My understanding of complex numbers is that it is the use of an extra axis of dimension. By plotting a signal on polar axis, the real number, x, is the sine of the signal, and the imaginary number, y, is the cosine of the signal. This understanding is derived from its use in the fourier transform. (By this understanding, one could take the cosine of a value to get its imaginary component, so my understanding it limited as it doesn't explain the logic behind √(−1)=i.) So, I have the question, is it possible to have three value complex numbers?
Since there are  complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions").   Yet no one uses these. Why is this?
What's the action of banging two beers together called? Let $p,q$ be primes, $p≠q$, then I have to show that $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$  So far I've tried a lot of things with minimal polynomials and bases, but I'm far from any argument that I find sufficient.  This is an example from my algebra course, first chapter about field extension. So far we know about basic definitions, minimal polynomials and their properties, algebraic elements and algebraic fields.  Could someone maybe give me a hint about how to proceed? Thanks a lot!!! I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a single qubit. I'm not sure I completely comprehend this.    Here's how I rationalized it: You take all the information you want to store, put it in binary form, and make it the real component of $\alpha $ or $\beta$ (the coefficients of the computational basis states).   Now I'm not sure if I've understood it right, but since it's still fuzzy in my head it would be great to get some kind of ELI5 explanation or possibly a more detailed picture of how this would, even theoretically, be possible.    Apologies if the question doesn't meet standards. I'm new to the forum and would be open to feedback regarding asking questions or answering them. Geocoding USA addresses that cannot be sent over internet? How much web can Spider-Man produce? Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb   How are these kinds of verb related to each other? Specially Non-Modals and Conjugated Verb. Is the "Auxiliary modal" just another name for "Auxiliary verb"? What's the added value of SD line over regression line when examining association between 2 variables? I'm trying to incorporate different practices to use when exploring a new data set. Especially, how to examine the association between two variables.  Steps for example (not necessarily by order):   plot a y-by-x scatter plot of the raw data to see the relationship visually. compute summary statistics for each variable (mean and sd) compute correlation coefficient r draw the OLS regression line, compute its slope and intercept etc....   I've come across the "SD line" in Freedman's Statistics book, which is defined as:     "the line that goes through the point of averages and climbs at the rate of one vertical SD for each horizontal SD" Freedman, D., Pisani, R., &amp; Purves, R. (2007). Statistics (4th edn).   Since this book ("Statistics") is a canonical textbook, I consider its choice to discuss the SD line as an indication for the line's importance. However, a simple google search for the term "SD line" doesn't yield as many independent results. Most of them come directly from Freedman's book. This tells me it's not a central concept in bivariate analyses in general.  When comparing the SD with the OLS regression line, it seems like the regression line is more informative (than the SD line) for predicting y from x. Therefore, I'm wondering if bothering to plot the SD line has any benefit or added value that I would not already have when plotting the regression line.  Example using mtcars dataset, focusing on association between weight and mpg  data(mtcars)  ## calculate means mean_wt &lt;- mean(mtcars$wt) mean_mpg &lt;- mean(mtcars$mpg)  ## calculate standard deviations sd_wt &lt;- sd(mtcars$wt) sd_mpg &lt;- sd(mtcars$mpg)  ## scatter plot plot(x = mtcars$wt, y = mtcars$mpg)  ## add the "point of averages" points(mean_wt, mean_mpg, col = "red", cex = 1.5, pch = 16)  ## calculate the slope of the sd line slope &lt;- -1*sd_mpg/sd_wt  ## plot the sd line curve(expr = x*slope + (mean_mpg - slope*mean_wt), add = TRUE, col = 'blue', lwd = 2, type = "l", lty = 2)  ## plot the regression line model &lt;- lm(mpg ~ wt, data = mtcars) abline(model, col = "orange", lwd = 2)  ## legend legend("topright",        legend = c("Regression line", "SD line"),        col = c("orange", "blue"),        lty = c(1, 2),        lwd = c(2, 2))     Thus, my question: how can the SD line increase one's understanding about the relationship between two variables, in a way that's either adding or complementing on what the regression line already tells? For a given hardware configuration, how do I find out if Ubuntu will run on it? What considerations should I take into account when choosing an Ubuntu version and  such as:    with a lighter desktop than the usual Gnome and Unity  with the even lighter LXDE desktop   Obviously Ubuntu does not run on some processor architectures. So how do I go about choosing the right version and derivate. How can I find out the minmal system requirements? How can I enable Hardmode? Access Denied using TakeOwn.exe I have got this file that I can't delete. It happened after a system crash, so the CHKDSK kicked in upon next reboot. After that, I can't delete the file.  THis is on Windows Home Server, and the file is one of those hidden Thumbs.db, and my WHS reports a "File conflict" on the file, the reason: Access Denied"  What I have tried so far, running as an Administrator: Delete: Access Denied TakeOwn.exe : Access Denied Attrib.exe -s -h : Access Denied cacls.exe : Access Denied   (even whe trying to display the current owner) Rename: File not found (its hidden) Re-boot in to safe mode and tried the above: Access Denied Running the above commands as Local System: Access Denied   Tizer Unlocker: Fail (Doesn't find anything locking the file) File Assasin: Crashes ("FindRemoteFileHandles returned NULL value.")  I have used the CHKDSK /f again, rebooted since some suggestions is that the file has been corrupted, but that didn't change anything. Any suggestions ?  