The Real Reason Complex Numbers are Useful!

Have you ever wondered why complex numbers are so effective? It's because of geometry!

00:00 Intro
02:06 How can numbers have "phase"?
05:19 Deriving the complex numbers
13:02 What about Euler's Formula?
14:18 Application: Fourier series
16:09 Closing remarks

This is a submission for 3Blue1Brown's 2nd Summer of Math Exposition contest (#SoME2).

[mikip]

That was outstandigly pedagogical. I think this is incredibly useful after you have done complex numbers in the usual learning path (which means accepting the square-root of one with faith and just by the growing number of consequences). This was a beautiful aproach.

[santoriomaker]

THANK YOU! For the past few months, I was trying to find an explanation in the internet on why specifically √-1 is so useful. Most of the time, I get the usual "oh, it's because it is used in formulas in *[insert certain field of science]*" explanation, which doesn't really help or explain why THOSE numbers are used instead of the usual real numbers.

It's really the only thing that's holding me back from fully understanding the potential of complex numbers and this video has perfectly helped me guide towards the right direction. :)

[abrahammekonnen]

9:00 I like that insight: The important idea of complex numbers is the connection between negation and rotation.

[tylerduncan]

Every time someone says complex numbers don't exist or dont matter, i show them this video. I have never seen a better demonstration of not only why they work but also how they are actually required and work properly. It's not just an extra bit we added to solve some specific equations, it's as important and real as the integers or reals.
No matter how hard you try to avoid it, you'll keep reinventing i.

[hexeddecimals]

This gave me a lot to think about!

My way of going about the effectiveness of _i_ lies in the definition of norm of a vector space (taking the square root of an vector dot-product'd with itself). The dot-product of any hypercomplex number is (a+bn)(a-bn) where n² is either -1, 0, or 1 which determines your flavor of hypercomplex number. In the case where n² = -1 you get that the norm is exactly the same as the 2D Euclidean norm, which gives you circles and rotations and all that jazz.

I think the math is cooler in my approach, but yours is a much better explanation. It's more intuitive and beginner friendly. Definitely the best SoME2 I've seen so far!

[mikewilliams]

Exactly this. The mystifying of i is fun for a time and helps us understand the beauty of it all, but it doesn't give further understanding. This was so clearly put and completely convincing. Bravo.

[joaovictorcarvalho]

you have no idea of how long i've been waiting for a video like this!! excellent

[wakeuppeople]

Bravo! You have a very good grasp of the phenomenology of complex numbers and a true gift explaining it to others! Keep it up, you are doing a fine job!

[isaaccastro]

This is by far the best math video I have seen… Please make more

[MattMcIrvin]

I think there's *one more* really important piece of the puzzle, which appears only in advanced calculus: Complex analyticity, and the integral theorems associated with it. Elementary functions on the real numbers--that is, the "nice" functions we spend most of our time learning about in primary and secondary school--all turn out to be analytic, or almost everywhere analytic, when you extend them to the complex numbers. That is, they are well-approximated by their Taylor series almost everywhere. That in turn means that they satisfy the conditions of certain absurdly strong theorems that make the complex versions really, really convenient to do calculus on. I've been staring at this for decades and it still seems close to wizard magic to me. It also seems to be closely related to a lot of deep facts in particle physics, in a way that I am not sure we entirely understand.

[buchelaruzit]

I've never learned about complex numbers (not even imaginary ones) but i understood everything in this (until the last part about the fourier series). i think it's my favorite some2 video yet!

[timpani]

Just thought I'd add a slight nitpick on an otherwise great video. At 9:55 we define the nth root of (1 angle theta), but in reality we need to be a bit careful. For instance, (1 angle 0) and (1 angle 2*pi) are really the same, but their nth roots (as defined here) are different (1 and 1 angle pi, respectively). While I do understand the reason for not going over this, it is definitely a pitfall that can be a bit tricky to handle (basically we have to make the nth root into a multi-valued function and "choose" one of them as a "principal nth root", which is a bit more intricate than it might seem at first glance).

[borges]

A video that perfectly explains the subject covered in 1 year of university in 17 minutes.

[Reashu]

This is how I was taught complex numbers in high school, and I've kind of missed it since then. Getting a refresher was great!

[tx]

Wow! This is by far the best explanation of complex numbers I've ever seen, very clearly explained and logically built-up. Great job and thanks for posting.

[AssemblyWizard]

I think another key property that makes complex numbers so useful other than rotations is that they are algebraically closed. They let polynomials have all their roots, enable analytic continuation, usual functions (sqrt, sin, exp, log, ...) have an almost entire domain & range (Picard's little theorem), and generally make math have less "undefined"s.
This is also the reason for their discovery, they were discovered as an algebra trick to make things work out.
I think rotations make them useful to introduce to a problem, but it's their algebraic necessity that makes them pop up everywhere.
Btw great video, and awesome sound quality!

[jakechapman]

This video gave such a clear explanation of complex numbers in a way that I could almost fundamentally understand! Phenomenal work!

[robharwood]

Wow! Super excellent. In terms of math exposition, this is one of the best I've ever seen. And it's also about a topic that is so widespread and usually confusing, that I think it definitely deserves recognition in SoME2.

[ellishawkins]

The best complex number description youtube I've came across yet, thanks

[divine]

I haven't learned anything about complex numbers in school, but this video cleared up everything I was confused about regarding imaginary numbers- and now I feel ready to take them on in a couple weeks. Cheers!