Community Connection

By Lal Balkaran
First of 2 Parts – The Hindu-Arabic Numerals
Our everyday life is intricately bound with numbers – time, dates, currencies, finance, temperatures, distance, passwords, personal identification numbers or PINs, telephone numbers, social insurance numbers, house numbers, lottery numbers, measurements, etc. Can you think of life without use of the digits 0 to 9? What if we were still using the Roman numerals (I, II, III, IV) linked to the alphabet which Europe was still using as late as the fifteenth century?
The numerals as we know them today (0,1,2,3,4,5,6,7,8,9) are often referred to as Arabic numerals and it was certainly through trading with Arab countries that they were introduced into the culture of Europe. However, in origin, the numerals derive from the symbols first used in India in the third century B.C.E. e.g., digits 1, 4, and 6 were current in India in roughly their present form at that time; numerals 2, 7, and 9 were to be found there a century later; while digits 3 and 5 were in use by the first century C.E. It is possible that the digit 0 was also used in India then although evidence for this has not survived and the term zero derives immediately (through Italian and French) from Arabic.
Digits 6 and 7 in use today resemble the same in the Brahminic script used around 300 B.C.E. in India while 2, 3, and 0 bear a striking resemblance to current Hindi and Devangari scripts.
Early forms of counting had no positional base value system. Around 1700 B.C.E. the positional base 60 evolved which was very helpful to the Mesopotamians who developed it to use in conjunction with their 360-day calendar. In the early C.E. or A.D. years the Greeks and Hindus began to use base 10 systems, but they did not have positional notation. They used the first ten letters of their own alphabet for counting. The Babylonians performed sophisticated mathematical computations with it, but no symbol for zero was devised. Instead, an empty space was left in the number, standing for zero. The Maya and Hindu number systems evolved after the Babylonian.
The merchants and traders in ancient Egypt and Mesopotamia (present day Iraq) used to set out pebbles in grooves of sand to calculate and add up accounts. There would be a “units” groove, a groove for “tens”, and one for “hundreds”. Such was a sample abacus (Greek for “tablet”). In Roman times, a calculating frame was made in which pebbles slid on wires and this was also called an abacus. The size of the abacus determined the size of the numbers which could be dealt with. The Roman numerals (I, II, III, IV, etc.) made simple addition, subtraction, and multiplication very complicated. It is interesting to note that the Roman word for “pebble” was “calculus” and here we have the derivation of our word “calculate.”
Though humans have always understood the concept of nothing or having nothing, the concept of zero is relatively new. Despite being highly skilled mathematicians, the Mayans never used zero in equations.
Zero first appeared in India around 458 C.E. Later, a Hindu astronomer and mathematician named Brahmagupta developed a symbol for it — a dot underneath numbers. He also developed mathematical operations using zero, wrote rules for reaching zero through addition and subtraction, and the results of using zero in equations. This was the first time in the world that zero was recognized as a number of its own, as both an idea and a symbol. With zero now in place, Brahmagupta went on to develop a positional notation for the base 10 system. He abandoned the letters which had been used for numerals past 9, and standardized the first nine symbols. The base 10 system found its way to Europe through the Moorish conquest of Spain. It is therefore not surprising that the earliest known European manuscript containing the numerals dates from 976, in Spain.
Europe was skeptical of the numerals and slow to change. But they became popular when merchants adopted it, since it proved so invaluable in their work and record-keeping. Before then, mathematicians struggled to perform the simplest arithmetic calculations.
The Italian government back then was suspicious of Arabic numbers and even outlawed the use of zero. However, merchants continued to use it illegally and secretively, and the Arabic word for zero, "sifr," brought about the word “cipher,” which not only means a numeric character, but also came to mean "code."
By the 1600s, zero was used fairly widely throughout Europe. It was fundamental in Rene Descartes’ Cartesian coordinate system and in Sir Isaac Newton’s and Gottfried Wilhem Liebniz’s developments of calculus. Calculus paved the way for physics, engineering, computers, and much of financial and economic theory.
Today, zero – both as a symbol (or numeral) and a concept meaning the absence of any quantity – allows us to perform calculus, do complicated equations, and to have invented computers. Computers write their numbers using only the digits 0 and 1 – the binary base or base two system. Their system is called base two because only the first two digits are used to form numbers and each place in the number is a power of two.
When we write our numbers we use the digits 0,1,2,3,4,5,6,7,8,9. This is called base 10 because we use ten digits to form any number. The placement of the digit in the number stands for that digit times a power of 10. A positional base value system is a number system in which the location of each digit influences the value of that digit, e.g., in base 10 for the number 375, the 3 is not merely worth 3 but because it is in the hundreds place, it is worth 300.
Geography and trade helped with the spread of the Hindu numerals. Through trade, the Arabs came in contact with Hindu merchants and so the numerals passed into Arabic usage. The Arab merchants had studied the number system in India by the ninth century and had been using it for centuries to calculate interest, convert currencies, and solve other problems of trade.
