "It's a matter of who does it and when."

## History: Roman numerals

The ancient Romans used a version of an additive number systems. The Romans represented numbers this way:

number | Roman Numeral |

1 | I |

5 | V |

10 | X |

50 | L |

100 | C |

500 | D |

1,000 | M |

So the number 2013 would be represented as MMXIII. This is read as 2,000 (two M’s), one ten (one X), and three ones (three I’s).

For any additive number system very large numbers become impractical to write. To represent the number one million in Roman numerals it would take one thousand M’s!

However, the Roman numerals did have one efficiency advantage: The order of the symbols mattered. If a symbol to the left was smaller than the symbol to the right, it would be subtracted instead of added. So for example nine is represented as IX rather than VIIII.

### Think / Pair / Share

If you don’t already know how to use Roman numerals, research it a little bit. Then answer these questions.

- Write the numbers 1–20 in Roman numerals.
- What is the maximum number of symbols needed to write any number between 1 and 1,000 in Roman Numerals? Justify your answer.

The earliest positional number systems are attributed to the Babylonians (base 60) and the Mayans (base 20). These positional systems were both developed before they had a symbol or a clear concept for zero. Instead of using 0, a blank space was used to indicate skipping a particular place value. This could lead to ambiguity.

Suppose we didn’t have a symbol for 0, and someone wrote the number

It would be impossible to tell if they mean 23, 203, 2003, or maybe two separate numbers (two and three).

Leonardo Pisano Bigollo, more commonly known as **Fibonacci** [4] , played a pivotal role in guiding Europe out of a long period in which the importance and development of math was in marked decline. He was born in Italy around 1170 CE to Guglielmo Bonacci, a successful merchant. Guglielmo brought his son with him to what is now Algeria, and Leonardo was educated in mathematics mathematics there.

At the time, Roman Numerals dominated Europe, and the official means of calculations was the abacus. Muḥammad ibn Mūsā al-Khwārizmī [5] described the use of Hindu-Arabic system in his book *On the Calculation with Hindu Numerals* in 825 CE, but it was not well-known in Europe.

Statue of al-Khwarizmi at Amirkabir University of Technology

Fibonacci’s book *Liber Abaci* described the Hindu-Arabic system and its business applications for a European readership. His book was well-received throughout Europe, and it marked the beginning of a reawakening of European mathematics.

## The Man of Numbers: Fibonacci's Arithmetic Revolution [Excerpt]

Before the 13th century Europeans used Roman numerals to do arithmetic. Leonardo of Pisa, better known today as Fibonacci, is largely responsible for the adoption of the Hindu–Arabic numeral system in Europe, which revolutionized not only mathematics but commerce and trade as well. How did the system spread from the Arab world to Europe, and what would our lives be without it?

*Reprinted with permission from* The Man of Numbers: Fibonacci's Arithmetic Revolution, *by Keith Devlin. Copyright* *©* *2011, by Bloomsbury Publishing.*

**Your days are numbered**

Try to imagine a day without numbers. Never mind a day, try to imagine getting through the first hour without numbers: no alarm clock, no time, no date, no TV or radio, no stock market report or sports results in the newspapers, no bank account to check. It&rsquos not clear exactly where you are waking up either, for without numbers modern housing would not exist.

The fact is, our lives are totally dependent on numbers. You may not have &ldquoa head for figures,&rdquo but you certainly have a head full of figures. Most of the things you do each day depend on and are conditioned by numbers. Some of them are obvious, like the ones listed above others govern our lives behind the scenes. The degree to which our modern society depends on numbers that are hidden from us was made clear by the worldwide financial meltdown in 2008, when over-confident reliance on the advanced mathematics of futures predictions and the credit market led to a total collapse of the global financial system.

