The first known use of numbers in India was in the time of the Harappans, about 3000 BC. Around this time, people in India began using the counting tokens that people were already using in West Asia. Soon afterwards, people changed over to writing their numbers down, using pictographs. The Harappans also developed standard weights (like our ounces and grams), and they were the earliest people to use base 10 for their weights.
After climate change caused the Harappan civilization to collapse, about 2000 BC, some invading Indo-Europeans ruled India. This did not stop mathematical progress, and may even have encouraged it, as the Indo-Europeans may have brought Babylonian mathematical ideas to India. By 1800 BC, Indian mathematicians were discussing the idea of infinity, pointing out that "if you remove a part from infinity or add a part to infinity, what remains is still infinity." As in Babylonia, Indian scientists made a lot of progress in geometry as a result of interest in astronomy, and by 1300 BC the Indian astronomer Lagadha used geometry to write a book of rules for the apparent movement of the sun and moon. Nobody knows whether Lagadha worked these rules out on his own, or learned about them from the Babylonians.
By about 400 BC, Indian mathematicians were doing more work on the idea of infinity. The Surya Prajinapti defines five kinds of infinity: an infinite line beginning from an endpoint, an infinite line going both directions, an infinite plane, an infinite universe, and the infinity of time. This was about the same time as the Buddha.
Around 300 BC, when Chandragupta was ruling India, Indian mathematicians began working on the mathematical idea of combinations. This is the study of how many combinations you can make out of the same group of things. For example, how many different poker hands can you make out of a pack of cards? Or, what are the chances that in any class of thirty kids, some of them will share the same birthday? They were working on how you could figure that out, and published their ideas in a book called the Bhagabati Sutra. Around the same time, Indian mathematicians worked out the first beginnings of our modern number system. By 100 AD, people in India were writing the numbers as in the picture here. The numbers 1,2, and 3 are pretty clearly taken from Chinese numerals. Nobody knows where the other signs came from. It may be that they were taken from letters of an Indian alphabet.
Indian mathematicians' biggest contribution was the use of zero as a placeholder, to make it easier to add and multiply numbers. Our word "zero" comes from the Sanskrit word meaning "nothing." Chinese mathematicians had already been using a blank space to mean "zero" for several hundred years, but the Indians created the empty circle to be clearer. In 458 AD, Indian mathematicians wrote a book, the Lokavibhaaga, that uses zero in this way. In 628 AD, Brahmagupta wrote a book explaining how zero worked, with rules like "The sum of zero and zero is zero" and "The sum of a positive and a negative is their difference; or, if they are equal, zero." With the formation of the Islamic Empire a few years later, the use of zero spread quickly from India to West Asia and Africa (by the 800s), and then more slowly to Christian Europe (not until the 1200s AD, and only specialists used it until the 1500s).
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