Mathematical Journeys by Peter D. Schumer

Author:Peter D. Schumer [Schumer, Peter D.]
Language: eng
Format: epub
Published: 2011-02-17T05:00:00+00:00

EPISODES IN THE CALCULATION OF PI

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that either 22 or 3.14 is the actual value of π —or even worse, that both are 7

exact values. Perhaps we aren’t so different from the ancient Babylonians and Egyptians who failed to distinguish between exact and approximate formulas.

Archimedes

The idea of using multisided regular polygons to either inscribe or circumscribe a given circle was used by many subsequent mathematicians to make improvements on Archimedes’s estimate. The Chinese scholar Liu Hui (ca. 260

C.E.) used a circle of radius 10 and inscribed polygons starting with a hexagon and working up to a 192-sided polygon. His work leads to π ≈ 3.1416, accurate to four decimal places. A couple centuries later, the Chinese mathematician Tsu-ching Chih (ca. 480 C.E.) proceeded from where Liu Hui had left off, doubling the number of sides six more times. With a 12,288-sided polygon, he was able to establish that π lay between 3.1415926 and 3.1415927. He stated a slightly weaker, but visually striking result: π ≈ 355 . This can be remembered by simply 113

writing 113,355 and separating the number in the middle. For nearly 800 years this was the most accurate value known for π.

The polygon used by Tsu-Chih had 3 · 212 sides. In 1430, the Arab scholar Al-Kashi continued with calculations working up to a polygon with 3 · 228 sides and derived 16 accurate decimal places for π. The most impressive feat in this direction was undertaken by Ludolph van Cuelen, professor at the University of

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