INTRODUCTION

After the early mathematicians from Egypt, Babylon, and Greece, mathematics still continued to pave its way toward so many ideas and discoveries. The successors of the Greeks in the history of mathematics were the Hindus of India.

The Hindu civilization’s record in mathematics dates from about 800 B.C., but became significant only after influenced by the Greek’s achievements. They contributed a lot and most of them were very useful in the development of the field. Most of their mathematical works were motivated by algebra, astronomy, and geometry.

PRESENTATION OF DATA

(AD 630) BRAHMAGUPTA: ALGEBRA & ASTRONOMY

Born in AD 598 in Northwestern India, Brahmagupta was the most prominent Indian mathematician and astronomer of the seventh century. He is often referred to as Bhillamalacarya or ‘The Teacher from Bhillamala’. Immediately after the sixth century, the Hindu algebra experienced its ‘Golden Age’ through the work of Brahmagupta in the early seventh century. His major work was the Brahmasphutasiddhanta (‘The Opening of the Universe’) or simply the Siddhanta which was written in AD 628 when he was 30. A corrected and updated version of the Siddhanta, the Brahma Siddhanta (‘The System of Brahma in Astronomy’) was a comprehensive treatment of the astronomical knowledge of the time (Katz, 1998).

Similar to other mathematical works of Medieval India, the mathematical ideas of Brahmagupta was imbedded as chapters in astronomical works since he applied his mathematical techniques to various astronomical problems. Nevertheless, his description of mathematical techniques was generally fuller with some examples (Katz, 1998).

The Brahmasphutasiddhanta was divided into two chapters namely: Ganitad’ haya and Kutakhadyaka. The Ganitad’ haya (‘Lectures on Arithmetic’) identified a ganaca, a calculator which is competent enough to study astronomy, as one ‘who distinctly and severally knows addition and the rest of the twenty logistics and the eight determinations, including measurement by shadow” It discussed arithmetic progressions, the rule of three, simple interest, the mensuration of plane figures, and finding volumes (Nowlan, n.d.).

Meanwhile, the most influential advancement of the Hindu algebra was Brahmagupta’s big step toward operational symbolism. The Brahmasphutasiddhanta was considered the first textbook ‘to treat zero as a number in its own right.’ In its second chapter, the Kutakhadyaka (‘Lectures on Indeterminate Equations’) defined zero as the outcome of subtracting a number from itself and used dots underneath numbers to express a zero. He gave some properties of zero as follows: ‘When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.’ Not only that, he also gave rules for operating with zero and the ‘rules of sign’ stated in a marketplace language, using dhana (‘fortunes’) to denote positive numbers ad rina (‘debts’) to represent negative numbers (Nowlan, n.d.).

The following rules should be familiar except for the terms used although Brahmagupta incorrectly claimed that ‘zero divided by zero is zero’:

‘A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multiplied by zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt’ (O’Connor & Robertson, 2000).

Another mathematical work presented by Brahmagupta was his algorithm for computing square roots. His most remarkable work was his solution for the complete integer of the equation ax ?? by = c, where a, b, and c are constant integers. Furthermore, he discussed the indeterminate equation ax + c = by and the quadratic indeterminate equation. Perhaps, Brahmagupta used the method of continued fractions in finding the integral indeterminate equation of the type ax + c = by. He also gave the rules for summing series such as the sum of the squares of the first n natural numbers as (n(n+1)(2n+1))/6 and that of the cubes of the first n natural numbers as ((n(n+1))/2)2 although no proof was found (Bell, 1945).

In terms of astronomy, Brahmasphutasiddhanta dealt with solar and lunar eclipses, planet positions and conjunctions. He believed in a static Earth and presented the length of year as 365 days 6 hours 5 minutes 19 seconds but later changed it into 365 days 6 hours 12 minutes 36 seconds in his second book (O’Connor & Robertson, 2000).

Brahmagupta wrote his second book named Khandakhadyaka, which literally means ‘sweet meat’ when he was 67. With it, he became the first to use algebra in solving astronomical problems. Note that this is not an improvement since the true length of the years is less than 365 days 6 hours. Moreover, he gave the sidereal periods for many heavenly bodies and anticipated the gravitational theory, writing: ‘Bodies fall towards the earth as it is the nature of earth to attract bodies, just as it is the nature of water to flow’ (Nowlan, n.d.).

(AD 710) BEDE: CALENDAR & FINGER ARITHMETIC

In the early medieval Europe, there was a commonly-held belief that academic pursuits, particularly science and mathematics, had collapsed into a dark age. The majority of learned scholars were churchmen. One of them was Bede who was called as ‘Bede the Venerable’. He was born on AD 673 in Northumberland, England and became one of the greatest medieval church scholars because of his numerous writings, including his treatises on the calendar and on finger arithmetic (‘European Mathematicians’, n.d.).

In the early 8th century, Bede had a problem. Each year, Easter had to be foreseen with accuracy because all other moveable feasts in the annual cycle leaned on its date and the opinion on when exactly that date might be was divided. It was very critical that was why an entire division of mathematics was assigned to the subject named computus. However, computus have to respect the church rules including the relationships among the celebrations like Easter, linked to Passover (derived from lunar cycles) and to solar calendar (since if it could not be celebrated on Sunday, it had to be on the first full moon after the spring equinox which in turn is based to solar cycles). To make matters worse, the lunar and solar cycles did not match very well (Mulcare, 2013).

In order to solve the problem, a cyclical table was needed to be developed based on a common multiple m of solar and lunar periods. The general concept was that m years after some reference period, Easter would fall on the similar date as in the reference period itself because a whole number or rural months would have passed. However, there would be an extra month which was then added to the lunar calendar. The closest approximate cycles were 3/8, 4/11, 7/19, 31/84 but these cycles were not universally accepted. Because of these, Bede started reviewing, evaluating, and analyzing the tables of the age and thus, developed the calendar (Mulcare, 2013).

