Splashes– 33/72 – Indian Contributions to Mathematics

Hindus have contributed most to the science of Mathematics. Our decimal system, place notation, numbers 1 through 9, and the ubiquitous ‘0’ zero, are the major contributions towards this fundamental science. In the Vedic age, India was ahead of the rest particularly, in mathematics and astronomy. Mathematics (Ganita) had been at the top of the sciences known as Vedanga. It is through the magic of Vedic Mathematics that Shakuntala Devi dazzled computer wizards by orally solving complex equations.

Combination of physical and Spiritual

Mathematics served as a bridge between understanding material reality and the spiritual conception. The mathematics of the Vedas contrasts the cold, clear, and geometric precision of the West and is cloaked in the poetic language that distinguishes the East. Vedic mathematicians devised sutras for solving mathematical problems with apparent ease.

Aryans were deeply interested in planetary positions to calculate auspicious times and they developed Astronomy and Mathematics side by side, towards that end. They identified various constellations (Nakshatras) and named the months after them. Indian system was far superior, to that of the Greeks. They could count up to 1012, while the Greeks could count up to 104 and Romans up to 108.

Ancient Literature on Mathematics 

Indians had adopted verse form to explain Mathematical concepts. The earliest available work in the field of Mathematics include Bakshali Manuscript that was discovered from Bakshali, a village near Takshashaila. Other texts are Ganitasara-Sangraham of Mahavira Acharya, who lived between Brahmagupta and Bhaskaracharya and Akshara Lakshana Ganit Shastra, containing 84 Theorems.

Invention of Numbers

Invention of numerals from 1 to 9 has been the major contribution of Indians to the science of Mathematics. Its importance is similar to the alphabets in any language, without which, the language cannot be visualized. Not only they designed the shape of numbers, but also assigned valuable meaning to each shape. By adding the system of placement to each number they expanded their expressive and application potential.

Discovery of Zero

The earliest use of the zero symbols is in one of the scriptural books dated about 200 BC. The zero is called Shunya or ‘nothing.’  It was initially represented by a dot and later it was replaced by a small circle. It was thereafter accepted as a numeral like others. There is a beautiful definition of the infinite in the following line of a Vedic mantra of Isavasya Upanishad:-

ॐ।।पूर्णमद : पूर्णमिदं पूर्णात् पूर्णमुदच्यते। पूर्णस्य  पूर्णमादाय पूर्णमेवावशिष्यते।।

It says: “Take the whole (Infinite Brahman) from the whole and the whole still remains”. This is almost like the mathematician, Cantor’s definition of infinity.

For example, in Roman Numeral System the figure ‘three thousand three hundred thirty three’ will be represented as ‘MMMCCCXXXIII’. The same figure can be easily expressed as 3333. For multiplying this figure by ten, all that is required under the Indian system is just to place a zero on the right so that all digits get shifted one step to the left and it is read as 33330. Performing such simple mathematical functions like addition, subtraction, multiplication and division was extremely tedious in Roman Numeral System.

Positional Notation System

Intellectuals all over the world had been grappling with numerical signs but till India supplied them the knowledge of positional notation system, they remained handicapped to utilize their application. If that had not happened, there would not have been any progress in the field of Mathematics as well as in Computer Science.

Our ancient mathematicians, by assigning positions to numbers extended the application of Mathematics to other branches of learning. The number on the right of digit raised the value of the first digit tenfold and the process continued till infinity. For example if we added one zero to the right of any digit, the value of the digit increased in multiple of ten.  Adding another zero would increase that to multiple of hundred and so on. Conversely if the digit is placed on the left side the value of the digit decreased in multiple of ten and so on. This revolutionary discovery goes to the credit of Indian mathematicians.

Decimal system

Not only zero denoted its own value, but the complete system based on its positional concept opened the gates for subsequent revolutionary system of decimal. Brhahmagupta pioneered the decimal concept about 1500 years ago. The Western world owes a great deal to India for this simple invention made by an anonymous Indian. Without it, most of the great discoveries and inventions (including computers) would not have come about. This invention was the decimal system of numerals – nine digits and a zero. Roman system of numerals even today is too clumsy to be used as a scientific tool.

Miscalled “Arabic” Numerals

The miscalled “Arabic” numerals are found on the Rock Edicts of Ashoka (250 BC), a thousand years before their occurrence as Hindsa even in Arabic literature implying imported from India. The Arabs carried this system to Africa and Europe.

