Are complex numbers just vectors defined by cosine (x axis) and sine (y-axis)? There is no clear and standard definition of a complex number, I recall a prof being at a loss when a freshman aske him what is a complex number.  I read somewhere that a great mathematicians was not ashamed to say, re the formula $e^xpi = -1$ I think, that nobody knows what it means , but they are proud of it, is that true?*  I suppose nobody has a clear answer, I am just asking to specify what is more or different from the regular operations on a vector field taken as a 360° circle, where points are identified by  cosine of the relative angle (real part) and sine (imaginary part)  Edit  The linked question/answer resembles only superficially my question. I stressed the crucial feature that the vector is defined by sin and cos and consequently obeys the rules of duplication etc of angles. I am not familiar with all details of that theory and I am asking , if any, about the differences. If there are none or minor ones, then complex numbers ar not numbers and not complex at all.  *I found the quote:      Benjamin Peirce, a noted American 19th-century philosopher,   mathematician, and professor at Harvard University, after proving   Euler's identity during a lecture, stated that the identity "is   absolutely paradoxical; we cannot understand it, and we don't know   what it means, but we have proved it, and therefore we know it must be   the truth   *
Relationship between complex number and vectors What is the relation  between complex numbers and vectors?  I have read in some places "a complex number a 2-dimensional vector".  One can easily observe that $i\cdot i=-1$ in complex multiplication like the cross product of vectors.    Is  vector only the generalisation of complex number to n dimensions?   Is there any fundamental difference between those two?
This question is essentially the same as one of my , except the only difference is that I am using "only" to change the text at the same location (i.e., ). In other words, I am destroying my earlier text as I go, and not adding new ones underneath the previous ones.   Since the text for the first "only" is shorter than the second "only", therefore the image will shift location. I would like for the image to remain at its location.   Here is an example     it may not be obvious but the second image is higher up than the first.  The code is here, can someone find a solution?  \documentclass{beamer}  \usepackage{graphicx}  \title[Title]{Presentation} \author{Sandro Botticelli} \institute{Italy} \date{1484}  \begin{document}   \begin{frame} \begin{figure} \includegraphics[scale = 0.8]{Venus}     \caption{Nascita di Venere} \end{figure}  \only&lt;1&gt;{In the centre the newly-born goddess Venus stands nude in a giant scallop shell. Its size is purely imaginary, and is also found in classical depictions of the subject. }    \only&lt;2&gt;{Alternative identifications for the two secondary female figures involve those also found in the Primavera; the nymph held by Zephyr may be Chloris, a flower nymph he married in some versions of her story, and the figure on land may be Flora. Flora is generally the Roman equivalent of the Greek Chloris; in the Primavera Chloris is transformed into the figure of Flora next to her, following Ovid's Fasti, but it is hard to see that such a transformation is envisaged here.} \end{frame}   \end{document} Did she judge him "wrong" or "wrongly"? Which one is the correct use?  She judged him wrong. She judged him wrongly.  Or, are both correct, but have slightly different meanings? Why can't energy be created or destroyed? My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed.  Why is that?  Is there a theory or scientific evidence that proves his statement true or false?  I apologize for the elementary question, but I sometimes tend to over-think things, and this is one of those times. :) I am using Google Chrome. I re-installed it today as it was crashing when I start it. I guess may be that this was due to any of my Extensions or Google Update. I disabled my extensions but it was still crashing. Now I installed an old version of chrome from . However, Chrome automaticallyuppdates itself (although Windows asks me before installing and I don't grant it permissions, its still annoying to get Google update asking me permission after a couple of minutes.   I saw the question at  however, it doesn't seem to hlp me. i disabled the service and I stopped it as well, but the problem still persists. I also tried to look into the registry according to the following link:  but the "Policies" section of my registry doesn't contain any folder called "Google". I use Windows 7 - 64 bit Home Premium Service pack 1.  Could anybody help? Thanks a ton! Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive Prove that $\phi(n) \geq \sqrt{n}/2$ I've got a situation where I have a 240V 30A dryer line, and I want to power something else from the same circuit, but NEVER at the same time as the dryer. I'm looking for a switch to use, like a DPDT ON-ON break-before-make switch. There are tons of DPDT switches I've found, and most seem to be ON-OFF. Hardly any show their wiring diagrams, so if they don't give more details, I don't know if they'll work. Note: the second circuit will not use more than 15A, maybe just 12A.  I'm hoping someone can recommend specific switches that will do what I want. Manufacturer and exact part# is what I'm looking for.   Personally, I don't care if it's a knife switch, something that looks like a standard light switch, two buttons, has relays, I don't care. The breaker box is full, it's a long and difficult wire run, and I do not need a separate circuit because I'll NEVER need to run both devices at once.  