The life of pi

The United States accords pi the ultimate accolade tomorrow, its own national day. Most recall it from their school days (hazily), but here Steve Connor charts its history and celebrates a number that is irrational, transcendental ... and unique

Published: 13 March 2006

In case it has escaped your attention, tomorrow is 14 March which, in American notation, is written 3/14. If you have a certain type of mind you will immediately notice that these digits bear a close approximation to one of the most important numbers in mathematics - pi.

Tomorrow has therefore been declared World Pi Day in honour of the mathematical constant that has beguiled and bewildered successive generations of numerate scholars since the days of ancient Babylon.

Every schoolchild is told 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 and you always get the same or "constant" number - pi.

It's a nice bit of trigonometry that we learn by gradual osmosis and forget by rapid diffusion. Yet its simple truth has provided mathematicians - ancient and modern - with a cornucopia of conundrums.

The first and most interesting is working out the precise value of pi. That has proved something of a challenge since the decimal places of pi can theoretically run on for ever. For the benefit of this short history of pi we can say that the value of the constant is 3.1416. A purist would of course argue that this is a gross estimation, preferring the more precise 3.14159265358979323846. Ultra-orthodox purists would add a few thousand more digits, but even they wouldn't be quite right.

A supercomputer in Tokyo once calculated pi to more than 2 billion digits. It could not, however, reach the final decimal place because as every mathematician knows, that lies somewhere beyond infinity, a place they go only in their dreams.

"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," Professor Stewart said.

The whole point about pi is it is both irrational and transcendental. Irrational because it cannot be written as a simple ratio of whole numbers and transcendental because pi is living proof you cannot square a circle.

If your concentration is beginning to wander a little, then let's start from the beginning.

When the town planners of Babylon began building that ancient city, they took a keen interest in geometry. It became evident to them as early as the 20th century BC that when any circle's circumference is divided by its diameter, the result was always going to be about three. In fact they calculated a value of this ration equal to 25/8 which comes within 0.5 per cent of the true value of pi. A less exact value was given by another early reference to pi, this time in the Bible (Kings 7:23), which described a round basin with the dimensions: 10 cubits in diameter and 30 cubits in circumference.

Scholars point out that, although this gives us a neat and tidy value to pi of exactly three, it is unfortunately quite inaccurate. (This is perhaps why Professor Frink in an episode of The Simpsons managed to gain the full attention of a hall full of babbling scientists when he shouted "pi is exactly three!")

It was, in fact, an Egyptian scribe named Ahmes who gave one of the earliest and most accurate values of pi. He documented it in a Middle Kingdom papyrus scroll written around 1650BC, which was in fact a copy of an even earlier scroll. Ahmes described pi as the result of dividing 256 by 81, or 3.160.

It was however Archimedes who is credited as being the first to elevate the calculation of pi to a more theoretical discipline. It is for that reason the number is sometimes known as Archimedes' constant.

Chinese, Indian and Persian scholars all had a go at calculating the constant but it was not until 1706 that someone gave it the name we know it by today. If William Jones, a Welsh mathematician, is remembered for one thing it is his suggestion to call Archimedes's constant "pi" after the Greek letter.

But the real work on pi had still not begun. In 1761, Johann Lambert demonstrated the irrational nature of pi. In its simplest terms, that meant you could not describe the number as a simple ratio of two whole numbers. Schoolchildren are told that pi is about 22/7, but that this is only an approximation because pi defies mathematical rationality.

The second major discovery came in 1882 when Ferdinand von Lindemann proved that pi had another unusual feature: it was transcendental. In mathematical terms, it means pi is not the root of any algebraic equation with rational coefficients.

In non-mathematical terms this means pi is proof of the old adage: you cannot square a circle. In other words it is not possible with a ruler and a compass alone to find a square with exactly equal area to a given circle.

But the more elegant nature of pi has been subsumed by the all important quest to crunch its numbers. The obsession perhaps began 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 was discredited when the first digital computers found that he had made a mistake at the 528th decimal place - rendering all subsequent digits meaningless.

