Consider the ring $(\Bbb{Z}, +, \cdot)$. Let $I$ be an ideal of the ring that contains the element $-1$. Show that $I = \Bbb{Z}$   Could some one please explain how to solve this.
Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains   a unit of $R$. Prove $J=R$, rigorously.   Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice.
Why does part of my model disappear when I zoom in on it in the 3D Viewport? Uninstall GRUB and use Windows bootloader I have Windows 8 pre-installed, and then I installed GRUB with Ubuntu. Ubuntu is not my thing, so now I want to remove it along with GRUB. From what I have learned, with UEFI, GRUB does not overwrite the windows bootloader in the EFI partition and is stored elsewhere. How would I remove GRUB and make my PC use the Windows bootloader instead? It should be noted that I created a separate /boot partition when installing Ubuntu. Do the laws of physics evolve? Probability expected value football problem The number of injuries at a college football game has the following probability distribution:      $$\begin{array}{|l:l|} \hline \textrm{Number of injuries} &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\ \hdashline \textrm{Probability} &amp; 0.14 &amp; 0.33 &amp; 0.23 &amp; 0.17 &amp; 0.09 &amp; 0.04 \\ \hline \end{array}$$      The coach must file a report whenever there is more than one injury in any game.      a. Calculate the expected number of reports filed in the next 20 games.      b. Calculate the standard deviation of the number of reports filed in the next 20 games.      c. Calculate the expected number of football games to be played before the first game in which there are at least four injuries. MAC OS for Intel based PCs I have an Intel dual core PC with 4GB of RAM and Graphics card. For on of my student Assignment I need to install Latest possible MAC OS on my system as a secondary OS.  Like on boot it asks me that which OS i want to boot with Win 8.1 or Mac Os..  I searched on the internet and found two conflicting opinion some people said there are few MAC OS Version which can be directly installed on Intel PCs some says there aren't?   I am here to get rid of the confusion that is an official latest possible MAC OS version for Intel based PCs? If not what are my options if I want to run MAC OS on my PC.   Please not that Virtualization options won’t work for me, it has to be working as full OS not an os inside another. Has the speed of light changed over time? Could someone judge my (stoner) hypothesis that the speed of light has changed over time -- i.e. as the universe has expanded in volume light has slowed down, perhaps going so far as back to the big bang when it was infinitely fast and there was no time because everything happened at once etc. Thinking that the speed at which information can propagate through the universe is linked to the size of it seems intuitive to me. My question -- is there an easy disproof of this? Would Einstein have to be wrong? Does it violate anything supposedly more fundamental such as quantum or string theories? Do any current experiments invalidate it? If not can you show me in any case why you think its unlikely.   Edit 8/24  I'm accepting Mark M.'s answer but will post this here because there is a character limit on comments  @Mark M thanks, good answer but as someone who has only read some popular physics, and should leave this to the experts, I'm still muddled in my personal theory. I don't see why you should need two units to measure the speed of light. The thing I have a hard time wrapping my head around is the relation of time and distance. They seem like they could fundamentally be the same thing. If you say time is measured fundamentally by the vibration of so and so quantum object in space... why can't we just measure that vibration distance as the constant... I'll repeat myself to try to be clear... there is a certain minimum distance that particles have to go to interact with each other... if it wasn't vibrating there wouldn't be time,its what creates the illusion of time...so instead of talking about speed or c as distance/time.... can't we simply talk about that distance a quantum object vibrates....I'll lead up to my point.....perhaps there can need be only one constant here, and that is the physical size of the universe. a tiny metal tuning fork doesn't appear to be vibrating at all but if you blew it up to the size of the empire state building the metal rods would move from window to window. perhaps as our universe expanded in size the length of that minimum vibration (perhaps infinite at point zero) would have expanded, and therefore created the illusion of time and the speed of light, which, as the universe expands, will continue slowing down. perhaps we are like a big balloon and we have been blown up and all the fields/particles-without-size are vibrating more and more in that space.. am i missing something obvious here? Why does $k/m=\omega^2$ for harmonic motion? Can anyone please give me a proof for $k/m=w^2$ in simple harmonic motion?  