I could resort to just using one plug and unplugging one then plugging the other one in, but I don't see the point of constantly exposing live wires (the plugs) where something could inadvertently fall on them or water could be sprayed, instead of keeping things all sealed in the wall with a safe, simple switch between two plugs. Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$ Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$    JUNK  Guessing asking for $8/9 * k = 0 $ (being a whole integer) for smallest integer so for this one is 9 . another guess would be the smallest element of $Q/Z$ being something like $9/8$  $14/5$ would be 5  $48/28=12/7$ would be 7? Sci-fi series: humans are immune to telepathy, become soldiers for an alien group fighting against invertebrate telepaths The series involves two groups of aliens who have been fighting an interstellar war for many years before the first book.  The war is started by a race of jellyfish-like telepaths who have used their powers to control several species which they use as soldiers.  The jellyfish aliens have declared a holy war against all sapient vertebrates.  The only species on their side that aren't being actively controlled through their telepathy are invertebrate species who have joined the holy war.  In this series, humans are immune to telepathy--what's more, a completely unconscious reflex tends to destroy the mind of anything even attempting to affect their minds.  One of the central themes of the series is how well suited humanity is to warfare, and all the sacrifices forced on humanity as they become nothing but soldiers fighting against the jellyfish, losing almost all culture not in some way related to being soldiers, as the other races in their alliance take over the other roles.  Other species I recall include a bird race that is gifted at linguistics but can be sent into shock by even seeing a low quality recording of a human walking around a room.  They just cannot handle the hide or flee reflexes triggered by seeing such an obvious predator.  Another race is a crab like group that is so antisocial they all keep suicide buttons at hand when working with other races, and only work to help the human alliance's side because the jellyfish won't leave them alone.  There is a race that bears a faint resemblance to large rats who were previously the best soldiers on their side, and another renowned for their intellect. Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot &amp; a &amp; b &amp; c &amp; d &amp; e\\\hline a     &amp; e &amp; d &amp; b &amp; a &amp; c\\ b     &amp; d &amp; c &amp; e &amp; b &amp; a\\ c     &amp; b &amp; e &amp; a &amp; c &amp; d\\ d     &amp; a &amp; b &amp; c &amp; d &amp; e\\ e     &amp; c &amp; a &amp; d &amp; e &amp; b \end{array}$$ it is easy to check that it is closed (no elements occur in the table which don't occur as row or column index), commutative (the table is symmetric), has a neutral element (the row and column of $d$ are copies of the index row/column), and has an inverse element for each element (there's a $d$ in each row and column). In other words, almost all important properties can immediately be seen. The only part missing is associativity.  Therefore my question: Is there a simple way to see directly from the operation's table (i.e. without doing explicitly all the calculations) if an operation is associative? Why was Sirius ever convicted? Sirius himself says at some point in Goblet of Fire that Crouch pushed through legislation to have him convicted and thrown in Azkaban without a trial.  There seems to be a general consensus, specifically voiced by Fudge but also implicit in Crouch pushing through this legislation, that Sirius was considered one of Voldemort’s innermost, most dangerous, and most trusted right hands. Easily as inner-circle as the Lestranges, for example.  It seems quite unlikely to me that someone so close to Voldemort should not be a Death Eater—apart from Fenrir Greyback (who I would think is much less inner-circle), I do not recall hearing of any of Voldemort’s important servants not being Death Eaters.  As far as canon describes, all Death Eaters have the Dark Mark magic-tattooed on their arm, to be used as a way for Voldemort to summon them to him. Considering that Karkaroff, who had definitely burnt all his Death Eater bridges, still had the Dark Mark, I am assuming that getting rid of the Mark is not possible: once it’s on, it stays on.  Yet, as Hermione points out in Order of the Phoenix (ch. 16, “In the Hog’s Head”), Sirius does not have the Dark Mark.  So why was he so swiftly convicted and locked up? I mean, I know they were eager to be seen to be doing something (as Fudge keeps pointing out is his excuse for most of the miscarriages of justice he commits)… but seriously, did no one think to check if Sirius actually had a Dark Mark on his arm or not? Did the Ministry of Magic not even know at the time that all Death Eaters had the Mark? Or were they just happy to accept—without the scrutiny of a full trial—that a non-Death Eater would commit mass murder for a Voldemort who was presumed defeated? I've noticed a MATCH SIMPLE and MATCH FULL, but I don't understand what they do. I see default is MATCH SIMPLE; but, how do the other MATCH clauses to the FOREIGN KEY constraint function? On vs At with date and time This must be a simple question for a native speaker.  I know that we use "on" with dates: I'll see you on January 1st.  And we use "at" with times: I'll see you at 17:30.  