Complex number fundamentals | Ep. 3 Lockdown live math

Intro to the geometry complex numbers.
Full playlist: • Lockdown math
Home page: www.3blue1brown.com
Brought to you by you: 3b1b.co/ldm-thanks
Beautiful pictorial summary by @ThuyNganVu:
/ 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.
itempool.com/
The graphing calculator used here is Desmos.
www.desmos.com/
The "Complex slide rule" came from Geogebra, via Ben Sparks.
www.geogebra.org/m/mbhbdvkr
Curious about other animations?
www.3blue1brown.com/faq#manim
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
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
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
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[@Dr.Kornelius]

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

[@appleslover]

Did you just use math and maths in the same sentence?

[@appleslover]

If you weren't a native speaker i wouldn't blame you, i myself can't make my mind on which accent to fellow so at the end i use Ameritish

[@zenbum2654]

In 2010 you were an American math student. Now, you're a British maths student. 😜

[@Dr.Kornelius]

@@zenbum2654 things can change a lot in 10 years 😄

[@luisabotero378]

So my life

[@elf7677]

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

[@W1ngSMC]

I did the same at 2 am as well.

[@nahometesfay1112]

I'm watching at 2am as well, but I didn't bother getting a paper and just did it in my head

[@harrybottero6756]

Nearly 2 too😁

[@Phenom5]

I did the exact same at 5 am bros

[@lyndonhanzpernites5860]

12:43 and I am blown away by the fact that doing it by hand is way, way easier in terms of complex analysis than the trigonometric formula.

[@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. ❤

[@raquelsanchez4129]

He did ignore the people that answered 69 lol

[@HimanXK]

@@raquelsanchez4129 I noticed that too

[@nirukumari7721]

Yes

[@509734]

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

[@omerdvir1709]

1,4Million now

[@muhamedbabe8924]

Amazing

[@Nick12_45]

you mean 1,544,213 right?

[@capilover1023]

"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

[@Nylspider]

Unit circle time

[@elyades2480]

I laughed, very hard.

[@mohammadazad8350]

*-Randy

[@appleslover]

" Newton lied about inventing calculus " _Einstein_ , _2030_

[@scepticusverisimillimenonm8450]

*supply of

[@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.

[@noahniederklein8081]

Thank you! I won't need it because I watched the stream, but this will help a lot of people. You might want to change 61:45 and under to 1:01:45 so the links actually work though, but that must've taken a long time.

[@JustTIEriffic]

@@noahniederklein8081 Will do, It's all in a spreadsheet so its an easy fix. Going through now and double checking them.

[@geoc2140]

I converted the bottom half to hours: 1:01:45 Q7: Prompt 1:03:00 Ask: Can we do without complex numbers? 1:05:10 Q7: Results 1:05:40 Q7: Solution 1:10:10 Q8: Prompt 1:10:50 Ask: sum/difference of angles 1:13:40 Q8: Results 1:14:50 Q8: Solution 1:16:10 Desmos Example 1:20:20 Bringing it all together 1:20:50 The cis shorthand explained 1:22:00 Q9: Prompt 1:23:45 Q9: Results (1:05:40 Closing Remarks)?

[@coolguy284_2]

Don't forget 31:44 : snarky remark.

[@DeepaKrishnanS]

Thanks.

[@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.

[@AzrealG42]

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.

[@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.

[@belalmohammed2265]

Try signal processing or control theory without complex number

[@cubing7276]

What does a resistor with imaginary resistance mean?

[@tommihommi1]

@@cubing7276 electrical signals are sine waves or can always be expressed as sums of sine waves, see the videos on fourier transformations. So you can express the signal at any point in time with amplitude and phase angle, which is extremely convenient to do as a complex number. You can think of a hand/needle/pointer/*phasor* that's spinning around in circles as time goes on. When the resistance is 1, voltage and current always have the same value at any point in time. If it's 2, the voltage will be 2x the current. So a non-imaginary resistance simply scales this complex signal. When you have a capacitor or inductor, the peaks of voltage and current are no longer at the same time, they're out of phase by (ideally) 90 degrees. So we just multiply our phasor by i, and there we go. Of course, any real-world part like a wire has a resistance, capacitance and inductance, and we can use complex numbers to describe this. hope this helps, I'm not studying this in English, so there might be some errors in translating to the correct technical terms.

