New lessons Tuesdays and Friday, noon pacific time.

All about ln(x).
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
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/1259288683489849344

Errors:
At minute 16, the sum should be written with a "..." to indicate going to infinity.
At minute 38, the exponent should have 1/(2s^2) instead of 1/s^2 for s to represent standard deviation.
At minute 54, an equal sign was mistakenly used in taking the derivative of x^3 / 3!.
At the end, it should be pointed out that the alternating series with x^n terms only converges for values of x between -1 and 1, so the values one can't be considered proven with values of x outside that range.  Everything with the argument here is fine, as it only deals with the convergent input, but that fact should still be mentioned.

Related videos.
Calculus series:
https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

The sum giving pi^2 / 6:
https://youtu.be/d-o3eB9sfls

The sum giving pi / 4:
https://youtu.be/NaL_Cb42WyY
https://youtu.be/00w8gu2aL-w (Mathologer)

-------------------
Video timeline (thanks to user "noonesperfect")
0:00:14 - Question 1
0:02:29 - Answer 1
0:06:27 - Prime nos. in Infinite Geometric Series (Basel problem) and their relationship with Natural logarithm
0:12:01 - More examples of prime numbers in infinite series and their relationship with ln
0:17:25 - Question 2
0:19:20 - Answer 2 and explanation using ln
0:22:25 - Question 3 and families of curves
0:26:37 - Answer 3 and explanation
0:28:50 - Imaginary exponential
0:30:57 - Derivatives of exponential terms
0:37:21 - Why derivative of e^t is the same as that e^t itself?
0:41:21 - Question 4
0:44:12 - Answer 4 and explanation using Python
0:46:02 - Taylor Series for e^x
0:48:29 - Derivatives of polynomial terms/Derivatives of e^x
0:50:56 - Derivative of natural logarithm using graph  
0:56:07 - Question 5
0:57:37 - Answer 5 and explanation 
1:02:15 - Euler–Mascheroni constant
1:08:37 - Question 6
1:12:41 - Connecting dots to the familiarity of different expression in math

------------------
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

OVERSEAS

What makes the natural log "natural"? | Lockdown math ep. 7

Intro to the geometry complex numbers.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
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)

------------------
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

Complex number fundamentals | Lockdown math ep. 3

Compound interest, e, and how it relates to circles.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks

Great Mathologer video:
https://youtu.be/-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://youtu.be/v0YEaeIClKY
https://youtu.be/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

Imaginary interest rates | Lockdown math ep. 5

Another view on the quadratic formula.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
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/1258217451323416577

Po Shen Loh on quadratics:
https://www.youtube.com/watch?v=ZBalWWHYFQc

Welch Labs on imaginary numbers:
https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF

Mistakes (there will always be mistakes):
At minute 22, I write "b' / 2" instead of "-b' / 2".

------------------
0:00 - Introduction
2:56 - "How often am I going to use this?"   ray tracing example
5:38 - Mental math tricks (factoring numbers, differences of squares)
13:36 - Properties of quadratic functions
18:40 - Deriving the variant quadratic formula
23:07 - Practice problems (ft. complex numbers!)
34:10 - Deriving the traditional quadratic formula from the variant
41:07 - Conclusion (key takeaways)
43:21 - Fun with joke poll questions
------------------

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

The simpler quadratic formula | Lockdown math ep. 1

Tips on problem-solving, with examples from geometry, trig, and probability.
Past episodes with integrated quizzes: https://itempool.com/c/3b1b
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks
Huge huge thanks to Ben Eater: https://www.youtube.com/user/eaterbc
And Cam Christensen, creator of ItemPool: https://itempool.com/

Notes by Ngân Vũ:
https://twitter.com/ThuyNganVu/status/1265480770832855040

Mistakes:
50:35, there should be a dx in the integral
54:40, if you notice the mistake here and are inclined to complain, keep watching


