Euler's formula with introductory group theory

[@3blue1brown]

For those who want to learn more about where the number e comes from, and why that constant 0.6931... showed up for 2^s, there's a video out it in the "Essence of calculus" series: kzbin.info/www/bejne/o2OweqN6p6tqe9U

[@reaniegane]

3Blue1Brown could you do simplectic geometry in classical mechanics ? Maybe a topic from Arnold's or mardsens book on classical mechanics. Those are graduate text to mathematics, but they cover physics

[@ev4_gaming]

3Blue1Brown If i^2 = -1, does i^4 (4x90° rotation) equal 1?

[@reaniegane]

EV4 Gaming yes. Multiply i by itself 4 times.

[@ev4_gaming]

Max Yurievich Okay, thank you

[@nirbhaythacker6662]

This is no proof, i really can't answer a teacher this way can I? You need to get mathematical with intuition.

[@vlogbrothers]

This is just so, so great. Approachable but rigorous, and in its way, kind of thrilling. -John

[@3blue1brown]

Thanks so much John!

[@fossilfighters101]

John! You watch this channel?! I didn't know there were other nerdfighters around here!

[@aaaab384]

But he didn't really prove anything whatsoever!

[@mitchbet]

aa That's not the point of the channel. Anyone who has done a maths degree has seen the proofs. This channel is to show maths to people who haven't done a maths degree

[@UtsavMunendra]

Hey nerdfighter!!!

[@arkapointer]

Final year math undergad. NOBODY (not even any top level book) has explained isomorphism/homomorphism to me the way you did. Much,much love and respect from India.

[@aryamanmishra154]

which uni?

[@JordanMetroidManiac]

Same!!!

[@aurelia8028]

Of course it's india...

[@notsoclearsky]

@@aurelia8028 no, he just got a shitty teacher. There are good and bad teachers everywhere. I'm in high school and my teacher introduced me to this channel. He has taught so many things that are out of my syllabus because he has real passion of teaching.

[@celestialrex551]

@@notsoclearsky you are lucky man , we got teachers who just want marks

[@aaronbrown5477]

Whenever I get demotivated with my studies, I watch your videos to remind myself the beauty of mathematics to motivate me to learn more of it. I know a lot of people say thank you, but through all my struggles, you make me want to continue to learn! So, Mr. Brown, thank you!!!!!

[@ahlamamr4659]

same

[@richardgalea9884]

As a 70 year old man, I thank you for at last you made me comprehend more the beauty of mathematics...

[@slinkywhite9255]

still alive?

[@ryan-tabar]

@@slinkywhite9255bruh

[@slinkywhite9255]

@@ryan-tabar I mean he's legitimate 70

[@hallowvalor3477]

@@slinkywhite9255💀

[@lilamjazeefa9466]

07:06 Sliiiide to the left! 06:57 Sliiiide to the right! 12:57 Criss cross! 10:47 Take it back now y'all! 13:37 Cha-cha real smooth! 02:30 Freeze! 24:28 Everybody clap your hands!

[@hoelefouk]

you are a genius.

[@Titurel]

Brilliant!

[@Reaction1s]

@19:07 - now it's time to get funky

[@ahlamamr4659]

that comment is under rated AHAHAHAHAHAHAHA

[@khiemgom]

Lmao can someone explain

[@geraldmerkowitz4360]

We will never say this enough : your animation is gorgeous

[@dbanetinc]

Archibald Belanus it is.

[@timh.6872]

I've seen both Essence of series multiple times, watched space filling curves and towers of hanoi, but this video, this one right here, is the most beautiful thing on this channel. Maybe I care too much about math, but the transition near the end from 2^x to 5^x to e^x, building up to that e^i*pi chokes me up every time. Pretty sure I had tears the first time I watched it. This video transitions from "why would this ever make sense" into "what else could it be" over the course of 20 minutes. Truly a masterpiece...

[@3blue1brown]

+Tim H. Wow, thanks so much! That's kind of you on so many levels, I'm glad you enjoyed.

[@wizardom]

His videos have the same effect on me!

[@saswatapatra5919]

Ohh i thought i was the only weird guy who cried over mathematics videos 😂

[@vaguebrownfox]

Wow, could have said it better myself. I never thought I'd cry over math---with happy tears.

[@zilanh3160]

if there was a bit of sad music i probably would've cried too

[@JovanaaaSK]

I am the most grateful human being after discovering this. Thank you so much. I was almost about to quit my master's thesis in maths (just because we never visualize anything, just learning stuff with the speed of light but no-one ever explains us how to imagine things and that really makes me nervous), but you're a part of the people/things that helped me get my thrill back. This is pure gold, thank you so much Sir, from the bottom of my heart.

[@TheGamshid]

I am in the same situation, feeling the same gratitude.

[@mysteriousman8576]

Jovana Krstevska Hi

[@gurumayummadan2646]

Jovana Krstevska I can relate. That is very true. Lesser and lesser imagination and visualization as we try to learn more. Kinda kills the fun. Fortunately, you have been saved. I am happy for you.

[@markopanev3317]

Prvpat gledam makedonec da komentira na vakvo video lol

[@seankelly8906]

No offense but I find it really surpising a master's student would say this. Don't get me wrong, I'm up late watching this for the hell of it because I like the teaching elements, but I'm just shocked someone in a graduate program needs to be taught to imagine things.

[@joshuaflackua]

I can say with utter confidence that this is the best KZbin channel I've ever come across. So incredible how much of my education clicks every time I watch one of your videos.

[@mikekeim3178]

I really wish I'd had a video of my class (a collection of 11th and 12th graders finishing up a multivariable calculus elective) watching this video just now. They broke out clapping and hooting when the big reveal happened at 20:50. You're doing something really special and greatly appreciated with these videos. Thank you for sharing this with all of us.

[@Cold_Ham_on_Rye]

I'm currently in grad school studying robotics. One of the cruxes of engineering is taking maths on faith. We don't necessarily have the time to make sense of these things. Your videos have helped me understand Linear Algebra and Diff Eq so much better beyond it's applications. I really appreciate what you do. e makes more and more sense every day.