[@jasonlee3247]

Chy 75 it’s a zero ohm resistor 👍

[@fitofight8540]

Chy 75 either Inductive or capacitive reatance.

[@beretperson]

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

[@kebien6020]

Check mate

[@shreechaturvedi4170]

@@kebien6020 1-0

[@medevon1]

A nice quote by Prof. E. J. Farell: "There are many infinities, and the one you're most likely thinking of is the smallest one."

[@heater5979]

We don't. There is only one unit circle. Your unit circle is actually the same circle as my unit circle. If that were not the case then mathematics would be different for the two of us. Which would not work out very well. But who said circles exist anyway? And what is that unit thing? Can you point out either of those things i the real world?

[@Mr.Nichan]

@@heater5979 I'm replying to your 1 comment. Does that count as a "unit" in the real world?

[@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.

[@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.

[@meta04]

Also Python complex literals (which exist; the floating-point properties are as a separate real and imaginary part each one being a double) also use a suffixed j.

[@sober4769]

but,j is current density

[@carultch]

@@sober4769 It is a lot more common that an electrical engineer uses current instead of current density, in their calculations. It if does end up mattering, capital J would be current density, and lowercase j would be the imaginary unit. Current density is for the physics behind a lot of electrical components and their theory of operation, but it is rare that you deal with the continuum mechanics of electricity as an electrical engineer. Current itself infinitely more common, than current density, for electrical engineers.

[@topilinkala1594]

@@carultch I've learned that u (or v), j, and i are used lower case when talking about alternating current and upper case when talking about direct current.

[@carultch]

@@topilinkala1594 I can't say that I'm familiar with that convention, since I've always used capital I and V for electrical calcs. I had wondered why they needed to use j instead of i, if capital I would stand for current anyway, and thought maybe it is just to avoid confusion when talking about the equation aloud.

[@ardiris2715]

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

[@matmagix3845]

look up his calculus series, it's fantastic

[@technoultimategaming2999]

His calculus series saved my grade and life. Not exaggerating

[@simoneyoung3984]

Teach me please pretty please

[@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.

[@GamerTheTurtle]

What did you study in University? I wish to pursue this kind of career, thank you

[@DragonKidPlaysMC]

Same

[@vamishhh]

Fax I totally understood it by the way

[@rnilu86]

@@GamerTheTurtle Electronics or Computer science may be. :D Just guessing.

[@RohitKumar-ns9cl]

Jack Martinelli do you mean the space shuttle simulator program?

[@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

[@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.

[@Hecatonicosachoron]

I mean, there's hardly a problem in maths that doesn't use this.

[@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! :)

[@vojtechhala5074]

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.

[@nishitha2685]

After watching your series, its as if all the math I've learned up until now was crap!! Your way of geometrical interpretation of all problems isn't something which is taught or rather known by many teachers. Great thanks for making this awesome series. This has increased my love for math to a higher extent.

[@angelowentzler9961]

The little mistakes make it better. Such things are very comforting to insecure students.

[@SreenikethanI]

hehe

[@furiousfajitaa2367]

Agreed. Whenever I make a mistake I immediately jump to the conclusion that I'm not good enough, that all my grades were luck and that I'm going to fail but seeing someone who is really good at maths make mistakes gives me more faith in myself

[@prawdziwamiloscistnieje]

You're a wonderful human being and a great teacher. I send you all the love in this world.

[@adityaagarwal636]

Lectures like these make me feel that all fields in math are connected very fundamentally, but since we learn then in different chapters, we just don't see it.

[@jorjc9573]

To say that I enjoy these lectures (and your channel) is a gross understatement. Thank you very much Grant for sharing your gifts so generously. You bring out the beauty in mathematics where so very few have been able to do so.

[@whatelseison8970]

I actually like the name imaginary. It makes them sound whimsical and interesting. The moment I heard of them I wanted to know more. On the other hand I think complex is a poor naming choice. It makes them sound complicated or hard to understand, which they really aren't. I'd prefer "compound" or "combined" numbers in that regard.

