I had to solve: $z^2 + 1 - i = 0$. What I did:  $$z_{1, 2} \frac{-0 \pm \sqrt{0 - 4(1 - i)}}{2} = \pm \sqrt{i - 1}$$  Now, I need to find $\Im(z_{1, 2})$ and $\Re(z_{1,2})$.  I think there is a way to solve it using the trigonometric representation, right? I somehow didn't found what I wanted so I'd love if someone can just tell me what I should look up. I'm okay with you people just giving me the proper term to google. :)  Thanks
If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
How can I control where smoke and fire are emitted from? Finding determinant of matrix without expanding show that determinant $$\left|\matrix{ x^2+L  &amp;  xy   &amp;   xz \\ xy   &amp;    y^2+L &amp;   yz \\ xz   &amp;    yz   &amp;   z^2+L \\ }\right| =  L^2(x^2+y^2+z^2+L)$$  without expanding by using the appropriate properties of determinant.  All i can do is LHS  $$x^2y^2z^2\left|\matrix{ 1+L/x^2  &amp; 1  &amp;   1 \\ 1   &amp;   1+L/y^2 &amp;   1 \\ 1  &amp;    1   &amp;   1+L/z^2 \\ }\right|$$ How do you count the lines of code in a Visual Studio solution? Showing $\left(1+\frac{1}{n}\right)^n$ is an increasing sequence by comparing the binomial expansions term by term So I know that showing $\left(1+\frac{1}{n}\right)^n$ is an increasing sequence has probably appeared on this site about 100 times, but my professor said he thinks the induction step is most easily seen if you expand this out and compare the summation term by term. I tried this and I did not what see why the inequality was obvious. Can somebody help me out here? Thanks! When I ssh into my Ubuntu 18.04 box I get the motd  1 update could not be installed automatically   Which goes on to tell me to check the unattended-upgrades log.  The log shows no problems. How can I clear this bogus message? I can't figure how to print the string "-n" with the over 30 years-old command echo.  I tried these:  Michaels:Scripts$ echo -n Michaels:Scripts$ echo "-n" Michaels:Scripts$ echo -e "-n" Michaels:Scripts$ echo -- -n -- -n Michaels:Script$ echo -- "-n" -- -n Michaels:Script$ echo "\-n" \-n   I use Mac OS X ML. I consider this behaviour very weird and unexpected. How do I safely delete all files with a specific extension (e.g. .bak) from current directory and all subfolders using one command-line? Simply, I'm afraid to use rm since I used it wrong once and now I need advice. From Awodey: In any category $\mathsf{C}$,  an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$,  an object $1$ is terminal if for any object $C$ there is a unique morphism $C \to 1$.  First I was puzzled by the fact that we can do mappings from a zero set to any other set in a category $\mathsf{Sets}$. But then after I accepted it conceptually, I do not understand why there is no unique morphism to an empty set from every object in a category $\mathsf{Sets}$. Is this because we cannot map a non-empty set to an empty set (since our mapping is total) although we can map an empty set to a non-empty set?  Please, note, I am familiar with the proposition saying: Initial (terminal) objects are unique up to isomorphism. Can I fetch the value of a non-standard CSS property via Javascript? If a sports team is down by 1-2 in a best of 7 series, what is their chance of winning (if both teams have a 1/2 chance of winning every game)? I know that the team has to win 3 of the 4 remaining games. So I thought that from the 4 remaining games, there are a total of $2^4 = 16$ total different outcomes.   Could someone point me towards the right direction to determine the probability of the team winning 3 of the 4 remaining games? Would it be 4C3/16 since the three wins can be chosen amongst the 4 games left? How do I prevent Eclipse from hanging on startup? I am using Eclipse 3.3 ("Europa"). Periodically, Eclipse takes an inordinately long time (perhaps forever) to start up. The only thing I can see in the Eclipse log is:       !ENTRY org.eclipse.core.resources 2 10035 2008-10-16 09:47:34.801     !MESSAGE The workspace exited with unsaved changes in the previous session; refreshing workspace to recover changes.   Googling reveals  that I remove the folder:  workspace\.metadata\.plugins\org.eclipse.core.resources\.root\.indexes   This does not appear to have helped.  Short of starting with a new workspace (something which I am not keen to do, as it takes me hours to set up all my projects again properly), is there a way to make Eclipse start up properly? Are two (or N) resistors in series more precise than one big resistor? Let's say I have one 2 k&ohm; resistor with 5% tolerance. If I replace it with two 1 k&ohm; resistors with 5% tolerance, will resulting tolerance go up, down, or remain unchanged?  I'm bad with probabilities, and I'm not sure what exactly tolerance says about resistance and its distribution.  I am aware that in the 'worst case' it will be the same; I'm more interested in what will happen on average. Will the chance of getting a more precise value increase if I use a series of resistors (because deviations will cancel each other out)?  On 'intuitive level' I think that it will, but I have no idea how to do the math with probabilities and find out if I'm actually right. all the LaTeX-tutorials tell me how LaTeX is an extention simplifying the building of TeX documents. Understandably, no one uses plain TeX anymore, right?  But I seem to learn better when I grasp the link between TeX and LaTeX. Does someone have a link to example files in plain TeX? I'd like to know which commands, macros etc. are LaTeX-specific. I only occasionally use the Fixed Zoom In/Out buttons on the Tools toolbar of ArcMap and had always assumed that they zoomed in/out by a set scale factor.  