According to some sources, Indians could perform a meticulous method of counting using their fingers because of their three-joint thumbs. The first recorded method was made by Bede. He illustrated a finger arithmetic method and showed how to represent numbers with the aid of fingers, proceeding from left to right (Fink, 2007).

(AD 750) FIRST USE OF ZERO SYMBOL

Zero is the ‘number for none of the thing being counted’ and the ‘number that links positive and negative number lines’. However, no one had thought of having a symbol for zero given that it was only represented by a space in its use as a place-value indicator. Fortunately, the Hindus were the first to recognize a mathematical representation of concept of no quantity.

At first, the Indians represented zero as a dot but later replaced it with the symbol 0. It had been believed that the Hindu zero used by India was a round goose egg-like shape similar to the one used today. There were many beliefs about the origin of this characteristic form. It might have been a Hindu invention or it might have been suggested by the Greek use of omicron (??) for zero. (Boyer, 1944).

The first indubitable appearance of a circle symbol for zero appears in India on a stone tablet in Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth century AD (‘Numbers’, 2004).

(AD 810) MOHAMMED IBN MUSA AL-KHWARIZMI COINS TERM ‘ALGEBRA’

Al-Khwarizmi, or perhaps his ancestors, descended from Khwarizmi, the region south of the Aral Sea now part of Uzbekistan and Turkmenistan. He was an early member of the House of Wisdom and one of the astronomers called to cast a horoscope for the dying caliph al-Wathiq in 847 although he failed (Katz, 1998).

Al-Khwarizmi wrote the earliest available arithmetic text that discussed Hindu numbers. In this text, he introduced nine characters to designate the first nine numbers and a circle to denote zero. He demonstrated a process of writing any number using these characters in a place-value notation. Then, he described ‘the algorithms of addition, subtraction, multiplication, halving, doubling, and determining square roots and gave examples of their use’ (Katz, 1998).

However, he made no advancement except for exhibiting a positive and a negative root for a quadratic equation without explicitly rejecting the negative (Bell, 1945).

The most important contribution of Al-Khwarizmi in the world of mathematics was perhaps his arithmetic text which contributed to the important mathematical words used today. He was best known for coining the term ‘algebra’ from the name of his book ‘Al-jabr’ which demonstrated simple algebra and geometry. Since he was believed to have presented the first algebra text that demonstrated general methods, he was often called the ‘Father of Algebra’. Not only that, the word ‘algorithm’ originated from Al-Khwarizmi’s name. Moreover, he also derived the word ‘zero’ from the Arabic ‘sifr’, which was Latinized into ‘zephirum’ (Allen, n.d.).

Al-Khwarizmi’s texts on algebra and decimal arithmetic were considered to be among the most influential writings ever.

(AD 810) HINDU NUMERALS

Aside from the important mathematical terms coined by Al-Khwarizmi, he had also made another contribution with his strong advocacy of the Hindu numerical system. He wrote a book about Hindu numerals discussing numbers 1 to 9, spreading the use of Arabic numerals. He described the system as ‘having the power and efficiency needed to revolutionize Islamic mathematics. It was soon adopted by the entire Islamic world and by Europe as well (‘Islamic Mathematics’, 2010).

(AD 850) MAHAVIRA: ARITHMETIC & ALGEBRA

The Indian mathematicians also handled equations in several variables. One of them is Mahavira. Mahavira, also called Mahaviracharya (“Mahavira The Teacher’), was a mathematician from Mysore in Southern India (O’Connor & Robertson, 2000).

Mahavira wrote the earliest Indian text, the Ganita Sara Sarangha (‘Compendium of the Essence of Mathematics’), which was created as revised edition of Brahmagupta’s book. It was devoted completely to mathematics where he presented a version of the hundred fowls problem: ‘Doves are sold at the rate of 5 from 3 coins, cranes at the rate of 7 to 5, swans at the rate of 9 for 7, and peacocks at the rate of 3 for 9. A certain man was told to bring at these rates 100 birds, for 100 coins for the amusement of the king’s son and was sent to do so. What amount does he give for each’? In turn, he gave a rather complex rule for the solution (Katz, 1945).

Also, Mahavira was one of the first to indicate an awareness of the problem involving the square root of a negative number by writing that ‘a negative number cannot have a square root because a negative cannot be a square’ (Groza, 1968).

Although a place-value system with nine numerals was always used in his work, Mahavira became interested in developing a new place-value system with his description of the number 12345654321 which he had obtained after a calculation and described the number as starting with one and then increases until six, then decreases in reverse order. This description was a clear indication that Mahavira was open to the place-value system (O’Connor & Robertson, 2000).

Operations with fractions including the methods of decomposing integers and fractions into unit fractions were also discussed in his work. An example would be 2/17= 1/12+ 1/51+ 1/68. He also used a method called kuttaka in order to test integer solutions of first-degree indeterminate equations. The kuttaka (‘the pulveriser’) method was established based on Euclidean algorithm with the method of solution resembling the continued fraction process of Euler (O’Connor & Robertson, 2000).

Furthermore, Mahavira completely omitted addition and subtraction from his discussion of arithmetic. Instead, he took multiplication as the first eight fundamental operations and filled the gap with summation and subtraction of series (‘South Asian Mathematics’, n.d.).

Mahavira really had so many contributions like giving special rules for permutations and combinations, describing a process for the calculation of a sphere’s volume and of a number’s cube root. He also attempted to solve some unaccomplished mathematical problems of other Indian mathematicians (O’Connor & Robertson, 2000).