The most beautiful equation in math, explained visually [Euler’s Formula]

[@pendragon7600]

You pretty much determined the course of my life. You released the imaginary numbers are real series when I was 14 and uninterested in math. That series kicked off my passion for mathematics and I'm now pursuing a PhD in complex analysis and algebraic geometry. My thesis focuses on modular forms. Seeing this video was like a flashback to where it all started. Eternally grateful to you and for the wonderful exposition you produce. I'll buy the book someday (PhD students have no money lol).

[@Orillians]

And now im the 14 year old watching his videos 🫠

[@WelchLabsVideo]

That's absolutely incredible - thank you so much for sharing & best of luck in your PhD!!

[@berlinisvictorious]

hey share arxiv for your thesis when youre done

[@EyalBarCochva]

Send me your address, I'll order it for you (Not joking)

[@ilevitatecs2]

@@WelchLabsVideo you should send him the book.......

[@Yotam1703]

Your complex numbers series was my introduction to higher math education! It’s a pleasure to see you continue it.

[@05degrees]

Yeah those videos are so neat! I haven’t watched all of them but it’s still evident.

[@Yotam1703]

@@05degrees oh you ought to! The entire series is super well executed and the fact that it follows the historical route (rather than a more straight forward, pedagogically informed route) gives it a really satisfying conclusion.

[@HuxleysShaggyDog]

lol forgot about it since it’s been so long

[@Waterwolf221]

same here. the 'imaginary numbers are real' series with the graphs got me obsessed with complex numbers like nothing else.

[@BooLightning]

Euler was like, nahh bro you’re both wrong

[@EconAtheist]

carnot: no euler: stfu and gbt thermodynamics

[@05degrees]

What I like about many mathematicians of the past, they were right about a lot and also wrong about a lot and sometimes had even plain petty debates (well, not those shown in this video). And also that there was place for people like Euler or Gauss who just plainly did magic trick after magic trick, having worked up incredible intuition through some unseen ton of work and acute observations. And it continues to this day but still it’s very understandable to look back and say just wow. Because all our newer inventions and clearer understanding indeed doesn’t diminish what was done. In times of realizations like these I feel grateful to be living on this planet. I hope we will be even more brilliant as time passes. And conquer our bad habits to a better degree, for there to be way more place for all that brilliance to emerge and fit.

[@CliffSedge-nu5fv]

Euler, the master of us all.

[@vinniepeterss]

😂😂

[@piepiedog1]

@@CliffSedge-nu5fv For those unfamiliar, this is an abridged quote from Pierre Laplace (himself one of the greatest mathematicians ever): "Read Euler, read Euler. He is the master of us all." Indeed, Euler is perhaps the greatest, most prolific, most intelligent mathematician of all time. He was the author of around a quarter of all works of novel mathematics, physics, and astronomy in the 1700s, as he cranked out extremely important work at an astounding pace. He contributed to, or outright pioneered, a wide range of areas of math and physics (including graph theory, algebra, real and complex analysis, among many, many others), demonstrating not only outstanding depth of knowledge, but also a wide breadth. It is hard to believe that such a person could actually even exist, but he clearly did. I wish he had the celebrity among lay-people like someone like Einstein has. Even when people have heard of him, he often gets the disrespect of being called "Leon-hard Yoo-ler." So yeah, I'm with Laplace. Read Euler.

[@manuelcastellanos122]

I'm from Dominican Republic and let's say math isn't much of a thing here. I initially watched your complex numbers series graduating from high school, and I knew a spark started there. Now I'm about to present my thesis for my MSc in Applied Mathematics and it is so rewarding to look back where it all started. Looking forward to get your book!

[@johnraffensperger]

Congratulations!

[@sagarbehera]

This is exactly why social media was here for. Congratulations.

[@davidjadunath1262]

A fitting testament to why knowledge should not be entirely proprietary, but made universally available, instead.

[@msergio0293]

Que historia tan genial ❤

[@dhhsscience4all]

felicidades

[@Kazner0h]

That final visualization is really stunning. It always amazes me how much a graphical representation can help make a complex subject feel intuitive.

[@abpdev]

I envy younger people who are just getting into math/science because they are starting of surrounded by such amazing free content to explain difficult concepts! I had a hard time understanding this equation in my first and second year. It just seem so odd and counterintuitive. It became one of those truths that I accepted because the textbook said so. The manner in which you presented it here just makes so much sense! The intro, the papers/conversations from the authors, notes, diagram... WELL DONE AND THANK YOU!

[@armands672]

I found the young generation has pretty much everything in their fingertip ... Google is way more powerful than yahoo of my time ... Then you have ChatGPT which can summarise / contextualise pretty much everything in Google or internet. But here is what I observe ... They routinely go to TikTok to search for their answer and there is no drive to dig whatever is out there supported by Google or GPT. I think it's both blessing and curse of young generations... You have all information at finger tip... Why bother studying it or even dig further the knowledge that is there.

