Infinite pi

The Indian mathematician Mādhava was the first to use infinite series to calculate pi, some time around 1400 CE.

fdecomite / CC BY

Yes, it’s time for yet another pi post. I’ll run out eventually, I promise!

When we last left off, Zu Chongzhi had found an algorithm to calculate pi that proved to be the most accurate for a thousand years. What happened a thousand years later? Two things: the birth of Mādhava of Sangamagrāma, a mathematician in Kerala, India; and his discovery of infinite series.

 Mathematicians used to have a real problem dealing with infinity. Specifically (but not exclusively), let us consider the nature of an infinite series. This is a chain of additions that carries on forever, for example adding up all positive odd integers (1+3+5+7+…) or adding up perpetually halving fractions (½+¼+⅛…). Now obviously that first series, positive odd integers, will just keep on getting bigger forever – in mathematical terms, the series is divergent. But that second example, perpetually halving fractions, is trickier. The longer you keep on going, the closer the sum of those halving fractions gets to 1. It is convergent; specifically, the infinite sum is converging onto 1. Put simply, in certain circumstances you can get a finite sum from an infinite series.

This is a real problem if you’re using to working with hard real solid easy numbers. How can you add up infinity? It’s counterintuitive. But then… a lot of mathematics is counterintuitive. And, it turns out, being able to add up convergent series is extremely useful. What Mādhava discovered was this: an infinite series exists that adds up to pi.

Start with 1. Then subtract a third. Then add a fifth. Then subtract a seventh. Add a ninth. Subtract an eleventh. Keep on going forever, bouncing back and forth, adding and subtracting increasingly small odd fractions. When you get to the end of infinity (ha), multiply the result by 4. The result is pi. Not an approximation of pi. Pi exactly. This discovery was revolutionary. Mādhava, or one of his followers, blew Zu Chongzhi’s pi record out of the water. 

Now, Mādhava is one of those historical figures we know mainly through his followers, so the specifics of when and how he found this are lost in the murk of time. The German mathematician Gottfried Leibniz later discovered the series independently, hence the name often used today: the Madhava-Leibniz series. And there are many more computationally efficient ways to calculate pi now. I’m still taken by the simplicity and elegance of this one.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s