Start with the pair of fractions (2/1)(2/3). Now increment each number by 2 to get the pair of fractions (4/3)(4/5). Repeat this to get the next pair, and repeat this infinitely many times. If you then multiply all the fractions, you get a marvelous value: it is equal to pi divided by 2. The Wallis product formula for pi was described in 1655. This video explains some of the historical motivation and then gives a modern proof using calculus of why the formula is true.
My blog post for this video
Factorial of 1/2 video
Wallis product historical motivation
Wallis pi and quantum theorem
Geometric intuition of Wallis product
Wallis's original book "Arithmetica Infinitorum"
Another proof of the Wallis product
Wästlund, Johan. "An Elementary Proof of the Wallis Product Formula for Pi." The American Mathematical Monthly 114.10 (2007): 914-17. Web. http://www.jstor.org/stable/27642364
If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions
Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.
My Blog: http://mindyourdecisions.com/blog/
Newsletter (sent about 2 times a year): http://eepurl.com/KvS0r
"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/