π: from Āryabhaṭa's approximation to Mādhava's infinite series

0:00Take any circle at all. Measure round it, measure across it, and divide the first by the second. Do that for a small circle and you get a number. Do it for an enormous one and you get the same number. Every circle, the same answer. That is the fact that makes this a number rather than a measurement, and it is why people chased it for thousands of years. Call it pi.

0:28It is worth pausing on how strange that is. Nothing about a circle tells you the two lengths should be related at all, and yet the ratio never moves. The question this video is about is not what pi equals. It is what kind of thing an answer to that question could even be. For a very long time, everyone assumed the answer was a fraction. So the work was to find a better one.

0:57Here is that search run properly. For every bottom number from one up to one thousand two hundred and fifty, work out the top that lands closest, and keep only the ones that beat everything before them. Fourteen bottoms set a record. Three over one. Thirteen over four. Twenty two over seven, the one everybody still carries around. And onwards, each closer than the last, ending at three hundred and fifty five over one hundred and thirteen. That is what centuries of effort looks like when you compress it.

1:37In the year four hundred and ninety nine, Aryabhata gave a value of their own. Three thousand nine hundred and twenty seven over one thousand two hundred and fifty. Divide it out and something unusual happens. It stops. Three point one four one six, and then nothing at all. That is not luck. The bottom number is two times five times five times five times five, and nothing else. A fraction whose bottom is built only out of twos and fives always has a decimal that stops. So this is a fraction you can write down completely, with no dots on the end.

2:21How good is it? Set it beside pi's own expansion and read across. Three, point, one, four, one. Both agree. Then the fourth decimal place. Aryabhata has a six. Pi has a five. That is where they part company. The difference is about seven millionths. To feel how small that is, compare it with twenty two over seven, which is the fraction most people were taught. That one is out by more than a thousandth. Aryabhata's value is more than a hundred times closer. Highly accurate is not flattery here. It is a measurement.

3:04And now the part that matters more than the accuracy. Aryabhata did not present this as the value of pi. They attached a word to it. Asanna. Approaching. Near. They were saying, in their own writing, that this is a good value and not the true one, and they indicated that they did not think an exact fraction would ever be found. Think about what that takes. They had the best number anyone had. And they labelled it as not the answer. That label is a better piece of mathematics than the number is.

3:43They were right to hedge, and in a way they could not have checked. Run the search again and that fraction does not even set a record. Three hundred and fifty five over one hundred and thirteen is closer to pi than it is. And look at the bottoms. A hundred and thirteen, against one thousand two hundred and fifty. A smaller fraction, doing better. That one is out by between two and three ten millionths. Better again by more than twenty five times. And still not pi. Every rung you climb, you are still on the ladder.

4:23Twelve hundred and sixty two years after that value was written down, the question was settled. In seventeen sixty one, Johann Lambert proved that pi is irrational. No ratio of two whole numbers is equal to it. Not that nobody had found one yet. That there is none to find. Which means the whole search was hopeless in principle. Every record-setting fraction on that list, and every one anybody might have gone on to discover, was always going to miss.

4:58Aryabhata's hedge was not modesty. It was correct, twelve centuries early. Now a reasonable next thought. Fine, one fraction cannot do it. What about several of them together? It does not help, and the reason takes one line. Add two fractions and you get a fraction. Subtract them, multiply them, divide them, and you still get a fraction. Every time. A hundred and forty four two step combinations were tried here. Every single one came out as a ratio of whole numbers, and not one of them was pi.

5:39So any finite pile of fractions, however you combine it, is just another fraction. And a fraction has already been ruled out. That argument closes off an entire strategy in a sentence, which is worth noticing. It does not say a clever combination has not been found yet. It says the whole shape of that answer is wrong. Which forces something uncomfortable. If a finite answer is impossible, the answer has to be endless.

6:14So if the answer is not a number you can write down, what is it? In the fourteenth century, Madhava of Sangamagrama, who founded the Kerala school of mathematics, changed what counted as an answer. Instead of a better fraction, they gave a rule. A recipe that never stops running, and whose output creeps closer and closer to pi without ever finishing. That is a strange kind of answer the first time you meet it. It is also, after Lambert, the only kind available.

6:51Here is the rule itself. Take one. Subtract a third. Add a fifth. Subtract a seventh. Add a ninth. Keep going through the odd numbers, flipping the sign each time, forever. Then multiply the whole thing by four. Two things are worth noticing before we run it. The bottom numbers are simply the odd numbers in order, so nothing here has been fitted or tuned to the answer. And the signs alternate. That is what stops an endless addition from running away to infinity, and it is why this settles instead.

7:31Now watch the running totals arrive. One term gives four. Far too big. Two terms give eight thirds, about two point seven. Now too small. Three terms, about three point five. Four terms, about two point nine. They leap over pi and back again, over and back, and the leaps keep getting shorter. Every odd numbered total sits above pi and every even numbered one sits below, checked here for the first forty. Every total after the second stays trapped between the first two.

8:09And the gap shrinks at every single step. Not sometimes. Every step. Be honest about the speed, though. After five terms the total is three point three, which is wrong in the very first decimal place, while one single fraction had already got three places right. To drag the running total inside five ten thousandths of pi takes two thousand terms. This is not the fast way to a decimal. It is an exact description, and that is a different virtue entirely.

8:47Which is what an infinite sum means here. Not a total you eventually finish. The value the running totals are approaching. One more thing worth seeing. Ask for three fractions between three point one four one five and three point one four one six, spaced evenly, and you get three perfectly good ones. All three of them land below pi. Not because of anything you did wrong, but because pi happens to sit in the last quarter of that gap.

9:21But you can pin it as tightly as you like from both sides. Three point one four one five nine two six is below, three point one four one five nine two seven is above, and neither of them is pi. The rule is.