I'm trying to teach middle schoolers about the emergence of complex numbers and I want to motivate this organically. By this, I mean some sort of real world problem that people were trying to solve that led them to realize that we needed to extend the real numbers to the complex.   For instance, the Greeks were forced to recognize irrational numbers not for pure mathematical reasons, but because the length of the diagonal of a square with unit length really is irrational, and this is the kind of geometrical situation they were already dealing with. What similar situation would lead to complex numbers in terms that kids could appreciate?  I could just say, try to solve the equation $x^2 + 1 = 0$, but that's not something from the physical world. I could also give an abstract sort of answer, like that $\sqrt{-1}$ is just an object that we define to have certain properties that turn out to be consistent and important, but I think that won't be entirely satisfying to kids either.
I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious.  So what practical applications of complex numbers exist and what are the ways in which complex  transformation helps address the problem that wasn't immediately addressable?  Way back in undergrad when I asked my professor this he mentioned that "the folks in mechanical and aerospace engineering use it a lot" but for what? (Don't other domains use it too?). I'm well aware of its use in Fourier analysis but that's the farthest I got to a 'real world application'. I'm sure that's not it.  PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That's going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?  The question:  is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO.
If $S_n$ has cyclic subgroups of these orders or not? This question was asked in my quiz and I am having difficulties in solving it .  Question : For a positive integer $n$ , let $S_n$ denote permutation group on $n$ symbols. Choose correct statements from below. (a) For a positive integer $n$ and for every $m$ with $1\leq m \leq n , S_n $ has a cyclic subgroup of order $m$; (b) For every positive integer $n$ and for every $m$ with $n&lt;m &lt;n!$, $S_n$ has a cyclic subgroup of order $m$; (c) There exists positive integers $n$ and $m$ with $n&lt;m&lt;n!$ such that $S_n$ has a cyclic subgroup of order $m$; (d) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of order $S_n$ .  Attempt: (a) is true as I can choose an element of length $m$ which will generate cyclic subgroup of order $m$. (b) Choose $n=4$ and $m=19$ to contradict (b). (c) I dont know if there can be found any such case and if yes how to do so . (d) I dont know about it . kindly tell. Answer:   A,C,D are true Can we switch terminals at Heathrow (LHR) for shopping only? Pressing Ctrl Alt NumPad 0 causes the camera to teleport over and align itself to the current view in the 3D viewport. However, it usually takes up only a small portion of the view, meaning that you have to move the camera back to fit the same view you had before. Is there a way to set the camera view to fill more of the 3D viewport when using this feature? Intuition for $\,c\mid a,b \iff c\mid \gcd(a,b)$ Retrieve malicious IP addresses from Apache logs and block them with iptables Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen? Why does the Hilbert curve fill the whole square? "Those who qualify will be awarded a certificate" or "those who qualify will be awarded certificates"? Drive from US to Canada with a dog Book about almost extinct warrior race with three castes: sen, kaa, ker The aliens in this book are an humanoid alien mercenary race, fighting for some other aliens, against humans?.   The Ker are the fighters, the sen are the wise ones, and the kaa the family/non-fighters.  Their clan gets almost wiped out because both their employers and humanity fear them, and the last survivors head towards their race' home planet, together with a human captive, who learns about their ways.  The other aliens are some blob-like aliens with little genetic variety.  Ring any bells? SQL Server FOR XML Enclosing Element? Given the following data:  972, 975, 985, 993, 993, 995, 998, 1001, 1004, 1007, 1008, 1009, 1011, 1015, 1016, 1020, 1022, 1032  I calculate the lower quartile as follows:  Number of values: 18, Number of 'gaps': 17.  Lower quartile is at 17/4 + 1 = 5.25th value. 5.25th value is 993 + (995 - 993) x 0.25 = 993.5  Upper quartile is at 3 x 17/4 + 1 = 13.75th value. 13.75th value is 1011 + (1015 - 1011) x 0.75 = 1014  Both the Excel QUARTILE function and the R quantile function agree with me. However the book I am using (Understanding Statistics by Graham Upton and Ian Cook) give different answers (993 and 1015 respectively). I'm confused. Which is correct? How to not count blank pages in frontmatter of two-sided document Concatenating a string and primary key Id while inserting Determine if Type is a pointer in a template function Is it possible to import css stylesheets into a html page using Javascript? If so, how can it be done?  P.S the javascript will be hosted on my site, but I want users to be able to put in the &lt;head&gt; tag of their website, and it should be able to import a css file hosted on my server into the current web page. (both the css file and the javascript file will be hosted on my server). Receiving commandline input while listening for connections in Python I'm not asking the diagonal values, but the length and breadth ones of the Nexus 7. And only of the LCD screen, not the entire hardware. Open more than one instance of Windows 10 settings How can I open more than one instance of the Windows settings?  It wasn't a problem with the old style, but with the new one I can't seem to open multiple settings simultaneously and side by side.  Example of the new settings window: I want to cd into /var/named but it gives me a permission denied error, and when I want to use sudo to do this I am not permitted. What is the technical reason for this, and is it possible to do this some other way?