[@nicholaskarras2759]

I think people hear imaginary and then their next thought is "doesn't exist,no application, how can this help me, they don't matter so I don't care" or they get confused on either how a number can be imaginary, or that all numbers are "made up and thus imaginary", a lot of ways hearing imaginary number can go wrong.

[@TrivialTax]

Thats exactly how they are called in polish language. 'Liczby zespolone' - ' Combined Numbers'.

[@buntythoidingjam]

All these teachers say terrible name for complex numbers or imaginary numbers. I also think it is interesting. I found it more interesting than real numbers or rational numbers!

[@DJ-yj1vg]

Composite numbers

[@krishnanair9390]

I like to call them dancing numbers personally, you know with Fourier Series and all

[@BlackEyedGhost0]

"Thank you for joining. Apologies for being mildly scattered throughou--" *video ends instantly*

[@nightmareshogun6517]

Yeah.. " i " noticed that.." i " wants to know what happened ...

[@milanstevic8424]

there's nothing important beyond that point. he just apologized for the sudden interruption that happened before that point in time.

[@thasyashetty3797]

@@nightmareshogun6517 i really likes what you did there.

[@yf-n7710]

The question about the non-real cube roots of 1 is always going to have a special place in my heart. I remember one time I was being difficult and pestering my parents, so my father gave me that question. He didn't tell me how many there were; he just told me to find them. I had only just learned about complex numbers, so I didn't know about the analog between multiplication and rotation, but that problem let me figure it out on my own. Also, I think we were hiking, so I didn't have any pencil or paper, and I had to visualize it in my head (but that's easier for me than algebra anyway). That was a really fun few hours. I didn't even realize until he told me years later that he was trying to get me to stop bothering him, but apparently giving me difficult math problems was a strategy he used to use. It certainly worked, and I got a few really fond memories out it.

[@joemiller8346]

I am a retired engineer (electrical and nuclear) and very much enjoy your lectures. You have an outstanding way of conveying understanding. Keep up the good work!

[@kek3324]

1:20:22 We are considering only the real part of doubling the angle, that is, cos(2Φ), but if we also take into account the imaginary part which is isin(2Φ) and compare it with the algebraic result, we get sin(2Φ)=2sinΦcosΦ ! Amazing how the familiar trig identities just pop out of complex numbers. Incredibly elegant!

[@SeeTv.]

Quaternions are complex numbers on steroids - my favorite quote of this lesson

[@AteshSeruhn]

I described quaternions with that exact phrase to a friend last year. It's great to see Grant's mind and mine have something in common .

[@nahometesfay1112]

If that's not a common way to describe them, then mathematicians need to bulk up on colloquialisms

[@bhanusri3732]

Can u tell me the timestamp where he talked about quaternoins.I watched the whole video already.Don't want to watch again

[@Tawnos_]

@@bhanusri3732 @40:55 or so

[@TechToppers]

I'm bad at English. What are steroids?

[@diegopablogordillovaras106]

I can't believe how many times I've studied complex numbers, and I've never realized why i^2 NEEDS to be -1. I thought it was a lucky definition that somehow turned out fine... So... thank you. I can't possibly express how grateful I am to you, for this moment of true insight, where everything makes sense. Mathematics is a beautiful imaginary world, but the way you commit to education and teaching makes our real world beautiful. Thank you!!

[@ferencszabo3504]

The most USEFUL math channel on the web! To know math is one thing ,and to teach it is another! This guy teaches it with the BEST possible method! Kudos to you, Grant!

[@profpoppinfresh]

Drink everytime Grant grabs a new unit circle.

[@smooooth_]

My liver

[@legendarysom5605]

Which pen he is using the black colored

[@carultch]

Would you prefer if he deliberately had circles with a radius other than 1?

[@RodelIturalde]

@@carultch aren't all circles unit circles. Just with a different type of length unit.

[@carultch]

@@RodelIturalde That would only be true if there were an infinite number of length units, which would be incredibly inconvenient. Also, it isn't necessarily length units that define circles in general. Mohr's circle for instance, has units of psi or Pascals, and the radius represents the physical quantity of maximum shear stress within the plane.

[@kashgarinn]

3b1b: get yourself some “frixxion” pens, they’re erasable.