Kate Bush should perhaps have learnt this lesson before she decided to sing the song "Pi" on her album Aerial. Ms Bush sings each number of Pi to 150 decimal places - or at least that was her claim until a rather sad obsessive type decided to check each digit. "All was well for the first 53 decimal places but then Kate sang 'threeeee oneeee' when she should have sang 'zeeeeeeroooo' instead," said blogger Chris McEvoy.

"She recovered for the next 24 digits but then it went to hell in a handbasket when she missed out the next 22 digits completely before finishing with a precise rendition of her final 37 digits."

The infinite nature of pi has also attracted the interest of science-fiction writers, such as the great American astronomer Carl Sagan who, in his book Contact, buried a hidden signature of alien intelligence within the seemingly random digits of Pi, which have no known pattern. "It was rather naughty, because you can't in fact do this," said Professor Stewart. "You can't arrange pi to have a pattern. It was a nice little conceit on Sagan's part. In a sense a pattern within pi is not something that even God could arrange," he said.

But God or no, that hasn't stopped pi from playing a central role in other science fiction plots. In one episode of Star Trek, Spock saves the Enterprise from destruction when he orders the spacecraft's computer, which has been taken over by aliens, to calculate pi to the last digit.

Terry Pratchett milked the irrational nature of pi for all it was worth in his novel Going Postal, where a wheel has a value of pi that is precisely three. This new pi triggers a chain of events that eventually leads to the destruction of the universe.

So as tomorrow approaches, think long, infinite thoughts of the number. Think about the actual value of numbers, and the approximate near-misses we see when we calculate pi. Pi, you see, is always going to be represented by an approximation because, like all irrational numbers, its digits never really end.

And just in case you miss out on tomorrow's festivities, you'll get another chance on 22 July. This is the day - 22/7 - when European date formats permit a celebration called Pi Approximation Day.

Another Life of Pi

Pi took on an entirely different meaning - and found a new audience - when it featured in the title of a Booker Prize-winning novel in 2002. Born the son of a zookeeper in Pondicherry, India, the hero of Yann Martel's fable finds himself saddled with the name of Piscine as a result of his father's admiration for a swimming pool he once visited in Paris. The young lad, tired of being teased, changes his name to Pi before finding himself adrift for 227 days in a small boat with a tiger following a shipwreck.

Numbers across the world

By Jerome Taylor

8 Chinese culture has long placed importance on the ability of numbers to predict the future and bring luck. Eight is auspicious for the Chinese because it sounds similar to the Cantonese word for prosperous " fa" and it is considered synonymous with the transformation of bad into good. Chinese businesses will pay top money for phone numbers, number plates and addresses containing the number 8. In 2003, an airline in China paid £160,000 for the phone number 88888888.

7 The king of auspicious numbers, seven has a long and significant history. For the Abrahamic traditions, the number is of particular importance and is often referred to as the perfect number. The Old and New Testaments are littered with references to the number, while The Book of Revelations mentions it 55 times. Similarly seven is a key symbol in the Koran where it is mentioned approximately 25 times and plays a central role in forming the Islamic belief system. At the height of the Haj, Muslims circle the Ka'ba in Mecca seven times.

13 Many cultures have associated the number with bad luck but perhaps none more so than modern day America. Cities lack 13th Avenues and many buildings in the States have no 13th floor. Conspiracy numerologists are quick to point out some of the world's most notorious killers, including Jack the Ripper and Charles Manson, have 13 letters in their name.

INFINITY: The ultimate impossible number, infinity has confounded mathematicians and philosophers. In Western tradition, Aristotle was one of the first to tackle this never-ending number-crunch, making the distinction between actual infinity and potential infinity. In 1895, German mathematician Georg Cantor expanded on the various theories surrounding infinity. The earliest known reference to infinity, however, appears in the Yajurveda - one of the four sacred Hindu Vedas written between 1500BC and 500BC - and was widely discussed by Jain mathematicians at least a hundred years before Aristotle.