[@sloppycee]

Euler's motivating example and insight that the i's should cancel; so log(-1) should have an I somewhere is such important knowledge. I hated how they taught math with no context or motivation. This is great!

[@benisrood]

We too. Math without context is a psychic crime against the student and against the beauty of the math itself.

[@JMAssainatorz]

@@benisrood It indeed takes the story out of it!

[@skfineshriber]

After studying engineering mathematics I gained an enormous respect for mathematicians such as Euler. Here I was, struggling like hell to understand their equations and transformations, while realizing these people had the insight and intellect to INVENT these mathematics. It was awe inspiring.

[@genekisayan6564]

I honestly don't know if they evented, I d rather say they discovered them given the contraints they have been dealing with for several years

[@Ken-no5ip]

You sure aint inventing anything

[@Mytubecleaner]

15:00 wow, that was an enlightement moment for me

[@k.nitishjoshi1875]

damn same here for me

[@fyu1945]

9:33 You define a conjugate as "flipping the sign of the imaginary component" but what you're doing here is not flipping the sign of the imaginary component, but flipping the sign of the exponent, which happens to be imaginary. Of course in reality it amounts to the same thing, but that's what we're trying to prove! We don't know yet if 2^(-bi) is the conjugate, so we can't use this fact to prove the magnitude of our number is 1.

[@benisrood]

Very true.

[@T3sl4]

Hmm, I don't know the history of the complex conjugate. It does arise pretty simply out of the geometry of complex numbers, and also of algebraic numbers (namely, radicals), or similarly just the basic factorization (a + b)(a - b) = a^2 - b^2, so should be pretty familiar or apparent to anyone working in the depths of mathematics at the time. The conjugate is then just whatever special operation has that geometric effect, which happens to be equivalent to flipping the imaginary sign. Perhaps that's enough justification to skip over it? Perhaps there's a deeper / more interesting history there, or indeed something to better define and then prove?

[@05degrees]

@@T3sl4 Yeah I also think conjugate feels pretty natural though _initially_ it can feel a bit puzzling, like what’s special about flipping the sign? What’s about it, why so arbitrary? But after seeing more examples like conjugate in quaternions, in split-complex numbers, in Clifford algebras in general and many more you start to feel what maybe wasn’t communicated too well: “a thing to make a real square”, “a thing to invert the transverse stuff in the number” and things like that, summarized by its property (A A*) is real and that it’s an (anti)homomorphism, that is (A + B)* = A* + B* and (A B)* = B* A* (anti and the order switching is for noncommutative cases and doesn’t matter if A and B commute). And then having (A A*) ∈ ℝ, we can define inverses by A⁻¹ = A* / (A A*), so far as A A* ≠ 0 (hello dual numbers and like). So we eventually also get an intuition that conjugate is sorta already a weak form of inverse (and happily it exists even when an inverse doesn’t). Then we can look at the identities above and say: look, if A is real then A* is also real: conjugation is an (anti)homomorphism that fixes reals! And it usually turns out it’s the only one such except the plain identity function. Which is a way to come at it too. And… yeah in the end a video could be useful.

[@T3sl4]

@@05degrees Indeed, the pattern runs deep; though (AFAIK) most of those applications came along much later, and it took Galois theory and other modern algebra topics to formalize and generalize it. There was probably some intuitive sense about it; perhaps that's part of why Euler was so confident, he had a subconscious and incomplete understanding of these relationships. Imagine how much math Euler might've uncovered if Galois had come first! (But historical hypotheticals being what they are, without the same background, Galois might've never thought about groups that way, or not before getting fatally embroiled in 18th-century politics, lol.)

[@tommyrjensen]

@@T3sl4 You are right to say that it looks like cheating. The fact that it works is because of the basic definition of i as solution of x² = -1. There are two solutions x. So we define that one of them will be named i, and the other is then -i. But there is no rule that explains which one of the solutions is actually i and which one is -i, since that is only up to our arbitrary choice. It means in particular that whenever you prove something about i that only depends on the fact i² = -1, then the same fact must hold when we replace i by -i.

[@ericdculver]

If I had the time to truly teach Euler's formula instead of just mention it in passing, this is exactly how I would want to teach it. Great job!

[@leonhardhoffmann4542]

Addition 2: I really want to give my biggest heartfelt respect and sympathy to you, you do a great job and seriously improved and still improve my life and that of others with your incredibly good videos. I am teaching extra lessons for math for 4 years now (on the side, on high school level) and know first hand how difficult it can be to explain and illustrate mathematical problems and formulas and do so whitout boring or overwhelming the learner, you do an extraordinarily well job here! Thanks!

