Euler's formula with introductory group theory

Intuition for e^(πi) = -1, using the main ideas from group theory
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There's a slight mistake at 13:33, where the angle should be arctan(1/2) = 26.565 degrees, not 30 degrees. Arg! If anyone asks, I was just...er...rounding to the nearest 10's.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: Filippo

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

[lilamjazeefa]

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[richardgalea]

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

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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

[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.

[geraldmerkowitz]

We will never say this enough : your animation is gorgeous

[mzrek]

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))

[johnchessant]

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

[timh.]

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...

[antonzavorotny]

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!

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Who is this? The perfect math teacher? Indeed.

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[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.

[munmunbanaerjee]

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...💙