[@hailmary7283]

The crazy thing is even in math programs (at least at my undergraduate institution), a lot of what we had to do was take math on faith. We were basically given the definition of a group as though they were handed to us on stone tablets like Mosses on Mount Sinai. Never asking "why the hell are we using this as our definition of a group?" or "what even IS a group and what is the broader concept it's supposed to represent?" It would be like teaching a six year old: multiplication is the process of repeating the addition property an indicated number of times. Sure it's correct, but you aren't really learning multiplication at a deeper level and it becomes harder to later generalize to multiplying by zero, fractions, or God forbid, negative numbers. Visuals of rotating a 4x5 grid to show that 4x5 = 5x4 are a much better way of understanding the fact that multiplication is commutative than simply being told it's an axiom and just something you have to accept. These 3Blue 1 Brown videos are a great example of the rotating the grid type of visuals that give one a much deeper understanding of what these concepts actually represent as opposed to some vague abstract construct with which to work.

[@samlefty6732]

So the tales were true: there really are friendly, constructive and informative comment sections on youtube. This is such a nice crowd of subscribers.

[@mohsinuddin7049]

Ur mom gay

[@ThichMauXanh]

@@mohsinuddin7049 no u

[@udith]

@@mohsinuddin7049 Lol!

[@tpat90]

A year ago, I wasn't able to find a single math student watching your videos. Nowadays this is a different story. And we even start to infect the new students with your amazing work. Your way to look at the topic is most of the time just ... amazing. And gives a perfect base to jump into even complex topics head on. Your videos led me to many really interesting discussions and that's why I always appreciate to see a new video of yours. If I had a wish, I would love to see your take on absolute continuous or smoothness (probably because I love the cantor function...) some day in the future. Have a great day!

[@3blue1brown]

I would definitely like to do an "Essence of real analysis" one day, and the topics you mention would be right on there.

[@kudancer]

This would be amazing.....I have thought the hardest part of math is analysis and you're just the guy to do it.

[@tpat90]

Evi1M4chine I study math and most of the people I know don't like to memorize something. We try to understand. And this counts for every single one of my lecturers and their assistances. Sometimes this goes so far down the rabbit hole, that we nearly commit an entire lecture, just to understand that one thing. Sure there are theorems you might only need to know exist in the first place, like stuff from Frobenius, fudge this dude and his proofs. Also there was a time in mathematics, where it wasn't accepted to use graphical assistance to help the understanding of a topic. Just look at most of Euler's and Gauß's proofs. Those feel like they had a completely different understanding for the topic, but every evidence of sketches or graphical explanation got deleted, since it was a 'no go' in the era. But this isn't the case anymore. Not to say that everything get's easier with graphical understanding of a simplified problem, but either way you have to put effort and a lot of work in understanding simple theorems to explain them in the way 3Blue1Brown does. And if it takes 10 hours just to get one idea, that's okay. And most of the fellow students I share my time with, love mathematics. Like two friends of mine stood in front of a chalkboard, trying to understand why you need certain properties in a theorem, if it would be enough to use more global statements. They spent 2 hours (after a complete day of lectures) just to understand this one bit. This is what studying mathematics is for. Understanding and expanding your knowledge on the topic of your choosing. BUT Wikipedia is a page of knowledge, not understanding. If you want to understand, you are welcomed to study mathematics. Get books about a topic or just search it up on Math Stackexchange. Wikipedia is to fast look up a certain theorem and it's not built for mathematicians, it's for everyone. To much information will sometimes lead to struggle, if certain people are unwilling to put in the said effort and time. They don't need an explanation how it works or why it works, but only that it will work and achieve the output needed. Nothing else. They simply don't have the time and energy to do so. Don't try to force them to understand those things.

[@firefly618]

Wikipedia decided long ago to be an _encyclopedia,_ that is, to collect knowledge supported by reputable sources. It's not the place to add your own explanations for complicated concepts. A blog would be a better place; or an homepage provided by your institution or company; or a KZbin channel like this one; or you could write a book if you have enough material. Either way, if and when your material gains acceptance and recognition, in a way that can be measured by that "cabal" of editors you despise, then it will be gladly accepted into Wikipedia.

[@tpat90]

Zaknafein Do'Urden A Zin-Carla in the comment section? Didn't thought you could type...

[@logixindie]

Who is this? The perfect math teacher? Indeed.

[@johnchessant3012]

So all e^(i*pi) = -1 is saying is that "rotation by 180 degrees is the same as a reflection". Neat!

[@UnordEntertainment]

not the same as a reflection, only the same as reflecting the point at 1+0i using the imaginary axis as a mirror line, note that such an action does however reflect the real axis across the "mirror point" of 0 because the real line is 1 dimensional and therefore has no orientation to help distinguish reflection and rotation

[@ugozahm4902]

It is the same as the central reflection centered on 0, but not as the axial reflection along the imaginary axis

[@randomname7918]

The one thing I didn't understand is what is the rule according to which the points on the vertical axis get mapped to the points on the circle at the end

[@chappie3642]

@@randomname7918 you take the imginary Number shown on the left and that's your x, then you raise the esponent (2, then 5 then e) to the x and that Is your result shown on the right

[@LydellAaron]

Very cool way to interpret it geometrically.

[@bishantadhikari3440]

"Those who know do, those who understand teach." And your are one of the best examples of the second one.

[@businesstuber3978]

The one I hear often is 'those who can, do. those who can't, teach' which is kinda funny. sorry about the four year old reply

[@antonzavorotny3537]

Please make more videos on group theory! Group theory in general is such a good topic to give intuition about because no one know what it is unless you study math. Also, I really need a good way to picture quotient groups!

[@3blue1brown]

I intend to, I agree that it's a really nice topic. For quotient groups, it helps a lot to contemplate of tightening up on the structure you care about for the thing being acted on, so to speak. For example, think of the map from D_8 -> Z_2, where for each square symmetry, you consider how it shuffles the two diagonal lines. That each, a symmetry maps to 1 if those diagonal lines are exchanged, and 0 otherwise. The Kernel of this map (which is a quotient D_8/Z_2) can be seen in all actions of D_8 that don't exchange those two diagonal lines, which consists of four actions, isomorphic to K_4. That would be better with some animations, but hopefully it helps a bit.

[@umbraemilitos]

3Blue1Brown Algebraic Geometry is a really interesting and visual discipline for Group Theory, Rings, Ideals, as well as concepts in Topology and Linear Algebra.