[@rishabhjohri4223]

I was 15 years old when I was introduced to this channel, which fuelled my interest in mathematics as well as artificial intelligence ( the "learning to see" Playlist on decision trees), now I'm 23 years old and pursuing Master's in artificial intelligence. Thanks for this amazing and free content!

[@Alexbaker698]

bro, is math is very difficult in BSAI?

[@vma011]

I discovered your videos on complex numbers 8 years ago, right when I was starting electrical engineering in college. I work with complex numbers and phasors basically daily, all of our electronics, automation mechanisms and electrical grid, work thanks to the properties of these unfairly-named "imaginary" numbers. Sometimes it's easy to get lost on why does this all magic work. So glad to be able to come back to your channel and find this ever growing content in the topic, now materialized in your own book. Congratulations for achieving that huge milestone, I will be definitely looking into getting a copy of my own!

[@kiko7247]

I just love these videos. The topic, animation, it's entertaining and educational at the same time. I love complex numbers and the history of math. Thank you. Your efforts are greatly appreciated

[@iveharzing]

This video is a beautiful explanation of Euler's Formula that I haven't seen before anywhere else. I subscribed to you after watching your first imaginary numbers video, and I keep being impressed by what you upload!

[@05degrees]

I’ve seen several nice videos about this, but each one is unique and there can’t be _too_ many. There should be even more. I’m thankful to each of their authors. 🧡 And it’s nice to know history better as well.

[@WelchLabsVideo]

Amazing, thanks so much!

[@JohnMartinIT]

The statement that "many of the equations and graphs we spend our time with are just shadows of a more elegant, powerful and higher-dimensional mathematics" reminds me of Plato's analogy of the cave, which I'm sure you appreciate has some interesting implications. Maybe that's why you say imaginary numbers are real; indeed, they may be more real than the more tangible shadows we deal with in this mundane world.

[@anastasijajamrik2978]

This is one of the best explanation of raising numbers to the power of a complex number and how e come up in this concept. Just so easy to understand!!! I had to rewatch this again because its so nicely explained! Thank you so much, this really helped me understand this weird concept!!!

[@thulioassis1023]

This reminded me of my teachers in high school. All brilliant teachers who explained the reasoning behind the formulas. Math can drive me crazy because I can easily become obsessed about a problem or a solution. Thank you for the video. It really sparked something inside of me.

[@punditgi]

Beautiful exposition. Can't wait to get the book! 🎉😊

[@akshatrai9007]

This channel is criminally underrated

[@michaelblankenau6598]

What kind of criminal penalties should be imposed ?

[@faisalahmed8098]

@@michaelblankenau6598 straight to gulak !

[@bender5835]

dude, I binged through your complex number series few years ago and still to this day I share that playlist with people. So looking forward to the book. Amazing visualizations btw. what a time to be be alive man. Ah, I am getting chills from excitement. Ty! Ty!

[@vari1535]

14:42 holy crap, i FINALLY understand the actual MEANING of this definition of e. thank you so much!!! (the video as a whole (and the poster summary, holy crap!!) was also AMAZING.)

[@jaewok5G]

i think this is the first video of yours i've ever seen. i understood every bit of this but only ever as separate bits. bridging the gaps of what was once left as "details that we won't go over in this class" gave me the small thrills of recognizing, "hey, that's practically e" and "hey, that's really close to a radian." fun stuff.

[@caio.tavares11]

Man that book looks amazing! Please consider adding international shipping :)

[@deancardno9246]

Oh no - I was just about to order it, and then saw your comment and realized it would only go to the US!

[@galileo3431]

Gosh, your flow of delivering information is so smooth! Subbed!

[@luisclovis09]

Thanks Euler for discovering the mathematical base for eletrical engineering. And thank you for making me able to visualize it 😄

[@GodzillaGoesGaga]

For engineering, physics, biology, chemistry and mathematics! How to make a system of solving periodic functions easy. Euler!

[@kushagra64]

This was the first time I felt e^i π = -1 is beautiful result...

[@jaewok5G]

i like it as -e^πi = 1 so that all of the mess is one side and absolutely simplest '1' is how it cleans up. just a personal preference but try it out, see how you like it.

[@ivocanevo]

Other aesthetics are also valid. I like the three unlikely heroes in the photo together, seen in the mirror with -1 in the reflection.

[@Meta7]

The "beautiful result" as it is typically written is e^(i*π) + 1 = 0. It has all the basic operations: Addition, multiplication, and exponentiation. It also has all the "basic" numbers: 0, 1, π, e, and i.

[@chaos.corner]

@@Meta7 I'm a Pi guy myself but the Tau guys might say that would also give you division and 2 also.