[@wierdalien1]

Or a laminated unit circle

[@smooooth_]

Or a whiteboard

[@jonnyp1340]

Or imaginay paper

[@hhaavvvvii]

@@jonnyp1340 He's already using imaginary paper in some areas though.

[@mmmfuhlendorf]

I'm LOVING these lockdown episodes. Thanks!

[@LostMekkaSoft]

"look, even python has complex numbers" - opens i-python

[@LostMekkaSoft]

dude, that was a joke ;)

[@hinkles73]

Imaginary numbers should be called "lateral." That name was actually proposed!

[@ricardo.mazeto]

By Gauss.

[@MrBorderlands123]

Imaginary numbers was a derogatory term that Descartes used that, unfortunately, has stuck hundreds of years later. Ironic that the father of analytic geometry thought that complex numbers were nonsense.

[@neonblack211]

Edward Hou that’s how they originally found a geometric interpretation for complex numbers though, rotations and scaling are a huge part of them

[@Leonardo-lb6rs]

Yes u r right.

[@nullplan01]

@@MrBorderlands123 Sticking with derogatory terms for the long haul happens often in maths and science. Big Bang cosmology was originally called that by detractors, who wanted to point out how absurd the idea is. Well, absurd or not, it appears to be factual. Or "climate change". They originally wanted to call that the "climate catastrophe", but some of the scientists involved with that felt that was too alarmist.

[@sbgx3]

Damn 8 years in engineering and I didn't got imaginary numbers, 1 hour online lecture later my mind has opened

[@ashwinjain5566]

that too was a primer. nice

[@kdawg3484]

Same. Completed full undergrad engineering curriculum. Just now learning that adding rotations equates to multiplying complex numbers. What a joke. Grant, you're a treasure.

[@michaeldonki9947]

@@kdawg3484 True which indicates how much of a bubble the current type of business aka universities are. Full of useless pHDs that can't teach (probably because they never understood it deeply either if they had same type of teachers? who knows), but yet you manage to get a bachelor and you seem like a scientist to the average popualtion :D .

[@gonzalochristobal]

I love how empathetic and willing to understand possible mistakes Grant is, made me feel not silly when failing :)

[@LokendraSD]

Thank You for keeping us involved and making our time productive. Thank you from the bottom of my heart.

[@luciavegaresto8522]

Hey Grant, thanks a lot for doing these live streams. You have a gift, really. It's amazing how well you're able to explain math to me and many others. And don't worry about your handwriting ^^

[@kademeyer9993]

this is my all time favorite thing to watch on youtube. i genuinely get excited to see that these have been posted an i get to watch them. thank you so much for this:)

[@Sofialovesmath]

Thank you for everything you do, Grant. You inspire me and make me love mathematics!

[@timsullivan4566]

(at 19:12) Reacting to the level of viewer participation: "This is genuinely delightful!" And u just know he MEANS that... and THAT is why I love this channel!

[@DROSTraceurADD]

YES! I specifically loved that phrase too hahahah

[@qumusabel]

This was an absolutely fantastic lesson! The explanations are so simple and elegant. Thank you very much for your effort. Can't wait for the next livestream!

[@shaldee6814]

русские вперед

[@qumusabel]

@@shaldee6814 wha

[@allurbase]

"(a,b) rotated 90° counterclockwise is (-b,a)" instant mindblow!!

[@VoodooBoot]

That is amazing, pure gold! Thank you so much for the great experience!

[@laiquocbao2565]

Honestly, can we do this format even after the pandemic? Maybe just once a month or twice a month, I find out that your vids about these fundamentals of math have taught me a lot more than what I learnt in university for entire 4 years. (I'm studying Mechatronics engineering, and the more I learnt, the more I realize the power of mathematics, but it's kind of too late/too hard for me to build my base mathematical knowledge now.) Anw, really good content, take care and keep up the good works.

[@Mageling55]

59:52 It takes some heavy abstract algebra to prove, and I only mostly understand the proof, but algebraic closure of a field is unique, so anything that would require access to all solutions over polynomials in the reals must use complex numbers, or a construction that is isomorphic to them, or something more complicated that has a subfield isomorphic to it. So, the answer to this question is effectively no. Using vectors and matrices is an extension that allows skews in additon to rotation and scaling, but if you limit it to only matrices that are shape preserving, then you reduce exactly to the complex numbers.

