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Thank you, well laid out and explained! I read the book 'Negative Math: How Mathematical Rules Can Be Positively Bent' by Martinez and was quite confused and amazed about how, as you say, 'aliens on another planet' might have done alternative maths where -1*-1=-1. Quite crazy

Thank you! It is pretty simple. I thought it would have been harder to understand.

Very enjoyable video. Thank you.

Fantastic video!

Amazing!

Definitely cool video but I think it is important to point out that complex numbers are in no way necessary for these proofs, and this isn't even really justification for the utility of them, since I'm pretty sure the proofs could just be phrased in terms of the basic rotation formulas and vectors. The complex numbers just provide a convenient and natural shorthand.

Very nice way to show different perspectives and their pros and cons.

Brilliant man, this is a great video.

Nice demonstration. But in case of Napoleon's Theorem, it is sufficient to prove that the lengths of the side of the triangle are equal, you don't need to introduce rotation over 120º.

What’s the Mac app you’re using?

What software are you using?

RIP Jim.

love the napoleon bit

I used to prove random common Euclidean geometry problems with complex coordinates. Works much better than real coordinates most of the time though not always better than using theorems

Beautiful!

Absolutely gorgeous video. We need more of this on RUclips.

The maths for napolean's theorem works out cleaner if you use w instead of i, see the wikipedia page on eisenstein integers for more info: en.wikipedia.org/wiki/Eisenstein_integer

I don't get the 1/sqrt(3) bit. Half the height of an equilateral triangle of side length 2(y-x) is [sqrt(3)/2]*(y-x)

do you have similar intuition for the product rule?

Nice video!

While napoleon waged war throughout Europe, hippolite Charles made love to Josephine in the imperial bed!! Napo couldn't figure it out!!!

I'm surprised the anti-science people who deny that the moon rotates haven't infiltrated this comment section.

I knew there is a better method for this! Thank you, that is really a great video and a much more powerful method than the normal one.

At the very start of the video you show two different "implementations" of Pythagoras's Theorem, which I took to be the same, as I didn't know that it was an evolution to joint geometry and algebra. If both representations are equivalent, I assume, that Euclidean Norm is implied. I also assume, that a right angled triangle is still the same if placed in a plane and moved around or rotated. Now the question: If I move the triangle and the shape changes, then the equivalence of both representations is no longer given. And is it right to say: an object changing shape when rotated is not an valid object under the algebraic and geometric relationship? (I know the wording is not "conventional" as math and English is not my native language ;-) )

Very interesting - I will use this method from now on. Thank you!

That is very cool

Some comment should mention the Petr-Douglas-Neumann theorem in this context. Allow me to. I knew about the theorem but had forgotten its name. Thanks for triggering me to look it up again. Wonderful little tidbit that!

Really appreciate your vdeos❤

Hello! Im from Brazil, and i am studying olimpiad's mathematic to get a medal at OBM and maybe, go to IMO. I was with difficult in geometry with complex numbers, but after your video (a video you sent 2 years ago) i understood! Dont stop making your videos, they are very good and could help another students with the same problem as me!

I believe the last theorem is where Symmetrical Components in Electrical Engineering come from. Representing a 3-phase unbalanced system by the superposition of three balanced systems.

Thank you.

That's really very neat, how you can use the half angle formula to carry the exact values all the way down as finely as you like. And then as soon as you get to a fine enough resolution to ensure the accuracy you want, you can use finite differences to interpolate in between that last round of exact values, to get your table entries on whatever spacing you want. I may have to write a Python program to do this. Maybe generate a nice LaTeX table set. 🙂 Or heck, even just a plain text set.

Well, it's more sensible for US to measure angles using decimal degrees, since we count in base 10. It made more sense for the Babylonians to do exactly what they did.

Ware the past scientists of Indian origin?

Thank you for a very dense and precise explanation of the equation of time. My interest stems from wanting to understand watches with the equation of time complication and how clocks are (or at least were) used for navigation. Having watched a couple of videos whose explanation didn’t really make sense to me, thank you for making a video that makes the issues crystal clear, though still not easy. It’s also nice to see the concept of precession applied.

This was fascinating - thank you!

Too bad, the youtube algorithm does not guage quality. This video should be top of the list when looking for chain rule.

Excellent! I really enjoyed this and definitely learned a lot. There were many, many pauses for the brain to catch up...

I always thought that Solar time = clock time. Maybe because I live near the equator where day = night throughout the year.

I used complex numbers miners to find all the 3rd points of similar triangles with two fixed points. It was a Riemann sphere.

It's funny to me that modular arithmetic was a major revelation to modern conputer scientists, whereas ancient babylonians were so comfortable with the idea of the modular multiplicative inverse that it was built right into the way they wrote numbers. 10 and 1/6 aren't just represented the same way, they act as the exact same number because 10 and 6 are each other's MI mod 60. That's hardcore AF.

Very nice geometric problems . It is also good to see ,especially for people who didn't like complex numbers , how much you can achieve by a clever application of them.

My opinion of this video can be summed up by the following: complex numbers fucking rock! Repurposing algebraic machinery to perform geometric calculations, the elegance of it is astounding.

brilliant! now how can one go about proving concurrency of lines using complex numbers?

😮😮😮 beautiful theroems

This is really interesting! Thank you for explaining. I want to get my hands on a 4 figure log table book now.

just found the channel, good stuff, please keep going

What was the software used in this video?

So cool ❤

Marvelous and memorable.