Indian Mathematics

An email discussion with my friend, Amit, who had just attended a lecture on the topic:

Yeah, I agree there is definitely some jewel of mathematics that was done in India from Aryabhatta to ramanujan. There was the vast period of dark ages in India though, from 0 AD to 19th century (right after Vikramaditya's Golden Age), coupled with Muslim and British invasion, which is said to be the Indian dark ages (except for the development of modern number system. Which makes me wonder, how can a development so profound take place in such a disconnected fashion. Do we then don't know about all the developments in that period?) It completely ruined much of our advancement as well as our self-confidence in ourselves. And it is so heartening to see us rising the wave again.

I am actually in the midst of reading a exceptional book on number thory and mathematics - The man who loved only numbers - a biography of Paul Erdos as well as a history of mathematics. Only ramanujan is mentioned amongst Indian mathematics. And that the number systems and zero was from India. A LOT of work was done in Egypt, Babylon, Persia and of course Greece over the last 2000 years, which India majorly missed out on. Or, maybe they are not as well-publisized, and that's why people why what Dinesh Singh are doing is so important.

Amit Kumar Saha wrote:

The talk was pretty good and if the speaker Dr. Dinesh Singh is to be believed than many of the great things in mathematics were found out by Indians much much before the European mathematicians found them out.
Examples are :

1. Pythagorus theorem - with constructive proof.
2. Pascal's triangle - almost the binomial expression. The actual
expression was later found when going through the notes on Pascal's
triangle by the Indian mathematician. It was of course found out by
European mathematicians later.
3. sine, cosine, differential of sin = cosine. Lots of trigonometry stuff
4. Calculus was very developed, much earlier than Newton.
5. Many numerical series analysis were done, in particular, 1*1 + 2*2 +
3*3 ... as well as 1³ + 2³ + 3³ +...
6. Value of PI was known to 5 decimal places and a series was given to get
the value of PI to as much detail as required.
7. Decimal notation and system of numbers (10 numbers)
8. Concept of infinity, even different levels of infinity and that
dividing by zero is not a good idea.
9. Sets, countable, uncountable.
10. That the earth is round, spins on its own axis, and that it goes round
the sun - Aryabhatta. He also gave an analogy - a person sitting on a boat
feels the bank of the river go past him, whereas actually the person is
moving.
11. Ramanujan gave some formulations of number of prime numbers which are
less than a certain positive integer and later gave an exact proof.

And all these happened over an era of around 3000 years and with many mathematicians involved. Basically, many of the fundamental advances of maths were already there in India, starting from the Indus Valley civilization. Makes you wonder, it is not for nothing that the Indian civilization has survived for so long.