e^(iπ) in 3.14 minutes, using dynamics | DE5

Euler's formula about e to the i pi, explained with velocities to positions.
An equally valuable form of support is to simply share some of the videos.

Not familiar with the calculus referenced in this video? Try taking a look at this one:

Another perspective on this formula, from Mathologer:

Another perspective from this channel:

And yet another from the blog Better Explained:

I'm not sure where the perspective shown in this video originates. I do know you can find it in Tristan Needham's excellent book "Visual Complex Analysis", but if anyone has a sense of the first occurrence of this intuition do feel free to share. It's simple and natural enough, though, that it's probably a view which has been independently thought up many times over.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: @ngvutuan2811

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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

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Various social media stuffs:

[bluebrown]

Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used.


As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!

[it_was_my_cat]

This is by far the most intuitive explanation of this identity I've ever seen!

[Peckingbird]

"Multiplying by i has the effect of rotating numbers 90°" - Both lost and mind blown.

[antonxuiz]

*uses τ*
Ah, i see you are a man of culture aswell

[liamsmith]

I have known that e^i*pi = -1 for many years.

This video was the moment I understood that fact. Thank you.

[LimeGreenTeknii]

2:32 Good thing you initially drew more than 4 arrows to illustrate this point XD

[jebbush]

Oh my god that makes so much sense.
Why was this never explained to me like this?

[Xidnaf]

i've watched other 3blue1brown videos about this but i never really got it until now for some reason. holy crap it's so intuitive now, i don't know what i didn't get it before

[kecpterexe]

Just wondering who came for Alan Becker

[andrea.dibiagio]

Kudos! This is pedagogy brought to the level of art. I hope your work, together with that of other excellent YouTube educators, will pave the way for a new generation of teaching.

[hiqwertyhi]

the e^iπ part comes right at 3.14 but also the e^i𝜏 part comes right at 3:14

nice

[gabrielz]

gotta click fast when 3blue1brown posts

[_Eamon]

The idea that multiplying a number by i rotates it 90 degrees into an imaginary axis is probably the most important thing I've ever gone my whole life without knowing.

[weebkien]

I actually just come here after watching "Animation vs Math" 😅

[basha____]

Who else got this recommended after seeing animation vs math?

[xyzct]

It's amazing how helpful Grant's use of color is. This was particularly pronounced on his superb Fourier series videos. The old analog chalkboards of my youth did not have this added dimension.

[thephysicistcuber]

I'm not first
and I'm not last
but when 3b1b uploads
I click fast.

[maacpiash]

I was afraid that by 3.14 minutes he meant 3 minutes and 14 seconds, which is, of course, inaccurate. I'm glad that didn't happen!

[TheTrexTeam]

I was comfortable with this identiy before, but I had a great “AHA!” moment. This is probably the most concise and surprising way to look at it I’ve ever seen. Thanks a bunch

[Szy]

Why am i coming to this after watching Animation vs Math?