Compound interest, e, and how it relates to circles. Full playlist:    • Lockdown math   Home page: https://www.3blue1brown.com/ Brought to you by you: https://3b1b.co/ldm-thanks Great Mathologer video: https://www.youtube.com/watch?v=-dhHrg-KbJ0 Beautiful pictorial summary by @ThuyNganVu: https://twitter.com/ThuyNganVu/status/1258221677990703105 https://twitter.com/ThuyNganVu/status/1258222002889875457 My other videos on imaginary exponents: https://www.youtube.com/watch?v=v0YEaeIClKY https://www.youtube.com/watch?v=mvmuCPvRoWQ Mistakes: In the off-handed remarks on quaternions, I mentioned rotation in 4d would require 10 degrees of freedom. That's wrong, what I should have said was it requires 6 degrees of freedom, and rotation in 5D is what requires 10 degrees of freedom. ------------------ Video Timeline (Thanks to user "Just TIEriffic") 0:00:00 Welcome 0:00:55 Q1: Prompt (Would you take an imaginary interest rate) 0:02:05 "e to the pi i for dummies" video shoutout 0:02:45 Q1: Results 0:03:30 Q2: Prompt (two banks, two rates) 0:04:55 Ask: Beauty of connections in math 0:06:00 Q2: Results 0:07:05 Desmos for Q2 0:09:10 Q3: Prompt (savings growth rate, 6% every 6mo) 0:10:35 Q3: Results 0:12:35 Desmos graph explored 0:14:45 Breaking down an interest rate 0:18:00 An interesting interest equation 0:19:20 Q4: Prompt (100*(1+0.12/n)^2 as n → ∞) 0:21:05 Ask: Quaternions 0:22:35 Q4: Results 0:24:50 Explaining Q4 0:26:40 Defining e 0:28:40 The definition of e from previous lectures 0:30:45 The imaginary interest rate 0:32:35 Graphing this relationship 0:33:50 The imaginary interest rate animation 0:37:55 Compounding continuously with i 0:40:45 The spring & Hooke's law 0:43:20 Q5: Prompt (Δx & Δv for a spring) 0:44:50 Ask: Rotation in for multiple dimensions 0:47:45 Q5: Results 0:49:50 Rewriting the spring's position 0:55:00 Bringing it all together 0:59:00 Ask: Hints on last lecture's homework 1:03:25 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