Intro to the geometry complex numbers. Full playlist:    • Lockdown math   Home page: https://www.3blue1brown.com/ Brought to you by you: https://3b1b.co/ldm-thanks Beautiful pictorial summary by @ThuyNganVu: https://twitter.com/ThuyNganVu/status/1258219199769440257 Errors: - On the first sketch of a complex plane, there is a "2i" written instead of "-2i". - At the end, in writing the angle sum identity, the last term should be sin(beta) instead of sin(alpha). - During Q9, the terms in parentheses should include an i, (1/2 + sqrt(3)/2 i) Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ The live question setup with stats on-screen is powered by Itempool. https://itempool.com/ The graphing calculator used here is Desmos. https://www.desmos.com/ The "Complex slide rule" came from Geogebra, via Ben Sparks. https://www.geogebra.org/m/mbhbdvkr 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. ------------------ Video Timeline (thanks to user "Just TIEriffic") 0:00:30 - W3 Results 0:01:00 - W4 Prompt 0:02:00 - Ask What would you call 'imaginary numbers'? 0:06:40 - Startingpoint & assumptions 0:10:25 - W4 Results 0:11:25 - Q1 Prompt 0:12:20 - Q1 Process 0:14:05 - RotatingCoordinates 0:16:40 - Q1 Result 0:17:40 - Q2 0:18:15 - Q3 Prompt 0:19:40 - Q3 Results 0:21:35 - RotationAnimation 0:22:35 - 3 facts about Multiplication 0:25:40 - Q4 Prompt 0:26:10 - Ask imaginary I vs physics i&j 0:28:15 - Q4 Result 0:31:00 - GeoGebraDemo 0:32:10 - Q5 Prompt 0:33:30 - Q5 Results 0:34:00 - Q5 Solution 0:35:55 - RotatingImages Example 0:37:10 - PythonExample 0:38:25 - PythonImage Rotation Example 0:40:35 - Ask Vectors & Matrices for rotation 0:42:40 - Q6 Prompt 0:46:55 - Q6 Results 0:47:25 - Q6 Solution 0:52:20 - RedefiningAngle Addition 0:57:20 - Q7 Prompt 0:57:55 - Ask Can we do without complex numbers? 1:00:10 - Q7 Results 1:00:55 - Q7 Solution 1:05:45 - Q8 Prompt 1:06:30 - Ask sum/difference of angles 1:09:25 - Q8 Results 1:10:25 - Q8 Solution 1:12:00 - DesmosExample 1:15:05 - Bringing it all together 1:16:25 - The "cis" shorthand explained 1:18:05 - Q9 Prompt 1:19:35 - Q9 Results 1:20:55 - ClosingRemarks ------------------ 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