Bigger picture

# Pinning down $\pi$ Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

People seem to have have known about $\pi$ for a very long time—after all, it’s quite natural to want to compute the area or perimeter of a circle. What they didn’t know, and what we still don’t know, is the exact value of $\pi$. Since it’s irrational, we can’t write it as a simple fraction, its decimal expansion is infinite and doesn’t end in a recurring block of numbers, so any attempt to write it down will be nothing more than an approximation. The Babylonians, whose culture flourished around 3000 years ago, seem to have mostly made do with a value of $3$, though one clay tablet gives the slightly better value of $3.125$ (written in modern notation). The Egyptians produced a closer estimate of $3.16$. The number $\pi$ even comes up in the Bible, implicitly, in a passage of 1 Kings 7 which sees Huram of Tyre build a circular “sea of cast metal” for King Solomon. It measured $30$ cubits in circumference and $10$ in diameter, giving the crude approximation of $30/10 =3$.

The first person to approach the problem of estimating $\pi$ systematically seems to have been Archimedes working in the third century BC. He fitted a regular polygon with $96$ sides inside a circle, so that its vertices were sitting on the circle. Then he drew a polygon with the same number of sides around the circle, so that the centres of its sides just touched the circle.

The areas and perimeters of these two shapes sandwich those of the circle, and Archimedes arrived at the estimate $\frac{223}{71} < \pi < \frac{22}{7}.$

The larger of these two numbers, $22/7$, might be familiar—even today some people think that $\pi$ is exactly equal to it. It’s an assertion that makes most mathematicians scream: writing $\pi$ as a bog standard fraction is denying its irrationality and thereby much of its mystery. Archimedes was out-performed, at least in calculational stamina, by the Chinese mathematician and astronomer Zu Chongzhi who in the fifth century AD worked out that $3.1415926 < \pi < 3.1415927.$ Zu Chongzhi’s original writings are lost, but if he used Archimedes’ method then he must have inscribed a regular polygon with $24,576$ sides in a circle and made a huge number of calculations to admirable accuracy.

Archimedes’ method for approximating $\pi$ is reassuring. Although we can never pin $\pi$ down exactly, it’s possible, at least in theory, to calculate it to any degree of accuracy by increasing the number of sides of the polygon you inscribe in a circle. But Zu Chongzhi’s effort also illustrates a limitation of this approach: you might have to make an enormous number of calculations to get a result that’s only accurate to a few decimal places.

Modern approaches to estimating $\pi$ use methods that are surprisingly un-geometrical. In the 16th and 17th centuries, in the age of calculus with its penchant for infinity, mathematicians came up with a number of expressions for $\pi$ using infinite sums and products. Perhaps the most beautiful result carries the names of Gottfried Leibniz, one of the inventors of calculus, and the Scottish mathematician James Gregory, although it had been discovered two centuries earlier by the Indian mathematician Madhava of Sangamagramma. It says that if we start with $1$, subtract $1/3$, then add $1/5$, subtract $1/7$, add $1/9$, and keep going like this forever, the sequence of numbers we get converges to $\pi/4$. That is, $\pi/4$ is the result of the infinite sum $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$ It’s an amazing result, linking geometry on the left-hand side of the equation to arithmetic on the right. But it is still useless when it comes to approximating $\pi$. If you add/subtract the first hundred terms in the Madhava-Leibniz-Gregory sum and multiply by $4$, the number you get is only accurate to only two decimal places.

However, in the early twentieth century, however, two Russian brothers, David and Gregory Chudnowski, came up with a complicated infinite expression for $\pi$ that converges quite fast. In 2011 Shigeru Kondo, a Japanese systems engineer, and Alexander Yee, a 23 year old graduate student who by his own admission “sucked” at school maths, used that formula to give the best estimate of $\pi$ to date. It was computed on Kondo’s home computer and gives $\pi$ to over ten trillion digits. The ten trillionth digit is $5$.

Incidentally, there are also people who prefer to memorise the digits of $\pi$ without the aid of a computer, a practice known as piphilology. The current official record stands at 67890 digits and is held by the Chinese Lu Chau, though the Ukrainian Andriy Slyusarchuk claims to have memorised all of 3 million digits.