[@stapler942]

I'm fond of Welch Lab's label "Lateral Numbers" mentioned in the first few videos in their complex numbers series.

[@josephcote6120]

Something you hear on Mathologer is "We've found that two completely different methods give us the same result. Do we just say 'Oh, that's nice'? Of course not, we try to find out why they are the same, what is the connection?"

[@ahlamamr4659]

wow, best math teacher ever!! I was stressing out a lot because I didn't understand anything in my maths class and my exams are coming up and my teacher keeps telling us how late we are and how bad we have managed our time and made this whole big deal out of it. Thank you for making it a lot easier and helping us connect the dots it's like everything starts making sense and I just realized how important are complex numbers. I started to really enjoy maths and see its beauty!!

[@IC-oo1mn]

Not typically a fan of livestreamed content, but this was done so brilliantly. Amazing effort by you and the team behind the camera

[@sarahp6512]

Thank you for this. Several of your videos use complex numbers to explore some concept or another, and there's always this assumption that the viewer is familiar with them (never stopped me from watching them anyway). Having never formally learned about them, I had this vague idea of what they were, but now I actually feel like I know enough that I can go back to those videos and get something new out of them. When I was trying to memorize the trig identities for my calc exam, I looked up methods to remember them, and the only thing I got was "rederive them using complex numbers" which was very unhelpful, since I had no idea that trigonometry had anything to do with complex numbers. It's cool to see how they actually connect in a really fundamental way, and where the identities come from, instead of the teacher going "here's an identity I just pulled out of my ass, now memorize it". I'm very excited to learn about how this all ties into exponential formulas and what Euler's identity is. Sorry for the long comment, I'm just really glad to finally actually learn what the hell is up with complex numbers after all these years of people talking about them without explaining them. You'd think they'd teach them (or at least mention them) earlier since they're so central to trigonometry and tie into a bunch of areas of math really elegantly. (for reference I just finished first year calculus)

[@1.4142]

hearted

[@TechToppers]

It's helping in AIME a little. Actually, a lot! Contest Math is different, but Pure Math has his own class.

[@sarahp6512]

Update: had a lecture on complex numbers in my mechanics I class (2nd year) because we needed to apply them and half the class hadn't learned about them yet. I'm noticing a pattern where my physics class applies a math concept before my math class teaches it, and it's very annoying (for example, vectors are used a lot in 1st year physics but not taught in math until 2nd year).

[@hybmnzz2658]

@@sarahp6512 that is indeed a common pattern in physics. Its kind of cool because physicists should have the intuition to use mathematics even if they don't understand the full rigour. Physics is also more rewarding when using math. I do agree that not teaching complex numbers by 2nd year uni is a big mistake from the math (?) department though.

[@corbinwhite4415]

I can't believe how complicated and obscure complex numbers seemed when I was first introduced to them in high school algebra. With this being just a primer, it's still difficult to see exactly how they apply to my area of interest, which is audio/DSP, but it feels like a much more approachable subject now. Thank you!

[@umedina98]

I have been follow you since undergraduate school. Now I am in graduate school and I am still learning from your amazing videos. You have made me see math from a different and beautiful perspective. That is simply priceless. I can not thank you enough.

[@PriyanshuSingh-uo6dr]

The part when he said 69 is close lmaoo I laughed so hard Edit : 28:34

[@Vodboi]

I also liked last time when he actually said a pretty interresting fact: squaring it gives 4761, and cubing it gives 328509, together they contain all digits from 0 to 9 exactly once (apparently its the smallest number with this property). A bit base 10-y tho but still. Funny when he went "because it has that property, thats probably why you submitted this"

[@alexgan3219]

Saame; 33:50

[@ornessarhithfaeron3576]

hehe 69 amirite xDDDDDD

[@fatimaezzahraemadrari4523]

I just like said close to whst???😕😕

[@BlackEyedGhost0]

Video was trimmed. This is now at 28:34

[@RahulAgarwal97]

