The imaginary number i is equal to the square root of negative 1. But i^i is a real number, approximately equal to 0.208. Why is this true? It is an incredibly tricky calculation if you want to justify the steps properly. All of the videos I've seen on i^i use an unjustified step, including mine. So what is the proper way to calculate i to the power of i? Special thanks this month to: Daniel Lewis, Kyle, Lee Redden, Mike Robertson. Thanks to all supporters on Patreon! http://www.patreon.com/mindyourdecisions 0:00 calculation 2:11 complex numbers 3:30 "proof" 1 6:21 "proof" 2 8:20 complex exp 9:12 complex log 12:32 the right way WolframAlpha https://www.wolframalpha.com/input?i=i%5Ei Math StackExchange https://math.stackexchange.com/questions/191572/prove-that-ii-is-a-real-number https://math.stackexchange.com/questions/1347504/for-which-complex-a-b-c-does-abc-abc-hold/ https://math.stackexchange.com/questions/1495532/when-is-abc-abc-true/1495550 https://math.stackexchange.com/questions/1347504/for-which-complex-a-b-c-does-abc-abc-hold/ https://en.wikipedia.org/wiki/Principal_value Complex Variables and Applications, James Ward Brown, Ruel V. Churchill, 7th edition, Chapter 1 Section 8, Roots of Complex Numbers The complex logarithm, exponential and power functions, Professor Howard Haber https://scipp.ucsc.edu/~haber/ph116A/clog_11.pdf Subscribe: null Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas! If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. Book ratings are from January 2023. My Books (worldwide links) https://mindyourdecisions.com/blog/my-books/#worldwide My Books (US links) Mind Your Decisions: Five Book Compilation https://amzn.to/2pbJ4wR A collection of 5 books: "The Joy of Game Theory" rated 4.3/5 stars on 290 reviews http://amzn.to/1uQvA20 "The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4.1/5 stars on 33 reviews http://amzn.to/1o3FaAg "40 Paradoxes in Logic, Probability, and Game Theory" rated 4.2/5 stars on 54 reviews http://amzn.to/1LOCI4U "The Best Mental Math Tricks" rated 4.3/5 stars on 116 reviews http://amzn.to/18maAdo "Multiply Numbers By Drawing Lines" rated 4.4/5 stars on 37 reviews http://amzn.to/XRm7M4 Mind Your Puzzles: Collection Of Volumes 1 To 3 https://amzn.to/2mMdrJr A collection of 3 books: "Math Puzzles Volume 1" rated 4.4/5 stars on 112 reviews http://amzn.to/1GhUUSH "Math Puzzles Volume 2" rated 4.2/5 stars on 33 reviews http://amzn.to/1NKbyCs "Math Puzzles Volume 3" rated 4.2/5 stars on 29 reviews http://amzn.to/1NKbGlp 2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter. My Blog https://mindyourdecisions.com/blog/ Twitter https://twitter.com/preshtalwalkar Instagram https://www.instagram.com/preshtalwalkar/ Merch null Patreon https://www.patreon.com/mindyourdecisions Press https://mindyourdecisions.com/blog/press