In case it has escaped your attention, tomorrow is 14 March which, in American notation, is written 3/14. If you have a certain type of mind you will immediately notice that these digits bear a close approximation to one of the most important numbers in mathematics - pi.

Tomorrow has therefore been declared World Pi Day in honour of the mathematical constant that has beguiled and bewildered successive generations of numerate scholars since the days of ancient Babylon.

Every schoolchild is told 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 and you always get the same or "constant" number - pi.

It's a nice bit of trigonometry that we learn by gradual osmosis and forget by rapid diffusion. Yet its simple truth has provided mathematicians - ancient and modern - with a cornucopia of conundrums.

The first and most interesting is working out the precise value of pi. That has proved something of a challenge since the decimal places of pi can theoretically run on for ever. For the benefit of this short history of pi we can say that the value of the constant is 3.1416. A purist would of course argue that this is a gross estimation, preferring the more precise 3.14159265358979323846. Ultra-orthodox purists would add a few thousand more digits, but even they wouldn't be quite right.

A supercomputer in Tokyo once calculated pi to more than 2 billion digits. It could not, however, reach the final decimal place because as every mathematician knows, that lies somewhere beyond infinity, a place they go only in their dreams.

"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," Professor Stewart said.

The whole point about pi is it is both irrational and transcendental. Irrational because it cannot be written as a simple ratio of whole numbers and transcendental because pi is living proof you cannot square a circle.

If your concentration is beginning to wander a little, then let's start from the beginning.

When the town planners of Babylon began building that ancient city, they took a keen interest in geometry. It became evident to them as early as the 20th century BC that when any circle's circumference is divided by its diameter, the result was always going to be about three. In fact they calculated a value of this ration equal to 25/8 which comes within 0.5 per cent of the true value of pi. A less exact value was given by another early reference to pi, this time in the Bible (Kings 7:23), which described a round basin with the dimensions: 10 cubits in diameter and 30 cubits in circumference.

Scholars point out that, although this gives us a neat and tidy value to pi of exactly three, it is unfortunately quite inaccurate. (This is perhaps why Professor Frink in an episode of The Simpsons managed to gain the full attention of a hall full of babbling scientists when he shouted "pi is exactly three!")

It was, in fact, an Egyptian scribe named Ahmes who gave one of the earliest and most accurate values of pi. He documented it in a Middle Kingdom papyrus scroll written around 1650BC, which was in fact a copy of an even earlier scroll. Ahmes described pi as the result of dividing 256 by 81, or 3.160.

It was however Archimedes who is credited as being the first to elevate the calculation of pi to a more theoretical discipline. It is for that reason the number is sometimes known as Archimedes' constant.

Chinese, Indian and Persian scholars all had a go at calculating the constant but it was not until 1706 that someone gave it the name we know it by today. If William Jones, a Welsh mathematician, is remembered for one thing it is his suggestion to call Archimedes's constant "pi" after the Greek letter.

But the real work on pi had still not begun. In 1761, Johann Lambert demonstrated the irrational nature of pi. In its simplest terms, that meant you could not describe the number as a simple ratio of two whole numbers. Schoolchildren are told that pi is about 22/7, but that this is only an approximation because pi defies mathematical rationality.

The second major discovery came in 1882 when Ferdinand von Lindemann proved that pi had another unusual feature: it was transcendental. In mathematical terms, it means pi is not the root of any algebraic equation with rational coefficients.

In non-mathematical terms this means pi is proof of the old adage: you cannot square a circle. In other words it is not possible with a ruler and a compass alone to find a square with exactly equal area to a given circle.

But the more elegant nature of pi has been subsumed by the all important quest to crunch its numbers. The obsession perhaps began 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 was discredited when the first digital computers found that he had made a mistake at the 528th decimal place - rendering all subsequent digits meaningless.

