The number "pi" has been a fascinating object for thousands of years. Intimately connected with a circle, it is not an easy object to get hold of completely rigorously. In fact the two main theorems associated to itthe formulas for the area and circumference of a circle of radius piare usually simply assumed to be true, on the basis of some rather loose geometrical arguments in high school which are rarely carefully spelt out.

Here we give an introduction to some historically important formulas for pi, going back to Archimedes, Tsu ChungChi, Madhava, Viete, Wallis, Newton, Euler, Gauss and Legendre, Ramanujan, the Chudnovsky brothers and S. Plouffe, and culminating in the modern record of ten trillion digits of Yee and Kondo. And I also throw in a formula of my own, obtained from applying Rational Trigonometry to Archimedes' inscribed regular polygons.

The lecture ends with some speculations about the future role that "pi" might play in our understanding of the continuuma huge problem which is not properly appreciated today.

This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.

Video Content:
00:00 Intro to the magic of "pi"
3:36 "Pi" is usually defined by area or circumference
7:27 Logical difficulty
10:22 Brief history of "pi"
13:58 The first formulas of "pi"
19:21 Formulas for "pi" discovered by Newton
23:45 "Pi" formula by S. Ramanujan (1914)
28:22 Page 269 of 'Divine Proportions'
30:59 Irrational real numbers
34:30 My attitude to "pi"
38:03 "Pi" is not a real number, it's a meta number

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My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon,   / njwildberger   . Your support would be much appreciated.

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