How is the number Pi, derived?

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The number Pi (3.14159...) is derived from the ratio of a circle's circumference to its diameter. Historically, Archimedes calculated Pi using circumscribed polygons, achieving three-digit accuracy with a 96-sided polygon. Modern calculations of Pi utilize infinite series, a method first formulated in the mid-1600s by Newton. The discussion highlights the evolution of Pi's calculation methods over time. Understanding Pi's derivation showcases both historical and contemporary mathematical techniques.
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How is the number 3.14159...etc, derived? I know it has been calculated out to very great lengths and I'm wondering how it's done.
 
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Just find the proportion of radius to circumferance, I believe. I know it is some kind of proportion.
 


My personal favourite:

\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots\!

:smile:
 


Modern calculation of pi no doubt use infinite series, first formulated in Newton's time (mid 1600's). Archimedes used circumscribed polygons w/ 96 sides to calculate pi with three digit accuracy.
 
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