[@ben1996123]

the quotient group G/H is a generalisation of "congruence modulo n". take 2 elements, a,b in G. in G/H, these elements represent the cosets aH and bH, which are equal if and only if ab^-1 is in H (or a+H = b+H iff a-b in H if you use additive notation). notice that this is basically the definition of congruence mod n: integers a,b are congruent mod n iff a-b is a multiple of n. example: suppose G = Z, H = 5Z. if a,b are elements in G, then the elements a+H, b+H are equal in G/H iff a-b is in H, i.e. if a-b is divisible by 5, i.e. if a=b mod 5. so G/H is the group of integers mod 5. another way to think about it is that to find the group G/H, you take G and replace all elements of H by the identity. so in the example Z/5Z, take the integers and replace all multiples of 5 with zero. consequently you have to replace all numbers of the form 5n+1 with 1, etc. so you are just left with 0,1,2,3,4 in the group. this idea is the same for quotients of rings by ideals. an example i like is R[x]/(x^2+1) where R[x] is the ring of real polynomials and (x^2+1) is the ideal generated by x^2+1, i.e. all real polynomials that have x^2+1 as a factor. the quotient ring R[x]/(x^2+1) is what you get when you take R[x] and replace all instances of x^2+1 by 0, so this ring is isomorphic to C. if f(x) in R[x] is a polynomial, you can write f(x) = (x^2+1)q(x)+r(x) where r has degree 0 or 1 by euclidean division, then x^2+1 becomes zero so f(x) is the same as r(x) in the quotient. example: take f(x) = 9x^3 + 2x^2 - x - 7. then f(x) = (9x+2)(x^2+1) + (-9-10x). replacing x^2+1 with 0, you see that f(x) is the same as -9-10x in the quotient, and indeed, f(i) = -9-10i

[@JM-us3fr]

Yes please! That's probably the closest he will ever get to number theory.

[@swiminbandgeek]

I absolutely love this I'm a chemist, and I love how this video in particular gets me thinking about how group theory works Some day I want to teach group theory for chemists too chemists so this mathematical understanding is really awesome

[@mz1rek]

Fascinating video, I really appreciate the visualization of the group theory. but I can't help but point out a minor ISSUE at 13:27 The rotation of 2+i is not 30 degrees, but approximately 26.565 degrees (arctan(1/2))

[@MatthewWroten]

Yeah this hurt me too

[@mz1rek]

@@MatthewWroten What hurts me is that my comment didn't get a heart. I want to assume that Grant didn't have a chance to read it! :)

[@bulubado]

thanks for point out error❤

[@_alyssarose]

After a bit of abstract algebra I finally understand the reason why you move zero in the additive group and one in the multiplicative group. They're the identities of that group! I'm still realizing things from these videos even after watching them in the past.

[@barth05]

Thank you for your work, I'm a PhD student in Electrical Eng. and wanted to go deeper in to the fundamentals of signal processing and linear algebra.. Your videos helped me think about these visualizations better, and you do a good job on them. I hope you continue inspiring and helping people all around the world. See ya. Kudos from Brazil.

[@itsiwhatitsi]

This Channel is better than University. You are the man 3Blue1Brown

[@jsutinbibber9508]

@Sophisticated Coherence but he does not teach everything else

[@AlchemistOfNirnroot]

@@jsutinbibber9508 What stage are you at in your education? Because these videos become absolutely indispensable at some point and fourth (not gonna get ya to the finish line - bad analogy - but gives some ideas about stamina (for example).

[@Artaxerxes.]

University has always been a joke. A scam

[@molomono9481]

Im sorry to hear you didnt go to a good one. In europe its very cheap and right now im studying at University of Twente in systems and control, it is very good. The teachers are competent and caring, our facilities are good. The campus promotes sports and social associations that help you develop skills outside of the core classes. I might be lucky but it is not a scam.

[@gabrielnettoferreira8452]

They really aren't a scam, lol ANY university in the world teach this kind of thing in mathematics related courses.

[@SpaghettiToaster]

Oh shit, that intuition leading to the conclusion i*i=-1 is just beautiful!

[@YitzharVered]

Well, i is DEFINED by being the square root of one. it would be the same thing as saying: "Wow! way to prove that the ratio between a circles perimiter and its diameter is exactly pi!"

[@SujithIshtar]

It wasn't a conclusion as such, but more of a process in the context of group theory of how i*i is -1. Well, i*i=-1 is the very equation by which i is defined, so he didn't prove anything new, but rather showed us the process of how it works. So people who would ask what's x*i or essentially, "what does multiplying by i do?" would find this illustration useful. Just saying i*i=-1 is a given is one answer, but more of a declarative one, and it raises plenty more questions than it solves. Giving an animation to explain the process helps people explore complex numbers. So it was the same thing as saying, "Just saying i*i=-1 is not satisfying. Here is something that you might find satisfactory and it will help you explore complex numbers as well."

[@standlegweak9854]

Trust me when I say this: do NOT make spaghetti in the toaster. Don't ask me how I know, it's not important. Just don't do it.

[@ZweiZombies]

But i^i is not - 1, it's ≈ 0.2

[@ZweiZombies]

Nevermind, I mistook the * for a ^

[@Sam_596]

So your explanation of complex numbers with shifting and stretching and rotating has taught me more about complex numbers, in an intuitive sense, than any teacher. So thank you

[@AdarshSingh-wv4ff]

The best thing about u is what you know?? U make an intuitive pictures of everything. U make us to think the abstractness of any mathematical topic.The same way I had learned maths in childhood and still learning. And I want to thank u a lot because this video helped me in writing my mathematical papers....

[@GelidGanef]

Has it really been only two years? Has it really been already two years! Congratulations! You are legitimately one of my favorite channels. It's been fun to be along for this ride, hope it continues for years to come!

[@peorakef]

If you were to start from square one and do a complete course covering math from calculus I, it would become the new standard reference of the modern world, replacing all those crappy books.

[@dez-m]

This channel started my love for calculus at a young age. Am currently pursuing Engineering at Berkeley. Thank you so much!

[@xyzct]

Good for you! I got my degree in geophysics from Berkeley, and I am so proud and grateful that I went there.

[@keeganmurphy2992]

Im mean and skip sponsors usally, but Cloud Lab is actually the first ad ive seen for something truly brilliant. cant wait to see what this idea becomes in the coming years

[@TheChondriac]

I just want to say, Grant, that I really love this revisiting of the Euler formula. I didn't quite understand the whole idea you were trying to get across in the first video, about stretching, moving, and rotating being related to certain operations. But the group explanation has actually really helped me to understand it.