Having already graduated, I still am sitting and watching your videos very actively. The different approach and thought process of thinking about the same concept is very surprising and interesting. Keep up the good work, and none of us mind the bugs :)

[@przemysawstarczynowski5514]

Great video, entire channel actually, helps a lot for your maths to sink in. And that handwriting with occasional slips, like that saying sin(β) but writing sing(α) at 56:03, brings human nature into maths, weirdly helps rather than distracts. Keep the good stuff coming! 👍

[@Mr.Nichan]

ca. 22:10 I just figured it out. Each complex number is written as the sum of a horizontal vector and a vertical vector. (The real and imaginary components.) Each of these lines is just one of the unit vectors (1 or i or -1 or -i) multiplied by a constant, so it's easy to see how the fact that multiplying by i rotates these four numbers (by virtue of multiplying i by i being how i is defined) would extend to rotating the horizontal and verticle components of any complex number. If you rotate these components than you rotate the number (ad makes sense visually). (That felt like "figuring it out" to me because it explains how the symbolic math is related to the visual rotations and how it all stems from "i×i=-1".)

[@Felipe-53]

Man how can you not love this guy, come on! I hope he becomes a billionaire. Grant, you're an inspitation, a person to look up to. Thank you so much!

[@empathematics8928]

He’s great, but he doesn’t need to amass a lot of private property to be happy. Sharing the love of math, I’m sure, is what makes life worth living for him.

[@manojkr9198]

Money not cool

[@alexsparks-bakota3099]

I can just see Cam and Eder sitting in tiny chairs with their legs tied up in the corner of the room furiously working on their laptops

[@JohnSmith-je7ip]

He was way too rude to them

[@erv993]

This is a masterpiece! Thank you so much, Grant!

[@nicholasn.2883]

This series is great. I get to fill in the gaps in my knowledge and bridge connections I never made before. Thank you

[@kruthikanj]

I was introduced to you channel by one of my friend almost an year ago, I've been hooked ever since, what I love is the amount of in depth information which is common in all your videos, keep up the good work guys, you're amazing, rock on 🤘

[@kaushallabba7219]

This was genuinely revelatory. Thank you!

[@alecyates3767]

Excellent stuff. Quite incredible the quality of this given it was all done live.

[@bryanroland8649]

Thanks for these classes. You're a great teacher who really understands the importance of an intuitive grasp of mathematical concepts. Speaking as someone who never got beyond basic arithmetic, I'm beginning to appreciate the beauty of higher maths.

[@yonasadiel]

19:30 "Writing is difficult" - Grant, 2020 edit: 19:08 after video trimmed

[@curiousmolar8104]

I don't get it. It seems like he is writing for the first time.😱

[@thigtsquare950]

7:35 “when you have a problem that you can solve you can just say ‘Oh I’ve define things so that we now magically have a solution’ “. Then you become part of the history of mathematics...

[@hassanakhtar7874]

Nope. You become a figure in textbooks or on calculators only when you show that these definitions have nice identities, interpretations, consistency, etc.

[@carultch]

I like Gauss's solution to renaming imaginary numbers. He suggested the term "lateral numbers", while real numbers would be called direct (+) numbers and inverse (-) numbers. Not entirely on-board with calling the negatives inverse numbers, because inverse more often means reciprocal, but I think lateral numbers gives a much better understanding about what they are.

[@MelodiCat753]

My mind is blown. Thank you for awakening me to what complex numbers are: vectors that have an awesome rotation definition for multiplication!

[@manos1779]

This is first time I have ever commented on a KZbin video, as I feel guilty that even after such an amazing lecture if I can't appreciate your effort to help the students to get better understanding on complex numbers. You videos are really amazing and wonderful sir. My thoughts about math has changed drastically because of your videos👏👏👏👏❤️❤️❤️

[@IElial]

THANK YOU ! The best explanation so far on quaternion = 3D rotation ! Even you just very very breifly talk about it, extrapolating from 2D rotation with complex number to 3D rotation with quaternion make finaly sense with this video !!!! Now I want to understand Gimbal lock in a first place.

[@DanielFrance81]

Good news: Grant made another video on the relationship between quaternions and 3D rotations! It is amazing. Enjoy!