How did we &mdash as a species and as a society &mdash become so familiar with and totally reliant on these abstractions our ancestors invented just a few thousand years ago? As a mathematician, this question had puzzled me for many years, but for most of my career as a university professor of mathematics, the pressures of discovering new mathematics and teaching mathematics to new generations of students did not leave me enough time to look for the answer. As I grew older, however, and came to terms with the unavoidable fact that my abilities to do original mathematics were starting to wane a bit &mdash a process that for most mathematicians starts around the age of forty (putting mathematics in the same category as many sporting activities) &mdash I started to spend more time looking into the origins of the subject I have loved with such passion since I made the transition from &ldquoIt&rsquos boring&rdquo to &ldquoIt&rsquos unbelievably beautiful&rdquo around the age of sixteen.

For the most part, the story of numbers was easy to discover. By the latter part of the first millennium of the Current Era, the system we use today to write numbers and do arithmetic had been worked out &mdash expressing any number using just the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and adding, subtracting, multiplying, and dividing them by the procedures we are all taught in elementary school. (Units column, tens column, hundreds column, carries, etc.) This familiar way to write numbers and do arithmetic is known today as the Hindu-Arabic system, a name that reflects its history.

Prior to the thirteenth century, however, the only Europeans who were aware of the system were, by and large, scholars, who used it solely to do mathematics. Traders recorded their numerical data using Roman numerals, and performed calculations either by a fairly elaborate and widely used fingers procedure or with a mechanical abacus. That state of affairs started to change soon after 1202, the year a young Italian man, Leonardo of Pisa &mdash the man who many centuries later a historian would dub &ldquoFibonacci&rdquo &mdash completed the first general purpose arithmetic book in the West, Liber abbaci, that explained the &ldquonew&rdquo methods in terms that ordinary people could understand &mdash tradesmen and businessmen as well as schoolchildren. While other lineages can be traced, Leonardo&rsquos influence, through Liber abbaci, was by far the most significant and shaped the development of modern western Europe.

Leonardo learned about the Hindu-Arabic number system, and other mathematics developed by both Indian and Arabic mathematicians, when his father brought his young son to join him in the North African port of Bugia (now Bejaïa, in Algeria) around 1185, having moved there from Pisa to act as a trade representative and customs official. Years later, Leonardo&rsquos book not only provided a bridge that allowed modern arithmetic to cross the Mediterranean, it also bridged the mathematical cultures of the Arabic and European worlds, by showing the west the algebraic way of thinking that forms the basis of modern science and engineering (though not our familiar symbolic notation for algebra, which came much later).

What Leonardo did was every bit as revolutionary as the personal computer pioneers who in the 1980s took computing from a small group of &ldquocomputer types&rdquo and made computers available to, and usable by, anyone. Like them, most of the credit for inventing and developing the methods Leonardo described in Liber abbaci goes to others, in particular Indian and Arabic scholars over many centuries. Leonardo&rsquos role was to &ldquopackage&rdquo and &ldquosell&rdquo the new methods to the world.

The appearance of Leonardo&rsquos book not only prepared the stage for the development of modern (symbolic) algebra, and hence modern mathematics, it also marked the beginning of the modern financial system and the way of doing business that depends on sophisticated banking methods. For instance, Professor William N. Goetzmann of the Yale School of Management, an expert on economics and finance, credits Leonardo as the first to develop an early form of present-value analysis, a method for comparing the relative economic value of differing payment streams, taking into account the time-value of money. Mathematically reducing all cash flow streams to a single point in time allows the investor to decide which is the best, and the modern version of the present-value criterion, developed by the economist Irving Fisher in 1930, is now used by virtually all large companies in the capital budgeting process.

What Leonardo brought to the mathematics he learned in Bugia and elsewhere in his subsequent travels around North Africa were systematic organization of the material, comprehensive coverage of all the know methods, and great expository skill in presenting the material in a fashion that made it accessible (and attractive) to the commercial people for whom he clearly wrote Liber abbaci. He was, of course, a highly competent mathematician &mdash in fact, one of the most distinguished mathematicians of medieval antiquity &mdash but only in his writings subsequent to the first edition of Liber abbaci in 1202 did he clearly demonstrate his own mathematical capacity.

