I've only recently started learning about imaginary numbers, and there is one thing I cannot really wrap my head around:  $i^2 = i*i = {\sqrt{-1}} * {\sqrt{-1}} = {\sqrt{(-1) * (-1)}} = {\sqrt{1}} = 1$  I'm aware that by definition $i^2 = -1$, but as far as I know, even definitions have to obey mathematical rules.  So where is the mistake? :)    EDIT:  Thank you all, seems like the wrong bit is ${\sqrt{-1}}*{\sqrt{-1}}$  Cheers :)
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
When I'm going through a whole bunch of shots to determine which to delete and which to flag, it's fastest for me to just keep both hands on the keyboard.  However, if I want to zoom in to 100% on a picture and then pan to a specific area, it seems I can only get halfway there -- 'Z' to zoom, but then I'm stuck having to pick up the mouse to pan around.  I was excited to find page-up/page-down/home/end can be used, but they seem to exist to provide an exhaustive, linear scrolling of the image, panning top-to-bottom then left-to-right, without unconstrained 2d movement.  Is there some way to toggle the arrow keys so they can pan instead of navigate to the next/prev photo?  Something else?  Or am I out of luck? What causes a force field to be "non-conservative?" Understanding DNA match of 1,723 cM I have a DNA match of 1,723 cM with a male 6 years younger than me. His father is one of a twin, could my match be not my half-sibling but my 1st cousin. SSH: How to disable weak ciphers? what is the difference between kind, type and sort passthrough graphics card to virtualbox So I'm trying to get my laptops NVIDIA fx 880m to pass to a virtual box running window seven on a linux mint 17 install.  So far everything seems to be (maybe) heading in the right direction:  the device passed through happily, when I booted the virtual box it installed a bunch of new drivers, but when I try to install the NVIDIA driver on the guest it can't find the card. I looked under the device manager and there's no listing for the nvidia card under the pci bus, my guess would be that I need to disable it in the host so that it can be passed through to the guest (the virtual box manual said they can't be shared) but I'm not sure how to do so. Can anybody help with this? This question does not seem to me to be a duplicate of , because here the given solution was that it was not possible with a windows host, however in this case the host is linux Radiation– white vs black house, hot or cool? In my book's thermodynamics chapter, it says that an      "object that radiates heat faster also absorbs heat faster. This means   that an object that is a more efficient radiator comes to equilibrium   with its environment more quickly. With this in mind, is it better to   paint your house black or white?"   I am confused which it would be. The book says white because      "in summer, you house is cooler than the environment and white   reflects away the heat. In winter your house is warmer than the   environment and white radiates away the heat" dotted line with appropriate length for answering a question? Kindly have a look at this code   \documentclass{article} \usepackage{fontspec} \defaultfontfeatures{Ligatures=TeX} \usepackage[small,sf,bf]{titlesec}  \setromanfont{Georgia} \usepackage{amssymb}  \newcommand{\checkbox}{$\square$} \begin{document}  If appropriate \checkbox  . Why appropriate, specify? ........... If inappropriate \checkbox . Why inappropriate, specify? .............. If appropriate \checkbox . Why appropriate, specify? ...........\\ If inappropriate \checkbox . Why inappropriate, specify? ...........  \end{document}   How can I make a dotted like, of appropriate length so that it is depressed, fainter, has consistent length so that it occupies the remaining white space in the  line or has specified length. Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions? Write down the set of distinct solutions and prove your list is complete. $x$ and $y$ are positive integers.  I have rewritten it as $\frac{(x+y)}{xy} = \frac1p$, but I don't understand where to go from here? The Complete Rate-Limiting Guide I know it is defined as $ dP \wedge dx  $ or explicitly,  $$ \frac{\partial P }{\partial y } dy \wedge dx   . $$  The question is, could it be $dx \wedge d P $?  Or  $$  \frac{\partial P }{\partial y }  dx \wedge dy ?    $$  I know the question must be very naive or even stupid. But I am indeed confused. Computing $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ What is the result of  $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ . I tried to do couple of algebraic manipulations, but I didn't reach to any conclusion.  Thanks a lot. Split a file by line and have control over resulting files extension What are review tests (audits) and how do they work? Why are my render times 30+ minutes, but other people's are ~3 minutes? I am currently using Sheep It! to render my blend (built on v2.68c) on multiple computers, and I noticed something rather strange. According to the log on-site, the frames rendered from other people's blends were completing in 3 to 4 minutes. This was on an 4 core i5. My main rig (with an 8 core i7) was completing frames in no less than half an hour. How is this possible?  To clarify, these people's frames are very complex, often with very detailed textures. I have 2 objects in my blend. Below are the options I changed for my blend:   Aspect Ratio: 1000px x 1000px Output: PNG Clamp Indirect: 0.98 Render Sampling: 128 Min Light Bounces: 0 Max Light Bounces: 4 Cycles Render   Is there something in there that I should change to make it go faster? I can't really comprehend how it's going to quickly.  EDIT: To answer what @Duarte asked...   System Specs: i7-920, 3GB DDR3, GT 630, Windows 10 I have two circles, extended to form a sort of coin with a star on it. It is not complex at all. I have Firefox and Steam running in the background, but not actively. I've been away from the computer for 8 hours and noticed this behavior. (I am running Blender from Steam.) I made sure when I began rendering that the blend was set for CPU rendering.          And to answer what @pycoder asked, yes, I am using Cycles Render. Let $a$, $b$, and $c$ be complex numbers with $a\neq0$. Show that the solutions of $az^2+bz+c=0$ are $z_1,z_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, just as they are in the case when $a$, $b$, and $c$ are real numbers. I am not talking about miracles, religious revelation, or artistic expression, but something more mundane. There is a lot of "empirical" evidence that the Riemann hypothesis is true, the scare quotes indicate that the source of the evidence is "mathematical intuition". In psychology there are mental states that no one sees, touches, hears, smells or tastes, qualia with their perceptual fullness that can not be conceptualized or directly communicated, and abilities/skills "know how" with the same problem. In linguistics Kripke's theory of reference relies on "modal intuitions" to decide the truth of counterfactuals. But an empirical counterfactual is an oxymoron, especially when it is embedded into an entire "possible world".  Freudian psychoanalysis and Husserlian phenomenology are attempts to deal with some of them systematically, but they are traditionally confined to the fringe of science at best. The mainstream approach is empirical second hand, but behaviorism in psychology and linguistics, and nominalism in mathematics, were not very productive.   According to the naturalized epistemology there is no a priori "first philosophy" of science, methodology is subject to the "tribunal of experience" and revision, just as the science itself. Should it apply to non-empirical experience? Sense empiricism has served science well for centuries, and was fully embraced by Quine.  takes him to task for it:     "Cognitive science is an empirical science working to unearth the mechanisms through which natural language constitutes itself. That account has had to recognize the indispensability of mental states, of beliefs, if it is ever to become adequate to the problem... There is still too much "first philosophy" in Quine. We must rescue naturalized epistemology from its own founder".    Note the use of "empirical" together with "mental states".  Questions: Does naturalized epistemology dictate that natural science should relax sense empiricism, and approach non-empirical phenomena partly first hand? If so, how can the scientific method be adapted to introspective/intuitive phenomena that are not readily reproducible, manipulable, measurable, and/or publicly accessible? What would play the selective role of empirical testing? 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