Complex number fundamentals | Ep. 3 Lockdown live math

Intro to the geometry complex numbers.

Beautiful pictorial summary by @ThuyNganVu:

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.

The graphing calculator used here is Desmos.

The "Complex slide rule" came from Geogebra, via Ben Sparks.

Curious about other animations?

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

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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Various social media stuffs:

[elf]

51:00 its 2am but this has made me go get a paper and calculate cos(75). That's how powerful this math series is

[blzKrg]

I love how he doesn't just ignore the minority of people that get the wrong answer even if they are very few. Instead he tries to understand why they got the wrong answer and what was could've been their thought process while answering and then he points out where the thought process went wrong and then gives the right idea to think about it. It's just lovely how great of a teacher he is. ❤

[Dr.Kornelius]

2010: watching youtube in math class 🥱
2020: watching maths on youtube 🤩

It’s nice to live in an age where 148, 000 people will sit and watch a 1.5 hour math lecture patiently

[capilover]

"Three things are infinite: the universe, human stupidity, and Grant's supply of unit circles; and I'm not sure about the universe."

- Albert Einstein

[overestimatedforesight]

Grant, I struggled hard with trig in school. It discouraged me so badly that I had left it as something I wouldn't understand and so I never moved on to higher math. Your lectures in this video and the last, and following along with the test questions, not only made me realize how beautiful and interesting trigonometry is, but also rekindled a long-forgotten love for math and confidence in my ability to do it. Thank you, thank you, for making these videos.

[beretperson]

Hey geniuses, if infinity isn't real, how can he have an infinite supply of unit circles?

[shahfenil]

Revisiting 3 years later, I am 100% convinced that in the whole world, math teachers in the school were completely incompetent. Most likely they never understood the math and didn't know how to teach the math subject. Majority of people went through hell. It took decades to appreciate math because of such beautiful math YouTube channels and kind teacher/professor. 3b1b channel is a gem. It teaches us how to think, how reason. If math is not directly useful in your life, don't worry, at least you will learn how to think, logical reasoning. It's very useful skills to have in your daily life.

[Garbaz]

Regarding 1:04:56 :
One my electrical engineering professors said that if mathematicians hadn't come up with complex numbers, electrical engineers would have.
Dealing with electrical circuits that involve capacitors, inductors (and alternating currents) without complex numbers is very difficult, having to deal with differential equations and trig identities, but if you interpret inductors & capacitors like resistors, but with an imaginary resistance, you get an incredibly beautiful and simple way to work with them.
In general, there is pretty much no area of electrical engineering that does not benefit greatly from using complex numbers. Especially everything involving AC.

[AzrealG]

Man, i'm almost in my 40's, and i just learned a new intuition behind a tool I know and use since 20 years. You're an awesome teacher.

[JustTIEriffic]

Video Timeline
0:00:30 W3: Results
0:01:00 W4: Prompt
0:02:00 Ask: What would you call 'imaginary numbers'?
0:06:40 Starting point & assumptions
0:10:25 W4: Results
0:11:25 Q1: Prompt
0:12:20 Q1: Process
0:14:05 Rotating Coordinates
0:16:40 Q1: Result
0:17:40 Q2
0:18:15 Q3: Prompt
0:19:40 Q3: Results
0:21:35 Rotation Animation
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 GeoGebra Demo
0:32:10 Q5: Prompt
0:33:30 Q5: Results
0:34:00 Q5: Solution
0:35:55 Rotating Images Example
0:37:10 Python Example
0:38:25 Python Image 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 Redefining Angle 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 Desmos Example
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 Closing Remarks
Edits: Changed timestamps to the hour format, moved them closer to event and updated them after video was trimmed.

[HeirofThorondor]

Grant, I've gotta say. What sets you apart from nearly every math teacher I've ever met is your presentation and humility. Despite the fact that you're unbelievably smart, you know exactly what kinds of logical questions that we who aren't as knowledgeable will ask. It not only makes us feel known and understood, but it significantly boosts your credibility and enhances your teaching. We're not simply taking things at face value because we know that the person teaching us has thought through things the same way we are and can address our concerns. Thank you.

[ardiris]

I learned all this 43 years ago, but this is the first time I saw the animations. Fascinating educational tool!

[luisfernandoff]

About conventions i or j:
In electric engineering the imaginary numbers are normally represented by “j”, instead of “i”. The reason is that the letter “i” is already used to represent current.

[KaiseruSoze]

I worked on the graphics engine for the space shuttle. And we used quaternions a lot. Mostly because we didn't have to worry about gimbal lock, they are much faster than matrix multiplication and make relative rotations more intuitive.

[SplendidKunoichi]

Working out cos(75) geometrically instead of plugging it into a calculator just singlehandedly allowed me to finally grasp quaternions
These streams are incredible

[Gallareton]

I'm already past my studies, I watch you only because I love math. It's really nice that you do what you do, keep it up, man! :)

[ASLUHLUHCE]

Don't know what's more incredible; the way imaginary numbers fit so well on the two-dimensional number line, or Grant's teaching.

[shumymikaball]

"Lets define x to be the answer of my question" - I love the applicability of these one.

[vojtechhala]

56:45 Good Lord, that's the defining property of exponential functions! Suddenly, I see the link between the two concepts. The click in my head was audible! Thank you, Grant, you're an awesome teacher.