Following the appearance of Liber abbaci, the teaching of arithmetic became hugely popular throughout Italy, with perhaps a thousand or more hand-written arithmetic texts being produced over the following three centuries. Moreover, the book&rsquos publication, and that of a number of his other works, brought Leonardo fame throughout Italy as well as an audience with the Holy Roman Emperor, Frederick II. Since the Pisan&rsquos writings were still circulating in Florence throughout the fourteenth century, as were commentaries on his works, we know that his legacy lived on long after his death. But then Leonardo&rsquos name seemed to be suddenly forgotten. The reason was the invention of movable-type printing in the fifteenth century.

Given the Italian business world&rsquos quick adoption of the new arithmetic, not surpisingly the first mathematics text printed in Italy was a 52-page textbook on commercial arithmetic: an untitled, anonymous work known today as the Aritmetica di Treviso (&ldquoTreviso Arithmetic&rdquo), after the small town near Venice where it was published on December 10, 1478. Soon afterwards, Piero Borghi brought out a longer and more complete edition, printed in Venice in 1484, that became a true bestseller, with fifteen reprints, two in the 1400s and the last one in 1564. Filippo Calandri wrote a third textbook, Pitagora aritmetice introductor, printed in Florence in 1491, and a manuscript written by Leonardo Da Vinci&rsquos teacher Benedetto da Firenze in 1463, Trattato d&rsquoabacho, was printed soon afterwards. These early printed arithmetic texts were soon followed by many others.

Though Liber abbaci was generally assumed to be the initial source for many, if not all, the printed arithmetic texts that were published, only one of them included any reference to Leonardo. Luca Pacioli, whose highly regarded, scholarly abbacus book Summa de arithmetica geometria proportioni et proportionalità (&ldquoAll That Is Known About Arithmetic, Geometry, Proportions, and Proportionality&rdquo) was printed in Venice in 1494, listed Leonardo among his sources, and stated:

And since we follow for the most part Leonardo Pisano, I intend to clarify now that any enunciation mentioned without the name of the author is to be attributed to Leonardo.

The general absence of accreditation was not unusual citing sources was a practice that became common much later, and authors frequently lifted entire passages from other writers without any form of acknowledgement. Without that one reference by Pacioli, later historians might never have known of the great Pisan&rsquos pivotal role in the birth of the modern world. Yet, Pacioli&rsquos remark was little more than a nod to history, for a reading of the entire text shows that the author drew not from Liber abbaci itself, but from sources closer to his own time. There is no indication he had ever set eyes on a copy of Liber abbaci, let alone read it. His citation of Leonardo reflects the fact that, at the time, the Pisan was considered the main authority, whose book was the original source of all the others.

Despite the great demand for mathematics textbooks, Liber abbaci itself remained in manuscript form for centuries, and therefore inaccessible to all but the most dedicated scholars. It was not only much more scholarly and difficult to understand than many other texts, it was very long. Over time it became forgotten, as people turned to shorter, more simple, and derivative texts. That one mention in Pacioli&rsquos Summa was the only clue to Leonardo&rsquos pivotal role in the dramatic growth of arithmetic. It lay there, unnoticed, until the late eighteenth century, when an Italian mathematician called Pietro Cossali (1748&ndash1815) came across it when he studied Summa in the course of researching his book Origine, transporto in Italia, primi progressi in essa dell-algebra (&ldquoOrigins, Transmission to Italy, and Early Progress of Algebra There&rdquo). Intrigued by Pacioli&rsquos brief reference to &ldquoLeonardo Pisano&rdquo, Cossali began to look for the Pisan&rsquos manuscripts, and in due course learned from them of Leonardo&rsquos important contribution.

In his book, published in two volumes in 1797 and 1799, which many say is the first truly professional mathematics history book written in Italy, Cossali concluded that Leonardo&rsquos Liber abbaci was the principal conduit for the &ldquotransmission to Italy&rdquo of modern arithmetic and algebra, and that the new methods spread first from Leonardo&rsquos hometown of Pisa through Tuscany (in particular Florence) then to the rest of Italy (most notably Venice) and eventually throughout Europe. As a result, Leonardo Pisano, famous in his lifetime then completely forgotten, became known &mdash and famous &mdash once again. But his legacy had come extremely close to being forever lost.

