3Blue1Brown | Intuition for i to the power i | Ep. 9 Lockdown live math @3blue1brown | Uploaded 4 years ago | Updated 20 minutes ago
i^i, visualized and explained.
Full playlist: youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: 3blue1brown.com
Brought to you by you: 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
twitter.com/ThuyNganVu/status/1263522876403011585
Video by Matt Parker:
youtu.be/9tlHQOKMHGA
Video by Red Pen Black Pen
youtu.be/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.
itempool.com
Curious about other animations?
3blue1brown.com/faq#manim
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
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: 3blue1brown.com
Twitter: twitter.com/3blue1brown
Reddit: reddit.com/r/3blue1brown
Instagram: instagram.com/3blue1brown_animations
Patreon: patreon.com/3blue1brown
Facebook: facebook.com/3blue1brown
i^i, visualized and explained.
Full playlist: youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: 3blue1brown.com
Brought to you by you: 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
twitter.com/ThuyNganVu/status/1263522876403011585
Video by Matt Parker:
youtu.be/9tlHQOKMHGA
Video by Red Pen Black Pen
youtu.be/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.
itempool.com
Curious about other animations?
3blue1brown.com/faq#manim
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
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: 3blue1brown.com
Twitter: twitter.com/3blue1brown
Reddit: reddit.com/r/3blue1brown
Instagram: instagram.com/3blue1brown_animations
Patreon: patreon.com/3blue1brown
Facebook: facebook.com/3blue1brown
