I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows:  $$ \sqrt{-1} = \sqrt{-1} $$  $$ \sqrt{-1\over1} = \sqrt{1\over-1}$$  $$ {\sqrt{-1}\over\sqrt{1}} = {\sqrt{1}\over\sqrt{-1}}$$  $$ \sqrt{-1}\cdot\sqrt{-1} = \sqrt{1}\cdot\sqrt{1} $$  Thus:  $$  -1 = 1  $$  My question is simple: how can it be? Where is the error? Is that a paradox?  Thanks in Advance!
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
join comma delimited data column Where can you ask about Career Advice? The  specifically says not to ask  for career advice but people do it anyway.  Should I consider the Help Center outdated or only loosely enforced?    And what about a career advice site on the network?  There are 70+ sites, and most of them are for relatively narrow interests (which is not a criticism, I enjoy looking through the many sections).  A career section seems like it would be very popular.  Also, a careers section would help keep career questions off the other sites. Enumerate or list all variables in a program of [your favorite language here] A friend asked me last week how to enumerate or list all variables within a program/function/etc. for the purposes of debugging (essentially getting a snapshot of everything so you can see what variables are set to, or if they are set at all). I looked around a bit and found a relatively good way for Python:   #!/usr/bin/python                                                                                                                                                                                                                            foo1 = "Hello world" foo2 = "bar" foo3 = {"1":"a",         "2":"b"} foo4 = "1+1"  for name in dir():     myvalue = eval(name)     print name, "is", type(name), "and is equal to ", myvalue   which will output something like:   __builtins__ is &lt;type 'str'&gt; and is equal to  &lt;module '__builtin__' (built-in)&gt; __doc__ is &lt;type 'str'&gt; and is equal to  None __file__ is &lt;type 'str'&gt; and is equal to  ./foo.py __name__ is &lt;type 'str'&gt; and is equal to  __main__ foo1 is &lt;type 'str'&gt; and is equal to  Hello world foo2 is &lt;type 'str'&gt; and is equal to  bar foo3 is &lt;type 'str'&gt; and is equal to  {'1': 'a', '2': 'b'} foo4 is &lt;type 'str'&gt; and is equal to  1+1   I have so far found a partial way in PHP (courtesy of ) but it only lists all variables and their types, not the contents:   &lt;?php // create a few variables $bar = 'foo'; $foo ='bar'; // create a new array object $arrayObj = new ArrayObject(get_defined_vars()); // loop over the array object and echo variables and values for($iterator = $arrayObj-&gt;getIterator(); $iterator-&gt;valid(); $iterator-&gt;next())         {         echo $iterator-&gt;key() . ' =&gt; ' . $iterator-&gt;current() . '&lt;br /&gt;';         } ?&gt;   So I put it to you: how do you list all variables and their contents in your favorite language?    Edit by : I propose this question follows the spirit of a little "". If you do not agree, just edit and remove the tag and the link. Compare means of group with a mean of a subgroup LOAD DATA INFILE easily convert YYYYMMDD to YYYY-MM-DD? Linear transformations map lines to lines I need help with this question:  Let $T$ be a linear map from $\mathbb{R}^n \to \mathbb{R}^m$, and let $L$ be a line in $\mathbb{R}^n$. Show that $T(L)$ is also a line (is it not enough to just consider lines in the plane). How to delete directories with path/names too long for normal delete In this  of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is said that the simply connectedness of Mandelbrot set is easier to prove.  However, I cannot find a proof through google. Can anyone provide me a reference or a proof of the simply connected of Mandelbrot set? How can we print line numbers to the log in java Ubuntu 20.04 Font and UI elements too big on login I'm running Ubuntu 20.04 on an ASUS X99-E-10G WS w/ an NVIDIA 1080Ti and an LG 4k 32&quot; monitor which persistently gets recognized as 27&quot;. When I log in, the text of certain UI elements is ~2x the normal size. After I go to settings/universal access and turn large text on and then off again, everything is the normal size I expect. Screenshot after logging in  After turning &quot;large text&quot; on  After turning &quot;large text&quot; off  How do I get the conditions I have after cycling the &quot;large text&quot; feature to be present immediately after booting? I have the 440 version of NVIDIA's drivers installed. How does one change the Attribute table view on QGIS 3? An initial answer says to use the Switch to Table view button but I can't seem to find this button. Is it possible to "upgrade" from a 32bit to a 64bit installation? Limit point versus limit and implications for convergence I understand that similar questions to this have been asked  and  but I believe my question is different because with regards to (1) I am not asking about the set $X$ that contains the points of my sequence and with regard to (2) because I have the distinct definitions of limit and limit point in mind. That said:  My Question: My textbook, Fundamental Ideas of Analysis by Michael Reed, defines a limit point as:     Definition: Let $(a_n)$ be a sequence of real numbers. A number $d$ is called a limit point if, given $\epsilon &gt; 0$ and an integer $N$, there exists $\color{red}{\mathrm{ an }} \ n \ge N$ so that $|a_n - d| \le \epsilon$.    The definition of a limit is as follows:     Definition: We say that a sequence $(a_n)$ converges to a limit $a$ if, for every $\epsilon \ge 0$, there is an integer $N(\epsilon)$ so that $|a_n - a| \le \epsilon \color{red}{\text{  for all  }} n \ge N$.    