[@wilh3lmmusic]

Except it's a much uglier version. Compare it to e^(iτ)=1

[@aglargalad]

Never appreciated that Euler's equation visualizes 2D exponents, sin, and cosine functions in 3D! This was an awesome video! Thank you!

[@gewinnste]

Not quite: It's the e-function, expanded to complex arguments that does all this. Euler's equation just represents the slice of that graph at x=1.

[@C5916C]

This is the first time I have been shown content from your channel and this was an instant subscribe. Hope to pick up the book soon!

[@sb-sm8ib]

This is wonderful, I loved it ❤. I always had this question about what it means to raise number to imaginary power. Besides i was hoping if you included interpretation of irrational powers.

[@05degrees]

Basically a limit as well, when it exists (if we don’t go to complex numbers yet, we’d have a hard time doing anything to negative bases; but with positive bases it will always work). This limit gets taken over rational exponents that approach a target irrational exponent X. So essentially the same as Briggs have done. Then it can be simplified to taking a limit over a sequence of rationals that approaches X, not all rationals. So when we need to know, say, A^ϕ where ϕ is the golden ratio, we can take a limit of A^(1/1), A^(2/1), A^(3/2), A^(5/3), A^(8/5), A^(13/8) and so on using successive Fibonacci pairs in the fractions (and there would be infinitely many other sequences that allow us to do the same).

[@sb-sm8ib]

@05degrees I, too, thought of such interpretation. I wanted to know if there are other ways to understand it.

[@05degrees]

@@sb-sm8ib I’d expect that as there is essentially just a single construction of reals from rationals (metric completion of sorts) then there can be (essentially) just a single extension of something defined on just rationals to something defined on all reals. There can still exist a completely unexpected thing, probably. What might be interesting to check out is that we can construct reals from just integers (called “Eudoxus reals” though Eudoxus didn’t invent them himself). Now is there a way to define powers with real exponents starting with just integer exponents (without defining rational exponents in the middle!), guided by this construction?.. IDK I’m lazy to think about that.

[@TazariaGaming]

The way that fully expanded form of the formula comes together as a combination of trig and exp functions is just a beautiful conclusion to this story. Your videos on imaginary numbers was what sparked my interest in them way back then. Thank you for making another video on the topic!

[@itscoolto1]

Nobody tell Terence Howard about this

[@QP9237]

Dude would be conjugating all over the place 😂

[@lailandadumbmathematician7747]

He’d have a stroke 😂

[@TECHN01200]

Your complex numbers series was how I came to learn and understand complex numbers. It is one of only a few, if not, the only one that really exists. It was phenomenally done. Unfortunately, I didn't receive any education on them in school, so I must be greatful to you for teaching me the math I would've never been able to find elsewhere put so well and intuitively. The image of you pulling a parabola out of the page was such a great visual.

[@blueslime5855]

At 9:10, the angle in blue should be arctan(2)≈63.4° You may have accidentally calculated arctan(1/2)

[@rmxevbio5889]

THANKS!!!!!

[@nsfa19]

This was the most well-explained video about Euler's equation I've ever seen and I've watched quite a few about this topic because I love it! Well done and good job! Keep up the good work and thank you very much for this video! 😃👏👏👏

[@supermuffinbros4797]

9:33 It's not clear why this makes the complex conjugate at this point in the explanation...

[@stefanbalauca7481]

The complex conjugate here is not to be considered just as a-bi is the conjugate of a+bi, but it actually has a much deeper meaning which stems from the symmetry of i. The equation x²=-1 has 2 solutions, out of which we arbitrarly choose to name one i and the other -i. However, this random choice means all maths has to be symmetric with respect to it so in a broader sense the complex conjugate of anything (number, equation, function) is just the "what could this have been if we chose -i instead". By this rule, if 2^(bi) = x+yi then the only option for 2^(b*-i) is to be equal to x-yi.

[@gdibble]

ᛣ *Well done and [well] said.* The elegance of this video and your style earned my Subscription. Just want to emphasize my appreciation for the time you took to produce and release this video. _Keep up the excellent work in maths and inspiring learners everywhere!_ 👏

[@mrolemartinruud]

Any plans on providing international shipping (Norway)? This series was a huge part of my understanding of the complex numbers, and I would love to support you through buying the book. Thank you for the amazing videos.

[@tedrex8959]

I would LOVE an international version, I was rather disappointed I couldn't order it in the U.k. Please think about doing a version for your over seas fans, I am sure we are not alone in wanting one.

[@primenumberbuster404]

This channel is the best thing on youtube right now.

