Sunday, March 14, 2010

A Brief History of Pi

The date March 14th, or 3/14, is Pi Day in the U.S., so I thought I'd commemorate by sharing a brief history of one of the most important and frequently occurring numbers in mathematics.

Pi (usually denoted by the Greek letter π) is the ratio of the circumference to the diameter of a circle.

π = C / d

This ratio is constant, regardless of the size of the circle. π can also be defined as the ratio of the area to the square of the radius of a circle.

π = A / r2

Time line

c. 2000 BC - Babylonians put the ratio of the circumference of a circle to its diameter at around 3. Later estimates (around 1900 B.C.) put the value at 25/8 (3.125). At around the same time in Egypt, the approximation 256/81 (about 3.16) was used, and in India, 339/108 (about 3.139).

c. 225 BC - Archimedes of Syracuse showed that the value of π was between 310/71 and 31/7. He used a method of inscribing a circle inside regular polygons, then estimating π based on their perimeters.

5th century BC - Chinese mathematician and astronomer Zu Chongzhi calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113, which is accurate to seven digits. This is the best approximation to the true value of π of any fraction with a denominator less than 30,000.

c. 1400 - Madhava of Sangamagrama used an infinite series method to estimate π to be 3.14159265359, which is accurate to eleven decimal places.

1424 - Jamshīd al-Kāshī gave an estimate π that is correct to 16 digits.

c. 1600 - Ludolph van Ceulen computed the first 35 digits of π, and had the digits engraved on his tombstone in 1610.

1647 - William Oughtred used π.δ as an abbreviation for "periphery-diameter" in his Clavis mathematicae (The Key to Mathematics).

1706 - William Jones was the first to use the symbol π on its own in his Synopsis Palmariorum Matheseos.

1737 - Leonhard Euler adopted the use of the symbol π, cementing its popularity.

1761 - Johann Lambert proved that π is irrational (π cannot be expressed as the ratio of two integers).

1789 - Jurij Vega calculated 140 digits of π, but only the first 126 were correct.

1881 - Ferdinand Lindemann showed that π is a transcendental number (there is no polynomial with rational coefficients of which π is a root).

1873 - William Shanks computed 707 digits of π, but only the first 527 were correct.

1949 - John von Neumann used ENIAC to compute 2037 digits of π, using about 70 hours of computing time.

1973 - The 1,000,000th digit of π was computed using a CDC 7600 supercomputer.

1999 - Yasumasa Kanada lead a group at Tokyo University that computed π to more than 206 billion places.

2002 - The Tokyo University group beat their own record by computing π to more than 1.24 trillion digits using 600 hours of computing time on a Hitachi SR8000 supercomputer.

2009 - Daisuke Takahashi at the University of Tsukuba in Japan, calculated nearly 2.6 trillion digits of π in 29 hours on a T2K Open Supercomputer. In the same year, Fabrice Bellard computed nearly 2.7 trillion digits of π in a total of 131 days on a 2.93 GHz Core i7 (desktop class) CPU.


The first 50 digits of π are:


Piphilology is the art of composing mnemonic phrases for remembering the digits of π. For example, the first nine digits are encoded in the number of letters in each word of the phrase
How I wish I could recollect pi easily today!
Sir James Jeans embedded the first 15 digits in the following phrase (sometimes attributed to Isaac Asimov, there are several variations):
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!
The first 740 digits of π are encoded in Michael Keith's 1995 variation of Edgar Allen Poe's The Raven. Here's the first section of 42 words:
Poe, E. Near A Raven

Midnights so dreary, tired and weary,
Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap - the weirdest tap!
An ominous vibrating sound disturbing my chamber's antedoor.
"This," I whispered quietly, "I ignore."

Keith followed this poem (or more appropriately, piem) in 1996 with Cadaeic Cadenza, a short story which encodes the first 3,835 digits of π, and in early 2010 he published Not a Wake, a novelette that recounts "A dream embodying π's digits fully for 10,000 decimals."

Additional References:

History of Pi by Petr Beckmann

The Universal Book of Mathematics by David Darling

For a much more detailed record of the computation of the digits of π, see the Chronology of computation of π.


Joe said...

That's all well and good, but how many digits can YOU remember Pi to, Bill?

Seriously, a good read. Lots of interesting facts of information to make my students find. ;)

Bill the Lizard said...

When I was in 8th grade, our teacher offered 100 extra credit points to anyone who could recite 50 digits of Pi in front of the class. (She gave us several weeks to prepare.) I didn't get the extra credit, but I did remember around 40 digits back then. I've forgotten most of them, but I can still remember 3.14159265 without using the first mnemonic. Plenty of precision for most purposes :)

The method Archimedes used of inscribing a circle inside regular polygons should be a fun exercise to show your students. Actually, that page on Archimedes is full of stuff that middle-to-high school students might have fun learning about. He lived in a fascinating period of history.

pbewig said...

At my blog Programming Praxis I showed a program that calculates π using both Archimedes' method and a monte-carlo method, in an another blog entry I gave a spigot algorithm for calculating the digits of π one at a time.

Jonathan Sampson said...

Excellent article, Bill. Like you, I've only ever really remembered the first few places of pi. Just enough to impress the ladies. Keep up the good work, you've got an excellent blog.

Bill the Lizard said...

Thanks for the additional links. The Monte Carlo method is an old favorite of mine. I haven't looked into the spigot algorithm yet, but maybe on next Pi day. :)

Bill the Lizard said...

If the ladies aren't impressed with how many digits of Pi you can recite, then they aren't worth impressing! :) Thanks for reading.

Brian R. Bondy said...

You can remember Pi more easily using a program I made about 10 years ago:

Cristobal said...

Hello Bill.
You wrote much more on pi than I did on my own blog :

I like 355/113 that I discovered on this occasion.

About piphilology... There must be "pipholologs" of all languages. Here is the most famous french one :

"Que j'aime à faire apprendre ce nombre utile aux sages !
Immortel Archimède, artiste ingénieur,
Qui de ton jugement peut priser la valeur ?
Pour moi, ton problème eut de pareils avantages.

Anonymous said...