Though the ruins at Harappa and Mohenjodaro in north-western India and Pakistan have given us many clues about the lifestyle of the Indus Valley Civilisation - probably the oldest civilisation that existed on the Indian Subcontinent - they have not as yet revealed any mathematical documents. The oldest records of Indian mathematics hence date back to the period of the Aryans. Indian mathematics has its roots in the Hindu religion, and has produced some great mathematicians, the importance of whose work is still being gauged.

In 476 CE (Common Era), the year of the fall of the Western Roman Empire, Aryabhatta, the author of one of the oldest mathematical texts, was born. His work, the Aryabhatiya, was written in 499 in the form of verse. It contains rules for various calculations in mathematics and astronomy. Aryabhatta put down methods of finding the square and cube roots of numbers, correctly stated the area of a triangle as half the product of its base and altitude and that of a circle as the product of its circumference and half its diameter. the Aryabhatiya also gives an approximation for the value of pi as 62832/20000, which is 3.1416, or correct to four decimal places (pi = 3.1415926535 up to ten decimal places; it is an infinite decimal) which, however, was possibly influenced by the value accepted by Greek mathematicians at the time.

The Decimal Number System

One facet of the Aryabhatiya which influenced later mathematicians was the assertion that each place in a number was ten times the preceding place, which defines the decimal place-value numeration (the tens place is ten times the units place, and so on.) From this initial breakthrough would later arise the development of using only ten numerals for the entire decimal number system, a norm in the modern era. (Previously, individual symbols were used for numbers above nine). Added to this was the adoption of zero as a placeholder; the Hindu notation used a round goose egg as the symbol for the “empty” places in the decimal system. Though the individual discoveries of the place system and a symbol for the ten numerals occurred concurrently (and was possibly shared mutually) in Greece and China, it was in India that they were combined to give rise to the modern decimal system.

The Origin Of Trigonometry

The Indians introduced the equivalent of the sine function linking the length of a circular arc and the angle subtended by it at the centre of the circle. In Aryabhatiya, the length of the arc of a circle of radius 3438 units is recorded for twenty-four different central angles from 3.75 degrees to 90 degrees. On dividing these lengths by the radius, we obtain a reasonably accurate approximation of the sine values of these angles. The tables also include an approximation for the versed sine (1 - cosine) of these angles. These tables replaced the Greek tables of chords and today, the sine function is the basis of trigonometry.

Brahmagupta lived in Central India about a century after Aryabhatta and put forward various concepts in mensuration and algebra, such as a formula for the area of a cyclic quadrilateral. He established a general form for the solution of a quadratic equation, and recognised the presence of two roots. Brahmagupta’s work is the first instance of operations involving negative numbers and zero (the Greeks could not represent their concept of nothingness as a number, hence the absence of zero in their early works.) Brahmagupta also recognised the irrational roots of numbers as numbers even though they seemed to be incommensurable. This lead was followed by mathematicians until the nineteenth century when the real number system was firmly established.

Brahmagupta also was the first to come out with a general solution for the indeterminate equation (one with no definite roots) ax + by = c (known as the Diophantine equation after Greek mathematician Diophantus). He stated that the roots would be in the form :

Interestingly, Diophantus himself could not arrive at this general form for his equation; he was content with just one solution for it.

Hindu mathematicians undoubtedly borrowed much from their contemporaries in Greece, Babylon and China, but they picked out those concepts that appealed to them, developed on them without apprehension and approached them with an open mind. Though their influence is rarely seen in analytical geometry and calculus, they gave the world the sine function and established the numeral system that is in use by most of the civilised world today.