Links to proofs and more below in the description.

1. The definition 00:10

Related: the area of circle intuitive explanation

2. Pi is irrational 00:47

Proof (video)

Proof (blog post)

Pi is transcendental (technical)

http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem

3. Archimedes estimated pi 01:42

BetterExplained post

http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/

4. Infinite series for pi 03:28

Madhava-Leibniz series

http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

Euler

http://en.wikipedia.org/wiki/Basel_problem

Chudnovsky algorithm

http://en.wikipedia.org/wiki/Chudnovsky_algorithm

http://www.numberworld.org/misc_runs/pi-12t/

5. Pi in probability 04:34

Normal curve

http://en.wikipedia.org/wiki/Normal_distribution

Buffon’s needle problem http://mindyourdecisions.com/blog/2013/05/27/buffons-needle-problem/#.VPo1S_nF98E

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My Books

“The Joy of Game Theory” shows how you can use math to out-think your competition. (rated 4/5 stars on 23 reviews) https://www.amazon.com/gp/product/1500497444

“The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias” is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 5/5 stars on 1 review) https://www.amazon.com/gp/product/1523231467/

“Math Puzzles Volume 1” features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.5/5 stars on 11 reviews. https://www.amazon.com/gp/product/1517421624/

“Math Puzzles Volume 2” is a sequel book with more great problems. https://www.amazon.com/gp/product/1517531624/

“Math Puzzles Volume 3” is the third in the series. https://www.amazon.com/gp/product/1517596351/

“40 Paradoxes in Logic, Probability, and Game Theory” contains thought-provoking and counter-intuitive results. (rated 4.9/5 stars on 7 reviews) https://www.amazon.com/gp/product/1517319307/

“The Best Mental Math Tricks” teaches how you can look like a math genius by solving problems in your head (rated 4.7/5 stars on 3 reviews) https://www.amazon.com/gp/product/150779651X/

“Multiply Numbers By Drawing Lines” This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 1 review) https://www.amazon.com/gp/product/1500866148/