For a polar representation of a complex number, why the principal argument is to be considered in $(-\pi, \pi]$ or how can it arise?   I read some books where they just maintained it, but they don't give any reason. Also I read in  where they maintain: Some authors define the range of the principal value as being in the closed-open interval $[0, 2\pi)$, but why? Further I search in  and get two same problems(links are , ). But i am not satisfied with such comments only. I expect for a detailed explanation. My special focus is how did it assumed.  Any help is highly appreciated.
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Why the principal value of complex number argument is taken within $(-\pi,\pi]?$ What is the harm in considering similar other intervals like $[0,2\pi)$ for the purpose?
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When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$.  What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
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Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a - bi$ does the image end up meaningfully different from the field I started with? Or when we write out complex numbers are we arbitrarily choosing which of the non-real solutions to $z^4 = 1$ to call $i$ and which to call $-i$?
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This question was inspired by an answer and some comments to .   Recall that the Champernowne constant is obtained by concatenating all natural numbers written in base 10 and then put $0.$ in front, that is, $$C_{10}=0.123456789101112131415161718192021222324\cdots$$  The question is does the decimal expansion of $\pi$ occur as a tail of this number? By which I mean is there some $n$ such that $$\pi=C_{10}\cdot10^{n+1}-10\cdot\lfloor C_{10}\cdot10^n\rfloor$$  Now it is obvious that any finite initial string of $\pi$ occurs in $C_{10}$. Not only that, it also occurs infinitely often (that's just because any finite string of numbers occurs infinitely often in $C_{10}$).  Hagen von Eitzen (who's answer inspired this question) also notes that proving that $\pi$ does occur as the tail of $C_{10}$ would imply that $\pi$ is base-$10$-normal which is an open question see e.g. , so it is very unlikely we can prove that. He also notes "that if such a position exists [one where $C_{10}$ starts giving the digits of $\pi$] (somewhere in the middle of an $n$-digit integer, say) then the first $10^{n/2}n$ or so digits of $\pi$ turn out nearly regular and this should give rise to an unusually good rational approximation."  I tried to consider if there might be some relationship with relative algebraicity, but even in that direction little seems to be known since it is unknown even whether $\pi$ is algebraic over $e$.   In conclusion it seems intuitively extremely unlikely that $\pi$ would occur in $C_{10}$, but I can't think of a proof that it doesn't. A good answer would also be a reduction of this problem to some known open problem (note that we need to reduce that $\pi$ is not the tail of $C_{10}$).  PS If someone can come up with better tags please have a go at it.
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If $a+ib$, $c+id$, $e+if$ are three complex numbers, than can we tell which one is greater or smaller between them? If yes, then how and if no then why not?   Can somebody give explanation on this.... I will be grateful to him.
50083
Disclaimer: this is a homework question. I'm looking for direction, not an answer.     Given a field $F$, show that $F[x,x^{-1}]$ is a principal ideal domain.   I'm unsure how to proceed. Would it be better to prove this directly? (ie, let $I$ be an ideal, show that $I = (f)$ for some $f \in F[x,x^{-1}]$). The proof that polynomials are a PID would involve (I imagine) division by remainder and use of degree, both of which don't seem to have obvious parallels for Laurent polynomials. Should I try and devise some parallels and mimic the proof for polynomials? Or is this overkill? (or just wrong?)  edit 1: I guess the approach I mentioned above amounts to showing that Laurent polynomials are a euclidean domain (we know that euclidean domain => principle ideal domain, so this would be sufficient) Are they a euclidean domain though? (it seems like this would have been the question if they were, instead of asking if they were a PID).  edit 2: I've spat this out, think most parts of it are correct, though it seems kind of ugly/cumbersome (but that might just be me trying to spell things out more than is needed):     Given a Laurent polynomial $f \in F[x,x^{-1}]$, define its "negative   degree" $\deg^-(f)$ to be the largest power of $x^{-1}$ that appears   in $f$.      Let $I$ be an ideal of $F[x,x^{-1}]$. Note that $\{x^{-\deg^-(f)}f \mid f \in I\} \subseteq F[x]$. Let $J$ be the   ideal in $F[x]$ generated by this set. $F[x]$ is a principal ideal   domain, so $J$ is a principal ideal and we have $J = (j)$ for some $j \in F[x]$.      We claim $I = (j)$ (now meaning an ideal of $F[x,x^{-1}]$).      Let $f \in I$. Then $x^{-\deg^-(f)}f \in J$, meaning $f = x^{\deg^-(f)}g$, where $g = x^{-\deg^-(f)}f$ is in $J$. Because $g \in J = (j)$, there exists $g' \in F[x]$ such that $g = g'j$, and thus $f = (x^{\deg^-(f)}g')j$ is a multiple of $j$, so $f \in (j)$.      Let $f \in (j)$. Then $f = gj$ for some $g \in F[x,x^{-1}]$. But note   that $j = x^{-\deg^-(f')}f'$ for some $f' \in I$, so $f = gx^{-\deg^-(f')}f'$, so $f$ is a multiple of an element of an ideal   $I$, so $f$ itself is in $I$.      This shows $I = (j)$. So an arbitrary ideal of $F[x,x^{-1}]$ is   principal, so $F[x,x^{-1}]$ is a principal ideal domain.   I kind of feel like I still don't "get" the proof (I more-or-less see how each part works with the others but I'm having trouble seeing the bigger picture), though this may be due to a poor handle on ideals in general.
