i^i, visualized and explained. Full playlist:    • Lockdown math   Home page: https://www.3blue1brown.com/ Brought to you by you: https://3b1b.co/ldm-thanks Mistakes: At 1:06:20, when changing r to equal 0.69*i, I said "this is what we might think of as (2i)^x", but that is not correct. It's what we'd think of as [Exp(ln(2)*i)]^x for whatever complex number Exp(ln(2)*i) is. Beautiful notes by Ngân Vũ https://twitter.com/ThuyNganVu/status/1263522876403011585 Video by Matt Parker: https://www.youtube.com/watch?v=9tlHQOKMHGA Video by Red Pen Black Pen https://www.youtube.com/watch?v=ABk1HK2AR2E Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ Video timeline (thanks to user "nooonesperfect") 0:18 Exponential function for i^i 1:26 Question 1 2:27 Plug-in imaginary number in exp(x) polynomial 3:38 Answer 1 and explanation 7:35 What it really means i^i? 9:14 e^it as a position vector 11:30 Question 2 12:39 Audience question from twitter 13:14 Answer 2 14:52 Where you get after traveling π/2 units of time for position vector e^it 19:48 Question 3 20:42 Audience tweets 23:34 Answer 3 35:50 Question 4 37:11 Answer 4 40:11 How exp(rx) or b^x really works? 46:28 Question 5 47:49 Audience tweets 49:26 Answer 5 57:06 Visualization of f(x)= exp(r*x) i.e. e^(r*x), where r= unique complex number 1:06:06 Questions to think about 1:08:51 Audience tweets 1:09:09 Power tower for i ------------------ The live question setup with stats on-screen is powered by Itempool. https://itempool.com/ Curious about other animations? https://www.3blue1brown.com/faq#manim Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com/ Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown