Why can't $\pi$ be expressed as a fraction?
Why is $\pi$ irrational if it is represented as $C/D$?
Add days to JavaScript Date How to add days to current Date using JavaScript. Does JavaScript have a built in function like .Net's AddDay? What is the precise definition of a rigid shape? Editing old questions Why aren't the humans wearing environmental suits in Star Wars spacecraft? Is there a way to make bash more TAB friendly? I use bash on both my Mac and as my cygwin shell. Overall, I'm quite pleased with it.  My one complaint however is that its TAB completion isn't as good as Windows Command Line TAB completion.   For example, if there are two files with the same prefix, then bash just beeps at me when I try to TAB complete. If I keep nailing TAB, then it will eventually show me a list of potential files.  I definately prefer how Windows will cycle through potential files right on the command line.   Is there a way to get bash to operate similar to Windows Command Line in this regard?  If there isn't (I do not want to switch shells), am I missing something about TAB completion in bash? Is there a more efficient way to use it? How to resolve the "GPT detected. Please create a BIOS-Boot partition" error when using boot-repair? After installing Ubuntu, Windows didn't boot so I installed boot repair to fix this problem. But I am getting the message displayed below.  I tried to make a new partition with unformatted file system but still receiving the same message.  How do I fix it? Ubuntu 13.10 Macbook Air Mavericks Parallels 9 Black Screen I can install Ubuntu 13.10 in parallels ok and then log in.  Problem is Ubuntu does an update and I get black screen / desktop such that I can't see any app windows that I open.  I've tried a few of the other suggestions and they don't work for me. Root filesystem check fails after power failure during installation During the "install" phase of the upgrade there was a power failure. After when starting up again the following errors are reported:  init: udevtrigger main process (420) terminated with status 1 init: udevtrigger post-stop process (428) terminated with status 1 init: udevmonitor main process (419) killed by TERM signal The disk drive for / is not ready yet or not present Continue to wait; or press S to skip mounting or M for manual recovery   Pressing M gives me the following message:  Root filesystem check failed. A maintenance shell will now be started. CONTROL-D will terminate this shell and reboot the system. Report sites that use Stack Exchange content without following attribution rules here NB The following is a community-generated list of websites that republish Stack Exchange content without attributing it properly. It is no longer being maintained, because the procedure for reporting such sites has changed; see the duplicate for more information. There are a number of license-violating clones of Stack Exchange sites popping up that use Stack Exchange's  data without following our Creative Commons attribution terms. Those terms are , and are also included as a .txt file in every data dump we produce.  Google now has  in search results. Also, you can :  The option to block a site appears when you click a search result and then navigate back to the search results page. Click the &quot;Block&quot; link next to that result to block all pages within the site's entire domain.   This post was created as an attempt to organize information originally posted . You may also be looking for . Is it possible to make an underfull tabulary spread out? I'm having some troubles understanding the solution (going from step $2$ to step $3$) of this linear congruence equation. Here's the problem with the solution:   Solve the congruence $2x + 11 \equiv 7 \pmod 3$.   First, we reduce all the coefficients $\pmod 3$:  $$2x + 2 \equiv 1 \pmod 3. \tag{1}$$  Next, we add $1$ to both sides, using the fact that $2 + 1 \equiv 0 \pmod 3$:  $$2x \equiv 2 \pmod 3. \tag{2}$$  Finally, we multiply both sides by $2$, using the fact that $2 · 2 = 4 \equiv 1 \pmod 3$:  $$x \equiv 1 \pmod 3. \tag{3}$$  That is, any number in the set $\{\ldots, −5, −2, 1, 4, \ldots\}$ will solve the original congruence.    Like I said my problem is the step between $(2)$ and $(3)$. If I multiply both sides of $(2)$ by $2$ I get:  $$4x \equiv 4 \pmod 3.