[@judsonkuhrman5577]

I think one of my favorite proofs I have found of Euler's formula was in a book I picked on relativity, where it starts with only the equations ds²=dx²+dy² and rθ=s, and ends up showing rθ=x+iy, without even defining the number π.

[@saras4123]

I love this. It's perfect. I'm lulled by numbers and finding order in disorder, patterns of beauty in a chaotic place. It's beautiful how he describes it to make sense... I don't have the education to do calculus now but I see numbers how he describes it, exactly. It's like 0 is middle C on the piano. I love patterns. Music is math. I'm in awe.

[@paulo9523]

This is by far is one of my favorite 3B1B videos, I come back to watch it again and again

[@munmunbanaerjee9572]

Cant wait for more complex analysis videos really...this video by the way was extremely thrilling to be watched and getting to atleast know that angle preseving or the analytic nature of the e to the z function isnt opaque but quiet enthralling for all if they stumble on this video and especially the point where the video closes off just vividly showing it...its a blessing to be a part of this channel...💙

[@berni1602]

I finally understood this concept. I'm a math lover, although I'm a biologist. I didn't quite undersand why, a teacher tried to explain it to me, but nothing; I read about the subject but nothing; I even watched other trying to explain it, but nothing. Finally I've found an explanation I can understand and it's beautiful, I mean it! Thank you so much!

[@helo3827]

This imo is the best Math channel on KZbin

[@chairmanmeow9110]

This is, hands down, one of my favorite video's on KZbin

[@farrankhawaja9856]

This needs to be shown to more people. I hate when they use the taylor series expansion to explain this. This is so much more intuative!

[@solidpython4964]

It is intuition, but it isn’t proof. Both should be considered.

[@Asteria32]

Hey Grant, this explanation is has really done it for me. I am a junior at uni studying physics and I am so glad that you have made this video because I finally understand things that I didn't even think I didn't understand in the first place. Not just with euler's formula but with math in general. Thanks a lot.

[@ColoredScreens]

I've read/watched lots of information about Euler's Formula and imaginary numbers in general. This was the first explanation that left no doubt in my mind as to why imaginary numbers are so intrinsically related to rotation. Thank you.

[@kummer45]

This man and his contributors deserves a worldwide recognition of mathematicians around the world. This channel is SERVING the mathematics to everybody.

[@blattsalat77]

Every time I learn something new and interesting in your videos I wan't to like it, but then I see that I already liked it. Keep up the good work!

[@garlic7099]

Currently studying for my complex analysis exam, and man am I ever grateful to you for this video! It was an 𝘪 opening experience for me!

[@sophiap.8859]

How dare you make me read that pun with my own two i’s

[@boston0086]

@@sophiap.8859 i cannot unsee that

[@boston0086]

Did you pass bro?

[@garlic7099]

@@boston0086 Hahah In the end I didn't even do it. The corona quarantine had just started and so I just got a passing grade. So yeah, I guess.. Sorta passed :)

[@boston0086]

@@garlic7099 Congrats bro:) the most important thing is that you you were interested in it! The topic about complex analysis is so much, you will always learn new stuff about that :D

[@g_vost]

polar coordinates are such a nice way to describe complex numbers, and better highlight how complex numbers, though described with 2 values, represent a single number. positive reals are rotated 0 radians, negative reals are rotated pi radians, making the real axis. these are all the numbers we know now. rotations other than 0 or pi radians are numbers that arent entirely on the real axis, and are called lateral numbers. in this system, numbers are defined by a magnitude and rotation. 1 is 1

[@cubicmetre]

It's amazing contrasting my understanding of this video last year before I started uni to just now where I literally just got home from a maths lecture explaining these complex numbers. I immediately recognized the content from the lecture being in this video. So first thing when I get home, study 3 blue 1 brown.

[@YoutubeDictatorship2024]

What are you doing now after uni?

[@xan1716]

I haven't watched the other videos, but the argument at 19:30 seems to be somewhat circular: If you are already at the point where you are comfortable enough with the definition of complex exponentials to be able to reason which base "makes sense", then you already know "why e^(i*pi)=-1". In other words, this video was less about 'the why' of this relation, and more about how it could make sense if one considers the exponential function to be a mapping between two groups on the complex numbers, e^(i*pi)=-1. Not to take away from the rest of the video, which was obviously great :) Edit: no pun intended with 'circular' :)

[@guilhermebotelho1615]

Agreed! The video is great, but this one step is not clear enough I think. The explanation at 21:06 kinda touches on it, because it hints that the derivative of different exponentials would mean a different rate of change around the unit circle, meaning that Euler's formula only works for e. That said, he still didn't elaborate on it enough I think. Edit: If you do the derivative with 2^x instead of e^x, you get i(ln(2)2^(i*t)) which might be the explanation we were looking for! Because the factor in the derivative is converted into a factor in the velocity around the circle. This explains the angle we get if we use 2^x instead of e^x, as well as e^x is special.

[@sharanmourya]

awesome !!!! when you uploaded the video about e^pi i two years ago, I couldn't understand it completely and this video helped me alot in gaining insight and complete my understanding of what "taking adders to multipliers" is-the phrase you used in that video. I am always thankful to you for making such amazing videos. I wish I could meet you and learn math directly from you.....keep up the great work of making people enthusiastic in math !! waiting for your essence of calculus series ........

[@reubenfrench6288]

At 13:33 that is NOT a 30 degree rotation. It is approximately a 26.57 degree rotation.

[@3blue1brown]

Ooh! Good catch, thanks for that. Silly me, mixing up my sides.

[@ZardoDhieldor]

Yup. Thought the same thing, immediately! :D

[@steventhijs6921]

Omg you're right, i measured it with my protractor (because that's quiqer than calculating it) and I got exactly that

[@deniskhryashchev7245]

Lol it's just arrcos(2/sqrt(5)). Awesome video

[@willferrous8677]

is arrcos the pirate version of arccos?

[@amatya.rakshasa]

You’re the best math teacher I never had. Thank you for making these videos. I can’t possibly ever express in words the true depths of my gratefulness. Thank you!!!!

[@prasadsawant1358]

group theory has bought us the solution of various problems which remained unsolved for hundreds of years...i didn't know it was that easy.Thank you very much 3Blue1Brown.

[@tux1468]

Group theory has a TON of potential when it comes to optimizing machine code, and I'm disappointed that not more people are looking into it.

[@HyrumCooper]

This really interests me as a student studying Computer Science. How can group theory be used to optimize machine code?

