Mathematics without history is soulless

π through the ages

I devoted some of my vacation to perusing the Rhind papyrus. It makes for fascinating reading. Dated 1650 BC (and now housed in the British Museum), this five-metre long scroll captures the rich mathematical legacy of Egypt. The pyramids of Giza stand tall as testimony to the Egyptians’ amazing skill and temperament for measurement. The Rhind papyrus gathers their broader contributions to arithmetic and geometry. It includes the Egyptians’ very own decimal counting system and a collection of problems that demonstrate an extraordinary flair for unit fractions.

Problem 50 of the Rhind Papyrus, which gives rise to an approximation of π

The papyrus also documents a primitive but elegant method of estimating the value of π. More precisely, the Egyptians approximated the area of a circle of diameter 9 by first chopping off a ninth of its diameter, then constructing a square with sides of length this reduced diameter, and finally calculating the area of the square.

‘Squaring’ the circle to approximate π (source)

Since the actual area of the circle is π*(9/2)² and the area of the square is 8², this comes down to estimating π as 256/81 — around 3.16, within 1% of its actual value. Not bad for 1650BC.

We have known since time immemorial that π is a constant — that is, the ratio of circumference to diameter of a circle is always the same, regardless of its size. Both your shirt button and the Earth’s equator (indulge me for a moment by assuming they are both perfect circles) will return the exact same ratio.

It was long suspected that π is an irrational number, so that its decimal expansion will never exhaust or repeat (this was finally proven in the 18th century). Approximating π has thus been a labour of love for every major civilisation. Archimedes made a quantum leap of progress by using an iterative method involving polygons of any size. The Chinese captured π to seven decimal places by the fifth century. Srinivasa Ramanujan — he who knew infinity (and π, it seems) — set the pace in the early twentieth century with outrageously fanciful representations of π in terms of infinite sums. Modern computational methods have perhaps taken the thrill out of the chase, reaching 22 trillion digits (yet they’re just as many decimal places away as the Chinese were).

Ramanujan’s approximation — outrageous isn’t the word

This is a mere glimpse into mankind’s eternal fascination with π. More than a number, it cuts across multiple fields — arithmetic, geometry, algebra and more– baffling and delighting mathematicians of all cloths to this day.