Euler's formula about e to the i pi, explained with velocities to positions. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: https://www.3blue1brown.com/ Not familiar with the calculus referenced in this video? Try taking a look at this one: https://www.youtube.com/watch?v=m2MIpDrF7Es Another perspective on this formula, from Mathologer: https://www.youtube.com/watch?v=-dhHrg-KbJ0 Another perspective from this channel: https://www.youtube.com/watch?v=mvmuCPvRoWQ And yet another from the blog Better Explained: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ I'm not sure where the perspective shown in this video originates. I do know you can find it in Tristan Needham's excellent book "Visual Complex Analysis", but if anyone has a sense of the first occurrence of this intuition do feel free to share. It's simple and natural enough, though, that it's probably a view which has been independently thought up many times over. Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. 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