[@ds08tf]

The number line transformations remind me strongly of 1D matrix transformations. Especially as you illustrated them in your Essence of Linear Algebra series. Thank you for helping me improve my understanding.

[@zairaner1489]

Fun fact: The transformations of the multiplicative group of th ecomplex numbers represents exactly those 2D lienar transformatiosn which are not just real linear transformations but even Complex linear transformations (meaning they are the linear transformations commuting with multiplication by complex numbers)

[@purplepenguin8452]

I think group theory and discussing graphs would fit well with your amazing visual style of presentation. I would love love love to see a series on group theory build up to giving a sense of why in the world checking if two graphs are the same is so freaking hard. For example touching on graph isomorphism and Babai's work. It sounds like it has something to do with the symmetric group just being so large it doesn't fit well with our intuition, but it's hard to see. It just hurts my mind that something can be "easy" enough to do in _linear time_ vs graph size for almost all graphs, yet no one knows how to do it in even an arbitrarily large polynomial amount of time vs size if it must work for _all_ graphs!

[@mr80s14]

I'd love that too. There are some textbook like introductions on youtube (check out Sarada Herke), but nothing to bridge jumping to any of Babai's lecture videos, and nothing really conveys to me the core difficultly they are fighting against. Linear in time for almost all graphs, I had no idea the gap was that big! I'd love to see a visualization of what that core difficulty is that somehow escapes almost all graphs. So mysterious, I bet there is something beautiful in there when explained well!

[@ganondorfchampin]

I think it has nothing to do with group theory, but instead just computer science. It's because the problem is NP-Complete as you can reduce the satisfaction problem to it somehow.

[@vector8310]

Your videos are treasures that should be championed by all who believe in the power and beauty of mathematics.

[@gabrielwu5787]

Wow, what an enlightening video! I had previously held a certain intuition about multiplication on the complex plane as scaling and rotation, but I didn't really understand WHY we choose to think of it like that. This video explains very well how exponentiation is simply a homomorphism between sliding and stretching. So when we extend into the complex plane, we turn vertical sliding into rotation to encompass all 2D transformations.

[@roshimap.b1833]

Wow.... this is the best introduction for group theory that i have ever seen. Thanks a lot.

[@johnchessant3012]

So in this view, adding 1 to both sides to 'beautify' the equation as e^(i*pi) + 1 = 0 actually obscures its true meaning? All e^(i*pi) = -1 is saying is "rotation by 180 degrees is the same as a reflection". And if we indulge the tau-ists, then all e^(i*tau) = 1 is saying is "rotation by 360 degrees is the same as doing nothing", which is pretty neat. Euler's formula almost seems like a tautology in this sense, in that this is the only natural way to define e^(i*x).

[@johnnicholson8811]

e^(i*tau) = 1 is the same as multiplying by 1. As for e^(i*pi) = -1, it is the same as e^(1/2*i*tau) = --1 or rotation of half turn. So, yes pi obscures its true meaning.

[@linus6718]

If e^(i*tau) is 1, doesn't that mean i * tau = 0?

[@Daniel_Lopes_13]

@@linus6718 No, because the exponential function in this case is periodic. You see, if e^0 = 1 then e^i*tau = 1 too because the exponential is periodic with period i*tau (the same as saying one rotation by 360 degrees does nothing), so: e^(k+tau*i) = e^k for every number k

[@Daniel_Lopes_13]

but you could say i*tau is congruent to 0 mod(i*tau)

[@FlyingSavannahs]

@Linus Yes, it is. x^0 = 1 for all x's that are real numbers. The reals are rational, irrational and transcendental numbers (think any point you can choose on a continuous numberline). π is a transcendental number so it's included. e too (también!). So, yep, i*tau = 0. Another way to verify is with e^iπ = -1 or e^i*tau/2 = -1 Square both sides and you get e^i*2π = (-1)^2 = 1 or e^i*(tau/2)*2 = e^i*tau = 1. Thus e^i*tau = 1 = e^0 Now take the natural log of both sides In(e^i*tau) = ln(e^0) i*tau * ln(e) = 0 * ln(e) and ln(e) = 1 so i*tau = 0 Booyah! @Daniel Lopes actually states the generalized formula for any value k an integer perfectly but denies his own equation for the valid k = 0. cred: I have an MS in Physics '85 and a Math minor, 30 years of scientific work experience, am left-handed, and have spoken with Doug Hofstadter personally about "Göedel, Escher, Bach, an Eternal Golden Braid."

[@dtstar331]

I just love how youtube sent me this video to recommended just to see that this 2 year anniversary happened 2 years ago

[@eswing2153]

The transformations you speak of are some of the most interesting things electrical engineers learn in university. You make this more accessible than how it was presented there. Generally these concepts come about in third year. The transformations allow engineers to get a certain intuition about how AC interactions work. Thanks for making such high quality visualizations and coherent explanations.

[@seanziesman602]

Ya I just realized rectangular form is addictive group of complex numbers and polar form is multiplicative group of complex numbers. It makes sense how you can represent them in both ways.

[@frtzkng]

*3b1b:* turns ads off for better video experience *KZbin:* alright lets turn ads back on and keep all the profits

[@twoli.dev.]

10:03 ill never forget when you called numbers schizophrenic

[@mekaindo]

Did you forgot?

[@renatofernandes1086]

sadly I can't like your video more than one time.

[@brikken1]

I agree, sometimes you just miss having the opportunity to show that extra gratitude. However, you could show monetary appreciation at patreon.

[@Talaxianer]

Just make sure that the total number of clicks on the like button is uneven

[@ir2001]

Lol

[@asterixgallier8102]

I desperately need the love button for this video!

[@msDanielp369]

share all of his videos on all your social networks and to your friends on messages

[@giacomolanza1726]

Thanks for this and other amazing videos. I really like to learn to do such animations. I found a small mistake at minute 13: the angle corresponding to 2+i is not 30° but rather arctan 1/2 = arcsin 1/√5 = arccos 2/√5 = 26°33'54"

[@vector8310]

Complex analysis is just glorious. Eternal gratitude to you for this and your entire KZbin corpus.

[@shmackydoo]

The group theory way of looking at things is almost as earth shattering to my mind as looking at the world and history through the lens of dialectical materialism. It's so important to understanding these things better.

[@frother]

18:51 "so wouldn't you agree that [completely unjustified leap in an argument that relies 100% on intuition]?"

[@paradox9551]

Completely agreed. It's so stupid how we just jump to that conclusion and then make more assumptions based on that with no explanations given.