[@Grateful92]

Fr

[@angeldude101]

We can go deeper. Complex numbers actually have 2 much lesser known cousins (lesser known because they don't actually expand what's algebraically possible). If we can define a number i such that i² = -1, why can't we define other non-Real numbers with Real squares? From this idea, we get the hyperbolic numbers (usually called the worse name of "split-complex numbers), and the dual numbers, defined respectively with j² = 1 and ε² = 0, though with j and ε not being Real despite their defining equations having Real solutions. Now what happens when we take _their_ exponents? e^j and e^ε? Can we do the exact same thing we did with imaginary numbers? _Of course we can!_ They even display similar "angle" adding behavior, though with different notions of "angles". I don't really see much magic in Euler's formula anymore... because the magic is entirely within the exponential alone, and Euler's formula is merely one manifestation of the beautiful exponential function. exp(φi) = cos(φ) + isin(φ), exp(φj) = cosh(φ) + jsinh(φ), exp φε = 1 + φε (wait a minute... that last one looks suspiciously similar to the the limit definition, but when N is set to 1. In addition, while I initially said that j and ε aren't Real numbers, the exponential _does not care_ and the three formulas I gave work regardless of what i, j, and ε are as long as they satisfy i² = -1, j² = 1, and ε² = 0. Even j = -1 works in the hyperbolic case.) Also do I need to mention it? Because Euler's formula works completely unchanged for quaternions. Just normalize and pretend it's i. The exponential might be my favourite function, and it bothers me how many sources try to explain it with the Taylor series, rather than the beautiful geometry of change proportional to the value, and the algebra of the bridge between the multiplicative and additive groups.

[@PeerAdder]

Which leads us into quaternions, and on and on...

[@DrunkenUFOPilot]

The case of ε² = 0 has become important in computational physics, optimization, machine learning and other hot areas. Clever application of this idea allows to compute derivatives of complicated multi-variable functions while computing the function's values at any point. Previously, one had to vary the input values by small amounts, re-compute the function, and take the differences. Or, sit and do all the algebra to find an expression for the derivative. Now several cutting edge areas of mathematics are making fast progress due to ε² = 0.

[@johnraffensperger]

Perhaps one of the best videos I've ever seen. Very well done. Thank you

[@cem_kaya]

i was playing around with numbers at a math class in high school. i accidently put i over 7. i could not understand how such a number would work on my own. i asked where is this number on a imaginary number plane to my math teachers. Some answered that is illegal, some admitted they did not know. One of my teachers (Ömer hoca) tried to explain it using the Cis function and a formulaic approach. To be honest, until that point i could derive the intuition behind most of the math and physics stuff on my own. This problem where is the number 7^i was impossible for me and no one around me cared about the intuition of things. One day YT recommended imaginary numbers are real to me. That video made me look at the internet from a new perspective. i learned that it can be a great tool for education and self learning. i thank you for widening my horizon. Evresince, the internet has been the most valuable resource in my life. Now i am studying ai at a masters level, thanks.

[@ArshiaSa-ku2qd]

I saw your masterpiece of a video on imaginary numbers 5 years ago and i still think about it some days. You were the only one that could explain the topic in a way that elevated my understanding of the topic.

[@erawanpencil]

This is the best video explaining logarithms of negative numbers I've ever seen... thank you for walking us through the steps, it helps for beginners like myself. Explaining our ordinary real number arithmetic as "shadows" of a truer reality in the complex space was evocative.... maybe you can make a video on the Riemann sphere some day or recommend some others. When Riemann put a light source at infinity and showed that Möbius transformations are just shadows or projections through that sphere, it seemed like it was telling us something about reality way beyond just complex analysis.

[@billynomates920]

that was one of the best maths youtubes i have ever seen.

[@TheSereneWanderer87]

After watching this beautiful video, I realize that I have studied nothing in school.

[@ryrylandcripps5811]

e^iPi=-1, my favorite. An exponential raised to the power of an imaginary number multiplied by an irrational equals an integer? Beautiful.

[@sss-chan]

At 9:33 why complex conjugate of 2^ bi would have a form 2^-bi? I mean it's clear from Euler's formula, but how to get it without proving it

[@CliffSedge-nu5fv]

-bi is the complex conjugate of bi, and taking the log of 2^bi = (log2)bi and -(log2)bi is its conjugate. Not a proof, but pumps the intuition. If the idea is that 2^bi is a point on some curve in the complex plane, why wouldn't 2^-bi be a symmetric point on the same curve or a point on a symmetric curve?

[@kristoferkrus]

@@CliffSedge-nu5fv Why would it be? The point of sss-chan's comment is that the video creator is using logic that hasn't been proven yet. I think most of us who watch this video already has that intuition, but we only have that because we have accepted things such as Euler's formula as facts. So you can't really build on that intuition here since we are trying to prove things that are more fundamental than that, and doing so would create a circular argument.