But what preposition has to be used when we speak for date and time:  I'll see you on January 1st at 17:30. looks ok.  But what in this case: It happened on 2014-01-01 17:30.  Is "on" correct when we are specifying the date and the time?   The date-time comes as a ready text as 2014-01-01 17:30 and I cannot modify it. I can only put text before the date-time string or after it. I needed some simple string encryption, so I wrote the following code (with a great deal of "inspiration" from ):      // create and initialize a crypto algorithm     private static SymmetricAlgorithm getAlgorithm(string password) {         SymmetricAlgorithm algorithm = Rijndael.Create();         Rfc2898DeriveBytes rdb = new Rfc2898DeriveBytes(             password, new byte[] {             0x53,0x6f,0x64,0x69,0x75,0x6d,0x20,             // salty goodness             0x43,0x68,0x6c,0x6f,0x72,0x69,0x64,0x65         }         );         algorithm.Padding = PaddingMode.ISO10126;         algorithm.Key = rdb.GetBytes(32);         algorithm.IV = rdb.GetBytes(16);         return algorithm;     }      /*       * encryptString      * provides simple encryption of a string, with a given password      */     public static string encryptString(string clearText, string password) {         SymmetricAlgorithm algorithm = getAlgorithm(password);         byte[] clearBytes = System.Text.Encoding.Unicode.GetBytes(clearText);         MemoryStream ms = new MemoryStream();         CryptoStream cs = new CryptoStream(ms, algorithm.CreateEncryptor(), CryptoStreamMode.Write);         cs.Write(clearBytes, 0, clearBytes.Length);         cs.Close();         return Convert.ToBase64String(ms.ToArray());     }      /*      * decryptString      * provides simple decryption of a string, with a given password      */     public static string decryptString(string cipherText, string password) {         SymmetricAlgorithm algorithm = getAlgorithm(password);         byte[] cipherBytes = Convert.FromBase64String(cipherText);         MemoryStream ms = new MemoryStream();         CryptoStream cs = new CryptoStream(ms, algorithm.CreateDecryptor(), CryptoStreamMode.Write);         cs.Write(cipherBytes, 0, cipherBytes.Length);         cs.Close();                     return System.Text.Encoding.Unicode.GetString(ms.ToArray());     }   The code appears to work fine, except that when decrypting data with an incorrect key, I get a CryptographicException - "Padding is invalid and cannot be removed" - on the cs.Close() line in decryptString.   example code:      string password1 = "password";     string password2 = "letmein";     string startClearText = "The quick brown fox jumps over the lazy dog";     string cipherText = encryptString(startClearText, password1);     string endClearText = decryptString(cipherText, password2);     // exception thrown   My question is, is this to be expected? I would have thought that decrypting with the wrong password would just result in nonsense output, rather than an exception. Site.createPortalUser() returning null in test context I need to convert an arbitrary amount of milliseconds into Days, Hours, Minutes Second.  For example: 10 Days, 5 hours, 13 minutes, 1 second. Sending email in .NET through Gmail Instead of relying on my host to send an email, I was thinking of sending the email messages using my Gmail account. The emails are personalized emails to the bands I play on my show.   Is it possible to do it? Query the post which have a thumbnail I've got a highlight section on the header of my website which shows the title and the thumbnail of the last X posts having a thumbnail.  Is there any custom query which allows me to retrieve only the last X post having a thumbnail?  At the moment I'm retrieving the last X * 10 posts (because not all the posts have a thumbnail) hoping that in this result I'll have X posts having a thumbnail. After doing my query I add a condition: if (has_post_thumbnail()).  This solution is really uncertain and ugly, a custom query with a thumbnail parameter would be great but I couldn't find any hint about it. Solve $x^2+5x+6 \equiv 187 \mod 187$ Solution $$x^2+5x+6 \equiv 187 \mod 187$$ $$ (x+\frac{5}{2})^2 \equiv \frac{1}{4}$$ $$ 4(x+\frac{5}{2})^2 \equiv 1$$ $$ y:= x+\frac{5}{2} $$ $$ 4y^2 \equiv 1 \mod 11 \wedge 4y^2 \equiv 1 \mod 17  $$ $$ ( 2y \equiv 1 \mod 11 \vee 2y \equiv 10 \mod 11 ) \wedge ( 2y \equiv 1 \mod 17  \vee 2y \equiv 13 \mod 17)  $$ $$ ( y \equiv 6 \mod 11 \vee y \equiv 5 \mod 11 ) \wedge ( y \equiv 9 \mod 17  \vee y \equiv 15 \mod 17)  $$ Combining that from CRT I got: $$ y \in \left\{49, 60,83,94 \right\} $$ and for example: $$ x+\frac{5}{2}  \equiv 94 \mod 187$$ $$ 2x  \equiv 183 \mod 187$$ some calculus and get... $$x \equiv 185 $$ And the same thing for each other case. Question Is there any faster (or smarter) way to solve equations like that? How could I retrieve the exact duplicate value from conditionally formatted list that highlight duplicates? I have imported a webpage table in to Google Sheets. It frequently being updated. A series of calculations has been conducted on the sheet named Shawn Of the link spreadsheet.  I have filtered the my desired results into the Calculation sheet of that linked document... which is in columns A5:A. I then applied =ARRAYFORMULA(if(A5:A200=&quot;&quot;,,A5:A200*{12,13,37,101,111})) in columns B5. The Results of the formula extends from B5:F. This results from a5:F was subsequently conditionally formatted to highlighted duplicates in the aforementioned, i.e. a5:f using the lists of values in K5:K and I5:5. In Columns A5:F, is it possible to generate/place values in the range A5:f with matching name of states and aligned values in J5:K and H5:I?