Defining roots of complex numbers I know that the equation $$ z^2=-8i $$ has two solutions, namely $z=\pm(2-2i)$. However, $\sqrt{-8i}$ is defined to be $2-2i$. Similar to the real case, $\sqrt{}$ selects the solution such that $\text{Re}(z)\geq 0$.  Why is this a convenient definition for the complex case? Is it merely the natural extension of the definition for the real case? Or is there a deeper reason? What about, for example, $$ z^4=-8i, $$ which has two solutions with positive real parts. What is the criteria here?
Principal $n$th root of a complex number This is really two questions.   Is there a definition of the principal $n$th root of a complex number? I can't find it anywhere. Presumably, the usual definition is $[r\exp(i\theta)]^{1/n} = r^{1/n}\exp(i\theta/n)$ for $\theta \in [0,2\pi)$, but I have yet to see this anywhere. Is this because it has bad properties? For instance, according to this definition is it true that for all complex $z$ it holds that $(z^{1/a})^b = (z^b)^{1/a}$?
I never really bothered to keep my Hotmail contact information up to date, so now that I connected it to Windows 8, suddenly I see all this outdated contact information or people I've long stopped talking to.    I don't see any buttons or options where I can click to delete someone.  So how do I clean up my Windows 8 People app and remove these contacts? $2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$ What’s the best way to reload / refresh an iframe? Is there any continuity between James Bond movies that have different actors portraying the title character? There was a foolish (in my opinion) question asked on Scifi.SE       James bond keeps returning with a different face, albeit usually a similar one, and often operates with high-tech gadgetry that seem to be on par with the Doctor's sonic screw driver.   In my opinion each new actor represents a reboot of the franchise using the new face, not a continuation of the story.   Is there any continuity between versions of James Bond which change actors? I'm not talking about characters like M or Q, I'm talking about events or past advisories being referenced in a different Bond movie where the main actor was someone else. Okay the scenario I'm thinking of is as follows:   I control a ,  and 3 untapped lands (2 swamps, 1 Island to be perdantic). In my opponents turn he casts a burn spell to kill my duskmantle.   I activate his first ability in response.   The way I am hoping the following interactions will go is:   Duskmantles ability will hit the stack at the top and resolve The burn spell resolves dealing its damage to Duskmantle The burn spell card hits the graveyard and triggers the infinite loop for duskmantle and mindcrank killing my opponent Duskmantle perishes   Is this the way that the stack would pan out? Do I win in this situation regardless of the stack being correct or not above? (I am not fully familiar with how the stack really works in magic) You can see the cards from my deck . Ideals in valuation rings How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion? "Baggage" versus "luggage" Dimensioning of a technical drawing in TikZ Is there is an easy way to mark dimensions in a technical drawing with TikZ?    Is there a library or something?  Edit  I am using XeLaTeX.  Update  I chose Martin's answer because it serves me well for the moment. Ultimately the best solution would be a library that, in an easy and consistent manner, would allow to change the arrow/dimension lines styles, would support polar coordinates, would allow to choose 2 nodes and the vertical distance that you want the dimension to be typeseted etc. How can I bring back the old menu style on Chrome 28 onwards, now that the --disable-new-menu-style switch has been removed? Rigorously proving equal tension on both ends There is always one point in introductory mechanics that has been continuously bothering me ever since I had taken my freshman physics course - Why does a massless rope have its tension equal on both ends?  I have been searching for a good explanation on various textbooks at different levels, but none of them provided me a satisfactory explanation. They either   State directly "we assume the string is massless and the pulley has no friction, therefore the tension must be the same throughout the string" without further any elaboration and continue the calculation, Claim without any reasons that "whenever the rope and the pulley are massless and frictionless, the tension will be the same on both sides", or Explain using a lot of hand-waving arguments regarding infinite acceleration, non-zero moment of inertia, etc. , in one line or two.   