Symbols in Trigonometry and Calculus.

Indians also added more branches to the field of Mathematics, such as Trigonometry and Calculus. They studied and applied this knowledge in Astronomy. The symbol for infinity is called the lemniscuses. English mathematician John Wallis introduced this symbol for the first time in 1655. Hindu mythological iconography contained a similar symbol representing the same idea. The symbol is that of Ananta, the great Adisesha of infinity and eternity, which is always represented, coiled up in a horizontal figure of ‘8’ just like the lemniscuses.

Algebra 

Aryabhatta, Brahmagupta, and Bhaskara conceptualized negative quantity in positional system. They found the square root of 2, and solved indeterminate equations of the second degree in 8 th Century AD. They expressed their science in poetic form and added grace to mathematical problems, a characteristic to India’s Golden Age. The same were unknown to Europe until Euler appeared on the scene thousand years later. Bhaskara invented the radical sign and many other algebraic symbols.

The largest numbers used by Greeks and Romans were 106, whereas Indians used numbers as big as 10 to the power of 53, as early as 5000 BC. The use of symbols-letters of the alphabet to denote unknowns, and equations are the foundations of the science of Algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of Algebra. Algebra went to Western Europe through the Arabs. To them Al-Jabr meant adjustment.

Ancient Indian Mathematicians

• Aryabhattalived during 475 AD – 550 AD and was born in Kerala. He had studied at Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 AD), he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of pi as 3.1416, claiming, for the first time, that it was an approximation. He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax – by = c, and also gave the table of Sines.
• Brahmagupta lived during the period 598 AD – 665 AD. He is known for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphuta Siddhanta. As a matter of fact, it is a corrected version of old astronomical treatise He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations.  He was the first mathematician to treat Algebra and Arithmetic as two different branches of Mathematics. He gave the solution of the indeterminate equation Nx2+1 = y2. He is also the founder of the branch of higher Mathematics known as ‘Numerical Analyses’. Brahmagupta‘s work Brahmasiddhanta was later translated into Arabic as Sind Hind.
• Mahavir Acharaya wrote Ganitasara Sangrahamin 850 AD, which is the first text-book on Arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse, and called it ‘Ayatvrit’.
• Bhaskaracharaya is the most well-known ancient Indian mathematician. He was born in 1114 AD at Bijjadabida (Bijapur, Karnataka). He was the first to enunciate that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. Bhaskara can also be called the founder of Differential Calculus. He gave an example of what is now called ‘differential coefficient’ and the basic idea of what is now called ‘Rolle’s Theorem’. Unfortunately, later Indian mathematicians did not take any notice of this. Five centuries later, Newton and Leibniz developed this subject further.  Bhaskaracharaya wrote Siddhanta Siromani in 1150 AD. It is divided into four sections as under:-

1. Leelavati – a very popular text-book on Arithmetic
2. Bijaganita – introduced chakrawal to solve algebraic equations. Six centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it ‘inverse cyclic’ equation.
3. Goladhayaya– chapter on sphere – celestial globe,
4. Grahaganita – mathematics of the planets.

Geometry

Vedic altars and sacrificial places were constructed according to strict geometrical principles. The Vedic altar had to be stacked in a geometrical form with the sides in fixed proportions. Brick altars had to combine specific dimensions with a fixed number of bricks. Again, the surface areas were so designed that altars could be increased in size without change of shape, which required considerable geometrical ingenuity. Geometry was known as Kalpa. Geometry was developed in India from the rules of the construction of the altars.

Geometrical rules found in the Sulbha Sutras, therefore, refers to the construction of squares and rectangles, the relation of the diagonal to the sides, equivalent rectangles and squares, equivalent circles and squares, conversion, of oblongs into squares and vice versa, and the construction of squares equal to the sum or difference of two squares.

Aryabhatta discovered the method of calculation for areas of triangle, trapezium and circle. In a verse (shloka), Arybhatta has explained the value for pi (tyajya) which is accurate to four decimal places.

We must appreciate the geometrical expression of Duryodhana in Mahabharata when he declared that he would not gift land to Pandavas even that measured equal to the land placed under a needle point. The statement expressed his acute sense of precision and measurement even as a layman.

Chand K Sharma

(Next: Splashes – 34/ 72 – Indian Contributions towards Physics)