The lack of biographical details make a straight chronicle of Leonardo&rsquos life impossible. Where and when exactly was be born? Where and when did he die? Did he marry and have children? What did he look like? (A drawing of Leonardo you can find in books and a statue of the man in Pisa are most likely artistic fictions, there being no evidence they are based on reality.) What else did he do besides mathematics? These questions all go unanswered. From a legal document, we know that his father was called Guilichmus, which translates as &ldquoWilliam&rdquo (the variant Guilielmo is also common) and that he had a brother named Bonaccinghus. But if Leonardo&rsquos fame and recognition in Italy during his lifetime led to any written record, it has not survived to the present day.

Thus a book about Leonardo must focus on his great contribution and his intellectual legacy. Having recognized that numbers, and in particular powerful and efficient ways to compute with them, could change the world, he set about making that happen at a time when Europe was poised for major advances in science, technology, and commercial practice. Through Liber abbaci he showed that an abstract symbolism and a collection of seemingly obscure procedures for manipulating those symbols had huge practical applications.

The six-hundred page book Leonardo wrote to explain those ideas is the bridge that connects him to the present day. We may not have a detailed historical record of Leonardo the man, but we have his words and ideas. Just as we can come to understand great novelists through their books or accomplished composers through their music &mdash particularly if we understand the circumstances in which they created &mdash so too we can come to understand Leonardo of Pisa. We know what life was like at the time he lived. We can form a picture of the world in which Leonardo grew up and the influences that shaped his ideas. (In that we are helped by the survival to this day, largely unchanged, of many of the streets and buildings of thirteenth century Pisa.) And we know how numbers were used prior to the appearance of Liber abbaci, and how the book changed that usage forever.

## 5.3: The Hindu-Arabic Number System - Mathematics

Music is an extremely subjective experience . Some of the sound waves that reach the human ear are perceived to be pleasant while others are unpleasant and merely termed as noise. Thus, music is the art of combining sounds with a view to beauty of expression of emotion. Musically good melodies are thus harmonious in character.

Mathematics is the basis of sound wave propagation, and a pleasant sound consists of harmony arising out of musical scales in terms of numerical ratios, particularly those of small integers. Mathematics is music for the mind while music is mathematics for the soul.

A suitable permutation and combination of some basic notes gives rise to melodious music which enthrals and transports one to a new enjoyable experience. While theoretically infinite possibilities can be thought of, only 279 had been in vogue. In Carnatic music, seven notes or *swaras* as they are called, form the foundation of the various permutations and combinations. Theoretically, one can mathematically think of 7! = 5040 possibilities. But only 72 of them, called Janaka ragas, have been analyzed and found to have practical usage from the melody point of view. A raga thus has a set of rules that specify what notes of the octave must be used under the given rule and how to move from one note to the other. A Melkartha raga must necessarily have Sa and Pa and one of the Mas, one each of the Ris and Ga's, and one each of the Dhas and Nis, and further Ri must precede Ga and Dha must precde Ni, and thus we have 2 x 6 x 6 =72 possibilities of the Melakartha ragas.

It was Venkata Makhi who first thought of classifying these 72 ragams or combinations of swaras, and later Govindacharya adopted a slightly different combination. The basic *swaras* in Carnatic music and those that correspond to the basic notes in Western music are as follows:

Western system: C D E F G A B

Carnatic music: Sa Ri Ga Ma Pa Dha Ni

In the equal tempered system of Western music, the successive notes have a ratio of twelfth root of 2 as shown below:

In the Carnatic music system also, just as in the Western music system, some of the *swaras* have small variants, and according to some scholars, 22 such notes are possible with the ratios as given below to the fundamental Sa.

Swara Ratio Frequency I Hz

But according to some scholars, Ma has only 2 variants while Ri, Ga, Dha and Ni have only 3 variants each while Sa and Pa are fixed, and Venkata Makhi has followed this system in his classification of the ragas in vogue. Although many ragas might have been in existence earlier, it was first Venkata Makhi who made a classification of the ragas by allotting them a number in what is called the Melakartha system, by adding prefix to the existing ones wherever necessary, following the *katapayaaadi* system of numbering he considered a raga to be a Melakartha raga only if all the 7 *swaras* were present in the regular order in the ascending as well as descending mode. Govindacharya, however, allowed some freedom in this respect, and permitted deviation of the order as well as absence of any particular *swara* in either mode provided all the 7 *swaras* were present.

