I was using Python to add some simple numbers together, and I got a strange result  &gt;&gt;&gt; 16+4.8+6.4+0.48+1.92+.48 30.080000000000002   That is clearly not the right answer, so why does Python add that extra bit? How reliable is Python for more complicated computations?
Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen?
Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real. Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.  We know $f(0)=0$, and if one assumes differentiability it is quite easy.      But how does monotone make up for differentiability? Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$  Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$.  I would however like to learn the technique of dealing with surds such as these, especially in trignometric problems as I have a lot of similar problems and I don't have a clue as to how to deal with those.  Hints please!  EDIT:  What I have done using half angles is this: (and please note, for convenience, I am dropping the degree symbols. The angles here are in degrees however).  I know that $$ \cos 15 = \dfrac{\sqrt3+1}{2\sqrt2}$$  So,   $$\sin7.5 = \sqrt{\dfrac{1-\cos 15} {2}}$$ $$\cos7.5 = \sqrt{\dfrac{1+\cos 15} {2}} $$  $$\implies \cot 7.5 = \sqrt{\dfrac{2\sqrt2 + \sqrt3 + 1} {2\sqrt2 - \sqrt3 + 1}} $$ No newline after (sub)section? I want my text to appear right after the call to \subsection{Name} but  I can't find how to do it. I'm using the article class.  An example of command:  \subsection{a)} Blah blah bla...   produces:     a)      Blah blah bla...   instead of what I want:     a) Blah blah bla...   Anything I could do? Do inequalities hold under differentiation? If I have the inequality $f \leq g$, does this imply $f'\leq g'$?  The reason I am asking is because I am using this fact to prove a question with Taylor theorem. Role of dependencies while installing software?? What is the role of dependencies while installing any software?I was installing vlc player and it were closed by giving me error something like check dependency .  What is to solution of this and how i'll get these dependencies??  Thanks in Advance. Choosing variables to include in a multiple linear regression model Abelian group (Commutative group) Prove that if in a group $(ab)^2= a^2 b^2$ then the group is commutative.  I am having a hard time doing this. Here is what I have so far: Proof:  $a^2 b^2= a^1 a^1 b^1 b^1$ =$aa^{-1}bb$ =ebb Hence,$aa^{-1}=e$  I am stuck, I do not know if this is the right process in proving this Why didn't Harry himself drink Polyjuice Potion instead of all others in Deathly Hallows Part 1? In Harry Potter and the Deathly Hallows Part 1, it is shown that members of the Order of Phoenix come to rescue Harry from his house at Privet Drive. There, they all drink Polyjuice Potion and assume the form of Harry to confuse death eaters in identifying who is the real Harry.  An obvious, more safe, better choice would be to change Harry into someone else. Maybe a muggle, like his cousin Dudley. And then, safely take him to the Weasleys, maybe apparating. That way, death eaters won't ever find out. Isn't that correct?  Why didn't they take this path, and follow a rather complicated and risky one? Is it possible to migrate programs from 32 bit Windows to a new 64 bit Windows machine (both Windows 7)? I've got a Windows 7 Pro 32 bit PC. Getting a new Windows 7 Home 64 bit.  Is there anything I can use to transfer my  programs (the ones that don't come in a 64 bit flavor)  to the new PC? There is a related  that doesn't address the programs part. Rotate only one screen If I have only the internal monitor (LVDS1) with an xrandr -o left command. Now I want to attach a second monitor, say to HDMI1. Then I'd like to rotate only the internal monitor, but not the external one. I tried the following (with no external attached though, since I am mobile currently), but that does not do anything:  xrandr --output LVDS1 -o left   How can I rotate only one monitor?  