Update: In a total separate scenario, I experienced a similar situation; Access Denied when trying to take ownership of a file. It turned out that the file was owned by TrustedInstaler, which is not possible to override just by being an Administrator. In this case, one can use psexec /ids cmd to elevate yourself to System and then TakeOwn.exe /a apache2 mod_rewrite redirect without http body Can "real" be used as an adverb to describe an adjective? Is this correct?     That is a real cool answer.   I learned that that was incorrect, since "real" is an adjective which can describe a noun, e.g. "real answer" but it is not an adverb which can describe an adjective, "real cool". Instead you would have to say:     That is a really cool answer.   Since "really" is an adverb. Why are site stats graph for beta sites not available to public? Currently, on all beta sites, the current site stats is placed prominently on the front page, which is great for knowing how the site is currently doing.  However, it doesn't say anything about the trend of the numbers - whether the site is sailing or it is sinking compared to the last month. According to Madara Uchiha, our moderator on Anime &amp; Manga SE, . Disregarding data which are not already public, , I can't see any reason why we are not allowed access to graphical representation of public site statistics. In fact, there is already a third-party app that scraps the information from the home page and . Can we have a beautiful visualization on how our beta sites evolve over time, from pony to unicorn?   Images stolen borrowed from Bulbapedia When certain nationalities of people want to travel in the Schengen zone, they are required to apply for a Schengen visa. This is a straight-forward process for most, but when an applicant has had problems with removals or refusals they may be reluctant to apply because they want to avoid the complexities of serial visa refusals. This reluctance may lead some into wondering how their history can be expunged, and this line of thought inevitably leads to the idea of getting their passport replaced with a new one.  This sounds like a great formula for people who have had prior refusals or removals.  In essence they can become 'reincarnated' as a first-time traveller and hence avoid the unpleasantness of revealing their prior interactions with the Schengen authorities.  The relevant parts of the application form are shown here...    Nothing asks about a prior removal or refusal, and the applicant is left with simple denials for 3 questions in order to present themselves as a first-time traveller.  Assuming the person is happy to set aside the moral issues associated with denying their history, are there any holes to this strategy?  This question is scoped to Schengen visa applications so that it will not be unmanageably broad, but answers from other regimes (particularly the affluent Commonwealth and the USA) are acceptable as long as the main points are relatively consistent with Schengen. Find Total number of ways out of N Number taking K numbers every M interval I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks. It seems that containing lava in solid glass, stone and iron blocks may not be sufficient to prevent wool from burning, even if the lava is completely encased.         After some delay, it seems that fire will jump onto the wool above the iron.   Are solid blocks more effective than airspace, or not? If so, Does it matter whether the corner block is filled in? What are the exact rules on distance that I need, and how can I prevent flammable blocks from being eliminated by nearby lava? Bonus: Does a flowing lava block behave differently than a lava source block when it comes to fire creation? I need to find node voltage at A. This seems like a trivial question, but when I set up the resistive voltage divider equations they are not equal to each other. Am I doing something wrong?  My equations are \$V_A = V_1(R3/(R3+R1)) = V_2(R3/(R3+R2))\$      I tried solving this problem with an LTSpice simulation and with the superposition method, but my answers did not match (although very close to each other). Is this because the simulator uses an approximation? Can someone compare results? Can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? At the moment I am running Vista 32 bit and use VPC 2007 a lot. I have several Windows 2000 Virtual Machines set up with complex software configurations. For me, one of the advantages of working with VPCs is that a Virtual Machine can easily be copied to another PC (like a notebook). Also, when I move to a new PC, I can immediately work with my existing Virtual Machines without the need to reinstall all applications.   I am thinking about buying a new pc with Windows 7 64 bit pro, and using the XP-Mode Virtual Machine that comes with Windows 7. The advantages would be support for more memory (so being able to run more Virtual Machines simultaneously) and having a licenced XP Virtual Machine.  I would like to know if you can transfer a VHD that is based on the XP-mode VHD to another machine without Activation issues. The other machine will of course, also be running Windows 7. I would guess that this would not be a problem.   However, I did read about a guy who had to reinstall his Windows 7 because of a harddisk failure, and could not access his XP-mode VHD anymore because it demanded re-activation.  So can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? Applied boolean modifier leaving shading artifacts with flat shading I tried to manipulate a cube with the Boolean modifier. I tried to cut out an edge of the cube with a cylinder. Also, the shading result looks awful in FLAT SHADING mode, so I turned on SMOOTH SHADING.   Has anyone an idea how to achieve a better result. (I also cut out a Sphere first, this made the result less awful).   I used Blender Internal and rendered with smooth shading for smoothness.  Using flat shading looks okay, except the "round" cutouts :/  Face normals are okay.  First image is smooth shading.    EDIT: User Pisurquatre suggested the EdgeSplit modifier, works perfect to me (Maybe a little tweaking here and there): In many scientific papers, the article before the word "Equation" is omitted. Is there any grammar statement behind this? For instance:     "Equation (8) contains various approximations, and we have to check their consistency."