Melakartha *ragas* have a *swara* pattern, with *arohanam* and *avarohanam*, the latter being the mirror image of the former, and both together make a musical palindrome!

(The list of the Melakartha ragas and the number of the Melakartha to which the raga belongs is decided according to the *katapayaadi* system of numeration as prevalent in ancient India, which is as follows:

The following verse found in *Śa* *ṅ karavarman's* * Sadratnamāla * explains the mechanism of the system.

नज्ञावचश्च शून्यानि संख्या: कटपयादय: |

मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर: ||

*nanyāvacaśca śūnyāni sa* *ṃ* *khyā* *ḥ* *ka* *ṭ* *apayādaya* *ḥ*

*miśre tūpāntyahal sa* *ṃ* *khyā na ca cintyo halasvara* *ḥ*

*Translation* : *na* ( न) , *nya* ( ञ) and *a* ( अ)- s i.e. vowels represent zero . The (nine) integers are represented by consonant group beginning with *ka*, *ṭ* *a* , *pa*, *ya*. In a conjunct consonant, the last of the consonants alone will count. A consonant without vowel is to be ignored.

The assignment of letters to the numerals is as per the following arrangement.

Consonants have numerals assigned as per the above table. For example, ba ( ब) is always three 3 whereas 5 can be represented by either *nga* ( ङ) or *ṇ* *a* ( ण) or *ma* ( म) or *śha* ( श).

All stand-alone vowels like *a* ( अ) and *ṛ* ( ऋ) are assigned to zero 0.

In case of a conjunct, consonants attached to a non-vowel will not be valueless. For example, *kya* ( क्या) is formed by *k* ( क्) + *ya* ( य) + *a* ( अ). The only consonant standing with a vowel is *ya* ( य). So the corresponding numeral for *kya* ( क्या) will be 1.

There is no way of representing Decimal separator in the system.

Indians used the Hindu-Arabic numeral system for numbering, traditionally written in increasing place values from left to right. This is as per the rule *a* *ṅ* *kānām vāmato gati* ( अङ्कानाम् वामतो गति) which means numbers go from left to right .

The moment the name of a raga is given, the above system is used to find the Melakartha of that raga. Sometimes to fix the correct number, the name of the raga is slightly changed, as for instance, Sankarabharanam is called Dheerasankarabhaaranam, and Kalyani is called Mechakalyani and so on.

Let us consider some examples. Take Mayamalava Gowla. Here, Ma stands for 5 and ya stands for 1. So, the number we get is 51, and as per the reversing rule, the number of the Melkartha is 15.

Then, consider Simhendra Madhyamam. Sa stands for 7, and Ma for 5. The number is 75 and on reversing it is 57, which is the Melakartha of this raga. (The second consonant ha has number 8, and on reversal would give 87 as melakartha raga which is nonexistent. Hence, ma is taken in *simha* as the second consonant).

Now, consider Vachaspati. Va stands for 4, and cha for 6, and the number is 46 which on reversing gives 64 as its Melakartha. Sa and Pa are taken as fixed for all the Melakartha ragas.

The question arises which variant of Ri, Ga, Ma, Dha, Ni figures in what parent raga.

For ragas whose Melakartha number is 36 or less, M1 is chosen and for ragas whose Melakartha number is 37 or more, M2 is chosen.

Regarding the choice of the variant of Ri, Ga, Dha and Ni, this is decided with a little bit of mathematics. This method, which was perhaps prevalent earlier was refined to generate the entire raga by Ajay Sathyanath in April 1999 and published under the title “Mathematical Fundas in Indian Classical Music” and this gives an elegant method to determine the variants needed. (cf. http://ajaysat.tripod.com/carnatic.html )

Step 1: First, find the number of the Melakartha raga using the *Katapayaadi* system. Suppose it is K.

