Junaid Mubeen


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.

Now consider the clean-cut form of π encountered at school. π is definitively introduced to students as the ratio of a circle’s circumference to diameter. There is no significance attached to the constancy of π; no intrigue. While students cannot be expected to grapple with the fiddly geometric proofs of this property, there is a teachable moment in probing that definition and exploring different circle dimensions. The string around the earth problem is a delight once π’s constancy has been critically accepted.

Instead, students are leashed to the drudgery of calculating areas, perimeters, arc lengths, volumes and more — marshalling prescribed formulas for π that they do not fully grasp or care about. The enthused among them will commit the first 10 digits of π to memory; the self-proclaimed genii will go further. They may catch wind of π’s infinite decimal expansion, without ever reflecting on how irrational numbers render digit memorisation futile. They may bask in Pi Day, missing the irony of celebrating this mathematical gem by reducing it to a rough two-decimal form.

March 14 is Pi day — good for puns if not mathematical understanding

This is the parody that ensues when mathematics is divorced from its history. We study history to understand how we arrived at the present. We probe the cause and effect of past human behaviours and we study counterfactuals to understand what might have been. This is how we progress as a species; we recognise that our historical trajectory is a contingent one. We do not accept our current state as immutable.

It should be no different for aspiring mathematicians. Students need to understand that mathematical ideas do not just spring into being. They develop gradually as humans explore and ask questions, often with immense struggle, reward and surprise (is it really obvious that π is constant?) The process of mathematical discovery is messy and uncertain, even if the end result appears clean.

What we now understand of π only comes from standing on the shoulders of past mathematical giants. They were all actors in mankind’s eternal quest to understand the language of the universe. Mathematics is an open invitation for students to carry on that journey, but first they must walk in the shoes of their predecessors, because that’s where our deepest mathematical insights reside. One cannot appreciate the terrifying beauty of π without chronicling past attempts to understand and approximate it.

Historical context gives mathematics (and mathematicians, dare I say) a rich personality that is all too often lost in formal study. It reveals the human side of mathematics; the pain and ecstasy of pursuing new mathematical frontiers. It normalises struggle and perseverence as traits of the common mathematician. It snips away the binary view that many students take towards maths and replaces it with a world replete with discovery and surprise.

No study of mathematics can be considered complete without attention to its history.

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