$G$ is set $\{x \in \mathbb{R} : x \neq -1\}$ with group operation $x\circ y=x+y+xy$.  Show that $f(x)=x-1$  is an isomorphism from the multiplicative group $\mathbb{R}^*$ to $G$.
So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity element, other than 0. Because if 0 is the identity element, then this group won't have inverses.  The set explicitly excludes -1, which I found to be its identity element, which makes going about proving that this is a group mighty difficult.
How to pronounce fractions with denominators larger than 20 where the last digit of the denominator is 1 or 2? eg 4/31 Disclaimer: I speak British English.  I've noticed a lot of differences between the way Americans and Brits pronounce numbers.1  Since the question concerns this, I thought it might be appropriate to draw attention to it case we inadvertently confuse each other.  My question is not about these differences, I just wish to highlight them in case they cause confusion.  1. You seem to happily call a quarter "a fourth" sometimes, we always call 131 "one hundred and thirty one", we pronounce double digits in phone numbers like 12449 as "one two double four nine", and I think we are much more likely to use expressions like "thirteen hundred" to mean 1300.    Question  When talking about fractions, I have frequently heard  1/2    a half 1/3    a third 1/4    a quarter 1/8    an eighth 1/64   a sixty fourth 1/56   a fifty sixth   etc.  Essentially the rule seems to be that, except for "a whole", "a half", and "a quarter", the word matches the ordinal number; that is to say:      Numeral       Ordinal         Fractional     one           first           whole     two           second          half     three         third           third     four          fourth          quarter     five          fifth           fifth     six           sixth           sixth     fifty-seven   fifty-seventh   fifty-seventh  Even though 1/4 is a quarter, 1/64 is a sixty-fourth.  So what’s 1/62?  A  sixty-twoth?  A sixty-second? Surely not a sixty-half!  I know that simply saying one over sixty-two can usually work, but I'm asking specifically for the word itself, i.e.  if I divide a huge pizza into 21 pieces, what are the pieces?  Other than baker’s twentieths.  Summary: Can anyone point me to any sources (whether style guides or common usage studies or anything else) that discuss the pronunciation of fractions; specifically one that discusses this separately from ordinal numbers, rather than how to form ordinal numbers in the first place. Preventing console window from closing on Visual Studio C/C++ Console application Why is find not accepting '-exec cp {} dir +'? I have a directory, dir1 which contains many files whose names end in either .jpg or .png. I want to copy all the .png files to dir2 which is empty.  This command works:  find dir1 -name '*.png' -exec cp {} dir2 \;   but this command doesn't:  find dir1 -name '*.png' -exec cp {} dir2 + find: missing argument to `-exec'   I also tried:  find dir1 -name '*.png' -exec cp {} -t dir2 + find: missing argument to `-exec'   and:  find dir1 -name '*.png' -exec cp {} dir2 \+ find: missing argument to `-exec'   After looking at , I even tried:  find dir1 -name '*.png' -exec cp {} dir2 {} + find: Only one instance of {} is supported with -exec ... +    says that:     -exec {} + was added in [version] 4.2.12 in 2005   My version of find is 4.4.2.  What am I doing wrong? This occurs in the  released recently to the public. When someone posts a code block in their question or answer, the code block is cut off on the right hand side, and I can't scroll to the right to view the rest of it. How can I view these code blocks? How do I reset the Windows XP Administrator password? I have forgotten the Administrator password of my Windows XP installation and locked out. Is there a way to reset it? I do have physical access to the PC. In the case of languages with a C-like syntax, we declare the main() method to return an int or float value (or void).  Is it possible to declare a non-void return type from main() in Java?  If not, then why not?  Does this mean that a Java program doesn't return any value to the OS? Minimum file permissions to rename file I put a 64-bit ELF file in my unprivileged user's home directory. I wonder which permissions do I need to have only possiblity to rename this file being logged in as unprivileged user. From what I understand I only need execute permission for user to whom file belongs to, set for parent's directory. Is that right?  So if path to my file is /home/Simon_dir/zip and I wrote chmod 0100 /home/Simon_dir and file permissions are set to 0000, should I have rights to rename the zip file being logged in as unprivileged user? How do I set an environment variable on the command line and have it appear in commands? If I run  export TEST=foo echo $TEST   It outputs foo.  If I run  TEST=foo echo $TEST   It does not. How can I get this functionality without using export or a script? False proof that $ρe^{iθ} = ρ$ and so complex numbers do not exist? My professor showed the following false proof, which showed that complex numbers do not exist. We were told to find the point where an incorrect step was taken, but I could not find it. Here is the proof: (Complex numbers are of the form $\rho e^{i\theta}$, so the proof begins there) $$\large\rho e^{i\theta} = \rho e^{\frac{2 \pi i \theta}{2\pi}} = \rho (e^{2\pi i})^{\frac{\theta}{2\pi}} = \rho (1)^{\frac{\theta}{2\pi}} = \rho$$ $$Note: e^{i\pi} = -1, e^{2\pi i} = (-1)^2 = 1$$ Since we started with the general form of a complex number and simplified it to a real number (namely, $\rho$), the proof can claim that only real numbers exist and complex numbers do not. My suspicion is that the error occurs in step $4$ to $5$ , but I am not sure if that really is the case. Can Mathematica do two evaluations at the same time? Why is the console window closing immediately once displayed my output? How to render a background image in 2.8? What did the public at large know about the assassination attempt on Palpatine and how he survived it? I mean, for the people of the galaxy, Palpatine was just a politician past his middle ages. How could this man have survived against four Jedi Council members in his office while “unarmed,” say that the attempt “left him scarred and deformed,” and people believe in this? Django template how to look up a dictionary value with a variable Can someone explain __all__ in Python? I have been using Python more and more, and I keep seeing the variable __all__ set in different __init__.py files.  Can someone explain what this does? Process substitution and pipe Stopping all automatic updates Windows 10 Text spirals with TikZ I just ran across this, and I'm curious about whether it can be replicated in TeX...    Ideally one would have the spiral getting tighter and tighter as one got to the centre, as in    I know TikZ has the ability to fit text along a path, but I'm not sure how one might make text get smaller and smaller.  The closest I can find is , but that uses manually inserted font commands.  Edit: I got what I wanted using mwibrow's answer: I'm using post thumbnails to link to a page.  Is it possible to add a class name to the post thumbnail image.  &lt;li&gt;&lt;a href="&lt;?php the_permalink(); ?&gt;" &gt;&lt;?php the_post_thumbnail(); ?&gt;&lt;/a&gt;&lt;/li&gt; Services Actions Missing in Central Admin