From these definitions, and given this question:     If a sequence converges it has exactly one limit point. Is the converse true? If not, provide a counterexample   Can I conclude that the converse is indeed true simply by appealing to the definitions, or do I need a tighter proof. And if I do, I'm a little stuck on how to refine this proof. Need a matrix to seventh power which is the identity but the original is not or the negative identity The question on the homework wants a $2\times2$ matrix $R$, but $R$ not equal to positive or negative identity, that when you raise to seventh power you get the identity.  $R^7=\operatorname{Id}$.  I thought it would have to be some ones and negative ones,  it no luck, then tried some ones and some $.5$.  Just using an online matrix calculator to do brute force.  We just started talking about eigenvalues with visual example, so assuming wouldn't need that since we have not talked about how to find them yet.  Any hints or ideas to get me back on track? I have played and tried to GM games since I was in elementary school, beginning with games like Shadowrun, Earthdawn, and Champions. Recently, I've gotten dack into gaming with a variety of other games (newer Shadowrun and Earthdawn, plus TOON! and Outbreak:Undead).  I've found examples of actual play and advice on how to actually GM a game in games to be lacking. I can only help but feel I am not alone as someone who isn't already a LITERAL dungeon MASTER.  I fully realize that there are an unlimited number of GM styles, personalities, etc. I do not think I am a very good GM though. No one else wants to be the GM, and few care enough to learn the complex rules that I often have to assist with and explain AS WE PLAY. Being left with the responsibility is quite overwhelming unless I create dynamic encounters or make it all up on the spot. I'm often left having no idea what to do.  Examples of Why I Think I Need Help  This question isn't about these examples so please don't dig into them in the answers - they're here to explain the basic things that leave me feeling like I don't know how to proceed as a GM.  I've read preparation is important, but last time I tried to make a story, the players immediately killed the primary character (a friendly person.) within about 2 minutes of starting the game. While I realize now that I let the dice take control over the story, it was rather difficult to continue a story that relied on this friendly character, with the players making enemies with the character's faction. It is still discouraging when my only other attempt at a prepared story ended with the players wanting to go in a complete opposite direction of where I planned. Both times, the hours of work I put into it were completely wasted, leaving me as ignorant as I always was before.  My group often wants to play Shadowrun, Earthdawn, or L5R- but I feel as though I am a horrible GM. The last time we played Shadowrun, I was bored out of my mind as we rolled hundreds of dice to do an obvious straight-forward "kick down the door" mission.   If I make security as I believe it should or as the forum threads describe their security or a few example missions I can scrounge up (examples of stories, not actual gameplay), my players literally feel completely hopeless as to how to achieve their mission without total failure.   One time when I was a teenager, I had players blow up at me saying "We have no idea what would work. You're telling us that our character's experience or the ally operative says that would get us killed, but what else can we do?" Since there are no examples of what players can do - I am left with the same feeling as the players. As an adult, I've had a similar occurrence where a player said to me, "I don't know what to do. I don't think we can even begin this mission." I had to make something up and just go with it, creating an NPC group (that fit in with the story) that goes in before the players to do something stupid so they don't have to. Mainly because I ALSO didn't know how to accomplish the mission, so I acted as the players would using my NPC group. Extremely boring and IMO a failed dungeon run.  I've had successful stories, but I've had more boring encounters than successful ones. Seeing as how roleplaying takes endless hours to complete a story, I do not see benefit in an activity that wastes more time than is enjoyable.  How Do You Learn To GM?  Where are the resources? Books, blogs, gameplay examples? Am I missing them? How did YOU learn to be a good GM? Or do you feel as bad at this as I do, even when it's a successful, fun encounter? Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card.  If you choose to swap, your expected value should be the same, as you still have a $50\%$ chance of getting the higher card and $50\%$ of getting the lower card.  However, the other card has a $50\%$ chance of being $0.5x$ and a $50\%$ chance of being $2x$. If we keep the card, our expected value is $x$, while if we swap it, then our expected value is: $$0.5(0.5x)+0.5(2x)=1.25x$$ so it seems like it is better to swap. Can anyone explain this apparent contradiction? Error installing Apache2 Server Google Chrome won't install on 17.04 I had to remove Chrome to have enough space to upgrade from 16.10 to 17.04.  I downloaded the latest version from Google.  But when it opens in the Ubuntu Software Manager, and I click on "Install", nothing happens. How to initiate a Full Reboot/Shutdown on Windows 10 Fall Creators Update Environment: Windows 10 Fall Creators Update / 1709  Background: There is a new feature in the Fall Creators update where post reboot applications and browser windows are reopened to the state they were in before the reboot.  Issue: The only time i really reboot is when I no longer want the applications to be open. Its driving me crazy that i cant fully shutdown/reboot the machine.   Solutions attempted I always reboot by using Win+R and running "Shutdown -r -t 0". the "-r" option should do a full reboot but that isnt working for me.   Any other suggestions or hacks to control via commandline on how to do a full reboot vs. this Zombie reboote (Which is a feature i like when i did not want to reboot and had to) Remove duplicate rows in MySQL