I want to prove that $\sqrt{p\over p+1}$ is irrational where $p$ is any prime number.   I have been thinking about this problem for several hours without any progress.  To show that $\sqrt{p\over p+1}$ is irrational, I assume that it is rational. So there are integers $m$ and $n$ (which have no common divisors) such that $\sqrt{p\over p+1}=\frac{m}{n}$. By squaring and rearranging, I get $pn^2=(p+1)m^2$ but I am not sure how to proceed from here. I can say that $(p+1)m^2$ is divisible by $p$ and then I do not know what to do!
I would like to show that  $$\forall n\in\mathbb{N}^*, \quad \sqrt{\frac{n}{n+1}}\notin  \mathbb{Q}$$  I'm interested in more ways of proofing this.  My method :  suppose that $\sqrt{\frac{n}{n+1}}\in  \mathbb{Q}$ then there exist $(p,q)\in\mathbb{Z}\times \mathbb{N}^*$ such that $\sqrt{\frac{n}{n+1}}=\frac{p}{q}$ thus   $$\dfrac{n}{n+1}=\dfrac{p^2}{q^2} \implies nq^2=(n+1)p^2 \implies n(q^2-p^2)=p^2$$ since $p\neq q\implies p^2\neq q^2$ then  $$n=\dfrac{p^2}{(q^2-p^2)}$$  since $n\in \mathbb{N}^*$ then $n\in  \mathbb{Q}$   I'm stuck here and I would like to see different ways to prove $\sqrt{\frac{n}{n+1}}\notin  \mathbb{Q}$
How to install a LibreOffice Language Tools (Spelling Check, Thesaurus...) What are the steps to install the Spelling Check and other language supports for LibreOffice. For example spanish, french, italian, etc.. apt: how to be sure that a specific package gets installed/updated from specific repository and not from the default one? Double Sideband Suppressed Carrier modulator List I am using ring modulator for DSB-SC  am modulation in a transmitter circuit. Can anyone please tell me about more DSB SC modulator for am modulation? I want to know which one to use where? Find the population size that maximizes the probability that two random samples of size $20$ will have exactly $2$ members in common Ten fish are caught in a lake, marked, and then returned to the lake. Two days later 20 fish are again caught, 2 of which have been marked.  (a) Find the probability of 2 of the 20 fish being marked if the lake has $k$ fish (assuming the fish are caught at random). (b) What value of k maximizes the probability?    Progress:     I got the (a), $$\binom{10}{2}\binom{k-10}{18}\bigg/ \binom{k}{20}$$ But how do I get part (b) where $k$ is to maximize the probability? I'm trying to use a mask to hide a part of a render layer, so that when the render layer is shown over a movie, some parts in the movie will appear to be in front of the render layer.  For example, assume I want to add a box to this image, so that it would look like it's behind the person:    I created the following simple object:    If I render the object, it looks great - the object is solid, everything else is transparent (I didn't use any 'sky'):     Now I create a mask to describe the standing person (or at least the part that would otherwise be covered by the box):    What I want now is to create a "hole" in the rendered box, so that when I use Alpha-over, the box look as if it's behind the person, but I can't find an easy way to do that. My first guess was to use 'Set alpha' to apply the mask (inverted) as an alpha for the rendered layer, but this doesn't look good:    The mask did create a "hole" where it should, but now all the other pixels are not transparent. How can I "retain" the transparent pixels while setting more pixels to be transparent based on the mask (use it as a matte)?  Note: I can do this with (I'm ashamed to say) some math work on the alpha channel, but it just doesn't seem right:    Thanks, and sorry for the [oh so] long question. Looking for a function $f$ that is $n$-differentiable, but $f^{(n)}$ is not continuous I am looking for a real valued function of real variable that is $n$-differentiable, but whose $n$th derivative is not continuous.  This is my function:  $f_n(x) = x^{n+1} \cdot \sin{\frac{1}{x}}$, if $x \neq 0$ and 0, if $x=0$. $n\in \{0,1,2,\ldots\}$ For example, if $n=0$, $f_0$ it's continuous and non diferentiable in $0$. When a 'blur' event occurs, how can I find out which element focus went *to*? Suppose I attach an blur function to an HTML input box like this:  &lt;input id="myInput" onblur="function() { ... }"&gt;&lt;/input&gt;   Is there a way to get the ID of the element which caused the blur event to fire (the element which was clicked) inside the function? How?  For example, suppose I have a span like this:  &lt;span id="mySpan"&gt;Hello World&lt;/span&gt;   If I click the span right after the input element has focus, the input element will lose its focus. How does the function know that it was mySpan that was clicked?  PS: If the onclick event of the span would occur before the onblur event of the input element my problem would be solved, because I could set some status value indicating a specific element had been clicked.  PPS: The background of this problem is that I want to trigger an AJAX autocompleter control externally (from a clickable element) to show its suggestions, without the suggestions disappearing immediately because of the blur event on the input element. So I want to check in the blur function if one specific element has been clicked, and if so, ignore the blur event. Can Windows use a Linux shell instead of cmd? I'm developing a multilanguage software. As far as the application code goes, localizability is not an issue. We can use language specific resources and have all kinds of tools that work well with them.  