[@AnonyoX]

This one is just too good a video! Grant, please keep these coming

[@davidjones5319]

Please keep up the fantastic content. Love it!

[@CaptainSpock1701]

"I'm gonna have a stern word with them *behind the scenes"* - I think you just did.

[@kirenireves]

Rule #1 for presenters: Never criticize the AV people during the presentation. They KNOW when there is a problem and - you can be sure - they are FRANTICALLY trying to fix it. Chastising them publicly is just a jerk move. Much better to say to the audience: "Please have patience. My AV folks are really great, so there must be something really unexpected for this to happen."

[@thasyashetty3797]

@@kirenireves I think it pretty clear that he was just being funny and the AV people probably knew this. But thank you for the insight, I'll certainly keep that in mind for the future.

[@kirenireves]

@@thasyashetty3797 He says it a few times, even encouraging the audience to give the AV folks a "stern word" over twitter. Then there is a tweet at 26:29 where someone compliments Ben Eater and says he doesn't "deserve the harassment" that Grant is throwing his way. (So it's not just me who noticed this.) Grant does *not* take that moment to say "I'm just joking...These guys are doing great." People who are presenting feel anxious when AV does not go smoothly and so they project it back on the AV folks so they are absolved of blame in the eye of the audience, but it just makes them look small. AV people get alot of abuse. Just try doing that job with an ungrateful set of presenters, and it'll drive you to drink. Thankless job.

[@ruhaanchopra8878]

@@kirenireves dude nobody really took I seriously. he was just joking, calm down.

[@kirenireves]

@@ruhaanchopra8878 I'm really calm. I am just pointing out that criticizing AV people in front of the audience is not a good move.

[@TheNKHome1]

We can call: Real numbers as "Horizontal numbers" Imaginary numbers as "Vertical numbers" Complex numbers as 'Circular numbers"

[@patrickcoffey5933]

I like both of your nomenclature but i think Sushant wins here on the amount of info you can derive from the name.

[@jgcooper]

@@Sushantgupta12 the rotational attribute is not really of i, if you really think about it, but of the operation. The problem is that we confuse symbols with operations and with the attributes of the numbers. Maybe we shouldn't talk about scaling, complex product, vector product, and all that as if they were the same operation (and often same symbol) with different kinds of numbers, but rather different operations with just groups regular numbers (where each number in a group might be orthogonal in some way to the others).

[@anshchaudhary986]

@@Sushantgupta12 Discussing tiny little facts with such a great community really brings up new radical thinking and analogy to simple things. That's the joy of it.

[@NitronNeutron]

I call complex numbers 2d numbers.

[@rafidkhalidnahi6601]

@@Sushantgupta12 Scalars and Rotators. This terminology is an absolute genius.

[@TheGmr140]

Dude you rock, your passion for mathematics is amazing, your videos are amazing and promote an intelligent culture. Math is fun.

[@itsmehere8285]

This was the thing I needed to hear!! The calculations you asked us to work out have been pretty easy for a JEE student in India but I really lacked the thoughts and idea behind it..more appropriately I didn't know what to think of while working with complex numbers- the only chapter I never felt but felt like could unveil a lot of mysteries!! This is my day 1 of unveiling that!!!

[@sevfx]

33:05 an addition to the i and j thing. In electrical engineering the i is reserved for a flowing current, hence the similar looking symbol j is used. (where i study in germany at least)

[@RunstarHomer]

I've probably left a similar comment on almost all of Grant's videos, but 3b1b is, by a mile, the best channel on youtube.

[@1997CWR]

Ben Eater has an awesome channel, too! If you haven't, check it out!

[@alopez545]

Grant, really enjoying your lockdown math series. Looking forward to more lectures!

