What does it mean to compute e^{pi i}? 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/1258220129327800320 https://twitter.com/ThuyNganVu/status/1258220541686628353 Not on the "homework" to show that exp(x + y) = exp(x) * exp(y). This gets a little more intricate if you start asking seriously about whether the series really converge, what they converge to, and how exactly you define a product with infinitely many terms. For anyone curious about the technical details, what you would want to show is that the Cauchy Product of the series for exp(x) and exp(y) converges to the product of the values exp(x) and exp(y) for any particular x and y. That requires the Merten's Theorem. Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ Video Timeline (Thanks to user "Just TIEriffic") 0:00:00 Welcome 0:00:20 Ending Animation Preview 0:01:15 Reminders from previous lecture 0:03:30 Q1: Prompt (Relationship with e^iθ=…) 0:05:40 Q1: Results 0:07:15 WTF, Whats The Function 0:10:00 Exploring exp(x) 0:11:45 Exploring exp(x) in Python 0:14:45 Important exp(x) property 0:15:55 Q2: Prompt (Given f(a+b) = f(a)f(b)…) 0:17:30 Ask: Which is more interesting, special cases or the general case 0:20:00 Q2: Results 0:23:50 Will a zero break Q2? 0:25:40 The e^x convention 0:27:10 Q3: Prompt (i^2 = -1, i^n = -1) 0:27:45 Ask: Zero does not break Q2 0:30:20 Q3: Results 0:31:05 Comparison to Rotation 0:33:00 Visualizing this relationship 0:36:50 The special case of π 0:39:20 Periodic nature of this relationship 0:39:40 Q4: Prompt (e^3i) 0:41:35 Q4: Results 0:43:55 Explaining the celebrity equation 0:45:55 Homework / Things to think about 0:49:15 Ask: Zero does break Q2. 0:50:30 Closing Remarks ------------------ 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