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

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Special thanks to the Patrons:
Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti

Tattoo by @themutemaker - thank you! Check out her awesome work here:

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References & Notes

Excellent History of Logarithms by Florian Cajori

Nice History of Euler’s Formula

Much of the visual approach presented here comes from Needham’s incredible book:
Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.

Other books referenced
Maor, E. (2011). E: The Story of a Number. Ukraine: Princeton University Press.
Penrose, R. (2021). The Road to Reality: A Complete Guide to the Laws of the Universe. United Kingdom: Knopf Doubleday Publishing Group.
Dunham, W. (2022). Euler: The Master of Us All. United States: AMM Press.
Wilson, R. (2019). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. United Kingdom: Oxford University Press.
Nahin, P. J. (2010). An Imaginary Tale: The Story of √-1. Ukraine: Princeton University Press.
Stillwell, J. (2013). Mathematics and Its History. United Kingdom: Springer New York.

Euler’s Amazing 1747 Paper
Euler, Leonard. *"Sur les logarithmes des nombres négatifs et imaginaires”* Written in 1747, but not published until 1862. Euler did publish a similar paper in 1749. See Cajori #3 above.

Note on Benroulli’s area of sectors
Euler’s counterexample using Bernoulli’s sector area comes in a couple flavors. The one presented in his 1747 paper "Sur les logarithmes des nombres négatifs et imaginaires” is a bit different than an earlier example in a letter to Bernoulli. I chose the earlier example for clarity. See Cajori vol 2 and Sandifer.

Feynman Lectures - Algebra

[pendragon]

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).

[Yotam]

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

[BooLightning]

Euler was like, nahh bro you’re both wrong

[manuelcastellanos]

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!

[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!

[Mytubecleaner]

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

[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!

[Kaznerh]

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

[kiko]

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

[fyu]

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.

[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.

[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.

[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.

[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

[DFeltran]

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

[itscoolto]

Nobody tell Terence Howard about this <.>

[Saki]

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.

[leonhardhoffmann]

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!

[evasuser]

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

[supermuffinbros]

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