[@AerosfilisOfficial]

I've always been interested in complex numbers because of games, as in, 3D rotation relating to quaternion which themselves are the next step up from complex numbers and just thinking about it differently helps so much! I've really been enjoying these lectures you've been doing, I graduated from high-school 6 years ago now, and while I did try to go through university, it just didn't fit me, I ended up enjoying learning from math papers and just playing around to understand more than just listening to the lectures they gave, so I dropped after changing uni once, then having to validate my first year in 2 times, so really not for me. Now I still love math but programming school I found doesn't have lessons as the school is more of a "figure it out yourself" (School 42 in Paris) which I love, but it has it's draw backs when it comes to trying to find new was to solve problems, you too easily get stuck into what your brain is used to, so all those math videos are always a really good way to just think differently. I'm a bit slow so I often have to pause to do the math but I can pull through. I know this format is more meant because of the virus and lock downs and I can't be surprised if the logistic makes it a bit hard to do regularly when all of this calms down buuuuuuut I would really love even just one like those every month :D And while I'm here, big up to Ben for all the interactive tool! Been enjoying his videos for a few months now, quite something too!

[@steelday]

I was asking myself what hell am I doing watching a lecture about complex number at 1:30am when I finished this video, the answer was "because it is fun". I didn't even notice the time passing. Thanks for all the work, everything on this channel is just brilliant

[@Kram1032]

The question you answered around 59:46 There's *also* ways to "get rid of" a lot of matrices by using extensions of complex numbers instead. Just use Geometric Algebra or Clifford Algebra. (Same thing, different names) There are good reasons why one would stick to matrices anyway I think, but a lot of things that are rather clunky and awkward using matrices work out extremely beautifully with Geometric Algebra.

[@laxmigurung2155]

Thank u as always, Grant Sanderson. You made my understanding of complex numbers better. During this lockdown, my interest in Mathematics has grown even more. Salute!!

[@boruiwang1738]

Every of this video reminds me the importance of a good teacher. Thank you, for keeping my interest in math.

[@orgadish]

Even as someone for whom high school math was 15 years ago, I still learned a lot from seeing this perspective connecting complex numbers to trig. Did you consider starting first with two dimensional numbers (without talking about imaginary numbers; eg “apples on the horizontal and oranges on the vertical”) and then show that i is the result of wanting to rotate in this plane. And only then that it turns out to be the sqrt(-1). And then wrap it back into the trig.

[@scfog90]

As an engineer I have to say that when its coming to dynamics, you`re lost without the understanding of complex numbers and their usage.

[@mostafaomar5441]

Thank you for the content. You make math feel so natural and intuitive.

[@stu007]

This guy is the CEO of maths

[@angelowentzler9961]

I am 48 years old and had quite a bit of math in my education but only now, after watching this, did the exponential vs cis() thing click for me. Thanks Grant!

[@ireallyhatemakingupnamesfo1758]

Why is he so pretty?? Like he’s amazing at maths, and more than that great at making it feel intuitive, but then on top of all that he’s properly gorgeous. Some people have all the luck in the world

[@theorange1729]

He doesn't write well though ;)

[@milanstevic8424]

NOT TO MENTION THAT HE HAS AN INFINITE SUPPLY OF UNIT CIRCLES oh man. I wish I was that guy.

[@manupeter8050]

Well he is passionate about sth which he absolutely reasonably believes helps us when he shares it with us. This makes people gorgeous.

[@milanstevic8424]

@@manupeter8050 your belief isn't very scientific, but then again it's very hard to argue against that! you might be onto something.

[@RodelIturalde]

@@milanstevic8424 maths isn't science in the first place either.

[@not_melkor]

Thank you. You have cleared a lifetime of angst. I was stuck with why and never got beyond. Thanks again.

[@cringeofficial6221]

I really love your videos. It helps me to remember that math is more than a bunch of formulas to memorize before the test!

[@kagankilic4905]

I'm a Uni student. And as i was watching this everything I've learned just... made sense. Thanks my dude

[@timh.6872]

I knew it was coming from the stream on Tuesday and was scheduled well in advace for completely different reasons. Nevertheless, today is my birthday and I'm definitely going to enjoy listening to Grant explain complex numbers again. Thanks for the stream!

[@pronounjow]

Today is my birthday, too!

[@timh.6872]

@@pronounjow HUZZAH!

[@jasonlee3247]

You knew your birthday was coming due to the stream earlier in the week?

[@timh.6872]

@@jasonlee3247 I knew this topic was coming due to tbe stream earlier this week.

[@Fraglers]

this series is actually useful for my studies, i've just started studying math for uni so thanks alot!

[@luisalbertotorrescruz422]

Grant, you wonderful human being!! THANK YOU for these amazing videos.