Step 2: Consider [[K/6]], called the ceiling function of K/6, that is, the integer which is equal to greater than K/6. e.g. Suppose K is 31. Then 31/6=5.1…, and [[K/6]] is 6. If K is 30, then [[K/6]] = 5. If [[K/6]] > 6, then take mod 6 of that number arrived at.

Step 3: Now, consider K modulo 6. Since this number will lie between 0 and 5 only, we make this lie between 1 and 6 by setting 0 as 6. If K is 31, then 31 = 1 mod 6. So, we consider only 1 for the procedure to be followed as outlined below.

Step 4: Consider a 3 x 4 upper triangular matrix, as shown below:

…. ….. 7, and we identify these elements as:

The matrix can now be written for convenience as :

In the example considered in step 2, we have the number 6 corresponding to (3,3), we have Ra3 and Ga 3.

Step 5: In step 3, we had 1 as residue after dividing 31 by 6. So, 1 corresponds to (1,1) as indicated above. So, the *swaras* chosen are Dha1 and Ni3.

31 < 36, and hence we have Ma1.

So, the raga characteristic of Melakartha raga 31 (Yagapriya) is:

Sa Ra3 Ga3 Ma1 Dha1 Ni1 and S in *arohana* and

Sa Ni1 Dha1 Ma1, Ga3, Ra3 and S in *avarohana*.

**Ex.** :Now, let us consider another example: consider Shubhapanthuvarali: *Sha* stands for 5 and *bha* stands for 4, so, the number is 54, which on reversal gives 45. 45 > 36, and hence we have Ma2.

To sum up, if [[K/6]] is as defined above, then in this case [[45/6]] =8 = 2 mod 6. 2 corresponds to (1,2), and hence we have Ri1 and Ga2.

45 = 3 mod 6. 3 corresponds to (1,3) in the matrix, and so, we have Dha1 and Ni3.

(i) Find the Melakartha number of the raga with the *katapayaadi* system. Suppose it is K.

(ii) If K = 36 or <36, then we have Ma1. If K >37, we have Ma2.

(iii) Consider [[K/6]] = a. If a is > 6, take mod 6 of a, suppose it is a*, then find the element in the matrix corresponding to a* which lies between 1 and 6 only. That will decide Ri and Ga.

(iv) Now, consider K = b mod 6. Find the element corresponding to b in the matrix, and that will decide Dha and Ni.

A list of the Melakartha (*Janaka*) ragas as well as the *janya* ragas under them and also those not associated with them or whose scales are not yet added is given in appendix 1.

## Positional Notation

Both the Babylonian number system and ours rely on position to give value. The two systems do it differently, partly because their system lacked a zero. Learning the Babylonian left to right (high to low) positional system for one's first taste of basic arithmetic is probably no more difficult than learning our 2-directional one, where we have to remember the order of the decimal numbers -- increasing from the decimal, ones, tens, hundreds, and then fanning out in the other direction on the other side, no oneths column, just tenths, hundredths, thousandths, etc.

I will go into the positions of the Babylonian system on further pages, but first there are some important number words to learn.

## Background

One of the essential requirements for any type of mathematics is a means of representing quantities. At first, tokens might be used—pebbles or small clay objects with one pebble representing each sheep in a herd, for example. Eventually, tokens of different shapes might be used to represent certain multiples, say five or ten sheep instead of one. With the invention of writing, it made more sense to use marks pressed into clay or made on paper or papyrus to keep track of one's possessions. The Babylonians, Egyptians, Indians, and Mayans all had developed elaborate systems to represent quantities by the first century a.d. The Babylonians developed a sophisticated number system based on the number 60, using it in commerce and for astronomy and astrology. By the last century b.c., this system included a symbol for zero, which was used as a placeholder in expressing quantities.

Of the several number systems, those that had the greatest effect on the development of mathematics in Europe of the Middle Ages were the Roman, the Chinese, and the Indian or Hindu, transmitted to the Western world by the Arabs and now known as Hindu-Arabic numerals. In the Roman system, still occasionally used today, letters of the alphabet were used to represent units and multiples of five or ten. In the Roman system the year 2004 can be written quite compactly as MMIV, with a hint of positional notation in that the "I" appearing before the "V" means that the one it represents is to be subtracted from the five represented by the "V." Roman numerals were adequate for record keeping and could be added and subtracted easily, but were far more cumbersome in multiplication and division and certainly not suited to the needs of modern science or commerce.

