If imaginary numbers bisect real numbers at zero, am I correct in thinking $0i = 0$?  And if so, does that mean $i = 0/0$?  EDIT:  I’ve just realised my careless error in my thought process.  I immediately just “removed” the $0$ from both sides without thinking about the fact that I was actually simplifying $\frac{0i}{0}=\frac{0}{0}$ to $i=\frac{0}{0}$ despite $\frac{0}{0}\neq1$, meaning $\frac{0}{0}i \neq i$
Why does $x$ divided by zero not equal $x$? After all, $x$ is not being divided by anything.
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help. Where's the problem in this equation? Resulting in $4 = 5$ I just saw this equation and I can't find out where's the problem:  $$25-45 = 16-36$$  $$25- 2 \cdot 5 \cdot \frac{9}{2} = 16- 2\cdot4\cdot\frac{9}{2}$$  $$25 - 2\cdot 5\cdot \frac{9}{2} + \frac{81}{4} = 16 - 2\cdot 4 \cdot \frac{9}{2} + \frac{81}{4}$$  $$\left( 5-\frac{9}{2} \right) ^2 = \left (4-\frac{9}{2} \right) ^2$$  $$5-\frac{9}{2} = 4 - \frac{9}{2}$$ $$5=4$$ Static nested class in Java, why? Pointwise convergent and total variation Prove that $n^3=n \text{ mod }6$ for every integer $n$. Prove that for every integer $n$ , $n^3=n \text{ mod }6$  I was having no clue how to do this, then I thought of case-by-case analysis and obviously it worked. The problem is that there were six case and together they are long.   Is there any shorter method of proving this. Kindly provide some hints. If APS-C and similar crop-sensor digital cameras have a focal length multiplying effect such that a 50mm lens has an apparent focal length closer to the field of view of an 80mm on a full frame camera, and yet at the same time the depth of field for the smaller sensor camera is more like the depth of field a 50mm lens would produce on a full frame camera (using the same aperture), then this would seem to suggest the concept of an "aperture dividing effect."  In other words, a 50mm f/1.8 lens on an APS-C camera would act more like a 80mm f/2.8 (approx. 1.8 * 1.6x) lens in 35mm equivalent — for depth of field, not considering exposure.  Can someone with a better understanding of the physics involved clarify this for me.  I've never seen this concept explicitly mentioned anywhere, so I am a bit suspect of it. After monitor calibration colors on my two monitors are still different, why? I have a pair of pretty nice monitors (HP ZR30w) which I calibrated with a pretty good calibrator (i1-Pro) and after I calibrated both they still show colors every so slightly differently.  My left monitor has a little more magenta color to it in photos.  Is monitor calibration only ment to get things 'close'?  Other people in the office have experienced similar issues with their pairs of monitors, both HP ZR30w and Dell 3007WFP-HC monitors.  If I can't even get colors to match between my two exact same monitors connected to the same computer I have little confidence that colors on my monitors will look the same on somebody else's calibrated monitor :p lcm versus phi in RSA In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) = (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the Carmichael lcm function $$\lambda(pq) = \operatorname{lcm}(p-1,q-1)$$ to define $d´$. It is clear that the $d´$ divides $d$, and therefore using $d´$ may be more efficient than using $d$. My question is: Are there any further reasons, e.g. regarding security, why one should not use $d$? I am using lightning:recordEditForm and lightning:inputField for creating a new record. this component is being used inside a lightning community builder. When i click the submit button, i am able to create a new record in my custom object. However i need the created record ID as i need to pass it in the URL for the next page. If $X$ is distributed $N(\mu_X, \sigma^2_X)$, $Y$ is distributed $N(\mu_Y, \sigma^2_Y)$ and $Z = X + Y$, I know that $Z$ is distributed $N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$ if X and Y are independent.  But what would happen if X and Y were not independent, i.e. $(X, Y) \approx N\big( (\begin{smallmatrix} \mu_X\\\mu_Y \end{smallmatrix}) , (\begin{smallmatrix} \sigma^2_X &amp;&amp; \sigma_{X,Y}\\ \sigma_{X,Y} &amp;&amp; \sigma^2_Y \end{smallmatrix}) \big) $  Would this affect how the sum $Z$ is distributed? It is stated that if $\psi(x_0)=\psi'(x_0)=0$ at some $x_0$, then $\psi(x)=0,\forall x$.  Hence, we cannot have the wavefunction and its derivative vanishing at the same point.  My question is, how to prove the above statement? Group of even order contains an element of order 2 I am working on the following problem from group theory:     If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.   That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$. This question dovetails with this post on .  