Swiss researchers have spent 108 days calculating pi to a new record accuracy of 62.8tn digits.
Using a computer, their approximation beat the previous world record of 50tn decimal places, and was calculated 3.5 times as quickly. It’s an impressive and time-consuming feat that prompts the question: why?
Pi is, of course, a mathematical constant defined as the ratio between a circle’s circumference and its diameter. The circumference of a circle, we learn at school, is 2πr, where r is the circle’s radius.
It is a transcendental, irrational number: one with an infinite number of decimal places, and one that can’t be expressed as a fraction of two whole numbers.
From ancient Babylonian times, humans have been trying to approximate the constant that begins 3.14159, with varying degrees of success.
The amateur mathematician William Shanks, for example, calculated pi by hand to 707 figures in 1873 and died believing so, but decades later it was discovered he’d made a mistake at the 528th decimal place.
In 1897, the Indiana Pi Bill in the US almost did away with fussy strings of decimals altogether. The bill, whose purpose claimed to be a method to square a circle – a mathematical impossibility – almost enshrined in law that π = 3.2.
What is it good for? Absolutely everything
Jan de Gier, a professor of mathematics and statistics at the University of Melbourne, says being able to approximate pi with some precision is important because the mathematical constant has many different practical applications.
“Knowing pi to some approximation is incredibly important because it appears everywhere, from the general relativity of Einstein to corrections in your GPS to all sorts of engineering problems involving electronics,” de Gier says.
In maths, pi pops up everywhere. “You can’t escape it,” says David Harvey, an associate professor at the University of New South Wales.
For example, the solution to the Basel problem – the sum of the reciprocals of square numbers (1/12 + 1/22 + 1/32 and so on) – is π2/6. The constant appears in Euler’s identity, eiπ + 1 = 0, which has been described as “the single most beautiful equation in history” (and has also featured in a Simpsons episode).
Pi is also crucial to something in mathematics called Fourier transforms, says Harvey. “When you’re playing an MP3 file or watching Blu-ray media, it’s using Fourier transforms all the time to compress the data.”
Fourier analysis is also used in medical imaging technology, and to break down the components of sunlight into spectral lines, de Gier says.
But, says Harvey, there’s a big difference between calculating pi to 10 decimal places and approximating it to 62.8tn digits.
“I can’t imagine any real-life physical application where you would need any more than 15 decimal places,” he says.
Mathematicians have estimated that an approximation of pi to 39 digits is sufficient for most cosmological calculations – accurate enough to calculate the circumference of the observable universe to within the diameter of a single hydrogen atom.
62.8tn digit accuracy – what’s the point?
Given that even calculating pi to 1,000 digits is practical overkill, why bother going to 62.8tn decimal places?
De Gier compares the feat to the athletes at the Olympic Games. “World records: they’re not useful by themselves, but they set a benchmark and they teach us about what we can achieve and they motivate others.
“This is a benchmarking exercise for computational hardware and software,” he says.
Harvey agrees: “It’s a computational challenge – it is a really seriously difficult thing to do and it involves lots of mathematics and these days computer science.
“There’s plenty of other interesting constants in mathematics: if you’re into chaos theory there’s Feigenbaum constants, if you’re into analytic number theory there’s Euler’s gamma constant.
“There’s lots of other numbers you could try to calculate: e, the natural logarithm base, you could calculate the square root of 2. Why do you do pi? You do pi because everyone else has been doing pi,” he says. “That’s the particular mountain everyone’s decided to climb.”