The Chinese system was not a decimal system, based on the number 10, but a centesimal system, based on separate symbols for the whole numbers between 1 and 9 and for multiples of 10 between 10 and 90. By alternating the pairs of symbols, the Chinese were able to represent numbers of any size. Because the Chinese number symbols were composed of single strokes, it was possible to represent them by short sticks, and to do arithmetic by moving sticks about according to preset rules. This led in the Middle Ages to the use of counting boards by merchants to do simple arithmetic. The Chinese were also responsible for the abacus, an arrangement of beads on wires that facilitated the ordinary operations of addition, subtraction, and multiplication.

The number system we use today, based on the numerals 0-9 and using position to denote different powers of 10, originated in India. Mathematical thinking in India dates back to at least 800 b.c. Number symbols first appear in the third century b.c., including among many alternatives the so-called Brahmi symbols, which include separate symbols for the numerals 1 through 9 and the multiples of 10 from 10 to 90. The Brahmi figures gradually evolved into the "1,2,3. " of today. By the year 600, they had come to predominate and to include a symbol for zero and for the use of positional notation. The *Brahamasphuta Siddhanta*, a treatise on astronomy written by the astronomer and mathematician Brahmagupta (598-c. 665) includes a treatment of arithmetic using the system of ten numerals, including zero, along with rules for fractions, the computation of interest, and rules for using negative numbers. Interestingly, Brahmagupta appeared to treat the fraction 0/0 as equal to zero, and avoided the question of dividing other numbers by zero.

Beginning in a.d. 632 Arab armies expanding from the Arabian peninsula established an Islamic empire that would stretch as far eastward as India and as far west as Spain. In 755 it split into two kingdoms, one with its capital at Baghdad. There, Hindu scientists and mathematicians found themselves welcome, despite their different religious beliefs. In Baghdad they could meet the descendants of the Greek scholars who had fled to Persia, bringing their mathematical interests, after the Emperor Justinian closed Plato's academy in a.d. 529. By 766 some Hindu mathematical work had been translated into Arabic. At Baghdad the Caliph al-Ma'mun (786-833) established a "House of Wisdom" modeled on the earlier Greek academy at Alexandria, with a library and observatory.

One of the scholars at the House of Wisdom was al-Khwarizmi (c. 780-c. 850). Al-Khwarizmi's book on algebra, popularly known as the *al-jabr*, was translated into Latin in 1145 by Robert of Chester (fl. c. 1141-1150), an English scholar living in Islamic Spain. Al-Khwarizmi would also become known to the Western world for a book known only in Latin translation as the *Al-goritmi de numero Indorum*, or *Al-Khwarizmi on the Hindu Method of Calculation*, in which he explains the Hindu number system and how it can be used in arithmetic calculations. It is from the title of this book that we obtain the word "algorithm" for any systematic method of calculation.

Among the readers of this book were Leonardo of Pisa (c. 1170- c. 1250), also known as Leonardo Fibonacci, an Italian who traveled throughout northern Africa and became familiar with the Arab system of numbers and methods of calculation. In 1202 he wrote the *Liber Abaci* or *Book of Calculations*, in which he described the Arabic system of numbers. Although the Hindu-Arabic system of numbers was not entirely unknown in Europe, it was Fibonacci's book that led to its widespread adoption in commerce and record keeping.

## Hindu-Arabic and Roman Numeral Systems

It was stated previously that the ancient Hindus are credited with discovering the decimal system of numeration we use today. This system was translated into Arabic prior to its introduction into Europe by traveling merchants around the 13th century. Hence it is also known as the Hindu-Arabic system.

Adoption of the Hindu-Arabic system met resistance due to the widespread use of the Roman numeral system during this period. Gradually, however, the superior Hindu-Arabic system was learned by the Europeans, and eventually it replaced the Roman system (*see* Roman numeral).