However, using ArcGIS for Desktop 10.2, I just set my scale to be 1:10,000,000 and then hit the Fixed Zoom In button 10 times while recording the subsequent scale each time and was surprised to see:  10,000,000 8,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,500,000 2,000,000 1,500,000 1,250,000 1,000,000   Dividing each scale denominator by its successor shows a sequence of: 1.25, 1.33, 1.2, 1.25, 1.33, 1.2, 1.25, 1.33, 1.2  Curious about what happened in between the scales above I experimented a little to discover that the first issuance of the Fixed Zoom In button simply takes the scale to the first of those listed above and then starts following the same pattern.  Try starting with 1:8,000,001 to see ...  8,000,001 8,000,000 6,000,000 5,000,000 4,000,000 3,000,000   This seems like quite a neat solution, but not one that I had realised was in place - does anyone know if it has been documented anywhere?    I've looked in the Online Help without success.  In answer to @DevdataTengshe, the scales currently in my available list (see below) seem to bear no relation to those used by Fixed Zoom In/Out.    However, when I add 1:7,500,000 into my available list it does seem to insert that as an additional scale step ...  10,000,000 8,000,000 7,500,000 6,000,000 5,000,000 4,000,000   This makes the algorithm being used even more mysterious to me. Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$ I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{\lambda ^2x^2+\lambda x+1}\text{d}x =\frac{1}{\lambda}\int_0^{\infty}\frac{(\ln x-\ln \lambda)^2}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\int_0^{\infty}\frac{1}{x^2+x+1}\text{d}x+\frac{1}{\lambda}\int_0^{\infty}\frac{\ln ^2x}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\frac{2\pi}{3\sqrt{3}}+\frac{2}{\lambda}\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x$$ But I hav no idea on process the remaining integral:( Anyone knows how to solve it?  THX guys! I came with another method might work for this: Recall the handy GF $$\frac{1}{x^2+x+1}=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ Then we have :  $$\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\frac{2}{\left(k+1\right)^3}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ With $$\sum_{k=1}^{\infty}\frac{\sin (kx)}{k^3}=\frac{\pi ^2}{6}x-\frac{\pi}{4}x^2+\frac{x^3}{12}$$ What can we arrived? I'm using the plugin  which registers it's own post type but the capability is assigned to "post".   I want to change it to "manage_options".  Is there a correct way without altering the plugin to do that? Calling a hook, a function or whatever?   public function register_post_type() {      $labels = array (         'name'               =&gt; _x( 'Custom Code', 'post type general name' ),         'singular_name'      =&gt; _x( 'Custom Code', 'post type singular name' ),         'menu_name'          =&gt; _x( 'Custom CSS &amp; JS', 'admin menu' ),         'name_admin_bar'     =&gt; _x( 'Custom Code', 'add new on admin bar' ),         'add_new'            =&gt; _x( 'Add Custom Code', 'add new' ),         'add_new_item'       =&gt; __( 'Add Custom Code' ),         'new_item'           =&gt; __( 'New Custom Code' ),         'edit_item'          =&gt; __( 'Edit Custom Code' ),         'view_item'          =&gt; __( 'View Custom Code' ),         'all_items'          =&gt; __( 'All Custom Code' ),         'search_items'       =&gt; __( 'Search Custom Code' ),         'parent_item_colon'  =&gt; __( 'Parent Custom Code:' ),         'not_found'          =&gt; __( 'No Custom Code found.' ),         'not_found_in_trash' =&gt; __( 'No Custom Code found in Trash.' ),     );     $args   = array (         'labels'              =&gt; $labels,         'description'         =&gt; __( 'Custom CSS and JS code' ),         'public'              =&gt; false,         'publicly_queryable'  =&gt; false,         'show_ui'             =&gt; true,         'show_in_menu'        =&gt; true,         'menu_position'       =&gt; 100,         'menu_icon'           =&gt; 'dashicons-plus-alt',         'query_var'           =&gt; false,         'rewrite'             =&gt; array ( 'slug' =&gt; 'custom-css-js' ),         'capability_type'     =&gt; 'post', // &lt;--- I want to manipulate this         'has_archive'         =&gt; true,         'hierarchical'        =&gt; false,         'exclude_from_search' =&gt; true,         'menu_position'       =&gt; null,         'can_export'          =&gt; false,         'supports'            =&gt; array ( 'title' ),     );      register_post_type( 'custom-css-js', $args ); }   The above code is in custom-css-js.php lines 200-239. A great Python IDE for Ubuntu I don't know if it's the right place to ask but can somebody advise me for buying a ram memory for my computer.  I have a Dell Vostro 200 with 1GB of RAM and 667MHz. (2 x 512MB)  I want to buy raise the amount of RAM but I'm confused whether to buy 1 x 2GB or 2 x 1GB  I know it's the same for the available amount of memory but is ther a difference maybe in the clock speed so that 2 x 667MHz will be better than 1 x 667MHz. or maybe accessing one single memory slot will be faster and better than accessing two slots.  Please anyone helps. First of all, I know that light does indeed travel slower in a medium like air or water, but that's because the photons are bouncing off of the medium's particles and in different directions so the light beam as a whole is slowed down (or at least that's the classical explanation I've read somewhere).  But my question is about light traveling freely in the vacuum. What if an astronaut during a spacewalk switches on a flashlight towards the opposite direction he is moving. Will that light travel slower than normal?  If not, is there another way to make light move slower? For example, can you make a beam of light move so slow that you can visually perceive it being affected by gravity? Adding tags to files in OS X: How to use, where are tags stored, and will they work on a network?