Pi: It Will Blow Your Mind

The history of Pi encompasses many centuries. An early approximation, from the Babylonians, is 3.14159… Another is from the Egyptians, 3 1/8 or 3.125, which could be concluded to indicate that they believed a circle with diameter nine has the same area as a square of diameter 8. The Bible was content to use 3, as demonstrated by I Kings 7:23:

Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.

Archimedes of Syracuse, 250 BC, computed pi to be

pi = 256/81 = 3.1604…

The astronomer Ptolemy, of Alexandria AD 150, used

3.1408 < pi < 3.1428

Also, in China in the fifth century, Tsu Chung-Chih calculate pi correctly to seven digits. Later in 1430, Al-Kashi of Samarkand computed pi to 14 places. Machin invented the formula

pi/4 = [(1/2) - (1/(3 * 2³)) + (1/(5 * 2^5)) - (1/(7 * 2^7)) … ] + [(1/(3 * 3³)) +

(1/(5 * 3^5)) - (1/(7 * 3^7)) … ]

Mathematicians used this formula to compute 707 digits in 1873. Later, they found the error after the 527-th decimal place. Newton developed a formula of his own:

pi/4 = 4 * tan^-1 * (1/5) - tan^-1 * (1/239)

Newton published 15 digits derived from this formula and later said

"I am ashamed to tell you how many figures I carried these computations, having no other business at the time."

Euler, in the 1700s, discovered many more formulas including

pi = (3 * square root of pi) / 4 + 24 * [(1 / (3 * 2^3)) - (1 / (5 * 2^5)) …}

and the simpler

pi² / 6 = 1 + 1 / 2² + 1 / 3² + 1 / 4² + 1 / 5² + …

and

pi^4 / 90 = 1 + 1 / 2^4 + 1 / 3^4 + 1 / 4^4 + 1 /5^4 + …

In 1897, Edwin J. Goodman, MD introduced and got passed a bill into the Indiana State House of Representatives which stated that

"the ratio of the diameter and circumference is as five-fourths to four;"

Other fallacies continued, as shown by a mathematician named Buffon’s needle experiment. He dropped a needle of k < 1 on a uniform grid of parallel lines. The probability of the needle falling on a line was 2 * k / pi. He and many other got the incredible answer of

pi = 355/113 = 3.1415929.

A mathematician named Gridgeman once did a completely facetious experiment in which he threw a needle of length 0.7857 and gave the value of pi as

2 x 0.7857 / pi = ½

Also incredibly he received the answer of 3.1428 = pi!

All this leads one to wonder what their fascination is with pi. There are no practical applications for pi at an industrial level. Simple approximations are suitable for engineering. So why do they insist on calculating pi to 6,442,450,938 decimals? One might argue science for the sake of science.

Pi has baffled and intrigued scientists and mathematicians for thousands of years. Perhaps they believe there is something about pi that would ease all other types of calculations. Perhaps they believe pi holds the key to future computation in areas of mathematics yet unexplored. Whatever it is, many more surprises and discoveries are to be made.