The Roman numerals are still sometimes used. Some examples of items on or in which Roman numerals still appear include clock faces and books, for numbering introductory pages and chapters.

The Roman system, like others that are not based on the principle of position, does not provide an efficient and easy method of computation. Here are some examples of computations using the Roman system. Equivalent computations using the Hindu-Arabic system are alongside.

## The Five Big Contributions Ancient India Made to the World of Math

As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of trigonometry, algebra, arithmetic and negative numbers.

Winston Mills-Compton teaches a class in mathematics at the Mfantsipim Boys School in Cape Coast, Ghana, June 20, 2006. Mfantsipim is one of the oldest schools in Cape Coast, a town that prides itself as the academic center of the country. UN Secretary General Kofi Annan is one of the school's alumni. Credit: Flickr

It should come as no surprise that the first recorded use of the number zero, recently discovered to be made as early as the third or fourth century, happened in India. Mathematics on the Indian subcontinent has a rich history going back over 3,000 years and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.

As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of trigonometry, algebra, arithmetic and negative numbers among other areas. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.

**The number system**

As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as the Vedas. In these texts, numbers were commonly expressed as combinations of powers of ten. For example, 365 might be expressed as three hundreds (3吆²), six tens (6吆¹) and five units (5吆⁰), though each power of ten was represented with a name rather than a set of symbols. It is reasonable to believe that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.

Brahmi numerals. Credit: Wikimedia Commons

From the third century BC, we also have written evidence of the Brahmi numerals, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics.

**The concept of zero**

Zero itself has a much longer history. The recently dated first recorded zeros, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from ten. Similar marks had already been seen in the Babylonian and Mayan cultures in the early centuries AD and arguably in Sumerian mathematics as early as 3000-2000 BC.

But only in India did the placeholder symbol for nothing progress to become a number in its own right. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the democratisation of mathematics.

These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though Fibonnacci’s book liber abaci.

**Solutions of quadratic equations**

In the seventh century, the first written evidence of the rules for working with zero were formalised in the Brahmasputha Siddhanta. In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.

**Rules for negative numbers**

Brahmagupta also demonstrated rules for working with negative numbers. He referred to positive numbers as fortunes and negative numbers as debts. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.

This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative.

For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that negative numbers were absurd. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.

For example, in a primitive farming context, if one farmer owes another farmer seven cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy seven cows and give them to the second farmer in order to bring his cow tally back to zero. From then on, every cow he buys goes to his positive total.

**Basis for calculus**

This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his development of calculus in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.

But Indian mathematician Bhāskara had already discovered many of Leibniz’s ideas over 500 years earlier. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Diophantine” equations, that would not be rediscovered in Europe for centuries.

The Kerala school of astronomy and mathematics, founded by Madhava of Sangamagrama in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would later be repeated in Europe including Taylor series expansions, infinitesimals and differentiation.

The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation suffers from the Eurocentric bias, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by providing key players at the forefront of every branch of mathematics.

*Christian Yates is Senior Lecturer in Mathematical Biology in University of Bath.*

## The Arabic and Hindu Numeral System

The Arabic and Hindu number system was developed around 800AD. Today this numeral system is very popular and widely used. Now we shall discuss the following four main attributes about this numeral system.

First, it uses ten digits or number symbols, and all the numbers we see around us are combinations of these ten digits. The digits are written as **0, 1, 2, 3, 4, 5, 6, 7, 8, and 9**. Secondly, the system contains groups of tens, and we usually have ten digits on our hands. It may be noted that the word **digits** mean the fingers. In the Arabic and Hindu numeral system, ten ones, ten tens, ten hundreds and 10 one thousand and so on are replaced by one ten, one hundred, one thousand and 10 thousands and so on, respectively.

Thirdly, starting from right to left, it uses place value:

- The first number shows the number of ones it has.
- The second number shows the number of tens it has.
- The third number shows the number of hundreds it has.
- The fourth number shows the number of thousands it has.
- And so on …

For example if we examine the numeral **7594**, there are **4** ones, **9** tens, **5** hundreds, and **7** thousands.

Finally, the value of a numeral is originated by multiplying every place value by its related digit and then adding the consequential products.