HN Theater @HNTheaterMonth

I expect that it is because in general, more useful functions and operations are closed in the complex plane. Things that may superficially appear complicated to a human being learning math, like analytic extension [1], from a theoretical perspective means that when you need the tool, it is there, and continues to expand the sets of problems you can address.

As you add restrictions to the set of numbers you want to use, you are importing those restrictions into every proof you want to do on those numbers. For instance, consider the question, "Does this polynomial of some high degree have roots, and if so, how many?"

The Fundamental Theorem of Algebra proves that the answer is yes, and the number of roots is the same as the degree (although some may be roots multiple times). See [2].

Now, as you watch 2, consider what happens to the proof if you confine it even to the reals, let alone the integers. While the complex plane is "more complex" from a human perspective, the proof of the Fundamental Theorem of Algebra using the complex plane is not that complicated. Consider trying to make equivalent statements about polynomials with only real roots, and not using the complex plane in the proofs. In cases where it is feasible, your proofs will inevitably be carrying around a lot more caveats about what polynomials it applies to, and there may be things that the caveats simply render infeasible. As you step down the number hierarchy, the caveats get worse and worse. There's more and more "holes" that every proof about those simpler numbers has to step around.

The same simplicity that makes it easier to start your education with just integers becomes a crippling limitation when trying to work with them, which is also in some sense the exact same reason why we have to step you up the number hierarchy even in non-specialist education. Math limited to just integers is so confining and difficult to work with that it isn't even enough for day-to-day life. The simplicity is a double-edged sword.

[1]: 3Blue1Brown on the topic: https://www.youtube.com/watch?v=sD0NjbwqlYw

[2]: https://www.youtube.com/watch?v=shEk8sz1oOw

That was a fantastic video. I was left wondering a bit about the difference between the Riemann Zeta function and the original Zeta function, but I found this 3Blue1Brown video which explains exactly that: https://youtu.be/sD0NjbwqlYw

I don't think it comes close to 3blue1brown's video on the same topic - https://www.youtube.com/watch?v=sD0NjbwqlYw

Still nice of the video in the original post to give a bit more of a historical context, instead of focusing entirely on the mathematics. It could have done without the building analogy around the beginning though, that didn't come up later and was just distracting.

⬐ pseudolus

+1 for the 3blue1brown recommendation. Between the two, a non-mathematician can acquire a layperson's comprehension of the Riemann Hypothesis. Additionally, John Derbyshire (of the unfortunate views regarding race) wrote a very good book "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics" that covered both historical and mathematical aspects of the Riemann Hypothesis [0].

[0] https://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mat...

⬐ EricMausler

That video made complex analysis click for me

ah, Mathologer video.

Have seen that.

Another one by 3b1b on that topic: https://youtube.com/watch?v=sD0NjbwqlYw

3Blue1Brow has a nice video on it https://www.youtube.com/watch?v=sD0NjbwqlYw

3B1B has a nice visualization of "Riemann zeta function and analytic continuation" which might help in further understanding of this proof.

https://www.youtube.com/watch?v=sD0NjbwqlYw

This is the most intuitive visualization of the Riemann Zeta function I've come across: https://youtu.be/sD0NjbwqlYw.

⬐ novalis78

Wow, thanks for sharing that. Very well done!

3Blue1Brown recently did a video on the Riemann zeta function: https://www.youtube.com/watch?v=sD0NjbwqlYw

It's not as in depth but it has some helpful visualisations that I've never seen elsewhere.

⬐ jorgenveisdal

Wonderful resource!

⬐ Cyph0n

Thank you, that was an amazing video. I literally hit the "Subscribe" button 2 minutes in!

⬐ db48x

His visualizations are great. If you've never had a mental picture of what a matrix does, I recommend his series of videos about linear algebra.

⬐ rahrahrah

I approve of this video.

(Seriously. There's so much flawed hand-waving going around. But this video is solid)