I do need some command line, since I want to put this into a script. Does a card that affects 'each' creature affect creatures that enter the battlefield after it resolves? My opponent casts , which resolves, then he attacks with his creatures.  In my hand, I have a creature card with Flash. If I summon this creature before the declare blockers step, will it be affected by Predatory Rampage's blocking requirement? Why can't I connect to Pokemon Go via mobile data? It connects to Pokemon Go via wi-fi perfectly fine, but when I try and connect to Pokemon Go via 3G it just gets stuck at the first bar on the loading screen with the eventual pop up saying "Failed to get player information from the server." My 3G connection works perfectly fine with every other app I have, and has been easily able to stream HD videos so I can't see connectivity with 3G being an issue.   It makes even less sense as once connected via wi-fi, the game easily continues to operate via mobile data once I leave the house. However once the app crashes (which it usually does after 15-30 minutes) I try to reload it on mobile data and it gets stuck on the loading screen. If I can find a local free wi-fi spot nearby I can re-connect easily, and then move back onto mobile data.  I've tried removing and reinstalling Pokemon Go and that didn't work. I also reset all my app preferences on my phone and that didn't work.  Not sure if this helps, my phone is a Huawei P8, Android version 5.0.1.  I signed up to Pokemon Go using my Google account. Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$. This integral is very closely related to the sophmores dream that states  $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$  For example here   Now I want to bound the integral, and showing that is less that 2.  For the interval $[0,1]$ a good bound is rewriting it to $\exp(x\log x)$ and using the expansion   $$ 1 - x \log(x) + \frac12 (-x \log(x))^2$$  but how does one handle $[1,\infty)$ ? In this answer here     gives bounds to the integral, but they are not tight enough.. So to taste my question again, how does one prove that   $$ \int_0^\infty \frac{\mathrm{d}x}{x^x} \leq 2 $$ How create Redirect pattern from old to new URLs? D7: I'm going to update an articles path pattern (/articles/[node:uid] → /post/[new tokens]) but I need redirect users from old path to new one using tokens. And I don't know how to do that.  I tried using the , but it allows set only separate URL for each node; no group redirect patterns are allowed. Also the  doesn't allow to set patterns. How can I achieve this?    The problem partially solves by applying "Update url alias" option in admin/content. I do not find out how to show all selected type nodes at one page without pages, because it is terrible slow applying option page by page. Anyway, by this way, old aliases still exist, work, and don't redirect to new aliases. I am trying to count the number of bit-strings of length 8 with 3 consecutive zeros or 4 consecutive ones. I was able to calculate it, but I am overcounting. The correct answer is $147$, I got $148$.  I calculated it as follows:  Number of strings with 3 consecutive zeros = $2^5+5\times2^4 = 112$, because the 3 zeros can start at bit number 1, 2, 3, .., 6  Number of strings with 4 consecutive ones = $2^4+4\times2^3 = 48$, I used the same reasoning.  Now I am trying to count the number of bit-strings that contain both 3 consecutive zeros and 4 consecutive 1s. I reasoned as follows:   the strings can be of the following forms: 0001111x, 000x1111, x0001111..thus there are $2+2+2 = 6$ possibilities for bit-strings where the 3 consecutive zeros come first. Symmetrically there are $6$ bit-strings where the 4 consecutive ones come first.  Thus the answer should be = $112+48-12 = 148$.  clearly there's something wrong with my reasoning, if someone could point it out, that would be awesome. Thanks I'm using Ubuntu 13.04 on a Lenovo X220.  Why do I have so many .Xauthority.