[@jjeastside]

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. The same way the conjugate of 2 - 3i is 2 + 3i. The conjugate is simply multiply the i by negative one. 2^bi is 2^b(-i) or 2^-bi.

[@jjeastside]

@@kristoferkrus Cliff is correct on the basis that is the definition of a complex conjugate. "Each of two complex numbers having their real parts identical and their imaginary parts of equal magnitude but opposite sign." What would of been nice would of been a visual to show that on the 2d plane that this was true and adding up their angles and magnitude does indeed add up to 1.

[@kristoferkrus]

@@jjeastside You're not conjugating 2^bi. You're conjugating the exponent of 2^bi. That's two different things.

[@pauls5745]

Why am I seeing aero/hydrodynamic shapes in the graphs of these? Too elegant and simple! There is deep beauty in math.

[@tedsheridan8725]

Even cooler - you can plot e^(x+yi) in 4D space, and see a continuous manifold that captures the entire complex function.

[@monishrules6580]

You mean in a 3d space with domain colouring?

[@tedsheridan8725]

@@monishrules6580 Nope, I mean proper 4D space. It's possible to plot four perpendicular axes on a 2D screen, plot a complex function, and rotate around in 4D space.

[@monishrules6580]

@@tedsheridan8725 do you have some video of that? I cant visualise it

[@tedsheridan8725]

@@monishrules6580 I will soon. I'm starting a whole comprehensive series on visualizing 4D on my math channel. The first video in this series should be up by next week. Subscribe and hit the bell so you won't miss it. www.youtube.com/@HyperCubist

[@Grateful92]

​@@tedsheridan8725I'm waiting! Thanks for the url, your channel is underrated as hell, I hope one day, just like Euler's case, you will be appreciated for your contributions to truly educate people. Thanks for what you do! Keep it up! Academic authority never likes someone who brings up something totally new that would troll their faulty logic. I thank youtube for giving people like you the platform to present things in a new and creative way. What a beautiful time we live in!

[@PASHKULI]

Great update on the series! I watched it back then and was so grateful someone (you) had taken the time to explain and visualise and elaborate on this amazing subject!

[@edwarding4355]

If I learned math in a historical way, I would have understood it better

[@tenma628]

such a great video !! i was astounded when you revealed that the exponent function's imaginary surface looks like a sin wave as it looked almost exactly like how i imagined it before you showed it. such a cool relavation !!

[@Vannishn]

9:33 It is really the conjugate you multiply ? I think it's the inverse ! 2^1 × 2^(-1) is one but the magnitude of 2 is 2

[@kristoferkrus]

For complex numbers of unit magnitude, their inverse are their conjugate. The problem here, though, is that it hasn't been shown that 2^bi has unit magnitude, and the fact that 2^(-bi) is the conjugate of 2^bi isn't proven in any other way either.

[@MaeleneReynolds]

I chose to help and share the video in the social media. I hope the popularity will grow.

[@fluffpuckot]

At 9:15min: in my world 18.4+26.6 = 45, not 55. And to think they taught us not to "drink and derive", but it is obviously good enough to spot a basic error.

[@revetastogne]

Spotted that error as well. Was going to leave a comment, but saw yours

[@pokemon9573]

Spotted the same thing, it made my little ADHD brain go insane.

[@DFeltran]

What a great video! Congratulations for the high quality content and great explanation!

[@abdulmoeed4751]

At 9:47 how can we assume that conjugate of 2^bi is 2^(-bi), in general conjugate of f(z) is not f(conj(z)). P.s love your videos tho ❤

[@rakshitgv]

Edit: I am starting think that you already knew this and might be questioning why this was assumed when this would (essentially) be a result of euler's formula (I am not entirely sure how one could rigorously prove this without using euler's formula). Notation wise, I will z* to represent the complex conjugate. Another thing to know would be cos(-x) = cos(x) and sin(-x) = -sin(x). Let z = 2^(bi) and z* = 2^(-bi) (we want to show that z* is the complex conjugate of z). Rewrite 2^(bi) as e^ln(2^(bi)) = e^(i*b*ln(2)), and similarly, z* = e^(-i*b*ln(2)). Using euler's formula, z = e^(i*b*ln(2)) = cos(b*ln(2)) + i*sin(b*ln(2)). Similarly, z* = e^(-i*b*ln(2)) = cos(-b*ln(2)) + i*sin(-b*ln(2)) => z* = cos(b*ln(2)) - i*sin(b*ln(2)). Since z* is of the form a-bi, and z is of the form a+bi (a = cos(b*ln(2)) and b = sin(b*ln(2))), the initial claim that z* is the conjugate of z is proved.

