The first known use of numbers in India was in the late Stone Age, about 4000 BC. Around this time, people at Mehrgarh 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 in the Bronze Age developed standard weights (like our ounces and grams), and they were the earliest people to use base 10 for their weights: to use weights in multiples of 10 and 100, rather than 12 and 60 as they did in Mesopotamia.
After climate change caused the Harappan civilization to collapse, about 2000 BC, some invading Indo-Europeans came to India. This did not stop mathematical progress, and may even have encouraged it. The Indo-Europeans may have brought new Babylonian mathematical ideas to India. But Indian mathematicians kept on contributing new ideas of their own. 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 movement of the sun and moon as they seemed to move around the earth. Nobody knows whether Lagadha worked these rules out on his own, or learned about them from the Babylonians.
By about 400 BC, a little after the time of the Buddha, 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.
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.
A generation later, about 500 AD, the Indian mathematician Arya Bhata worked to improve on Hipparchus and Ptolemy‘s trigonometry, developing the ideas of sines and cosines, tangents and cotangents, and writing the first table of sines in order to help with his work in astronomy. Arya Bhata also worked on ways of measuring time. And he worked on improving our understanding of place value and zero. 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.”