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Jul
28
comment Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$
I've edited my post and added the surjectivity argument.
Jul
28
answered Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$
Jul
28
comment Find equation that represents $m$ as the subject
@Nick The answer is, essentially, that you need to plug numbers into formulas, or perhaps write a computer program to do so if you can't face doing it yourself. If you try to solve it algebraically, you'll end up with something disgusting like George's formulas above no matter how you spin it (Wikipedia's methods are all different ways of saying the same thing). I like George's recursive algorithm, but of course it (almost certainly) won't give you exact answers. Still, let me know if I can help with anything.
Jul
27
comment Find equation that represents $m$ as the subject
Perhaps the most appropriate tag is "algebra-precalculus". (Though I wouldn't expect anyone to learn how to solve a cubic equation until well after calculus, admittedly, but it has the same flavour as a quadratic equation, just much uglier...)
Jul
27
answered Find equation that represents $m$ as the subject
Jul
27
comment Why do I disagree with my calculator?
I agree with GEdgar, but if remembering that numbers have to be packaged with the sign to the left of them is hard, it's enough to remember that + and - are to be given the same priority, and so you should work from left to right, which always gives the right answer.
Jul
27
comment How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole
Ah, sorry - my three-dimensional mental picture confused me. Thanks for the clarification.
Jul
27
comment How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole
But all rotations that send P to S do arise in this way, right?
Jul
27
comment Euler's phi function $\phi(n)$ is even for all $n \geq 3$; when is it not divisible by $4$?
What do you know about $\phi$? Can you calculate, for example, $\phi(27)$?
Jul
27
comment Why do real powers need positive bases? (and to handle them)
You are absolutely right. It is not true that the power $a^b$ is only well-defined when $a\geq 0$. You can ascribe a well-defined value to $a^b$ for any $a$ when $b$ is a positive integer, for example; in fact, you can extend the function far more than that. But you lose a lot of its nice properties (e.g. $a^{b_1} a^{b_2} = a^{b_1+b_2}$) when you do so. It's fairly safe to say that the power $a^b$ is only defined *for all $b$* and *in a nice way* when $a\geq 0$.
Jul
27
awarded  Quorum
Jul
27
comment how to interpret theorem about polynomial factorization over modulo ring?
Have you misunderstood the result? You are given a formula - $a_n(1 + \sum a_i)$ - and told that, if it's $0$, then your polynomial has a factor, and if it's not $0$, then it doesn't. There is no need to distinguish between trivial and non-trivial linear factors, because the result is true in both cases. $X+1$ is a factor of $X+1$, even if you want to think of it as somehow "trivial".
Jul
27
comment how to interpret theorem about polynomial factorization over modulo ring?
(What is a "trivial" linear factor anyway? There are only two linear polynomials - $X$ and $X+1$ - and neither satisfies the condition $a_n(1 + \sum a_i) \neq 0$.)
Jul
27
comment how to interpret theorem about polynomial factorization over modulo ring?
The claim is that that polynomial has no linear factors if and only if $a_n(1 + \sum a_i) \neq 0$. In the case of $X+1$, $a_n(1 + \sum a_i) = 1(1+1) = 0$. So you should expect a linear factor. This linear factor is $X+1$. Where's the confusion?
Jul
27
answered What character can replace word “let” in proofs?
Jul
27
comment Arithmetic Base Conversion
If x is an even number (except 2), write 1 + (x-1). If x is an odd number (except 1, 3, 5, 7), write 1 + 3 + (x-4). Or, if you like, x is already a sum of one odd number (x).
Jul
23
comment Prove $\sqrt{k}$ is not a rational number.
Sorry, but this is nonsense. The English doesn't even make sense. Can you rewrite your proof carefully?
Jul
23
comment The union of a sequence of countable sets is countable.
You don't need to pinpoint duplicates. Here's another argument (I'm going to take $\mathbb{N} = \{1, 2, \dots\}$): enumerate all the elements of each $E_n$; send the $m$th element of $E_n$ to $(p_n)^m$, where $p_n$ is the $n$th prime. This is an injection of $\bigcup E_n$ into $\mathbb{N}$. (If the $E_n$ are allowed to overlap, then for each $x\in\bigcup E_n$, just consider $x$ as an element of $E_n$ for the smallest valid $n$, I suppose.)
Jul
23
comment Is this infinite series a Fourier series?
Noted. You're absolutely right, and it's sloppy of me to talk about "convergence" - I simply wasn't prepared to copy out half a textbook about when a fourier series is useful and when it's not. To the best of my knowledge (though correct me if I'm wrong), an arbitrary choice of coefficients $a_i$ is more often than not meaningless - I was simply trying to get across this idea.
Jul
23
answered Is this infinite series a Fourier series?