Pi with my little eye

Pi with my little eye
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First Published: Thu, Dec 08 2011. 08 42 PM IST
Updated: Thu, Dec 08 2011. 08 42 PM IST
Everyone has a p (pi) story. Mine involves slide rules. Remember slide rules? If so, I know your age. If not, know this: they are slick little tools that aided calculation in pre-calculator days (there were such days).
The rule I owned, I always stared at one mark on it in fascination. I had learnt about this peculiar number pi. I knew nobody could tell me its value. Yet between “3” and “4” was this tiny line with the Greek letter pi below. If nobody knew its value, how did the guy who made the gadget know exactly where to put that line?
So what is this exotic number anyway? No more, and no less, than the ratio of a circle’s circumference to its diameter, but it has fascinated mankind forever. You can imagine a prehistoric man idly drawing a circle in the sand, imitating the shape of our sun and moon, or flowers. Then he measures it, and finds that its circumference is about three times greater than its diameter.
About, but not exactly.
Thus was born a long romance with pi. As mathematics grew more sophisticated, people found better ways to calculate pi than measurements in the sand. In AD 499, the 23-year-old Indian prodigy Aryabhata wrote in his Aryabhatiya:
“Add four to 100, multiply by 8, then add 62,000. This way we approach the circumference of a circle whose diameter is 20,000.”
This implies a value for pi of 3.1416, off by about (but not exactly) 0.0002%. Accuracy like this, achieved 1,500 years ago, is more than enough for a slide rule. So much for my wonder at how its maker knew where to put the mark.
For centuries, we’ve known pi accurately enough for pretty much every purpose we might imagine. Getting even closer has, for centuries as well, been “purely a matter of computational ability and perseverance”, says Petr Beckmann in his A History of pi. That is, the more work you are willing to do, the nearer you can get to the truth of pi.
Therefore, simply calculating pi isn’t a particularly interesting exercise.
Far more interesting are the schemes people have devised to do the calculation. The Greek genius Archimedes spelled out one that used polygons and a circle. Amazingly, mathematicians used his method for centuries. Not till 1671 did someone discover a totally new approach: the Scottish mathematician James Gregory found this “infinite series”:
p = 4 x (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 …)
It’s infinite because it doesn’t end. But it “converges” nevertheless to pi, that is, the more terms you add and subtract, the closer you get to the value of pi. Try it yourself.
As you probably found out, Gregory’s series is nearly useless for actually finding a value for pi. The problem is that it converges far too slowly; meaning, it takes too long to get close to pi. Accuracy to just two decimal places (digits after the decimal point)—which Archimedes had 19 centuries earlier, which Aryabhata easily surpassed— needs over 300 of Gregory’s terms.
Still, Gregory had opened the floodgates. Charging through soon after was the great Isaac Newton, with a series for pi that was a Lamborghini to Gregory’s bullock cart. With just 22 terms, he had pi correct to 16 decimal places: and this was only a side result from the virtuoso’s other efforts. As Beckmann observes, even “the crumbs dropped by giants are big boulders”.
Through the 17th and 18th centuries, infinite series for pi popped up like so many mushrooms. For example, several mathematicians massaged Gregory’s series in ingenious ways to produce their own series. That way, Abraham Sharp calculated pi to 72 decimal places in 1705, which must have seemed quite a feat. Except that the following year, John Machin raised the bar to 100 places, and he ceded ground in turn some years later to Thomas de Lagny’s 127 places (though his 113th digit turned out to be wrong).
And then there was Leonhard Euler, for whom there’s an excellent case to be made that he was the greatest mathematician of them all. He produced formulae for pi like the rest of us tell jokes that nobody laughs at. One of his more famous series is:
p²/ 6 = 1 + 1/2² + 1/3² + 1/4² + ...
With another one he discovered, the story goes that he used it to calculate pi to 20 decimal places in one hour. You think that’s a trivial thing? Maybe so, in these days of calculators and computers. But Euler’s only tools were pen, paper and his remarkable mind. That speaks of the kind of man he was.
Quite a story, pi. These days, someone makes news every now and then by reciting from memory some phenomenal number of digits of pi. Give it a shot if you like, though I won’t say it’s easy as 3.141592653589793238462643383279...
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers explores the joy of mathematics, with occasional forays into other sciences. Comments are welcome at dilip@livemint.com
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First Published: Thu, Dec 08 2011. 08 42 PM IST