[@abdulmoeed4751]

@@rakshitgv your edit is exactly my take, our reasoning would be circular if we use the euler formula in it's own derivation

[@MrCoreyTexas]

haven't gotten the chance to watch your video yet, but your thumbnail in the list of related videos really stood out visually to me!

[@Cpt.Zenobia]

If the graph you drew @3:39 is of log(x), then the slope at x=-1 is not 1. looks like d/dx at -1 is -1 and at 1 is 1. does not make sense! looks more like graph of 1/-x^2 + 1.

[@joaopaulolyrafialhobreda5872]

I actually noted the same thing. In the video they show a graph where the functions are opposite, df(-x)/dx = -df(x)/dx, while they claim they have the same derivative.

[@ferriswhitehouse1476]

I took some courses on complex numbers for electrical engineering 8ish years ago and this was such an awesome refresher for me. I remember thinking that complex number math was one of the greatest achievements of humanity. This video is an absolute masterpiece, I followed perfectly start to finish through a whole bunch of stuff that I thought I forgot. The context, the timeline, the visualizations, the narration, everything is just so awesome. Thank you!

[@GodzillaGoesGaga]

After 30+ years after graduating I still think Euler, Laplace and Fourier are the most significant contributions to Mathematics and Science. But then again I also have Newton and Lebnitz as heroes too!! Calculus is cool AF !!

[@ferriswhitehouse1476]

@@GodzillaGoesGaga ya, absolutely agree with those 🙂All of those topics were like learning magic

[@rmxevbio5889]

SO... You both do not have a use for complex number anymore?? Im currently studying Control system Eng

[@GodzillaGoesGaga]

@@rmxevbio5889 You need complex numbers for Laplace transforms and Z transforms which are fundamental in control theory.

[@ferriswhitehouse1476]

@@rmxevbio5889 I still use them regularily

[@spitsmuis4772]

9:35 Wait, how do you know that 2^(-bi) is the complex conjugate of 2^(bi) ?

[@Nitiiii11]

At this point, you don"t. But it doesn"t really matter, you can just omit this premature conclusion and continue the same way.

[@Sagitarria]

I want to say two things one your work is phenomenal. And you are maybe the only KZbin creator that I have ever given constructive criticism to who has respectfully taken it. I have so much respect for you and your work

[@kristoferkrus]

9:28 Why do you say that 2^(-bi) is the complex conjugate of 2^(bi)?

[@egemenyaln]

I was thinking the same

[@xyzest6492]

Indeed proving they're complex conjugates without using Euler's formula seems quite non-trivial

[@vascomanteigas9433]

You need a Theorem of Complex Analysis, called the Reflection Propriety. Let f(z) a Complex function of Complex Variable. If f(z) are analytic (see Cauchy Integral Theorem and friends), and also when restricted to real variables, the same function had real results. Then the function had Analytical continuation by Reflection: f(conj z) = conj f(z), for every Complex values z. 2^x is a real function, then the analytical continuation 2^(I*x) for conjugate z makes the function conjugate.

[@kristoferkrus]

​@@vascomanteigas9433 Good point.

[@CherryNerd27]

Thank you so much for making this beautiful video! You made all the different pieces of maths fit together so effortlessly. I hope that I never forget this explanation in my life.

[@yash1152]

4:03 okay whits. stay bound.

[@PuHaPro]

Thanks for breaking it all down so clearly. This is really going to help!

[@angrydachshund]

@1:01 Congrats on your book. From the subject matter, does that make you a fiction author?

[@NetTubeUser]

Your video is exceptionally slick, clean, professional, and impressive... and informative, indeed! You did an outstanding job. Congratulations.

[@fullfungo]

9:30 unfortunately, you are wrong here. You previously said that if (a+bi)•(a-bi) = c Then |a+bi| = sqrt(c). But now you are saying that 2^bi and 2^-bi are conjugates without any explanation, except that you replaced bi with -bi. This does not work in general. Consider f(x) = 2i•x. Then f(bi) is -2b and f(-bi) is 2b, but they are not conjugates. You got lucky here. If 2^x was equal to, for example, 1+2i•x-x^2 (or some other expression) Then 2^bi and 2^-bi would not have been conjugates. You need to prove that 2^bi is decomposable into f(b)+i•g(b) where f is an even function and g is an odd function first.

[@shurturgal]

Amazing video, I really liked the ones you made some time ago, and even made a playlist out of it so that I can watch them easily whenever I want too. Thank you for your work!

[@siddharth-gandhi]

I’ve a playlist called great videos where i save videos which really resonate. So far i think all your videos I’ve watched have gone to that playlist. Stellar job good sir! Hoping you keep at this for many more topics

[@prabodhshejwalkar]

Your complex numbers series was my introduction to higher math education! This changed my life!

