The Big Question: How close have we come to knowing the precise value of pi?

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The Independent Online

Why are we asking this now?

A French computer programmer has calculated pi to a world record of almost 2,700 billion decimal places. Fabrice Bellard said he did it for the programming challenge rather than any particular interest in this infinitely long number. All the more remarkable is that he did it on a desktop computer costing less than £2,000. It took 131 days to complete the calculation and the resulting number took up more than 1,000 gigabytes of memory on his hard drive. Downloading it would take 10 days and reciting it aloud would take about 49,000 years.

What was the old record?

It was achieved using a multi-million pound supercomputer at the University of Tsukuba in Japan last August. Mathematician Daisuke Takahashi calculated pi to 2,577 billion decimal places. But his computer was roughly 2,000 times faster at processing data than Bellard's machine. The Japanese supercomputer calculated pi at a speed of 94.2 teraflops, which is 94.2 trillion floating point operations or calculations per second. It took 29 days.

What exactly is pi?

As every schoolchild should be able to describe, pi is the ratio of the circumference of a circle to its diameter. In other words, divide the distance around the edge of a circle by its diameter – a straight line through its centre connecting one edge to another – and you always get the same constant, denoted by pi. The problem is that this number is infinitely long. For most purposes the value of pi can be estimated to two decimal places, i.e. 3.14. But keep dividing the circumference by the diameter and you soon get to 3.14159265358979323846. Keep on dividing and you will eventually get to the number achieved by Bellard.

Is there another way of describing this number?

Sometimes it's denoted as 22/7 [i.e. 22 over 7] which is an approximation. In Europe this is celebrated by assigning 22 July as pi Approximation Day. In the US, however, they celebrate pi on 14 March – World pi Day – which of course in American date notation is written as 3/14. Either way, you have two opportunities in the year to celebrate the cerebral pleasures of pi.

Why does pi fascinate people so much?

It fascinates experts because it is a number that is considered both irrational and transcendental. It is irrational because it cannot be written as a simple ratio of whole numbers and transcendental because pi is living proof that you cannot square a circle. "The mathematics of pi is often rather pretty," explained Ian Stewart, professor of mathematics at Warwick University. "All numbers are interesting but some are more interesting than others and pi is the most interesting of the lot."

When did we first learn of pi?

It seems to go back to the ancient Babylonians. When the town planners of Babylon began to build the city they took a keen interest in geometry and it became evident to them as early as the 20th century BC that when any circle's circumference was divided by its diameter the resulting number was always about three. In fact they calculated the value of this ratio as equal to 25/8 [25 over 8] which is 3.125 – within 0.5 per cent of the true value of pi.

Interestingly, a less exact value of pi is given in the Bible (Kings 7:23) which described a round basin with the dimensions: 10 cubits in diameter and 30 cubits in circumference. This conveniently results in pi being a nice round number – three – but it is quite inaccurate. Mathematicians would reel back in horror at such a tidy calculation, which is why Professor Fink in an episode of The Simpsons managed to gain the full attention of hall full of babbling scientists when he shouted "pi is exactly three!"

What do we know about the history of pi?

Much has been written about the history of pi. An Egyptian scribe called Ahmes who lived at the time of the Middle Kingdom documented one of the earliest and most accurate accounts of pi in a papyrus scroll written in around 165BC, which was in fact a copy of an earlier scroll. He described pi as the result of dividing 256 by 81, or 3.160.

Archimedes elevated the study of pi to a more theoretical level, which is why it is sometimes called Archimedes' constant, and a succession of Chinese, Indian and Persian scholars all had a bash at calculating the ratio. But it was a Welsh mathematician named William Jones who in 1703 suggested that the constant should be called pi, after the letter in the Greek alphabet.

What do more recent studies tell us?

Two main milestones stick out. The first occurred in 1761 when the Swiss mathematician Johann Lambert demonstrated the irrational nature of pi. We were told in school that pi is about 22/7 but that is only an approximation because pi defies mathematical rationality. Pi is an irrational number, meaning that it cannot be expressed as a fraction of two whole numbers.

The second breakthrough occurred in 1882 when the German mathematician Ferdinand von Lindemann proved that pi was transcendental, meaning that it is not the root of any algebraic equation with rational coefficients. In geometric terms this means that it is not possible with a ruler and a compass alone to find a square with exactly equal area to a given circle – you cannot square a circle.

When did the obsession with calculating pi begin?

The number crunching of pi started in earnest perhaps with the German mathematician Ludolph van Ceulen who, in about 1600, computed pi to the first 35 decimal places. He was so proud of his accomplishment that he had the digits inscribed on his tombstone. A life-long obsessive called William Shanks spent 20 years calculating pi to 707 decimal places. Unfortunately, his achievement in 1873 was subsequently discredited in 1944 when the first digital computers found that he had made a mistake at the 528th decimal place – rendering all subsequent digits meaningless.

The first general computer, the Electronic Numerical Integrator and Computer, calculated pi to more than 2,000 digits in 1949 and in 2005 a Chinese student, Lu Chao, memorised and recited pi to 67,890 digits, without making a mistake.

Can we be sure Mr Bellard hasn't made a mistake?

We'll have to take his word on it for the moment. He said he has spent 16 days verifying the number but other mathematicians are now furiously cross-checking the digits to try to find any flaws. Apparently they have already done the first 2.6 trillion digits and so far have not reported any mistakes. Already there is talk of further number-crunching challenges to break Mr Bellard's record, including one by himself "depending on his motivation".

Why would he or anyone else want to do this?

Don't ask. People who crunch pi are almost as unfathomable as the number itself.