But what is the best approach in defining a multilanguage database schema? Let's say we have a lot of tables (100 or more), and each table can have multiple columns that can be localized (most of nvarchar columns should be localizable). For instance one of the tables might hold product information:  CREATE TABLE T_PRODUCT (   NAME        NVARCHAR(50),   DESCRIPTION NTEXT,   PRICE       NUMBER(18, 2) )   I can think of three approaches to support multilingual text in NAME and DESCRIPTION columns:   Separate column for each language  When we add a new language to the system, we must create additional columns to store the translated text, like this:  CREATE TABLE T_PRODUCT (   NAME_EN        NVARCHAR(50),   NAME_DE        NVARCHAR(50),   NAME_SP        NVARCHAR(50),   DESCRIPTION_EN NTEXT,   DESCRIPTION_DE NTEXT,   DESCRIPTION_SP NTEXT,   PRICE          NUMBER(18,2) )  Translation table with columns for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains a column for each language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID,   TEXT_EN NTEXT,   TEXT_DE NTEXT,   TEXT_SP NTEXT )  Translation tables with rows for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains only a key, and a separate table contains a row for each translation to a language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID )  CREATE TABLE T_TRANSLATION_ENTRY (   TRANSLATION_FK,   LANGUAGE_FK,   TRANSLATED_TEXT NTEXT )  CREATE TABLE T_TRANSLATION_LANGUAGE (   LANGUAGE_ID,   LANGUAGE_CODE CHAR(2) )    There are pros and cons to each solution, and I would like to know what are your experiences with these approaches, what do you recommend and how would you go about designing a multilanguage database schema. I'm still learning, but I'm trying to follow a tutorial to apply texture to a model. I only want to apply the texture to one object in the scene, so I selected it, unwrapped it, and painted over it. Now I selected it in the Edit Mode (I can't select it in the Object mode because all is locked together) and went into Node Editor, but the nodes are greyed out.  Why? Is every solution of Einstein field equations unique? Einstein's equation is  $$8 \pi T_{ab} = G_{ab},$$  where the left side contains the stress-energy tensor and the right side contains the Einstein tensor.  Is there exactly one unique stress-energy tensor corresponding to a given spacetime curvature? Or is it possible for one curvature (i.e. one spacetime metric) to be produced by multiple different mass configurations? What is a word that means "someone who pretends to be your friend but is actually your enemy?" Why is ATP the most prevalent form of chemical energy storage and utilization in most cells? Unexpected voltage with switch turned off When umpire change the cricket ball, was the old ball thrown as waste or keep for reuse? I no longer need an answer to a question; what should I do to the question? What is the most appropriate way to configure biblatex for use with an unsupported language? Does either pill actually do anything? When Morpheus offers the pill choice to Neo, they are still in the Matrix. That leads me to belive that none of the pills are real thus having no particular effect in the real world. Was it some sort of placebo effect that freed Neo from the Matrix or do the pills have an actual effect? What is a catchy word that means (non-)self-descriptive I'm having fun with paradoxes, and I'm trying to come up with a catchy word for non-self-descriptive.  Maybe "autosemantic" fits the bill for self-descriptive (definitely open to other suggestions!), but I'm having a hard time negating it in a natural way.  For an example of the sentence I'm trying to construct:     Is "inautosemantic" inautosemantic?   It just isn't flowing. Select Permissions Across Multiple Schemas Within the Same Database I have been asked by a client to implement some views for reporting purposes, the views will be accessed via PowerBI, Excel and SSRS. The designated user will only have access to the view and no underlying tables must be available to the user.  The problem I have encountered in that the SQL within the view touches 3 different schemas (All within the same database):   Pupil Provider  Security   The view is:  CREATE VIEW dbo.vTestPermissions AS SELECT a.Column1,        b.Column1,        c.Column1 FROM Pupil.Table1 a JOIN Provider.Table2 b ON a.Column1 = b.Column1 JOIN Security.Table3 c ON a.Column1 = c.Column1   The owners of the Tables/View are as follows:   Pupil.Table1 - Owner Pupil Provider.Table2 - Owner Provider Security.Table3 - Owner Security vTestPermissions - Owner dbo   When I select from the view I get an error:     The SELECT permission was denied on the object 'table3', database   'TEST', schema 'Security'   I have tried giving SELECT permission with and without the GRANT option to the schema, and the tables, but that makes the underlying tables available to the user.  Any help on this will be greatly appreciated.