What Is The Factorial Of 1/2? SURPRISING (1/2)! = (√π)/2

535
by Super User, 6 months ago
0 0
Type 0.5! in your calculator to see what the factorial of one-half is. The result will be 0.886..., and the exact answer is the square root of pi divided by 2--amazing! How is this possible, when the factorial of a number n is defined as n! = n(n-1)(n-2)...1 and this definition only makes sense for whole numbers?

The calculator result is not an error, and in this video I explain how the factorial can be extended beyond the whole numbers for all real numbers by the gamma function. Once we extend the factorial function beyond whole numbers, you can see why the factorial of one-half is equal to the square root of pi divided by 2.

Bohr-Mollerup theorem
http://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem

Applications
https://www.math.washington.edu/~morrow/336_10/papers/joel.pdf

Numerical computation
http://www.rskey.org/CMS/index.php/the-library/11

Alternative ways to extend the factorial function
http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html

Support me at Patreon: http://www.patreon.com/mindyourdecisions

MindYourDecisions on the web:
Books: http://goo.gl/BDlEkB
Blog: http://mindyourdecisions.com/blog/
Twitter: http://twitter.com/preshtalwalkar
Facebook: https://www.facebook.com/pages/Mind-Your-Decisions/168446714965
Google+: https://plus.google.com/108336608566588374147/posts

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/