Complex number fundamentals | Ep. 3 Lockdown live math
Intro to the geometry complex numbers.
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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
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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.
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.
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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.
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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
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Video Transcript
Today we are going to talk about one of my absolute all-time favorite pieces of math.
It's incredibly fundamental to engineering, to mathematics itself,
to quantum mechanics, but it's something that has a terrible, terrible name.
We call them complex numbers.
And worse than that, the things that bring about
complex numbers we call imaginary numbers.
And before we get into any of it, what I want to do is start with kind of a poll,
just to poll the audience on seeing what you guys can consider to be, well, real.
What do you consider to exist when it comes to numbers?
So we've already been doing a couple polls in the warm-up animations,
but as a serious poll of sorts, one that's actually going to help me see where
you're coming from before we begin the lesson here,
I just want to ask you a very simple question.
Okay, so let's go ahead and pull it up here.
Pull it on up.
And, oh, for whatever reason it seems like we're having trouble polling this one up.
Okay, there we go.
Took a little delay there.
Among the values 2, square root of 2, square root of negative 1, and infinity,
which would you personally consider to really exist,
whatever really existing means to you?
So in theory, if you guys go to 3b1b.co slash live, you should be able to answer this,
and then the statistics based on your answers are going to start populating the screen.
We won't know what those answers refer to.
At the moment, all we know is that, you know, someone who has a lead,
another one that's pretty close behind it, and as your answers come in and
as the servers kind of digest them, we'll start to see some of the stats here.
If you go to that page, by the way, 3b1b.co slash live, which redirects to itempool.com,
what you're going to find is at the very top, you can ask a question on Twitter,
and all that's going to do is basically open up a tweet that's going to have the hashtag
in it, lockdown math, and that's the way that we're going to be doing questions to this.
Instead of a live chat, anytime you have a question or a comment that you want to,
you know, insert into the lesson and let that be part of the discussion,
it's going to be pulled up here, and it looks like we already have one.
This is from Yash Dave, who asks, if you could rename the complex numbers and the
so-called imaginary numbers to something more intuitive,
to a name that conveyed the fact that they have numerous applications in the real world,
what would you name them?
I couldn't be happier that you asked, Yash.
So I have one friend who seems very passionate about
the fact that we should call them sneaky numbers.
Personally, I'm very fond of trying to connote spinning and rotation,
and this is one of the things we'll talk about in today's lesson,
is the fact that what we call complex numbers, what we call imaginary numbers,
some of the main uses that they have come from very elegant descriptions of
how to rotate stuff, and I hope you'll kind of see what I mean as we proceed with that.
So on our poll, asking you guys which numbers you consider to really exist,
which is of course a subjective question, there's no right or wrong answer here,
I am genuinely curious, actually, how this breaks down,
because it's not at all a strong consensus in one direction.