[@TheBookDoctor]

Excellent. My only quibble is that in discussing multiplication on the number line, you only referred to the "multiplicative group of POSITIVE numbers." And while it's pretty obvious what multiplication by negative numbers should do to the line, you didn't spell it out. And yet, when talking about i*i = -1, you referred to -1 as the "unique action" without having earlier made clear that negative numbers are also a valid part of the multiplicative group.

[@zairaner1489]

I actually found that quite genius of him. The point was that positive real numbers are pure stretchings while negative numbers also include rotations, which are a complex thing (and notice that the real exponential function only puts out positve numbers, so the real exponential function only gives you a relation between the additive real numbers and the multiplicative positve numbers)

[@zairaner1489]

And he did spell it out when he explained the complex multiplication and how it relates to rotation ;)

[@gerhetTD]

uhm.... it's genius of him because frankly multiplication by real negative number is just stretching and flipping through 0, it's not really a complex thing. In 1D, there is no need for rotation to involve. It happens that in complex, "flipping" through y-axis is 180 degree rotation.

[@zairaner1489]

Exactly. If he had talked about multiplying by negative numbers in 1D, it could have looked like flipping. But that would have created a misconception when extending to the complex plane because multiplying by negative 1 is not flipping in the complex plane but just rotation.

[@columbus8myhw]

It's 'cause only the positive reals are in the range of the exponential map e^x (where x is real), but all complex numbers other than zero are in the range of the exponential map e^x (where x is complex).

[@RickyMud]

I genuinely started laughing at the animation for ixi =-1 and e^pii =-1 such a cool way of showing it

[@josejimenez850]

Phenomenal. I've been subscribed to your channel for two years now and every time I watch a new video of yours I am absolutely astonished at your work. Math made beautiful. Just wow. You are a gold mine of youtube

[@tylerm442]

I love your channel. I usually end up watching your videos right when they come out and I tend to just go with the flow bc I hv no idea what's going on lol. But then I watch them a few more times, maybe not to completion, or with huge gaps of time in-between viewings, and I suddenly feel like switch go off in my head. And then once that happens and everything falls into place, it just feels beautifully awesome and I can extend my gratitude enough for making such abstract concepts relatively easy to digest and really comprehend. Because at the end of the day, I could read a textbook, but this gives so much more depth which is extremely helpful especially when starting out.

[@Patsoawsm]

"They are one example in a much larger category of groups" Literally, haha

[@zairaner1489]

Is that a category theory joke? xd

[@umbraemilitos]

Patrick hehehehe The Joy of Cats.

[@HilbertXVI]

@@nasajetpropulsionlaborator8727 Lmao, projecting much?

[@ZardoDhieldor]

0:48 "so here, two years later" *whince* This much time already? D: After checking, I'm relieved, that for me at least, it's only a little bit more than one year. But still, time passes! :/

[@3blue1brown]

You're telling me. I just kept thinking "2 years? That can't be right, surely that's not right".

[@sofia.eris.bauhaus]

for me it was more like: _only_ 2 years? this channel feels like such a classic and is has so many great things in it already.. :D

[@KevinHallSurfing]

Odd how your vocalization has slowed slightly over two years. Is the aging process exponential or linear? Will your voice be twice as slow in another 2 years or slowed to a barely audible rumble? ;-)) Wish we had vids this back in the Sixties. Nice!

[@cbhorxo]

10:51 Here, you say you're squishing by a factor 0.5, but you're actually *stretching* by a factor of 0.5, i.e., squishing by a factor of 2

[@markhughes7927]

I am not a mathematician only an appreciator of clear and logical thought - I found that exposition absolutely breathtaking for clarity and logic. Arithmetic actively (verbally) conceived: slide, stretch and rotate and refer to the constant pi. The next stage I guess is to use descriptive language which extends this to plottings within the volumetric spaces of polyhedra? My thanks to you for unblocking something in mind.

[@joetursi9573]

You should market these videos, especially this one, to all colleges offering group theory. Just wonderful. Thanks so much!

[@torinstorkey]

This is Euler's identity. Euler's formula is e^θi = cos(θ)+i sin(θ)

[@luisdias5246]

You lost me when you brought up exponentials in, I can't see the relation between the top and bottom at 17:21

[@pyrodynamic4144]

Three years later, but here's how I understood it. I don't think there's necessarily a direct relation between the operations themselves, meaning a way to deduce the operation that must be performed below by seeing the operation performed above, but what happens is: On the top, you slide by whatever the power is. So if the power is 3, you slide by 3. If it's -1, you slide by -1. On the bottom, you squish by 2 to the respective power. So if it's -1, you stretch by 2^(-1)=0.5. If it's 3, you stretch by 2^3=8. This is because exponentiation is repeated multiplication, so if you raise 2 to the third power, it's the same as stretching by 2, then again by 2, then by 2 once more.

[@cliffordwilliam3714]

Isn't it trying to say that both additive and multiplicative actions have similar result for the same expression? Sliding 1 unit left and then 2 unit right lands on the number 1. Squish by 0.5 and then stretch by 4 lands on number 1 also right?

[@youtubecensorpolice9112]

@@cliffordwilliam3714 If you start the arrow on 0, squishing and stretching by any amount shouldn't move the arrow away from 0. I don't understand at all what he's trying to get at in that part of the video.

[@6s6]

I can't wait until you hit 1 million subscribers.

[@eccentricOrange]

Well you should stop waiting now

[@blackcat5771]

@@eccentricOrange lmao

[@willh9104]

Fantastic video. I dealt a lot with group theory in inorganic and quantum chemistry - this was a much better refresher than going through my old notes!

[@bayleev7494]

It's difficult to appreciate just how amazing this video is until you've actually done a course on group theory. When I watched this a couple of years ago, I got the main gist of the idea and I was impressed. However, even though the idea in itself is beautiful, I appreciated how succinctly you summarized the actual rules of group theory so much more. Also, in my education, I didn't really look at group theory as symmetric in nature, so when you showed the different numbers as actions that preserve symmetry it blew my mind. Great video, probably one of the best on this channel (and that's saying something!) :)

[@rusgon]

A little bit confused 19:43 why 2^i gives rotation 0.693 20:01 why 5^i gives rotation 1.609

[@marche45]

Ruslan Goncharov When you map the exponentials in the complex plane it corresponds to rotations around the unit circle. So 2^x can be written as e^xln(2). Taking the rate of change (i.e derivative) of the rotation, you get the original function times ln(2). So ln(2)2^x. Where ln(2) is approx 0.693. Same with say ln(5), as 1.609.

