We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?
I've had  on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers.  I'm frequently asked that question.  This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful.  It is a question about what makes them numbers.  It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.  Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number.
Is every open subset of $ \mathbb{R} $ uncountable? Is every open subset of $\mathbb{R}$ uncountable?  I was crafting a proof for the theorem that states every open subset of $\mathbb{R}$ can be written as the union of a countable number of disjoint intervals when this question came up.  I feel like the answer is yes, but I'm not sure how to go about proving it or whether there is a crazy construction (like the Cantor Set)  that creates a countable, open subset of $\mathbb{R}$. Any ideas? Is verifying ISOs downloaded from the official website worthwhile? I downloaded the ISO from , selecting the default "Ubuntu Desktop" option.  The website links the page  which gives instructions how to verify ubuntu.  This seems pretty tedious, and I am wondering how realistic it is that there is a problem with ISO downloaded from the official website. I note that the process of verification itself requires me to download software that is new to me, thus introducing another attack vector on me even as I am closing another one.  For what it's worth, I am planning to use Live USB only and not to fully install Ubuntu. Does that make a difference? Why doesn't the Sun appear green to our eyes? Can less retain colored output? Can I get less not to monochrome its output?  E.g., the output from git diff is colored, but git diff | less is not. How can I disable the "no backup in x days" from TimeMachine? (Updated answer choice for High Sierra. See the old answer if you are still on Yosemite)  I'm running OS X Yosemite, and I want to disable the notifications from Time Machine. I rarely do a hard-drive backup because all of my essential files are in the cloud. I don't need to run timemachine as often as it suggests/warns/annoys me about.  How do I disable the notifications? Time Machine is not in the Notifications preferences, and I don't want to stop using it entirely. Let $G = (\mathbb{C} - \{0\}, \cdot)$, a subgroup $U = \{x+yi \mid x^2 + y^2 = 1\}$. Use the Fundamental Theorem to show that $G/(\mathbb{R}_{&gt; 0},\cdot)$ is isomorphic to $U$ where $\mathbb{R}_{&gt; 0}$ denotes the positive reals.  I am struggling to find the correct mapping from $G$ to $U$ that results in the the kernel being the nonnegative reals under multiplication. I'm hoping to get help with this mapping then I can show it is a homomorphism and onto and can use the Fundamental Theorem. How to choose trace width for very high current PCB? I am currently designing a PCB layout for my team's high current, high voltage project and am having trouble with choosing my trace width.  The circuit is composed of 12 inductors placed in series with short traces connecting them together (about 5mm max length). We are feeding 10μs pulses at ~300A/1000V at a maximum rate of 10 pulses per second by discharching capacitor banks into the coils. The capacitors are recharged by an external source.  I have looked for resources on the subject, but nothing I could find applies to the level of current and voltage we are applying.  Is there a rule for determining trace width for very high current pulses? Flag abusive users I think the community does a great job policing questions/answers, etc. and the community has some good tools available to do this.  Between flagging questions with some reasons behind it and editing posts, closing them, and deleting them, the only thing that seems to be missing to me is the ability to flag a user.  Reasons to flag a user as abusive:   Duplicate User Abusive Behavior Offensive user name or profile User profile consists of spam Others? Sending and Parsing JSON Objects in Android A subspace $X$ is closed iff $X =( X^\perp)^\perp$ Let $X$ be a subspace of a Hilbert space $H$.  prove:  $X$ is closed iff $X =( X^\perp)^\perp$   I do not know how to proceed. any hints would be appreciated.  thanks.  $X^\perp = ${ $y \in H$:$\langle x,y\rangle = 0 \forall x \in  X\}$. Someone mentioned that "veiling glare" or "veiling flare" was to blame for reduced contrast in a photo. What does this mean exactly?  What causes veiling glare, and how can it be avoided? How does it relate to those floating polygonal lens flares one often sees in movies? "have" vs."have got" in American and British English I have looked through several questions and answers on EL&amp;U, and often there is an indication that American English prefers "have" while British English prefers "have got". In addition, there are several references to "have got" being more informal than "have" (e.g.: , ). But where is it considered more informal? In American English alone or both American and British English?   