It shouldn't matter @MaryStar, so long as you know you can take roots. But if you want to think about that, $b$ will be the root of the polynomial $x^2 - (1-a^2)$, which you can consider over $\mathbb{Q}[a]$. Oh, and I guess you swapped the $a$ and $b$ above. We know $a$ is constructible, and so to must $\sqrt{1-a^2}$.
@KajHansen I think she is worried about a third degree quantity, for example showing up as the result of adding numbers whose degrees are a power of $2$.
Since the constructible numbers are a field, the constructibility of $\sqrt{1-a^2}$ implies the constructibility of $1-a^2$ and thus the constructibility of $a^2$. Since we can take square roots of constructible numbers to get more constructible numbers, this would imply the constructibility of just $a$.
Therefore, if $a$ is not constructible, then $\sqrt{1-a^2}$ cannot be.
@KajHansen Do we not have to suppose that $\sin \theta $ is constructible then we concluse that $a$ is constructible and that means that $\frac{p-1}2$ must be a power of $2$, or is this wrong??
@DanielFischer Am I the only one having major issues with the new version of MathJax? On my computer it's significantly less responsive than the last version. I had thought a couple of times when editing an answer today that my computer had frozen. And the titles of some of the questions I've answered are now not being displayed correctly on my profile page. They are being "shrink-wrapped", so to speak.
Crap. I accidentally gave away the answer above since I misread the problem statement. Suppose that $\sin(\theta)$, which is equal to $\sqrt{1-a^2}$, then apply that the constructible numbers are a field to conclude that $a$ itself is constructible, which forces that $2^k + 1$ condition @MaryStar
Ugh. Embarrassing. I was just a bit hasty in reading the problem earlier.
Once you have that $\sqrt{1-a^2}$ constructible $\implies a$ constructible, then you've done the rest of the work above in your reply to robjohn.
@RandomVariable I don't know. I haven't been using MathJax these days, catching up with moderation and all, so I have no direct experience. You should probably mention it here.
@DanielFischer I mentioned the shrink wrap thing, and a developer said it has always been an issue. But it didn't seem to be much of an issue before this update. A solution would be to edit the titles of the those questions. But I'm not going to bump questions for that reason.
@RandomVariable The title of this looks borked to me. It's the only one on the first page of your answers (sorted by votes) that doesn't look right for me. Is it the same for you?
Three on the second page.
Wait, there's a second on the first page. Overlooked it first.
@DanielFischer I see 3 on the first page. There is one that has $- \alpha$ on the 2nd line. I saw a strange on in sos440's profile where it starts on the first line, goes to the second line, and then goes back to the first line.
What do you want to know @MikeMiller? I don't have super high hopes for complex because it's a large class that's popular amongst more of the "casual" math majors, so I feel like it'll be watered down.
Dr. Clark's topology, on the other hand, looks like it's going to be very fast-paced and nontrivial.
Right now, I'm working on a 50-ish problem problem-set for topology that is supposed to help him "gauge where we are". Lots of set theory and the like. Nothing difficult so far, but he said the length on this set should be indicative of future sets.
This question was asked here, where the answer uses this description.
The last line reads:
"The special case of $0$ does not have a unique representation in scientific notation, i.e., $0=0×10^0=0×10^1=..."$
My question is the value of $a$ cannot be $0$ since, as they state:
$a$ is a $\color...
@Nick Right, so that refutes the stupid plum-pudding model. but how do you deduce from the results of Rutherford's experiment that electrons circle around the nucleus?
@skullpatrol i can't, cause it's not true that i learn fast :P
@BalarkaSen Very good question. Geiger–Marsden experiments conclude this from the size of the atom and the fact that it's nucleus is so small. I'm not sure how they thought the electrons were moving but I'll tell you as soon as I figure it out.
One reason might be that it was the most reasonable way for the electrons to be held to the nucleus. The centripetal force from the orbits held them together. I'm not sure, maybe you should enquire at the h Bar. They have a good handle on this.
So I was thinking about this : If you have some gas on a container and you increase the pressure while leaving the volume of the gas fixed, then temperature increases super-fast.
But can you even increase pressure but leave volume fixed?
If you can, then increasing pressure very fast should literally burn your gas up.
@BalarkaSen I can't help you with that right now. sorry. Nothing comes to mind. On the bright side, I've progressed 8 levels on Dink Smallwood since we've started chatting :D
why not just use "elementary results" other than a fancy term like "mathematical atom". you guys are talking like those philosophers who like to complicate words to make a simple sentence look complicated :P (which is a bit stupid, IMHO).
philosopher : "the great galactic light is emerging from skullpatrol along with a vibrant noise of the ultra-universal intersection"
@skullpatrol either way, i don't agree with what you'd call mathematics. it's not about connection of different branches. most of mathematics is about interpreting bad pictures using good logic, as Poincare said ;)
@Chris'ssis it contains lots of difficult integrals, series, limits ... only a handful number of people present in this chat can tackle them, you're of course one of them : )
Study of prime numbers is huge, from elementary properties to analytic perspectives and algebraic generalizations. Unless you be more specific, it's hard to tell what you're looking for.
OK, @Eka. Look at Crandall & Pomerance. I see from your profile that you are interested in computer programming, so the book would be more to your taste.
Hi all, I was wondering if someone could give me a hint for this problem. If f(z) is holomorphic on the punctured unit disk D\{0}, then I want to show 0 is a removable singularity of f(z) if and only if 0 is a removable singularity of f'(z)f"(z).
Now, derivatives of meromorphic functions and products of meromorphic functions are both meromorphic, so I only need to consider what happens when 0 is an essential singularity of f. This is where I'm not sure how to proceed. It's clear that f' and f'' have essential singularities as well, but products don't necessarily preserve essential singularities (exp(1/z)*exp(-1/z) for e.g.).
@PhilipHoskins Suppose that $f$ has a removable singularity of $f$. Then $f$ is holomorphic in your disk $D$ so $f'$ and $f''$ are, and $f'f''$ is, i.e. $0$ is a removable singularity of $f'f''$.
Well, yeah, I can always contract the equator by contracting in through the equatorial plane and then projecting it onto the surface of the circle but how does that tell us anything about $S^\infty$?
I need to read questions more carefully. I wrote a long, detailed answer to what I thought a question was asking, but it wasn't what it was asking at all. :(
It's not worth posting on its own; it was about the difference between the homotopy groups of a 1-skeleton of a complex and higher skeleta. I've saved the answer for future reference.
This question was asked here, where the answer uses this description.
The last line reads:
"The special case of $0$ does not have a unique representation in scientific notation, i.e., $0=0×10^0=0×10^1=..."$
My question is the value of $a$ cannot be $0$ since, as they state:
$a$ is a $\color...
