Calculating the decimal expansions of the first values for $\frac{1}{n}$ with $n \in \mathbb{N}, n&gt;0$, we observe no period longer than the denominator: $$1, 0.5, 0.\overline{3}, 0.25, 0.2, 0.1\overline{6}, 0.\overline{142857}$$ Is it always true? Another observation is that prime numbers seem to yield longer periods than composite numbers.
I am currently working my way through "A Concise Introduction to Pure Mathematics" by Martin Liebeck, and have been stuck on an excersise for a couple of days now. The question reads:  "Show that for an integer n $\geq2$, the period of the decimal expression for the rational number $\frac{1}{n}$ is at most $n-1$."  I've been trying to solve this without making any substantial progress. And yes, I have seen explanations that when performing a long division the decimal expression either terminates to $0$ or eventually repeats. But I haven't seen any proof of this that I've been satisfied with, even less do I know how to formalise this type of proof. I've tried to write out long division with arbitrary digits, but that obviously turns out to be too much of a hassle and basically impossible to draw any conclusions from.
How do I Uninstall Windows 10 or Windows 10 preview? Sound of a next scene starts one moment earlier than video $$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$   by directly substituting $x=0$ i got $\infty$ by using L-H's rule, i got $-1/8$   the given options are  $a)\frac{24e}{11}$    $b)\frac{11e}{24}$    $c)\frac{e}{11}$    $d)\frac{e}{24}$    may be the question would be  $$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$ I found it in an old dictionary and I'm not sure what it means. It looks like the number &quot;3&quot;, but the top of the three has been flattened(and slightly curved). I've only seen this in three or four words. [] Vacuum sealing vs. water displacement method when doing longer sous vide cooking Modernist Cuisine recommends that for longer sous vide cooking (times greater than 36 hours) that bags are vacuum sealed rather than just using the water displacement method to remove the air from a ziploc bag.  What is the reason for this? Will I be losing something essential if I just use a regular ziploc bag without vacuum sealing? I'm cooking two different dishes in the oven at the same time that each require different temperatures. One is a roasted vegetable dish that says to cook at 475 for 35-40 minutes and the other is parmesan crusted pork chops at 350 for 40-45 minutes. What do you suggest I do? My laptop has a function key which turns F keys into keys which change volume or brightness. I would like to swap a number of them so that they perform that function under normal operation and are an Fx key while the function key is being held.  I assumed that the function keys were just normal keys codes with the set of keys being large enough to include all the possible speical keys. I tried to so as this tutorial suggests () and use xev to obtain the key codes and then would use xmodmap to swap their value.  This does not seem to work though. The F12 key gives the xev output:  KeyPress event, serial 41, synthetic NO, window 0x4a00001,     root 0xb1, subw 0x0, time 468005, (23,-15), root:(89,37),     state 0x10, keycode 96 (keysym 0xffc9, F12), same_screen YES,     XLookupString gives 0 bytes:      XmbLookupString gives 0 bytes:      XFilterEvent returns: False   Which I think is to be expected. However, the Increase Volume button give this response:  KeymapNotify event, serial 41, synthetic NO, window 0x0,     keys:  65  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0               0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0    And so does the Decrease Volume key. How can I perform the swapping of the keys that I would like?  Thank you Can anyone tell me how to get my caption on side of my figure, and at the same time aligned with the top of my figure? Possible to bake texture to new UV map? How should we cope with 12 month old waking in the night? We are trying to get our 12 month olds sleep routine improved now that both of us will be returning to work.  We've been working on getting the settling routine sorted out each night, and haven't quite yet got to him falling asleep on his own, but he does now settle quickly at the end of the routine in our arms.  We will be moving to trying to get him to settle in his cot soon.  Once settled he will tend to do about 3-4 hours in his cot before waking screaming.  When cuddled he will settle down in about 10 minutes and can generally be put back in his cot.  However often he will wake several times.  Last night he went down at 8, woke at 10, woke at 12, woke at 1am.  Each time putting him back in his cot.   At that point we gave up and brought him into our bed where he slept through to 6am.  When waking up he is often on either all fours or standing in the cot whilst screaming.  The main problem is that he's no longer sleeping "solidly" and will often poke and prod his mum in his sleep, and she therefore doesn't get a great nights sleep.  And with us both back to work we both need our sleep as much as possible. RegEx for Javascript to allow only alphanumeric Separate long math text under sum symbol into different lines? I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines? How to modify an array's values by a foreach loop? I've always had a LOST.DIR folder on my external SD card, which until now has been empty.  Today my phone complained that my SD card was damaged and wouldn't mount it, so I pulled it out and looked at it on my computer.  There's now a 70237 file inside LOST.DIR which is 128 MB.  It starts with something like "&euml;&lt;MSWIN4.1", seemingly to indicate that it's a dump of a  (which makes no sense).  What did Android dump here and and why?  (I think I may have disconnected my phone from my PC yesterday without ejecting the drives first, so that might be the "why", though it was from Windows XP and not Windows 98.)  Is there a simple way to recover the information in this file? Why does my query run fast in Environment A, but slow in Environment B? I have a piece of SQL that seems to run really fast in Environment A, but the exact same query runs really slow in Environment B!  Environments are supposed to be the same, so what should I do and/or where should I look in order to see why the query doesn't perform the same? Is there a way to locate diagrams and move them wherever you want in QGIS 1.7.4/ 1.8.0? I created a pie chart under the tab "diagram". I can mark those diagrams with a green box and move them via the label-tool, but they won't change their position in the end. When I create diagrams with the "diagram overlay" tab they cannot even be marked.  Also, what is the difference between "diagram" and "diagram overlay"? I just attempted to fix a typo on a question on cooking.stackexchange.com and saw this     Oops! Your edit couldn't be submitted because:         Your edit summary must be at least 10 characters   This edit is less than 6 characters; surely more can be improved with your edit?      So, misspelled words have to just be left misspelled?  That could prevent the post from being found in a search. How to pass argument to script which is input to bash How can I retrieve mp3 metadata in Python? Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?