Hi, everyone. We're continuing this series about Ramanujan.
Last time, I'd left you with Ramanujan working as a clerk at Madras Port Trust.
Now in the local area, he was starting to gain a reputation as being a mathematical genius.
So his friends were encouraging him to write to Cambridge mathematicians.
Now these letters would have looked like the letters of a crank.
They would have been long lists of formulas, and no proofs, right..
Some of these letters were being ignored, and I don't blame them.
So, mathematicians get letters like this all the time, right.
Outrageous claims and no proofs.
They should be grateful they didn't have e-mail in those days.
You should see my inbox!
Now, G.H.Hardy recognized some talent there.
So in this long list of formulas, some of them were just rediscoveries of old results.
But some of them were new, and some of them were new results, and important results.
So G.H. Hardy and Ramanujan had arranged a scholarship so Ramanujan could come to Cambridge to work
for, effectively, his Ph.D. This was in 1914, when Ramanujan was 26.
And the second formula that I want to show you is one that he came up with during this Ph.D.
Now, just before I show you this formula from Ramanujan,
I want to compare it with a formula from classic mathematics.
This is a formula by Leibniz, the 17th century mathematician
who came up with this sum. So we've got here one minus a third, plus a fifth, minus a seventh, plus--and this is something that goes on forever,
so this is an infinite sum.
So the first question is, does this infinite sum have a value?
Now, it does, and the value is kind of surprising.
So, the value of this infinite sum is pi over four, which is amazing!
It's a classic formula, it's a great formula, and in theory, you could use this to calculate pi.
So Ramanujan decided to come up with his own pi formula.
So I'm going to show you this. Ramanujan's pi formula was this here.
9801 divided by 1103, square root of 8, and this is equal to pi up to six decimal places.
In fact, this is just the first term in another infinite series.
And if you take that as a whole, you get pi.
Now, six decimal places might not sound like a lot, but it's enough to accurately calculate the circumference of the earth.
You could use it to calculate the circumference of the Earth to within one meter.
So, with an error of less than this.
Somewhere less than one meter.
Now, to do the same with Leibniz's formula, to calculate pi to six decimal places,
you would have to calculate Leibniz's formula up to five million terms!
Ramanujan's formula can do it in one term,
and each successive term adds on another eight decimal places.
Although Ramanujan proved this to be true, no one knows how he actually got there in the first place,
especially these values like 1103.
But unlike other mathematicians, Ramanujan would deduce his results from numerical examples.
And now mathematicians have used Ramanujan's formula, or formulas inspired by that,
to now calculate pi to record breaking numbers of digits.
So Ramanujan's pi formula was something he came up with during his time at Cambridge,
although it wasn't his only result--he came up with far more important results during that time as well,
in things like hypergeometric series, elliptic functions, and partitions.
But everyone loves pi, so I thought I'd show you Ramanujan's pi formula.
Tomorrow's formula will be the third in this series.
And tomorrow's formula is something that he noticed before he arrived at Cambridge,
but only something that he perfected during his time working with G.H. Hardy.
It's going to be slightly more controversial, so I'm going to look forward to that.
And I'll see you next time.