This point may seem obvious to those who made the above mentioned references, but the fact is that even the most fluent users of English in my country will say without a doubt that 'have got' is the preferred usage in the UK and, therefore, it must also be more formal. Why? Because in an academic environment, the more informal structures are generally frowned upon, outside some specific spoken exercises. And if the teachers insist on using "have got", it follows it isn't informal.  So, is the teaching of (British) English in our schools transmitting the wrong idea?  EDIT: I'm adding some information in answer to the comments.  I'm mainly concerned with the idea of possession:      I have got a cat vs. I have a cat and She has got a dog vs. She has a dog   As mentioned in the comments, every (Portuguese) student will be told that "have" and "have got", when it comes to the idea of possession, are absolute synonyms, the only two differences being:    "have got" is preferred by British English, "have" is preferred by American English   the negative and interrogative structures differ    I have never heard of any distinction in formality. In fact, I was rather surprised when I went through the EL&amp;U archives and read about it.  So, would "I have got a cat" be as formal/informal as "I have a cat"?   And how do British and American English look at its formality/informality? Is it the same? I am using k-means clustering algorithm to cluster a given dataset into two. I took two different initial points and run k-means clustering algorithm for each initial points. With the first initial points I got “within cluster sum of squared distances” (error) =23874.341. With the second initial points I got “within cluster sum of squared distances” (error) = 42913.684. 23874.341 is less than 42913.684. So, can we say the first one is the better clustering quality? If yes, is it always hold true? Or can we say less within cluster sum of squared distances is always the better clustering quality? How do you tell what brand/make/model of bike I have? Inner join vs Where Is there a difference in performance (in oracle) between  Select * from Table1 T1  Inner Join Table2 T2 On T1.ID = T2.ID   And  Select * from Table1 T1, Table2 T2  Where T1.ID = T2.ID   ? I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it equivalent to find a bijection from $\mathbb{Q}$ to $\mathbb{Z}$? I want to update ubuntu 11.04 throught apt-get. So I added the following lines in apt.conf  export http_proxy=http://username:password@proxyserver.net:port/ export ftp_proxy=http://username:password@proxyserver.netport/  export http_proxy=http://deepak:Deepak@123@12.1.1.1:3128   My question is: how do I insert special characters to a username or password?  For instance: my password for proxy is Deepak@123 and it is getting an error. MacVim -- how to make Ctrl+A go to beginning of line, Ctrl+E to the end of the line I'm trying to switch to MacVim as my main editor, and I'm liking it so far, except for one issue:  I got really used to the Mac shortcuts control-A to go to the beginning of a line, and control-E to go to the end of a line.  However, these two don't work in MacVim (control-A does something weird -- it seems to increase the last number in a line by 1 or something, and control-E seems to scroll down the screen by one line).  Any way to make these two shortcuts go to the beginning of a line / end of a line respectively instead? Two Answers, One Question Sometimes, when I answer a question on Stack Overflow, I have two different answers that are equally valid. This happens most when answering architectural questions where there are many valid ways to solve a problem.  Is it better to put each alternative into a single answer, or put each alternative in its own answer?  Posted each alternative separately will allow the voting system to float the best answer to the top, but I don't want to be accused of rep whoring by "flooding" a question with answers.  What's your opinion? One or two answers? Just curious whether the process takes a while, and whether it scrapes through history to determine if it was once valid or not. Currently, I have 6 0-vote accepted answers and 20 1+-vote answers (), and that makes it > 5 and > 20% but not for long if voting goes up or down from here. ;-P  Update:  Special thanks to Grace Note for pointing out the  to follow. I upgraded this to a bug because it's been 48 hours during which the criteria supposedly were met. 6 0-votes, 20-23 1+ votes during the whole period. The criteria only last so long, so it may go out of scope soon. If it only checks every 3 days, I'm not sure if that's intentional or a bug.