I am familiar with some of larger, paid-for business registry's like  and .  While the detailed information their business listings database include (like # of employees, annual sales, industry code, etc) is fantastic, if you are on a limited budget, sometimes the cost of purchasing the data outweighs the value you can get from it.  If you have geocoding capabilities in house, one could potentially create a reasonable surrogate by spatially-enabling (geocoding) a publicly available business listing on your own.    If you were looking to take on this type of project, where would you start your research? Do you know of US agencies at a State or National level that maintain this type of information, AND are publicly available for download or minimal purchase? I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1}, $$ where $|x|&lt;1$. bash regex with quotes? How can I make Devanagari fonts render properly in Ubuntu 16.04? Devanagari fonts appear broken like this:     It should appear this way    I have the same problem in browsers and office applications.  Is there any way to fix it? Why are image thumbnails cropped proportionally (not per dimensions)? For some reason WordPress is ignoring my thumbnail sizes and keeping thumbnails cropped proportionately instead of specific dimensions. Here is what I have in functions.php:  add_image_size( 'main-thumbnail', 728, 410, true );   and in content.php:  &lt;div class="thumbnail"&gt;     &lt;a href="&lt;?php the_permalink(); ?&gt;"&gt;        &lt;?php the_post_thumbnail( 'main-thumbnail' ); ?&gt;     &lt;/a&gt; &lt;/div&gt;   I added these two things and regenerated my images with the  plugin, but it doesn't seem to matter. The images don't change.  For some reason WordPress isn't recognizing the hard crop and is keeping the images proportional, so some of my images are way taller than 410 pixels. Why is this? Difference between declaring variables before or in loop? How can I tell Ubuntu to do nothing when I close my laptop lid? There has been some discussion about getting notified on comments to questions or answers that don't belong to you, and the long-standing wish to be able to keep track of new answers or developments in questions asked by other people.  Previous discussion in regards to subscribing to questions:                  (very similar to this one)      In regards to subscribing to comments:                    The general feeling - at least here on meta - seems to be against broadening the notification rules, out of fears of useless notifications to flood users' inboxes, turning SO into more of a discussion forum than it should be, etc.   Here's a suggestion that would keep the user 100% in control what they get notified about, not generate any notifications by default, and require only minimal modifications to the User Interface.  Each question and each comment area would get an envelope symbol.    it is inactive (=white) by default.  When those envelopes are hovered over, the tooltip says click to subscribe or something.  When clicked on, the envelope lights up in green or some other colour (but different from the actual notification envelope's orange to avoid confusion.) You are then subscribed to any updates to that question or comment thread.    For each subscription, you receive notifications in the "responses" tab, as if you were the owner of the comment or question.     To unsubscribe from a question or comment thread, you open the page, and click the green envelope. It turns back white, and the notifications disappear from your responses tab (if technically possible - otherwise, there just won't be any new ones added.)    No further functionality, like managing subscriptions or anything, is necessary IMO. If the flood of notifications from a subscribed question starts to annoy you, you just open it and unclick the envelope. Done.   It would make it possible to leave a comment to a newbie's question and make sure their answer reaches you even if they don't know they need to address you using @username You could subscribe to those "oh, I'd like to know the answer to that" questions and receive active notification, instead of them getting forgotten in your favourites list You could subscribe to discussions you are interested in the outcome of, but don't want to participate in. It would (in my opinion) sort out a number of woes aired by many users in a good way, without adding annoyances to those who like the system just fine the way it is. It would not require a change in the visible layout. As far as I can see, the space the envelope would need to be is empty everywhere.   I don't know SO under the hood of course, but technically, the system would be doing what it's doing already, notify question owners of updates - just that the circle of internal "owners" would grow.