Kate Bush should perhaps have learnt this lesson before she decided to sing the song "Pi" on her album Aerial. Ms Bush sings each number of Pi to 150 decimal places - or at least that was her claim until a rather sad obsessive type decided to check each digit. "All was well for the first 53 decimal places but then Kate sang 'threeeee oneeee' when she should have sang 'zeeeeeeroooo' instead," said blogger Chris McEvoy.

"She recovered for the next 24 digits but then it went to hell in a handbasket when she missed out the next 22 digits completely before finishing with a precise rendition of her final 37 digits."

The infinite nature of pi has also attracted the interest of science-fiction writers, such as the great American astronomer Carl Sagan who, in his book Contact, buried a hidden signature of alien intelligence within the seemingly random digits of Pi, which have no known pattern. "It was rather naughty, because you can't in fact do this," said Professor Stewart. "You can't arrange pi to have a pattern. It was a nice little conceit on Sagan's part. In a sense a pattern within pi is not something that even God could arrange," he said.

But God or no, that hasn't stopped pi from playing a central role in other science fiction plots. In one episode of Star Trek, Spock saves the Enterprise from destruction when he orders the spacecraft's computer, which has been taken over by aliens, to calculate pi to the last digit.

Terry Pratchett milked the irrational nature of pi for all it was worth in his novel Going Postal, where a wheel has a value of pi that is precisely three. This new pi triggers a chain of events that eventually leads to the destruction of the universe.

So as tomorrow approaches, think long, infinite thoughts of the number. Think about the actual value of numbers, and the approximate near-misses we see when we calculate pi. Pi, you see, is always going to be represented by an approximation because, like all irrational numbers, its digits never really end.

And just in case you miss out on tomorrow's festivities, you'll get another chance on 22 July. This is the day - 22/7 - when European date formats permit a celebration called Pi Approximation Day.

Another Life of Pi

Pi took on an entirely different meaning - and found a new audience - when it featured in the title of a Booker Prize-winning novel in 2002. Born the son of a zookeeper in Pondicherry, India, the hero of Yann Martel's fable finds himself saddled with the name of Piscine as a result of his father's admiration for a swimming pool he once visited in Paris. The young lad, tired of being teased, changes his name to Pi before finding himself adrift for 227 days in a small boat with a tiger following a shipwreck.

Numbers across the world

By Jerome Taylor

8 Chinese culture has long placed importance on the ability of numbers to predict the future and bring luck. Eight is auspicious for the Chinese because it sounds similar to the Cantonese word for prosperous " fa" and it is considered synonymous with the transformation of bad into good. Chinese businesses will pay top money for phone numbers, number plates and addresses containing the number 8. In 2003, an airline in China paid £160,000 for the phone number 88888888.

7 The king of auspicious numbers, seven has a long and significant history. For the Abrahamic traditions, the number is of particular importance and is often referred to as the perfect number. The Old and New Testaments are littered with references to the number, while The Book of Revelations mentions it 55 times. Similarly seven is a key symbol in the Koran where it is mentioned approximately 25 times and plays a central role in forming the Islamic belief system. At the height of the Haj, Muslims circle the Ka'ba in Mecca seven times.

13 Many cultures have associated the number with bad luck but perhaps none more so than modern day America. Cities lack 13th Avenues and many buildings in the States have no 13th floor. Conspiracy numerologists are quick to point out some of the world's most notorious killers, including Jack the Ripper and Charles Manson, have 13 letters in their name.

INFINITY: The ultimate impossible number, infinity has confounded mathematicians and philosophers. In Western tradition, Aristotle was one of the first to tackle this never-ending number-crunch, making the distinction between actual infinity and potential infinity. In 1895, German mathematician Georg Cantor expanded on the various theories surrounding infinity. The earliest known reference to infinity, however, appears in the Yajurveda - one of the four sacred Hindu Vedas written between 1500BC and 500BC - and was widely discussed by Jain mathematicians at least a hundred years before Aristotle.

## Sunday, March 12, 2006

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