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Ted Shifrin on this site posed an interesting problem to me: show that   $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta &gt; 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta &gt;0$$  I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get   $$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$  where $(*)$ represents some trig stuff that is bounded near zero.   Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.  The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas?
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Consider the function $f(x)=\sqrt x$.  Pretty much every resource I can find gives its domain as $\{x\in \mathbb{R} \mid x \ge 0\}$ and its range as $\{y\in \mathbb{R} \mid y \ge 0\}$  I don't see why the domain needs to be restricted to positive numbers. Isn't the square root of a negative number defined? That is, the square root of a negative number is a complex number of the form $a + bi$.  So why isn't the domain of $f(x)=\sqrt x$ instead $\{x\in \mathbb{R}\}$ and its range $\{y\in \mathbb{C} \mid y \ge 0\}$?  What am I missing?
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Proposition: Characteristic of an integral domain must be either $0$ or prime number.   I'm confused by this proposition.  I think the characteristic of an integral domain should be always $0$. Suppose it has characteristic $n$. Then $n * a = 0$ for all a of the integral domain.  since n is not $0$ and, if $c * d = 0$ in integral domain, it means $c=0$ or $d=0$, a should be $0$. Hence $n * a$ is not $0$ when $a$ is nonzero. Therefore, characteristic should be always $0$.  What's wrong with my thought?
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Where $i$ is the square root of negative one.  And is there a generalization of the $n$th root of $i$? Also how would this look graphically on the real number axis? Thanks
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How is $3\pi-10$ element of $[-\pi/2,\pi/2]$? According to me this means that $530$ is an element of $[-90,90]$. i.e. Range of $\arcsin$ function.  (Note:- I was trying to find the value of $\arcsin(\sin 10))$)
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Say a function's domain is $\{-\pi/2, \pi/2\}$. How should I interpret this interval?  It starts from where? To where? In what direction?  And if it starts from $3\pi/2$, would the next one be $-5\pi/3$.(because it starts from negative, $-\pi/2$).
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in our online lecture we had the following statement but I don't see exactly why it's true , if anyone could explain me why  it would be appreciated.          We have  $a \in \mathbb{C}$ a complex number and $r \in \mathbb{R}$ with $r&gt;0$. Let $U=\mathbb{D}(a,r)\ _{\diagdown \left\{a \right\}}$ and $f \in O(U)$ a holomorphic function such that $Re(f(z)) \geq 0 \; \forall z \in U$.      And it says that $f$ can be extended ( in a holomorphic function )  all over the disk $\mathbb{D}(a,r)$ and that $a$ isn't a essential singular point.   Thanks in advance for your help .
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I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$).  I say this is practically the case, so in every calculation you just assume that "$a + 0i$" is equal to the real number "$a$" and you always get the right results (this might not even be true, but I am no expert and as far as I can see this is the case). But then again, I believe that just cutting away "$+ 0i$" makes this number is not the same kind of number anymore, it's changed its structure and is not totally equivalent anymore. This is where he says I am absolutely wrong.  So, is it mathematically (strictly speaking!) perfectly fine to say that every complex number with an imaginary part of $0$ is just a real number? Or does this change the mathematical structure so much that it cannot be mathematically-valid but just something that happens to work (which is what I believe)?  I must say that I believe a complex number is one entity, whilst he believes that it is just a conglomerate of other entitites. So, as one said here, "$(a + bi) - bi = a$", I believe is barely a good argument. But then again: is it correct to view $a + bi$ as a single entity (like "$5$" or "$2$", just consisting of multiple symbols)? I'm (as a non-mathematician) not sure there either.
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I'm a bit embarrassed that I've had difficulty on getting around this one:  $$e^{2\pi i x}$$  Solving it by itself, we can reduce it down to $(e^{2\pi i})^x = 1^x$ such that $e^{2\pi i x} = 1$ for all $x$.   However, directly plugging in non-integer rational numbers, chiefly $x = 1/2$, we get results that does not stay true to the above equality.   I'm a bit perplexed of trying to explain the different results for this one. I admit that I haven't had too much experience with complex numbers compared to other fields.
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Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be rational, or at least irrational? Thanks.
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The SR invariance formula makes space-like intervals imaginary (e.g., the distance $x$ in a given frame has interval $ix$). Yet modern physicists consider it bad form to define the distance itself as $ix$, as Minkowski did. It seems to me that this distinction is trivial—what’s wrong with defining the distance as the value of its interval? An answer to a previous question of mine on this topic referred to a paper by Visser showing that there is a way to do this that can be generalized to GR.  So what’s the fuss about? Even if there weren’t such a way, I don’t see how that would invalidate Minkowski’s definition for SR. Since the interval measure is what’s important, what fundamental difference can it make whether one defines the distance as x or as the imaginary interval $ix$?  The whole idea of a LT being an imaginary rotation (important, because phi is additive) depends on the use of the invariant measure, which is the complex length between two points in C2, the space in which complex rotations exist, and so the space in which the x-t plane is being considered to be embedded when we talk of complex rotations. In C2 it can easily be shown that the angle between any two axes, real or imaginary, is a real right angle, and this explains the invariance formula—it is simply the Pythagorean theorem for that complex triangle. So all this ‘invariant interval geometry’ is an integral part of special relativity. If you insist that distances are real and use a Pseudo-Euclidean metric, which would imply entirely different values for these angles, you are throwing out an enormous portion of SR. It is hard to see how that can be billed as an advance in our understanding.