[@ganondorfchampin]

Because calculus.

[@PancakeDoesGaming]

+Marcel.M But, why take the derivative? Shouldn't it be that if I were to have, say, x = 1, mapping it by 2^x would just directly rotate it by 2 radians...? On which part did I misunderstand?

[@ganondorfchampin]

Doesn't work like that, at all. You need to take the derivative in order to rotate correctly to get the precise map with complex numbers.

[@PancakeDoesGaming]

+ganondorfchampin Ah... but, why? (I know barely anything about this, thank you for your patience)

[@sameerdeshmukh5050]

Thanks for the video. The analogy is really helpful. I have just one question: In 20:00, you explain that the additive action maps to the multiplicative action, and that the action of vertical sliding maps to rotation as a result. How do you come up with the numbers for the rotation in radians? What is a more intuitive way to understand this scaling apart from using Euler's formula? (which is what we are trying to prove so it does not make sense to use the formula before proving it).

[@davidkoenig8822]

I have a similar question. Would love to see if anyone responds to this.

[@guilhermebotelho1615]

Me 3! At 19:40, he said "it happens to be 0.693 radians", which is the moment that made me really want to now *why*! The explanation at 21:06 kinda touches on it, because it hints that the derivative of different exponentials would mean a different rate of change around the unit circle, meaning that Euler's formula only works for e. That said, he still didn't elaborate on it enough I think. Edit: If you do the derivative with 2^x instead of e^x, you get i(ln(2)2^(i*t)) which might be the explanation we were looking for! Because the factor in the derivative is converted into a factor in the velocity around the circle. This explains the angle we get if we use 2^x instead of e^x, as well as e^x is special.

[@pocikprostoi7172]

+

[@Maurycy5]

I am still confused. All is good until 19:00, where we jump to exponentiating by complex numbers. What was covered: adding as shifting and multiplying as stretching. Also, exponentiating by WHOLE numbers as translating a shift to a multiplication. But we still do not know how to exponentiate by all real numbers, not to mention complex numbers. Did I miss something?

[@bernardkho]

agreed, got stuck at the same point

[@MithileshHinge]

Me too, got stuck there. Why does exponentiation by complex number correspond to rotation of the plane? It can be verified mathematically, but I'm not getting the intuition behind it.

[@absurdengineering]

You can think of exponentiation by reals as a way of turning a discrete function into a continuous one - it still has to go through the same discrete points. That can be one starting point towards understanding. It turns out that when you do a series expansion of a real exponential then it usually works just fine in the complex domain, and you can then look up the sine and cosine in the same series expansion and you’ll see the Euler’s formula that way as well. Again: that’s just one of many approaches. But going from discrete to continuous exponentials requires a better definition of an exponential than the elementary one - whether you look at a series expansion or something involving logarithms doesn’t quite matter.

[@tonylopez5937]

He didn't actually explain it, he just asserted it. At 18:49 he said "So wouldn't you agree that it would be *reasonable* for this new dimension of additive actions, slides up and down, to map directly to this new dimension of multiplicative actions, pure rotations?" I completely understand how multiplying by 2^(real number) will stretch the space, but he never said why multiplying by 2^i would rotate it, he just made a comparison to how multiplying by i made the space rotate. I agree with @Mithilesh Hinge, I can see it algebraically, using Euler's formula ( e^ix = cos( x ) + i sin( x ) ), but I don't get the same intuitive feel as Grant/3B1B provided for how multiplying by a complex number can rotate a space in the complex plane. Can someone give an intuitive, geometric reason as to why 2^i causes a rotation? Maybe there isn't one, which is acceptable, but I was hoping for something as elegant as his earlier explanation of rotations due to multiplying by complex numbers. And, as some others have mentioned, even if 2^i = 0.769 + 0.639 i, that looks like a stretch and a rotation, which is not the "pure rotation" he mentioned. Otherwise, though, this video is wonderful. 😊

[@Ishwar_Das]

@@tonylopez5937 if you multiply the complex number z= r*exp(it) by the complex number exp(ix), you get r*exp(i(t+x)). You are just rotating z by x radians because the angles add. The author of the video is not explaining what is really happening here. He is assuming Euler’s formula and the polar form of a complex number. He is just saying that the image of pi*i under the exponential map (homomorphism) is -1 because Euler’s formula is true. His graphics are nice, but he is not using group theory to explain why Euler’s formula is true. So the title of this video is a bit misleading.

[@stochasticxalid9853]

Young man, you are one of the most creative mathematicians I have ever seen. Chapeau for your work. I am definitely a fan of yours.

[@ofelia829]

Notes in music work like this. There’s a relationship amongst the degrees of a key and all the possibilities of neighbours. No matter if counterpoint or harmony,they do function symmetrically. Even inversions in micro . You can actually see the actual distance depending on the inversion. Thank you for uploading.

[@MrCTruck]

13:14 it clicked so profoundly I actually uttered "oh shit"

[@mina86]

Coincidentally, I just got my Tau t-shirt in mail…

[@ZardoDhieldor]

Tau FTW! :D

[@dev02ify]

Tau makes everything better except for euler's formula. e^(Ti/2)=-1 D: I guess you could write e^Ti = 1

[@SirBastian200]

e^(Ti) = 1 :D

[@SuperBonobob]

Tau makes Euler's formula much better: e^(i*τ) = 1

[@LoLrand0mness]

i lately pondered about this e^Ti = 1 i've put this formula into the "triangle of power" and that kinda confuses me... example of my inline notation: (2^3\8) -> 2^3=8, 3\8=2, 2_8=3 now to my confusion: (e^Ti\1) -> this would conclude: e_1=Ti, but also there is: e_1=0 -> Ti=0 ?!?! i surely must have made some mistake... e^i != e^|R (?) -> a different action, that i can't simply "equal" (?)

