History about Ancient Pi Approximations and a Joke

April 25, 2012

A bit of History about Pi

Mainstream historians believe that ancient Egyptians had no concept of π

As early as the 19th century BCE, Babylonian mathematicians were using π ≈ 25/8, which is about 0.5 percent below the exact value

The Indian astronomer Yajnavalkya gave astronomical calculations in the Shatapatha Brahmana (c. 9th century BCE) that led to a fractional approximation of π ≈ 339/108 (which equals 3.13888…, which is correct to two decimal places when rounded, or 0.09 percent below the exact value).

Recall an n-gon is a polygon with n sides.

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach utilizing polygons which was used around 250 BC by Greek mathematician Archimedes.

Archimedes of Syracuse

(Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC)

This polygonal algorithm remained the primary approach for computing π for over 1,000 years. Archimedes computed upper and lower bounds of π by drawing regular polygons inside and outside a circle, and calculating the perimeters of the outer and inner polygons as you can see below.

By using the equivalent of 96-sided polygons, he proved that :

223/71 < π < 22/7.

Archimedes’ upper bound of 22/7 may have led to widespread belief that π was equal to 22/7. Around 150 AD, Greek-Roman scientist Ptolemy, in his book Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.

(From Wikipedia, several articles)

Nowadays we know hundred of thousands of decimal figures of PI, but using computers!

A  joke

Time to relax

See you tomorrow!


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