The story of pi

by Lazarus Mudehwe

Undoubtedly, pi is one of the most famous and most remarkable numbers you have ever met. The number, which is the ratio of circumference of a circle to its diameter, has a long story about its value. Even nowadays supercomputers are used to try and find its decimal expansion to as many places as possible.

For pi is one of those numbers that cannot be evaluated exactly as a decimal --- it is in that class of numbers called irrationals.

The hunt for pi began in Egypt and in Babylon about two thousand years before Christ. The Egyptians obtained the value (4/3)^4 and the Babylonians the value 3 1/8 for pi. About the same time, the Indians used the square root of 10 for pi. These approximations to pi had an error only as from the second decimal place.

```	(4/3)^4	   =	3,160493827...

3 1/8	   =	3.125

root 10	   =	3,16227766...

pi	   =	3,1415926535...
```

The next indication of the value of pi occurs in the Bible. It is found in 1 Kings chapter 7 verse 23, where using the Authorised Version, it is written "... and he made a molten sea, ten cubits from one brim to the other : it was round about ... and a line of thirty cubits did compass it round about." Thus their value of pi was approximately 3. Even though this is not as accurate as values obtained by the Egyptians, Babylonians and Indians, it was good enough for measurements needed at that time.

Jewish rabbinical tradition asserts that there is a much more accurate approximation for pi hidden in the original Hebrew text of the said verse and 2 Chronicles 4:2. In English, the word 'round' is used in both verses. But in the original Hebrew, the words meaning 'round' are different. Now, in Hebrew, etters of the alphabet represent numbers. Thus the two words represent two numbers. And - wait for this - the ratio of the two numbers represents a very accurate continued fraction representation of pi! Question is, is that a coincidence or ...

Another major step towards a more accurate value of pi was taken when the great Archimedes put his mind to the problem about 250 years before Christ. He developed a method (using inscribed and circumscribed 6-, 12-, 48-, 96-gons) for calculating better and better approximations to the value of pi, and found that 3 10/71 < pi < 3 10/70. Today we often use the latter value 22/7 for work which does not require great accuracy. We use it so often that some people think it is the exact value of pi!

As time went on other people were able come up with better approximations for pi. About 150 AD, Ptolemy of Alexandria (Egypt) gave its value as 377/120 and in about 500 AD the Chinese Tsu Ch'ung-Chi gave the value as 355/113. These are correct to 3 and 6 decimal places respectively.

```	377/120	       =	3,14166667...

22/7	       =	3,142857143...

355/113	       =	3,14159292...

pi	       = 	3,1415926535...
```

It took a long time to prove that it was futile to search for an exact value of pi, ie to show that it was irrational. This was proved by Lambert in 1761. In 1882, Lindemann proved that pi was more than irrational --- it was also transcendental --- that is, it is not the solution of any polynomial equation with integral coefficients. This has a number of consequences

• It is not possible to square a circle. In other words, it is not possible to draw (with straight edge, compass and pencil only) a square exactly equal in area to a given circle. This problem was set by the Greeks two thosand years ago and was only put to rest with Lindemann's discovery.
• It is not possible to represent pi as an exact expression in surds, like root2, root7 or root5+root3/root7, etc.

From that time on interest in the value of pi has centred on finding the value to as many places as possible and on finding expressions for pi and its approximations, such as these found by the Indian mathematician Ramanujan:

```		(1 + (root3)/5)*7/3	 =	3.14162371...

(81 + (19^2)/22)^(1/4)   =	3.141592653...

63(17+15root5)/25(7+15root5) =	3.141592654...

pi			 =	3.141592654...
```

The last approximation is so good (9dp) that my ancient Casio calculator tells me it's the same as pi! (Sadly, many people would believe my calculator)

Currently the value of pi is known to 6.4 billion places, but I won't attempt writing them out!

Finding info on the web is one of the easiest tasks in existence. Steve Berlin has a nice article, and this site offers software that can be used to get pi to plenty decimal places. Want to change the value of pi? Sorry, the voting is over, but the results are here. I could go on and on, but instead I'll just leave you with The Albany Pi Club which has several links, including the brilliant Uselessness of pi page and the recently started Joy of Pi page.

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