About 825 AD, a Persian mathematician, Mohammed ibn-Musa al-Khowarizmi wrote an enthusiastic book about the Hindu numerals entitled Al-jebr we’l mukabala in which he shows how to treat equations by (a) taking quantities from one side to the other, (b) uniting similar terms into one term. From the word “Al-jebr” our own word algebra is derived. Zero was crucial to al-Khowarizmi, who used it to invent algebra. He also suggested that a little circle should be used in calculations if no number appeared in the tens place. The Arabs called this circle "sifr," or "empty." Al-Khowarizmi also developed quick methods for multiplying and dividing numbers, which are known as algorithms – a corruption of his name.
In the twelfth century, mathematical works in Arabic were translated into Latin, thus giving Europe the tools for further progress.
Leonardo da Pisa, better known as Fibonacci (1170-1240), was a widely travelled Italian mathematician and merchant, who was captivated by the Hindu numerals brought out by the Arabs through trade in bazaars across the Mediterranean. Fibonacci saw the value and beauty of the numerals and discovered that these numerals could be applied with ease such as addition, subtraction, multiplication and division. This simple Arabic arithmetic, made possible by the Hindu numerals we all use today, was mostly unknown in Europe at the time. Instead, Europeans used Roman numerals to record numbers and a counting board, or abacus, to add and subtract, then the most prevalent tool for doing arithmetic.
In 1202, Fibonacci brought these Hindu-Arabic numerals to Italy where he wrote several books about them in Latin and became a celebrated mathematician. Fibonacci’s Liber anaci (book of calculation) became the most influential book on Hindu-Arabic numerals and arithmetic in Europe although it would take three hundred years for the mathematical system it promoted to become widely used there – checked at every point by vehement resistance from the Church and the guilds. However, this development was highly popular among merchants, who used Fibonacci’s equations involving zero to balance their books.
When, in the same century that Fibonacci’s Liber abaci was published, the English philosopher Roger Bacon (1214-1292) attempted to promote Hindu-Arabic mathematics as the foundation for all science, the key to a university education and a necessary study for theologians, he was accused by the Church of magic and condemned to life imprisonment. In Italy at that time, Roman numerals were still being used for recording and calculating. For example, 1776 would be denoted as MDCLXVI.
It should be noted that Fibonacci is best known for the Fibonacci series in which every term after the first two is the sum of the preceding two terms. Thus the series consist of the numbers 0,1,1,2,3,5,8,13, 21…The sum of the second and third terms equals the fourth term, and the sum of the third and fourth terms equals the fifth term, and so on all the way through the series.
But while scholarly Europe rejected and even outlawed Hindu-Arabic learning, the new mathematics found a ready audience among merchants of Italy, where it became known as ‘abacco’ mathematics – abacco being a corrupted form of ‘modo arabico’ (Arabic style). Thus the stage was set for Hindu-Arabic numerals to be the pillars for double entry accounting (pioneered by Hindu merchants) and introduced in Italy through Venice by trade with these Indian merchants.
Only by the end of the fifteenth century did Hindu-Arabic numerals begin to appear in accounting records. This was done through the pioneering efforts of Fra Luca Bartolomeo de Pacioli (1445-1517), a Franciscan monk, Renaissance mathematician, and magician and constant companion of Leonardo da Vinci. Pacioli’s 1494 landmark book entitled Summa de arithmetica, geometria, proportione et proportionalita forever changed the world of commerce and accounting. Pacioli codified the state-of-the art bookkeeping practices of Venice using the Hindu-Arabic numerals and laid the foundation for today’s bookkeeping and accounting. More on this in my next article.
An outstanding classical scholar, Cuthert Tunstall (1474-1559) who later became Bishop of London, wrote his Latin arithmetic, De arte supputandi (1522), which contained some interesting information about the “new” Hindu-Arabic number system and arithmetic in England. In his book, he shows place value by means of dots or points above figures. We have now become used to commas being placed in such a way as to group the figures in threes. Thus the number 235644563 is now written as 235, 644, 563.
The word ‘mathematics” came into use in the late sixteenth century and it is derived from a Greek word meaning “something learned” or “science.” Broadly speaking, it is divided into sections: (1) pure mathematics, which is the abstract science of space and number, (2) applied mathematics, which deals with the application of mathematics to all branches of science and engineering.
Today, what we failed to accomplish since the dawn of civilization were actually done under a mere 100 years – largely due to use of the digits 0 to 9. The greatest discoveries and accomplishments including advances in knowledge could not have been achieved without these numerals, e.g., aviation, computers, television, telecommunications, space exploration, technological advances including medical technology, construction, rise of mega cities, rise of multinationals, rise in commerce and trade, etc. They were all done with the use of the numerals 0,1,2,3,4,5,6,7,8, and 9 which are now taken for granted but which define today’s modern world and who we are.

(Lal Balkaran, MBA, FCPA, FCGA, FCMA, CGMA, CIA is self-taught and an award-winning auditor who has been widely published across the globe with seven reference books on business and another six on Guyana including five documentaries and scores of articles in professional journals. Lal is the founder of IIA-Guyana and currently an independent audit consultant. He can be reached at lalbalkaran@rogers.com)