[@badlydrawnturtle8484]

So watching this again two years later, it makes a bit more sense... except for one thing. I actually came back to this video because I've been searching for an explanation for this particular part that I can't make sense of: (Starting at timestamp 18:32) "We already know that when you plug in a real number to 2^x, you get out a real number, a positive real number, in fact. So this exponential function takes any purely horizontal slide and turns it into some pure stretching or squishing action. So wouldn't you agree that it would be reasonable for this new dimension of additive actions, slides up and down, to map directly to this new direction of multiplicative actions, pure rotations?" No, I can't say that's reasonable to me. The hangup is with the "pure" qualification of the rotation. A pure imaginary number as an additive action is a pure vertical slide... but a pure imaginary number as a multiplicative action is a slide AND a stretch, for every point on the imaginary axis except for i itself. So I don't see how it follows that this stretch should be dropped when discussing a transformation from additive to multiplicative actions in regards to i.

[@solewalk]

when you multiply something by a non-zero imaginary number, it should cause rotation.

[@jadezhang4102]

I think the situation here is slightly different from the one around 12:21. In that case, "taking the number 1 and dragging it" to something like 2i would involve both a rotation and a stretch, and it is true that in this case only i would be a pure rotation. He uses that to demonstrate the multiplicative group of complex numbers. However, the two screens at 18:32 are showing the connection between the two additive and multiplicative groups in exponentials, where the property relates them with the inputs being additive and the outputs being multiplicative. I think the point he is making there is that there is a relationship between the respective components too (see 8:49 for addition and 13:29 for multiplication). Addition has two, purely horizontal and purely vertical, and multiplication has two, purely stretching and purely rotating. His argument is that since purely horizontal (addition) --> purely stretching (multiplication) then also purely vertical (addition) --> purely rotating (multiplication). In this way, he's talking about the relationships between the respective components of each group, not the definition of the multiplicative group itself. Disclaimer: I don't really know group theory, this is just how that part of the video made sense to me. Hope it helped, would love to discuss it with you more!

[@solewalk]

You should focus on the mapping of the purely imaginary numbers, through e to the x, to the points in the unit circle centered around the origin. So, a pure imaginary number has a corresponding point on the unit circle, and the point on the unit circle represent the multiplicative action. Since the corresponding point is on the unit circle, it doesn't change magnitude of whatever is multiplied by it. Look at the formula: e ^ ( i * x) = sin(x) + i * cos(x) where ^ means power. This is the unit circle centered at the origin.

[@felixacool]

@@solewalk being multiplied by i is different from multiplied by 2^I in the current context where 2^I hasn't been articulated (geometrically)

[@PhysicsHigh]

Where have you been all my youtube life. Subbed as I refresh my mathematical skills

[@christinosim]

You have taught me in a way no other person has ever before. The best way.

[@jacobhelbig6967]

Please do another video about the Riemann Hypothesis and Ramanujan Sums!

[@brucefrizzell4221]

Thank you for the many subtitles .

[@amdreallyfast]

Your thoughts sound not far off from transformations in graphical math, such as moving and rotating a point in 3D space. Have you examined quaternions? They follow a similar line of thinking to this 2D complex plane, except that they're 4D numbers: a real number line and 3 imaginary number lines, all orthogonal to each other. I wrote an extensive essay for myself on them a couple years ago and their big brother, the dual quaternion (the dual operator was interesting).

[@jasondoe2596]

John Cox interesting!

[@zairaner1489]

The Quaternion group (the multiplicative group consisting of the quternions +-1,+-i,+-j,+-k) is one of the smallest non-commutatative groups.

[@brandonbocklund]

John Cox they don't sound far off because they are not. Crystallography is the direct application of group theory in the graphical math sense. It can be used to describe the symmetries of the atomic positions in crystalline materials.

[@riccardoorlando2262]

Smallest in what sense? There are finite non-commutative groups...

[@Agrantar]

Thanks for this, man! I was wondering if there were higher-dimensional analogues to complex numbers - now I have a name to dig deeper.

[@jacklee5895]

After reading a basic algebra book at least one month, I was blocked by the content of the group theory and could only drink ... , but this video made me start enjoying the world of group theory. Thanks for the excellent presentation!

[@stefanoctaviansterea1266]

We need a series for Essence of Group Theory. Students in high school are taught this subject in a purely numerical and definitional way, nothing to give them intuition or any representation of what they learn about at all. Everything is just symbolic algebra thing without sense, created to torture students. We need light like this in a dark educational system.

[@MasterHigure]

10:03 Schizophrenia is not split personality. It's not even related. It's more of a hallucinatory disease, mixed with psychosis. (A person with schizophrenia imagine things, and cannot tell what's real and what's not.) Conversations with people who aren't there, or general paranoia are some manifestaions.

[@guineapig55555]

schizophrenic: "a state characterized by the coexistence of contradictory or incompatible elements."

[@riccardoorlando2262]

Well, confusing what is real and what is not and being stuck in a rather complex state of mind seems perfectly fitting for this video. Now, let's ask that group of people who aren't there whether they are interested in switching places, and whether they care about keeping the same neighbors.

[@rogeroney3]

Does this mean the vertical axis of the additive group undergoes modulus arithmetic when its transformed? Like with e as the base, a vertical transformation of 2pi maps to a 360 degree turn, or the same as doing nothing? Do you "lose" information then when this is done?

[@zairaner1489]

Wow thats a pretty good catch. Indeed if you look back at the group of rotations of the square, this group is the same as the integers modulo 4, and the circle group (group of all rotations) is just the Real numbers modulo 2*Pi. The loss of information you describe comes from the fact that exp(2*Pi*i*k)=1 for all integer k, so multiple points are mapped to 1 (adn thus multiple points are mapped to any point). In general, "loss of information" like this are so important they are described through the fundamental theorem on homomorphisms. the most important theorem of grouptheory

[@LesSpins]

I don't get why symmetry is a condition for defining a group, after all, for a purely asymmetric object (especially for such an object actually) you can still say that : some transformation plus another transformation is equivalent to doing a third transformation only. Why bother with the symmetry ?

[@zairaner1489]

In order to get a group this way, you need this functions to be invertible. And invertible functions are in a way symmetries on the set you are looking at.

[@LesSpins]

But an asymmetric object can be rotated 90° to the right and then 90° to the left and still be asymmetric. The symmetry argument apply to member of the set and not the transformations on them, right ?

[@LesSpins]

Great explanation, quite intuitive actually.

[@paulfoss5385]

@@LesSpinsSo in the case of the asymmetric object that you're rotating, there is still an underlying symmetry in your rotations, namely the symmetries of space itself.

[@jb_kc__]

Finally the intuition of Euler's formula clicked... much love man you are a credit to society

[@sergiocardenasreyes7418]

You just simplified so much... as an Engineer during finals, I salute you.