I have tried energy conservation and Newton's laws as follows :  In the case of a mass-spring system,  $$F=ma =-kx\\ F=ma = mr\omega^2 $$  hence   $$\frac km  = \omega^2$$  Or   $$\frac12 mv^2 = \frac12kx^2$$  $$\frac km  =  \frac{V^2}{x^2}   =   \omega^2$$  therefore $k/m  =  \omega^2$  Are these valid and correct? I have been stuck in this for the  entire day. How can I split multiple joined words? I have an array of 1000 or so entries, with examples below:  wickedweather liquidweather driveourtrucks gocompact slimprojector   I would like to be able to split these into their respective words, as:  wicked weather liquid weather drive our trucks go compact slim projector   I was hoping a regular expression my do the trick.  But, since there is no boundary to stop on, nor is there any sort of capitalization that I could possibly key on, I am thinking, that some sort of reference to a dictionary might be necessary?    I suppose it could be done by hand, but why - when it can be done with code! =)  But this has stumped me.  Any ideas? How does $e^{i x}$ produce rotation around the imaginary unit circle? Euler’s formula states that $e^{i  x} = \cos(x) + i \sin(x)$.  I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power of $ix$ produces rotation.  Can anyone give me an intuitive understanding? TikZ Angle Label - Help it "fit" better? I have the following drawn with TikZ    using the code  \documentclass{article} \usepackage{tikz} \usepackage{tkz-euclide} \usetkzobj{all} \begin{document} \begin{tikzpicture} \coordinate (O) at (0,0); \draw (O) circle (3); \coordinate[label = above left:$M$] (M) at (130:3); \coordinate[label = above:$N$] (N) at (80:3); \coordinate[label = below:$A$] (A) at (240:3); \coordinate[label =below right:$B$] (B) at (300:3); \draw (A) -- (M) -- (B); \draw (A) -- (N) -- (B); \tkzMarkAngle[draw = black, fill = white, opacity=1](A,M,B) \tkzLabelAngle[pos = 0.8](A,M,B){$x$} \tkzMarkAngle[pos = 0.2,draw = black, fill = white, opacity=1](A,N,B) \tkzLabelAngle [pos=0.8](A,N,B){$27 ^{\circ}$} \end{tikzpicture} \end{document}   Notice however, tht the 27 degrees does not fit in very nicely. How can I adapt the code to try and move the 27 a bit "lower" so that it fits in better, as well as the angle mark?  Also, how can I get two "double arrows" in the middle of lines AM and BN, to indicate that they are 'parallel'? How do I change the GRUB boot order? Ubuntu 14.04.5/16.04 and newer on AMD graphics Why are my RAW images corrupt but the previews okay? Evaluate $\int\frac{1}{1+x^6} \,dx$ I came across following problem     Evaluate $$\int\frac{1}{1+x^6} \,dx$$   When I asked my teacher for hint he said first evaluate     $$\int\frac{1}{1+x^4} \,dx$$   I've tried to factorize $1+x^6$ as  $$1+x^6=(x^2 + 1)(x^4 - x^2 + 1)$$ and then writing  $$I=\int\frac{1}{1+x^6} \,dx=\int\frac{1}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx=\int\frac{1+x^2-x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$ $$I=\int\frac{1}{x^4 - x^2 + 1} \,dx-\int\frac{x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$  However $$x^4-x^2+1=\left(x^2-\frac12\right)^2+\frac{3}{4}$$ But I can't see how it helps  I've also tried to reverse engineer the   And I need to have terms similar to $$\frac{x^2-1}{x^4-x^2+1} \quad , \quad \frac{1}{1+x^2} \quad , \quad \frac{1}{(x+c)^2+1}\quad , \quad \frac{1}{(x+c)^2+1}$$ in integrand, How can I transform my cute looking integrand into these huge terms?  Since in exams I will neither have access to WA nor time to reverse engineer the solution moreover it does not seem intuitive,is there any way to solve this problem with some nice tricks or maybe substitutions? Why do "df" and "du" commands show different disk usage? I cannot access ubuntu system on my pc, the error message: "the system is running in low graphics mode" and I tried some commands I searched from internet.   I found out a problems seems there is no available disk space. I used "df" and "du" commands to check, the results are as follow:  du -j --max-depth=1 23G  ./home 3.3G ./usr  ......  28G   and  df -Th  filesystem   Type   size    used   available   use%    mounted on  /dev/sda5    ext4   68G     68G    0           100%    /   how would I clean up the system for more disk space? options for appearance of links in hyperref How many arguments does a function require, and how to use that in `Manipulate` How do the tens­es and as­pects in English cor­re­spond tem­po­ral­ly to one an­oth­er? Non-na­tive speak­ers of­ten get con­fused about what the var­i­ous tens­es  and as­pects mean in English. With in­put from some of the folk here I've put to­geth­er a di­a­gram that I hope will pro­vide some clar­i­ty on the mat­ter.  I of­fer it as the first an­swer to this ques­tion. Con­sid­er it a liv­ing doc­u­ment. In­put is wel­come, and good sug­ges­tions will be  in­cor­po­rat­ed in­to the di­a­gram.    No­ta bene: What this is not is a dis­cus­sion of whether there are  more than two tens­es in English. , to which this ques­tion is not in­tend­ed to sup­ply ar­gu­ments one way or the oth­er.  Here, the aim is to pro­vide an overview of what con­struc­tions English-speak­ing peo­ple use for con­vey­ing in­for­ma­tion about ac­tions re­fer­ring to past, present, and fu­ture, and to pro­vide it first and fore­most to pre­cise­ly the peo­ple who are like­ly to use "tense" as a catch-all term in their search, rather than to lin­guists who know bet­ter.  Break­ing News There is now an ex­cel­lent ELU blog ar­ti­cle ti­tled  . It is high­ly rec­om­mend­ed read­ing. How to differentiate formally? Why 220V and 110V? Why did the industry choose these seemingly unintuitive values of 110V and 220V for the mains AC? Is there a physical reason why they aren't simply 100V and 200V?