Although those hand-waving arguments of type $(3)$ do sound quite reasonable to me and will convince me for a while every time I review them. But after a while, I will start feeling not well about this whole tension concept again.  Therefore, I am looking for a rigorous (in the physicist's sense) proof of exactly why and when do the tension in rope will be uniform, and hopefully from that I can clean up all of my conceptual difficulties. Errors in matrix equation? I'm getting errors in a matrix equation. When I compile it looks fine, but I'd like to fix them so I don't have to press return for every error at compile time. The code for the equation is:  \begin{equation*}  \begin{bmatrix} U(1,t)  &amp; \cdots &amp; U(n,t) \\  \vdots &amp; \ddots &amp; \vdots \\ O_{n1} &amp; \cdots &amp; O_{nn}  \end{bmatrix} \times \begin{bmatrix} W(1,1)  &amp; \cdots &amp; W(1,n) \\  \vdots &amp; \ddots &amp; \vdots \\  W(n,1) &amp; \cdots &amp; W(n,n)  \end{bmatrix} = \begin{bmatrix}  D(1,t)  &amp; \cdots &amp; D(n,t) \\ \vdots &amp; \ddots &amp; \vdots \\ I_{n1} &amp; \cdots &amp; I_{nn} \end{bmatrix}  \end{equation*}   The errors I'm getting are related to missing $'s inserted. What to do if I can't get a visa to attend a conference that accepted my paper? My work got accepted into a very good category in a very good conference in my field. Due to complicated visa issues it will be very hard for me to attend that conference.   If I don't attend the conference they are certainly going to remove my work from the proceedings.  I am worried I won't be able to claim any sort of achievement if I don't attend this conference.  Can I still add it to my CV and specify that my work got accepted in this conference in a very good category? Does any skill have a global cooldown? I am new to this game, and will try to optimize my game-play as much as possible. I will do this by making macros for my spells and abilities.  Are there global cooldown on spells and abilities in League of Legends?  Since this is important regarding macros to trigger spell and ability combos.  A Global Cooldown(GCD) is a cooldown which affect most if not all abilities, spells or items when using an ability, spell or item.   In most games it is there to prevent macroing/spamming of multiple instant abilities, spells or items, and usually has a set short duration independent of the ability, spell or items normal Cooldown. Do elevator upgrades increase your passive credit earnings, too? $\lim_{x\to 0} \frac{\sin x - x}{x^2}$ without L'Hospital or Taylor Start script on open blender file Possible Duplicate:          I sometimes work outside and are using a 3G usb connection with quite a limited data allowance.  I want some fast and easy way to temporary block all incoming and outgoing traffic in windows firewall except for chrome to be sure that Dropbox or something else is not using up my data.  I've tried to set Windows Firewall Properties to "Block" which gives "Outgoing/Incoming connections that do not match a rule are blocked".  Problem: There are many allow-rules like Dropbox that would "override" the block. And I don't want to disable all rules and then enable them, since there are some that I want disabled even when I get to the office...  Any suggestions?  Thanks! I see pictures online of everyone having all these Pokémon to catch being everywhere around them, but it seems like I can only ever see one or two. I live in a rural area, but even walking miles through the farmland, it's rare to find any Pokémon.  Why is this? How can I find more Pokémon? The various modern revolutions in physics, in psychology, in politics, even in literary style, have not escaped his intelligent notice, but they can scarcely be said to have influenced him deeply.    Could you please explain the following?  They can scarcely be said to have influenced him deeply.  Or, I can not understand such a structure like the following:  In contrast, prices set by a central planner are necessarily arbitrary, since there is no standard against which they can be said to be correct or incorrect. d'oh! If $E$ has Lebesgue measure $0$, must there exist a translate such that $E\cap E+x=\varnothing$? If $y\in E\cap E+x$ then $y=h=h'+x \implies x=h-h'$ for $h,h'\in E$. Likewise, if $x=h-h'$, then $h\in E\cap E+x$. So that $E\cap E+x$ is non empty iff $x=h-h'$ where $h,h'\in E$. Thus, if there exists a real $x$ which cannot be expressed as above, the result is proven. The problem is I cannot prove that such an $x$ must exist. I have tried assuming that all real $x$ are of that form, but that gives me nothing except that $E$ must be uncountable and unbounded. Any hints?