The formula at 7:17 is wrong, my mistake. I forgot to say that e^(z + bi) = e^z is required as well. Correction: kzread.info/dash/bejne/eJuqzqWihbbVcdI.html

Tau looks more like it should be 0.5 pi.

@Fox0fNight

Күн бұрын

pi/2 is usually given the name gamma. It's just more convenient to use in trigonometry, ex: sin(x+pi/2)=cos(x). It really depends on which formulas you're working with, but it is way more often to see 2pi or pi/2 than pi. If you're looking at branches of functions or integral bounds, maybe you'll see (-pi, pi], but that's really just [0, tau).

not sure how this got in my recommendations but now i know what a tau is thank you

Well, I always write my 2pi as tau in my undergrad, until it becomes proper time in General Relativity.

The reason I like τ is because it forces people to evaluate where π is used and how π is used. You can't just look at formulas, you have to evaluate the usefulness of all these different formulas as well.

Couldn’t we decide to take 2πi as a new constant for some situations (especially complex exponentiation) and use π in other situations (like real trigonometry)? You make a good point about how the period of the exponential function should be recognized, but it’s not 2π, it’s 2πi and I’d be happy to see more popular non real constants!

@astraea-paradoxesreasoning6498

2 күн бұрын

Yeah that's a good argument. Just because 2pi*i has a really nice property, does that mean 2pi gets credit for that as well so to speak? Maybe not. Maybe it's really the number 2pi*i that is really special, like you point out.

@Fox0fNight

Күн бұрын

I usually write tau with a dot on top to denote 2πi. When I'm doing trigonometry I find that π/2 is way more useful, so I denote it as gamma, like I've seen in some formulas.

Astraea speedrunning a degree in mathematical semantical aesthetics: xD So I agree that tau has more meaning, but I don't think bringing in complex exponential identities and seeing how pi and tau fit in has much more argumentative power one way or the other as compared to the laymens argument: pi is the half circle, but tau is the _full_ circle! Tbh I think the reason people still use π is just because 2π is simply just easier to format and the elegence of using τ doesn't quite overcome the annoyence you will encounter putting τ/2 wherever you had just a π before...

@astraea-paradoxesreasoning6498

3 күн бұрын

I agree with that. Tau would be a little more annoying than pi to actually use all the time. And yeah, I've also been pondering about the different definitions of the complex exponential.

@Fox0fNight

Күн бұрын

We can use both symbols for sure, just like we have floor(), round() and ceil() that are the same things but shifted. Though I do still believe 2π or π/2 are way more used in math than π is. And are also helping more with the intuition. Do you have any examples in which π would make sense to be used?

@astraea-paradoxesreasoning6498

Күн бұрын

@@Fox0fNight Yeah pi/2 is a great one to bring up. Between pi/2, pi, and tau, it looks to me like pi/2 is the most convenient to have a symbol dedicated to. Of course pi/2 being an angle between orthogonal things gives it a direct connection to a very powerful mathematical concept. I actually wasn't able to find a formula involving pi (multiplied only a rational number of 1 or -1) that felt especially significant compared to some formulas with pi/2 or tau. I would be curious to hear about one though.

Personally, I prefer the real circle constant pie= pi*7/22

nice video but you're wrong

No.