------------------
Video timeline (thanks to user "noonesperfect")
0:34  9-Problem Solving Principles/Tip 
1:15  Question 1 (Probability)
2:08  Who is Ben Eater? 
4:25  Inscribed angle theorem, θL=2*θs
5:58  Tip-1
7:48  Tip-2
8:16  Question 2
9:34  Answer 2 
10:29 Tip-3
15:17 Tip-4
22:48 Question 3
25:56 Answer 3 (Marked incorrectly, Answer: Option D) 
26:31 Answer 1
27:28 Explanation for Q1, Floor function
30:38 Tip-5
32:53 Tip-6
33:36 Question 4
34:43 Answer 4
36:37 Question 5
38:10 Answer 5
41:48 Probability graph in Desmos 
44:08 Revisiting an alternating series sum for ln(2)
47:29 Tip-7
51:08 Tip-8
55:23 Audience questions through tweets
57:28 Tip-9
58:29 Python programming for various probability question
1:04:31 Arts created using Desmos graph tool with mathematical expressions
1:05:54 Thank you, appreciation to the team and all. 
------------------
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

Tips to be a better problem solver [Last lecture] | Lockdown math ep. 10

What does it mean to compute e^{pi i}?
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
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


------------------
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

What is Euler's formula actually saying? | Lockdown math ep. 4

Back to the basics with logarithms.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks

Triangle of power:
https://youtu.be/sULa9Lc4pck

Beautiful pictorial summary by @ThuyNganVu:
https://twitter.com/ThuyNganVu/status/1258222677573001219

------------------
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

Logarithm Fundamentals | Lockdown math ep. 6

A fun puzzle stemming from repeated exponentiation.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks

Notes by Ngân Vũ:
https://twitter.com/ThuyNganVu/status/1261014161464516608?s=20

Play along on Desmos:
https://www.desmos.com/calculator/nul32eaaa9

Related videos.
Calculus series:
https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

In particular look at:
https://youtu.be/CfW845LNObM

Numberphile on Grahm's constant:
https://youtu.be/XTeJ64KD5cg

------------------
Video timeline (thanks to user "noonesperfect")
0:36  Question 1
1:13  Answer 1
1:29  Introduction to tetration
3:37  How exponentiation works in tetration
6:10  Python program for power tower iterations
8:40  Question 2
9:32  Python Program regarding question 2
10:37 Answer 2 and explanation
13:18 Power tower for infinite size converges or not? (Thumbnail question) 
15:21 Question 3
16:28 Footage of Grant's setup arrangement problem due to construction-work back at home.
16:49 Answer 3 and explanation
17:40 Checking logic behind 2 different problems of power towers whose answer converges to the same value (Is it even possible?)
19:42 Checking same logic using Desmos graph tool i.e. Cobweb Graph (Desmos graph link in description) 
28:12 Question 4
28:51 Questions from audience tweets
29:15 Knuth's Up Arrow Notation and Graham's Number (Check Numberphile video in description) 
32:32 Answer 4 and explanation
37:29 Homework/Challenge Puzzle 
39:20 Thumbnail question power tower logic
40:55 Audience questions from twitter
41:45 Power tower for complex numbers/Fractal set
45:19 Brainteaser 
48:06 More questions from tweets
53:17 Notes for lock-down series in Grant's Tweeter 
------------------
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

The power tower puzzle | Lockdown math ep. 8

i^i, visualized and explained.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://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 
https://twitter.com/ThuyNganVu/status/1263522876403011585

Video by Matt Parker:
https://youtu.be/9tlHQOKMHGA

Video by Red Pen Black Pen
https://youtu.be/ABk1HK2AR2E

------------------
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 
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Intuition for i to the power i | Lockdown math ep. 9

Intro to trig with a lurking mystery about cos(x)^2
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Brought to you by you: https://3b1b.co/ldm-thanks

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Contents:
Introduction  - 0:00
Q1 Graph of (cos θ)²  - 2:14
Q2 Translations of cos θ to (cos θ)²  - 5:34
Q3 f(2x) = f(x)²  - 10:54
Intro to Trig  - 13:14
Q4 sin(3) & cos(3)  - 16:44
SohCahToa - 20:44
Q5 Leaning Tower  - 22:29
How to Compute Trig Functions?  - 30:59
Q6 sin(π/6)  - 33:34
Q7 cos(π/6)  - 36:19
Q8 Trig of -θ  - 43:34
Computing Trig Functions  - 47:44
Q9 cos(π/12)  - 0:49:54
Adv Trig Functions  - 0:56:44
Q10 Graph of tan(θ)  - 1:00:30
JSON comment  - 1:02:24
“The most exciting part of the lecture” –1:05:34

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The live question setup with stats on-screen is powered by Itempool.
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The graphing calculator used here is Desmos.
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Curious about other animations?
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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

Lockdown math