$$  Now, I understand that $4$ is congruent to $1 \pmod 3$ so the RHS "simplifies" to $1 \pmod 3$ but I don't understand why the LHS "simplifies" to $x$. Can someone help me with this? Question concerning linear combinations of vectors and linear independence in Linear Algebra. Derivative of determinant of a matrix I would like to know how to calculate:  $$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big).$$ I have this function $$f(x) = x\cos(1/x)$$ defined for all $x \geq 1$.      How do I prove that    $$f(x+2)-f(x)&gt;2$$    for all $x\geq 1$?   I found $$f'(x)= \cos(1/x) + \frac{\sin(1/x)}{x}$$ and $$f''(x)=-\frac{\cos(1/x)}{x^3}.$$ How do i use this information? Update 02/02/2019 I appreciate all the responses and help but unfortunately i must have not seen the emails stating there were responses! I am now looking into a NEST thermostat and was curious if they too, use the same connections. Lastly, looking for a quick recommendation on which WiFi Compatible therms are preferred by the memebers that respond! Thanks again your knowledge is highly appreciated! Bob  I Purchased the Honeywell WiFi Prog Thermostat to replace my Round Digital Honeywell Thermostat. I have separate Heating and Cooling (Heating is a Weil McLain Boiler for Hot water baseboard and Hot water(no tank) and Cooling is a separate unit that was added later.)  Wires:  RC Cool R Heat W Heat Y Cool G Cool   3 wires from AC unit and 2 from Heater.  If I were to go the separate transformer route to power the thermostat, where would I land the 2 wires?  C and ??? In the Tiny Tina DLC "The Beard Makes the Man" quest, the objectives are (1) Crush five drunk dwarves, (2) Pick up five beards, and (3) Get five dwarves drunk. (The last one's optional.) I found the miner dwarves in the mine on the minimap, but they won't follow me into the crusher machine. How do I lure them? Does it have to do with getting them drunk, and how can I do that with no alcohol around? I thought getting drunk was optional? What happens 'internally' on a unstable/overclocked/undervolted CPU? When you undervolt or overlock an CPU, when it passes a certain limit it starts to make the system unstable, reboot, etc. But these are the effects of it.  What I'm curious about is what happens on the CPU. What happens? It gives wrong answers to instructions? Bits on cache flip? Count number of increasing functions, nondecreasing functions $f: \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, m\}$, with $m \geq n$. I stumbled upon a question given like:  Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.   Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following two types:  (a) strictly increasing; i.e., whenever $x &lt; y, f(x) &lt; f(y)$, and  (b) non-decreasing; i.e., whenever $x &lt; y, f(x) \leq f(y)$.  I tried in the following way.  For (a). We can say $f(n)&gt;f(n-1)$ AND $f(n)&gt;f(n-2) \ldots f(n)&gt;f(1)$ (total $n-1$ elements)  Similarly for $f(n-1)&gt;f(n-2), f(n-1)&gt;f(n-3) \ldots f(n-1)&gt;f(1)$ (total $n-2$ elements)...  and $f(2)&gt;1$  Like this $$(n-1) + (n-2) + \ldots + (1) = \frac{n(n-1)}{2}$$  Is this correct?  For (b). Since there's equality, it will be $$n + (n-1) + ... (1) = \frac{n(n+1)}{2}$$  Is my approach correct? Any help is appreciated. Thanks in advance. Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A&#39; \mathrel{\overset{f&#39;}{\to}} B&#39; \mathrel{\overset{g&#39;}{\to}} C&#39; \to 0 $$ and morphisms $a : A \to A&#39;, b : B \to B&#39;, c : C \to C&#39;$ making the obvious diagram commute. The snake lemma states that there is then an exact sequence $$0 \to \ker a \to \ker b \to \ker c \to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c \to 0$$ where the morphisms between the kernels are induced by $f$ and $g$ while the maps between the cokernels are induced by $f&#39;$ and $g&#39;$.  It is not hard to show that the morphisms induced by $f, g, f&#39;, g&#39;$ exist, are unique, and that the sequence is exact at $\ker a, \ker b, \operatorname{coker} b, \operatorname{coker} c$. With the use of a somewhat large diagram shown , we can even construct the connecting morphism $d : \ker c \to \operatorname{coker} a$. However, I'm stuck showing exactness at $\ker c$ and $\operatorname{coker} a$. I thought Freyd might have had an element-free proof in his book, but it turns out he proves it by diagram chasing and invoking the Mitchell embedding theorem [pp. 98–99]. Is there a direct proof? I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this?