The Endless Pursuit of Pi

How many times does the diameter fit into the circumference of a circle? Wind a string around it and you quickly find it is "about $3$ and a bit." But the moment you try to pin down exactly what that "bit" is, you set off on a chase that lasted humanity four thousand years. That "and a bit" is the circle ratio $\pi$ ―― the most famous number in mathematics, one whose digits never come to an end no matter how far you compute.

When you write $\pi = 3.14159\ldots$, those dots go on forever. Its decimal expansion never ends and never settles into a repeating pattern (it cannot be written as a ratio of two integers ―― it is irrational). That is precisely why computing the circle ratio is always a problem of approximation ―― how close to the true value can you get? In this article we follow the story of that endless chase, and glimpse the very origin of what we call numerical computation.

Squeezing the Circle Between Polygons

What tormented the ancients was the circle ―― a curve. You can measure the length of a straight line, but a curved circumference resists any ruler. The first brilliant answer to this puzzle came from Archimedes, a Greek mathematician of the third century BCE.

His idea was this: consider a regular polygon that fits snugly inside the circle, and another that wraps snugly around it from the outside. A polygon's sides are straight, so their total length can be computed. The circumference is then longer than the inner polygon and shorter than the outer one ―― that is, taking a circle of diameter $1$ (whose circumference is exactly $\pi$), $\pi$ is squeezed between two numbers.

Add more sides and the polygons hug the circle ever more closely (Figure 1). Archimedes patiently worked all the way up to a 96-sided polygon and derived the splendid range $\dfrac{223}{71} < \pi < \dfrac{22}{7}$, that is $3.1408\ldots < \pi < 3.1428\ldots$ (the lower end lies about $0.024\%$ and the upper end about $0.040\%$ from the true value). With no computer ―― not even today's written notation for arithmetic ―― he reached this far by hand alone. Approximating a circle with polygons lets you savor this "squeeze" a little more slowly.

Infinite Sums ―― a New Weapon

The polygon method is reliable, but more digits require doubling the number of sides again and again, computing square roots to high precision at every step ―― brutally laborious. What changed the game was the infinite series, a new weapon pioneered by the Kerala school in India (14th–16th centuries) that flourished in Europe from the seventeenth century onward. Formulas that express $\pi$ as a sum of infinitely many terms were discovered one after another.

The famous Leibniz formula (also called the Madhava–Leibniz formula, since Madhava of the Kerala school already knew it), for instance, is strikingly simple.

Just add and subtract the reciprocals of the odd numbers, alternating signs, and $\pi$ appears ―― a beautiful formula. Yet it approaches its limit painfully slowly: even after $100$ terms the error is still about $0.01$, after $1000$ terms about $0.001$, and after $10{,}000$ terms about $0.0001$ ―― every extra correct digit demands ten times as many terms (Figure 2) ―― slower, in fact, than polygons, which gain a digit for roughly $3$ times as many sides. So people searched desperately for series that converge faster. Using cleverly engineered formulas like the one in computing π with Machin's formula, virtuosos of hand calculation wrote out hundreds of digits of $\pi$ on paper.

For example, a series left by the Indian mathematician Nilakantha uses the same simple adding and subtracting of fractions, yet approaches $\pi$ far faster than Leibniz's formula.

Since you are using a series, you must stop adding somewhere. The part you "cut off" becomes the error ―― a feeling you can get a grip on in the feel of truncating a series. A fast series, then, is simply one that "achieves a small error with few terms." Without anyone quite noticing, the race for digits of $\pi$ had become the central problem of numerical computation itself: how to approximate cleverly.

How Computers Changed the Race for Digits

When computers arrived in the twentieth century, the pursuit of $\pi$ entered a new phase. Machines blew past in an instant the digit counts that humans had spent lifetimes obtaining. But the fascinating thing is that the race did not end there ―― it accelerated. A million digits, a billion, a trillion ―― and now it reaches hundreds of trillions (Figure 3).

Why compute so many? In practice a dozen or so digits of $\pi$ is more than enough ―― even handling a circle the size of the observable universe to the precision of a single hydrogen atom takes fewer than forty digits (NASA JPL estimates $37$ decimal places). We keep extending the count anyway because computing $\pi$ serves as a yardstick for the power of new algorithms and computers. Think of a fast new method? Try it on $\pi$. Build a new supercomputer? Compute $\pi$ to check that every corner of it works. Over a long computation, it can even happen that a cosmic ray strikes memory and a single bit flips for a moment (a transient error, with the hardware undamaged, called a "soft error" ―― here "soft" does not mean software, but "not a permanent failure"). Such a chance corruption shows up, from a discrepancy in even a single digit, just by running the same computation again and comparing. And by deliberately also computing with a different formula and cross-checking, you can catch even nastier faults ―― a bug in the program, or a CPU design flaw that "returns the same wrong answer every time." A $\pi$ that matches to an enormous number of digits is the surest proof that the machine ran to the end without a single mistake. The circle ratio became, somewhere along the way, the arena for testing one's technical mettle.

Living With "Never Divides Evenly"

What the story of $\pi$ teaches is an obvious yet profound fact: mathematics has partners that never divide neatly. $\pi$ cannot be written as a fraction (it is irrational), and as a decimal it never ends. There was never any "last digit" to begin with. So when we use it as a number, all we can do is use a value brought close enough to the precision we need, knowing it carries an error.

This is no sour grapes. It is the very starting point of the field of numerical computation. Most quantities in the world, like $\pi$, refuse to fit into neat forms. Most square roots and logarithms, and most numbers in nature ―― none of these can be written out in finitely many digits. That is exactly why we need techniques for obtaining not "the true value itself" but "a value close enough," together with a justified error. The circle ratio is the oldest and most famous exercise on which humanity has honed those techniques over two thousand years.

Coda ―― Endless, and All the Better for It

From Archimedes' polygons, through the virtuosos who wielded infinite series, to supercomputers racing for digits ―― the tools changed, yet humanity kept chasing one and the same number. Knowing full well that no end will come, still pressing for one more digit ―― this seemingly pointless obsession is exactly what forged the algorithms of approximation and raised the field of numerical computation.

"Deal wisely with numbers that never divide evenly, all the while estimating the error." That is the wisdom the circle ratio left us. Carry this feeling with you to the basic level next, and look a step deeper into floating-point numbers and the true nature of error. You should see the endless story of $\pi$ transforming into solid computational technique.

Summary

Further Reading

• Gottfried Wilhelm Leibniz (Wikipedia) ―― the seventeenth-century German mathematician and philosopher after whom the "Leibniz formula" in the text (the Madhava–Leibniz formula) is named.
• Nilakantha Somayaji (Wikipedia) ―― the mathematician of the Indian Kerala school, Nilakantha, who left a series that converges faster than Leibniz's.
• How Many Decimals of Pi Do We Really Need? (NASA JPL) ―― an estimate of how many digits of π practical and cosmic-scale calculations need.