* files in my home directory? I.e  .Xauthority .Xauthority.0JW6UW .Xauthority.2HWRVW .Xauthority.2TA6VW .Xauthority.49F8VW .Xauthority.4E27UW .Xauthority.7CLTVW .Xauthority.8JGGXW .Xauthority.96SAVW .Xauthority.AOUFVW .Xauthority.ASDOVW .Xauthority.CGVJXW .Xauthority.E0ZSVW .Xauthority.E5VZWW .Xauthority.FE64UW .Xauthority.FIHDWW .Xauthority.HL45VW .Xauthority.HYCCVW .Xauthority.ILUEWW .Xauthority.JXJHVW .Xauthority.K1QJXW .Xauthority.KXSOVW .Xauthority.LEKEVW .Xauthority.M48WVW .Xauthority.M6QTVW .Xauthority.MW6NWW .Xauthority.N57TWW .Xauthority.O5HAWW .Xauthority.OIC5VW .Xauthority.P2RPVW .Xauthority.Q0FRVW .Xauthority.QW01WW .Xauthority.RDW5WW .Xauthority.T0TLWW .Xauthority.TSO4WW .Xauthority.U6R9VW .Xauthority.V3KAWW .Xauthority.W8Z1WW .Xauthority.XDLGVW .Xauthority.XOOHWW.Xauthority.XUICVW  They are all zero bytes except the first, with timestamps all over the place.  What are these, and are they safe to delete? If a two vectors $\bf u$ and $\bf v$ in linearly independent, can their images be linearly dependent given they were transformed by a linear transformation $T$? I think not. My reasoning If the images are linearly dependent, then $c_1T({\bf u})+c_2T({\bf v})={\bf 0}$ for scalars $c_1$ and $c_2$, not both $0$. Then $c_1T({\bf u})+c_2T({\bf v})={\bf 0}$ $T(c_1{\bf u}+c_2{\bf v})=T({\bf 0})$ $c_1{\bf u}+c_2{\bf v}={\bf 0}$ Therefore, ${\bf u}$ and $\bf v$ are linearly dependent. My reservation A homework problem seems to contradict the above. Here's the problem: Let $\mathbb R^n\rightarrow\mathbb R^m$ be a linear transformation. Suppose $\{{\bf u},{\bf v}\}$ is a linearly independent set, but $\{T({\bf u}),T({\bf v}\})$ is a linearly dependent set. Show that $T({\bf x})={\bf 0}$ has a nontrivial solution. I've found a nasty VBS way to do this, but I'm looking for a native PoSh procedure to edit the properties of a .LNK file.  The goal is to reach out to remote machines, duplicate an existing shortcut with most of the correct properties, and edit a couple of them.  If it would just be easier to write a new shortcut file, that would work too. Setting background-image using jQuery CSS property In order to better understand the SQL Server query processor, I've been thinking about how the ORDER BY clause works, and how SQL Server provides results.  It appears SQL Server will provide results in exactly the same order for any given set of data that remains the same and never, ever changes.  Once you introduce any kind of uncertainty such as developers changing something, the order of results can no longer be expected to be the same.  Simply seeing the results in the same order, each time you press F5 does not guarantee the results you expect.  Try this:  USE TempDB; DROP TABLE TestOrderBy; DROP TABLE TestOrderBy2; CREATE TABLE TestOrderBy (     ID INT NOT NULL CONSTRAINT PK_TestOrderBy PRIMARY KEY CLUSTERED IDENTITY(1,1)     , SomeData varchar(255) ); INSERT INTO TestOrderBy (SomeData) VALUES ('E'); INSERT INTO TestOrderBy (SomeData) VALUES ('D'); INSERT INTO TestOrderBy (SomeData) VALUES ('C'); INSERT INTO TestOrderBy (SomeData) VALUES ('B'); INSERT INTO TestOrderBy (SomeData) VALUES ('A');  CREATE TABLE TestOrderBy2 (     ID INT NOT NULL CONSTRAINT PK_TestOrderBy2 PRIMARY KEY CLUSTERED IDENTITY(1,1)     , SomeData varchar(255) ); INSERT INTO TestOrderBy2 (SomeData) VALUES ('E'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('D'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('C'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('B'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('A');  SELECT SomeData  FROM TestOrderBy UNION ALL SELECT SomeData FROM TestOrderBy2;  CREATE INDEX IX_TestOrderBy_SomeData ON TestOrderBy (SomeData); CREATE INDEX IX_TestOrderBy2_SomeData ON TestOrderBy2 (SomeData);  SELECT SomeData  FROM TestOrderBy UNION ALL SELECT SomeData FROM TestOrderBy2;   The results:    As you see, adding a simple index on the fields selected in the query alters the order of the results.  From here on, I'm adding ORDER BY unless I truly don't care.