[@gtagamerm4169]

was waiting since i've seen your contents on Reimann surface

[@RakibHossain-mq7qv]

In the era of short videos for all those who watched this video to the end, you have a beautiful mind.

[@luigiwilkins8806]

Awesome vid. I learned of Euler's formula in my circuit analysis class, but it was not really explained, so I find this video really insightful. I love your videos, and I always leave with something new.

[@QP9237]

I always enjoyed your complex numbers videos. I saw them right after I took complex variables in undergrad, and got to say it's what kept pushing me forward to reading and collecting complex analysis textbooks (single and several variables), and I love it so much. I even am starting to apply some of the logic into my medical career for research/modeling purposes. Also my all time favorite number is Gelfond's constant, because it's the power relation of the negative real and imaginary units producing a positive transcendental number!

[@evasuser]

We are lucky that videos like Welch (and 3b1b and others) exist and fortunately yt still allows them to exist.

[@aglawe1]

I was taught this in high school but never was able to understand how exponentials had to do anything with trigonometry. The way you explained in the video both historically as well as mathematically blew my mind. Surely it is hard to accept this equation without being able to visualise it. Thank you so much for your effort and keep up the good work 👍

[@johnferrara2207]

The hardcover is at the top of my Christmas list. REALLY looking forward to this.

[@Saki630]

your 2016 video series on Imaginary Numbers was critical to my understanding the subject during my advanced mathematics courses for my M.S. ME. I never forget the hard work you did showing the mapping of the complex plane.

[@girishrajgor9791]

You are Flashlight on Path of Math....No words to explain for your hard work...carry on..

[@hichams4445]

People like you make me love the internet and math. thanks a lot

[@iccuwarn1781]

Fantastic video as always! You're really setting the bar high for all the other math youtubers :)

[@underfilho]

your playlist in complex numbers was my insight that I really love this, and now I'm doing my undergrad thesis in Complex Analysis, thanks. I'm doing it by connecting Complex Analysis and Harmonic Funtions, using it to solve the Dirichlet Problem by the Riemann Mapping Theorem.

[@wacharaboy]

No wonder you have so many compliments in the comments. You are really GOOD explaining. Never thought I could be even able to grasp some of the knowledge you provided. I've subscribed and I haven't even finished the video: with experience now I know when a teacher is actually able to make me learn something...

[@andrashorvath2411]

Amazingly great job at explaining it at the right pace with the right info and nice animations, cheers.

[@EconAtheist]

Geez I've seen all sorts of Euler's Formula breakdowns on KZbin -- this one is the most fleshed out, me likey! New subscriber here. /also, early in the video, you immediately started rotating your 3d extrusion so i didn't have to burn brainpower thinking of how it should look //yes!

[@kevinboles3885]

kudos on the book! you picked a fascinating topic for sure

[@phrozenwun]

Even though you touch on differentials, I think you omit one of the more elegant demonstrations; the idea that e^x is its own derivative allows you to move to an expanded representation of e^x, cos and sin demonstrating equivalence. Beautiful presentation none the less, thank you for sharing.

[@kaabirali1]

I love this explanation. Fundamental questions are truly fascinating.

[@belg4mit]

This is a really good explanation of why these things are connected, so often one only is told/roughly shown that they are.

[@xfry]

OMG I love your Imaginary Numbers are real and I go to buy it. Thank you for promoting love for math with your videos!

[@InterestsInEverything]

I have followed this channel since its inception, and it has been wonderful! But I want to point out one small thing: You said (at 2:42) that we broke away from the "multiply by itself" definition. This is actually a misconception based on what people think "negative" means. Since "negative" simply refers to "opposite" (negatives are the numbers opposite the positive on the number line), then a negative EXPONENT just means a negative operation (the opposite operation of multiplication is division). Exponents raise the POWER, and operations are an increased power of a stagnant object. Thanks for all your hard work, Stephen!

[@popcorn485]

Your original video was a huge hit! This is a fantastic follow-up! Absolutely excellent! (Edit: And heaps of great content in between I should add.)

[@WelchLabsVideo]

Glad you enjoyed it!

[@4m0d]

I don't yet fully understand the series but I am grateful for your effort, and it is so cool you wrote a book so we can get an intuition behind the beautiful complex numbers.

[@GoldenAgeMath]

Preordered the book immediately! The original imaginary numbers are real series is one of my favorite pieces of math exposition and one of my favorite things on KZbin of all time! I hope to someday create something half as good!

[@carolinalp]

Amazing video!!! So clear, so beautiful as Euler's equation itself. Thank you!

[@SreenathSreekrishna]

After working with complex numbers for over a year, even using euler's identity, I never really understood it completely until this video. Thank you for a fascinating explanation!