# Mathematics in Medieval India

# Mathematics in Medieval India

*Overview*

Indian mathematicians developed some of the most important concepts in mathematics, including place-value numeration and zero. By developing new techniques in arithmetic, algebra, and trigonometry, medieval Indian mathematicians helped make modern science and technology possible. Their innovations were brought to the West when treatises by Muslim scholars were translated into Latin.

*Background*

The years from a.d. 320 to about 500 were critical in the development of Indian civilization. In the north, under the Gupta dynasty, Sanskrit culture thrived, great universities were founded, and the arts and sciences flourished. In the south, where Hindu and Buddhist dynasties reigned, merchants seeking new trade opportunities started colonies and spread Indian culture throughout surrounding regions, especially Southeast Asia.

During the Gupta period, the observatory at Ujjain in central India was the heart of mathematical scholarship, and many mathematical techniques were developed to meet the needs of astronomers. The astronomical text the *Surya Siddhanta,* written by an unknown author some time around a.d. 400, contains the first known tabulation of the sine function. Indian mathematicians also developed the concept of zero, the base-10 decimal numeration system, and the number symbols, or *numerals*, we use today.

The entirety of Indian mathematics were compiled by the mathematician Aryabhata (476-550) in a collection of verses called *Aryabhatiya *in 449. The book describes both mathematics and astronomy, covering spherical trigonometry, arithmetic, algebra and plane trigonometry. Aryabhata calculated π to four decimal places, computed the length of the year almost exactly, and recognized that the Earth was a rotating sphere.

The famous astronomer Brahmagupta (598-670) wrote important works on mathematics and astronomy, including *Brahma-sphuta-siddhanta *(The opening of the universe) and *Khandakhadyaka*. He studied solar and lunar eclipses, as well as the motions and positions of the planets. Unlike Aryabhata, Brahmagupta believed that the Earth was stationary, but he, too, calculated the length of the year with remarkable precision.

*Impact*

The numeration system developed in India facilitated further advances in mathematics. Earlier ways of writing numbers, such as Roman numerals, used symbols to represent individual quantities, and these were added to determine the value. For example, X was the symbol for 10, and XXX was the symbol for 30, and 50 was L. Numbers expressed this way can be lengthy: 1,988 is MCMLXXXVIII. More to the point, there is no convenient way to do computations with them. People who used Roman numerals and other similar systems did their calculations with counting aids such as the *abacus.*

In contrast, Hindu arithmetic used number symbols that went only from 1 to 9, and instead of using more symbols for higher numbers, they introduced a place-value system for multipliers of 10. Each place had an individual name: *dasan *meant the tens place, *sata* meant the hundreds place, and so on. To express the number 235, the Hindus would write "2 sata, 3 dasan, 5". Seven hundred and eight would be "7 sata, 8".

Toward the end of the Gupta period, Indian mathematicians found a way to eliminate the place names while keeping the advantages of the place-value system. They used a symbol called *sunya,* or "empty" to designate a place with no value in it. This is equivalent to the symbol we call zero. With this they could write 708 for "7 sata, 8," and easily distinguish it from "7 dasan, 8," or 78. The physical alignment of tens, hundreds, etc. in columns resulted in the development of new arithmetic techniques for working with numbers.

About 800, the Hindu mathematician Mahavira demonstrated that zero was not simply a placeholder, but had an actual numerical value. His tenth-century successor Sridhara further recognized that the zero was as meaningful a number as any of the others. Without the zero, modern mathematics, and therefore most of modern science, would have been impossible.

The twelfth century mathematician Bhaskara (often called Bhaskara the Learned) was, like many of his predecessors (such as Brahmagupta), head of the Ujjain observatory and a gifted astronomer. His two mathematical works, *Lilivati *(The graceful) and *Bijaganita* (Seed counting) from the series *Siddhantasiromani*, were the first to expound systematically the use of the decimal system, based on powers of 10. He compiled many problems with which earlier mathematicians had struggled, and presented solutions.

Bhaskara was starting to understand the special nature of dividing by zero, as he specifically noted that 3/0 is infinite. He was, however, unable to generalize this to any number divided by zero. He enumerated the convention of *signs *in multiplication and division: two positives or two negatives divided or multiplied yields a positive result, and a positive and a negative divided or multiplied gives a negative result.

In algebra, Bhaskara built on the work of Aryabhata and Brahmagupta. He used letters to represent unknowns, as we do in algebra today. Bhaskara developed new methods for solving *quadratic equations,* that is, equations containing at least one variable raised to the second power (*x*^{2}). He studied regular polygons with up to 384 sides, in order to calculate increasingly precise approximations of π.

One of the first Muslim mathematicians to write about Indian techniques was Muhammad ibn Musa al-Khwarizmi, a teacher in the mathematical school at Baghdad. His book *Al-Khwarizmi Concerning the Hindu Art of Reckoning* was translated into Latin as *Algoritmi, de numero Indorum.* The Latinization of his name from "al-Khwarizmi" to "Algoritmi" eventually became our word for a mathematical procedure, *algorithm*. When his book on elementary mathematics *Kitab al-jabr wa al-muqabalah* (The book of integration and equation) was translated into Latin in the twelfth century, the term *al-jabr* became *algebra*.

Indian mathematical techniques were disseminated in the West through texts such as these. They were first brought to Moorish Spain, where they then spread to the rest of Europe. However, they did not come into common use there until the digit symbols were standardized after the invention of the movable-type printing press in the mid-1400s.

During the thousand years that followed the Gupta dynasty, successive waves of invaders poured through India. First came the Huns, who descended from central Asia beginning around 450, finally conquering the Gupta empire 50 years later. Muslims arrived from Arabia in the 700s, and from Afghanistan and Persia at the turn of the millennium. Muslim sultans ruled from Delhi between 1206 and 1526, helping to disseminate Indian mathematical advances throughout the Islamic world. Because Indian symbols were introduced to the West by Muslim mathematicians, they came to be known as "Arabic" numerals. Today scholars generally refer to
our way of expressing numbers as the *Hindu-Arabic numeration system*.

**SHERRI CHASIN CALVO**

*Further Reading*

Bose, D.M., S.N. Sen, and B.V. Subbarayappa (eds.). *A Concise History of Science in India*. New Delhi: Indian National Science Academy, 1971.

Ibn Lablan, Kushyar. *Principles of Hindu Reckoning*. Madison, WI: University of Wisconsin Press, 1966.

Murthy, T. S. Bhanu. *A Modern Introduction to Ancient Indian Mathematics*. New Delhi: Wiley Eastern Ltd., 1992.

Rao, S. Balaachandra. *Indian Mathematics and Astronomy*. Bangalore, Jnana Deep Publications, 1994.

Sarasvati, T. A. *Geometry in Ancient and Medieval India*. Delhi: Indological Publishing, 1979.

Schulberg, Lucille. *Historic India*. New York: Time-Life Books, 1968.

Srinivasiengar, C. N. *The History of Ancient Indian Mathematics*. Calcutta: World Press Private Ltd., 1967.

Swetz, Frank J. *From Five Fingers to Infinity. A Journey through the History of Mathematics*. Chicago: Open Court, 1994.

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