though in physics I attribute it to 1) we prefer using angular frequency $\omega$ rather than ordinary frequency $f=\omega/(2\pi)$ 2) we don't like using sqrt(2pi) for normalization, so we give the 2pi to the k-space integral
Suppose I do this: @MaryStar -I tell one guy to go stand on a circle of radius $r$, walk around it for $\theta$ degrees, and then tell him to jump (fly?) up a distance $z$. I then give another guy the same starting instructions, but tell him to go DOWN $z$.
What instruction would you give to the first guy to reach the second?
@ABeautifulMind I'm sorry, Jasper. People here would miss you, you know.
This is what I would do: I'd put an "infinite 2-sided mirror" on the $(r,\theta)$ plane, and tell each guy to move to the spot he sees his reflection at.
(that's practically giving away the answer-think about it for a little while)
You might use a soda can to help you visualize this.
Sun's up...mhm....looks OK, and the world survives into another day. Had another dream about lions at the door, and they weren't quite as frightening as they were before. @ABeautifulMind
Huge orange flying bowl rises off the lake, thousand year old petroglyphs doin' a double take, pointing her finger at eternity...
The reference to the lions is actually a reference to the book of Daniel-in it King Darius, enraged by Daniel's refusal to worship him, seals him in a lion den overnight. But when they check on him, he is unscathed.
So I take the "lions" as being a symbol of our fears-"the things that will eat us"
I found the following:
Cylindrical coordinates $(\rho , \theta , z)$.
This system consists of the following coordinate surfaces:
Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ constant }$
Semiplanes that passes through the $z-$axis: $\theta=\arctan \left (\frac{y}{x}\right )=...
> I would like to argue here that there is nothing wrong with thinking of them as infinite decimals: indeed, many of the traditional arguments of analysis become more intuitive when one does, even if they are less neat.
@columbus8myhw To amplify: to make clear how you define addition, for example, you usually need to talk about limits of some kind, and then you're pretty close to cauchy-sequence land
I don't understand ho we have found these surfaces... For example, when $\theta=\arctan \left (\frac{y}{x}\right )=\text{ constant } $ how do we conclude that the surfaces are semiplanes that pass through the $z-$axis??@DavidWheeler
so, if we hold $\theta$ constant, for example (like $\pi/3$), that tells us we're somewhere in that "direction" (but we don't know "how far out" or "how far up/down").
@columbus8myhw, one can construct $\mathbb{R}$ artificially as well. I.e. take $\mathbb{R}$ to be a totally ordered field with the least upper bound property. It can be shown that a unique set satisfies these stipulations.
By the way, I thought of something: One way to construct the complex numbers is to say that they're ordered pairs $(a,b)$ of real numbers, with certain laws saying how to add and multiply, right?
Is there? I can draw my Argand diagram with my $i$ at the top, and you can draw your complex plane with $i$ at the top. We'd be using different diagrams, in a sense, and we'd never know.
(Every once in a while, posts crop up on Math SE asking what the difference between $i$ and $-i$ is...)
@infinitesimal Yeah, but what if, when I write "$i$", I really mean what you'd write as $-i$? Then there'd be no way to tell, since both of our ways of writing end up working exactly the same.
@columbus8myhw It's even more interesting: Suppose $\omega = -\dfrac{1}{2} + i \dfrac{\sqrt{3}}{2}$. There's no way to tell apart the fields $\Bbb Q(\sqrt[3]{2})$ and $\Bbb Q(\sqrt[3]{2}\omega)$ until we embed them in $\Bbb C$.
You can think of it as "polynomials in $\sqrt[3]{2}$", which only go so high, because when we get to the cubes or higher, we can "knock the degree back down"
Question: We know that, with respect to the reals (is that the right way to phrase it?), there's a difference between $\sqrt2$ and $-\sqrt2$, because one of them has a square root and the other doesn't. Right? (cont'd)
So, what about with respect to the complex numbers? Is there a difference between $\sqrt2$ and $-\sqrt2$ then?
I was thinking, there's got to be some property that $\sqrt2$ has that $-\sqrt2$ doesn't. With the reals, we have $\exists c(c^2=X)$; it's a property that's true when $X=\sqrt2$ and that's false when $X=-\sqrt2$.
Can we have something similar for the complex numbers? Obviously, with the complex numbers, the above example doesn't work.
(Is the formal phrase "formula"? Is $\exists c(c^2=X)$ a "formula"?)
I found the following:
Cylindrical coordinates $(\rho , \theta , z)$.
This system consists of the following coordinate surfaces:
Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ constant }$
Semiplanes that passes through the $z-$axis: $\theta=\arctan \left (\frac{y}{x}\right )=...
