What's wrong with this demonstration? (1 = -1) What's wrong with this demonstration?:  $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = (-1)^\frac{2}{2}$$ $$A \implies 1 = (-1)^1 = -1$$  So finaly we get  $$1 = -1$$
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? 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}
Get list of ALL offers from Amazon Product Advertising API Transforms moving around the 3D cursor? I think I accidentally pressed something to make it like this:    See how the XYZ arrows are at the 3D cursor instead of those vertexes I've selected?  How do I fix that? Whenever I select on a vertex or something, those blue, red, green arrows would always be where I selected, now it just keeps going to the 3D cursor. Finding net magnetic and electric force on charged particle This is from my textbook, it is not an assigned problem, but I want to understand.   It says:      Consider the situation in the figure, in which there is a uniform electric field in the x direction and a uniform magnetic field in the y direction. For each example of a proton at rest or moving in the x, y, or z direction, what is the direction of the net electric and magnetic force on the proton at this instant?   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(Normally you only get one land per turn so a max of 4 mana on turn 4.) Regard the diff equation:  $mϕ′′+aϕ′+(mg/L)ϕ=0$  $ϕ(0)=0.1$  $ϕ′(0)=0$  where $m=0.1,L=1,a=2,$  1) Rewrite the second order diff equation as a system of first order linear equations.  2) What is the matrix that comes from the rewritten equations? Why isn't this sudo mv operation with wildcard working? restore table from .frm and .ibd file? I have previously saved a copy of /var/lib/mysql/ddms directory ("ddms" is the schema name). Now I installed a new MySQL on a freshly installed Ubuntu 10.04.3 LTS by running apt-get install mysql-server, I believe version 5.1 was installed. After I copy the ddms directory under /var/lib/mysql, some of its tables work fine, these are the tables with an associated set of three files: a .frm file, a .MYD file and a .MYI file.  However, there are two tables with a different set of files: a .frm file and a .ibd file. These two tables didn't show up in the table list in phpMyAdmin. 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Now, I understand the logic behind the one-liner:  each input line is used as a key in a hash array, thus, upon completion, the hash contains unique lines in the order of arrival.  What I would like to learn is how exactly this notation is interpreted by awk. E.g. what the bang sign (!) means and the other elements of this code snippet.  How does it work? Lookup City and State by Zip Google Geocode Api I am trying to make a floor plan for my office, have managed to make all of the rectangular components (e.g. the rooms) however some of the desk layouts are circular and I haven't been able to work out how to make exact and measured circles. They need to have a radius of 1.45m. Repeat copies of array elements: Run-length decoding in MATLAB I have a custom xmodmap file I use, including useful multi-language diacriticals, english quotes, dashes &amp;c. I want to use this map with kmscon, so I need to create a xkb configuration from it. Is there an automated method to do it? 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