If $x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x }  \right\rfloor =1$, then $x$ is irrational.  I am thinking of using the contrapositive: If $x$ is rational, then $x-\left\lfloor x \right\rfloor +\frac { 1 }{ x } -\left\lfloor \frac { 1 }{ x }  \right\rfloor \neq 1$  However, I don't know how I would approach this after that point. My first guess it to create a rational number, $\frac { a }{ b } $ such that $a,b\in\mathbb{Z}$, but I don't know how to manipulate it from there to get to the right hand side. I guess the floor function is throwing me off.   I would appreciate a hint/nudge in the right direction so that I can arrive at the solution by myself.
Prove : $$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$  I think the way to go here is to falsely assume that $x$ is rational, thus having $x=\frac mn$ with $m,n$ integers and go with that, we also know that $x$ with the celling belongs to the integers and that $\frac 1x$ with the celling gotta be a $0$. Still can't prove it. Thank in advance !
Is it okay to vectorize this image to make a poster out of it even though the image is not Creative Commons or anything like that? I'm not using any part of the image directly and only going to use it as a guide to make my own SVG version.  This is the image:    Here is .  It says that it is CC licensed but the original NatGeo one isn't, I don't think.  I don't plan to make money off it or anything, I just want a nice poster and the quality of the original isn't that good. In what kind of rings a divisor of a product is a product of divisors? In a , if $a|bc$, then $a$ can be written as $a = a_1a_2$ so that $a_1|b$ and $a_2|c$.  Is this property of a commutative ring strictly weaker than the property of being a UFD?  If so, does it have a name, or is it equivalent to some other well-known property? Why is /tmp a symlink to /private/tmp on Mac OS X? In other words, why isn't /tmp just a regular directory, like on Linux or BSD? I understand how it works and I don't mind it, I'm just interested in the (historical?) reasoning behind it. If I change the password for the root user is that going to automatically change the password for the user account?   I found this link for root:    How do I change the user account password? help with some Hilbert style proofs in a propositional logic axiom system. I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. What is $\sqrt{i}$? If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary?     Is it used or considered often in mathematics? How is it notated? got me wondering about the differences between these three ways of measuring size: a kibibyte, a kilobit, and the conventional kilobyte.  I understand that these measurements have different uses (data transfer rate is measured in bits/sec), but I'm not quite sure if I can tell the difference between Mb and MB and MiB.  Here is a comment, reproduced below, taken from  (emphasis mine).     The C64 has 65536 bytes of RAM. By   convention, memory size is specified   in kibiBytes, data transfer rates in   kilobits, and mass storage in   whatever-the-manufacturers-think-of-now-Bytes. Harddrives use T, G, M and k on the   label, Windows reports the size in Ti,   Gi, Mi and ki. And those 1.44MB   floppys? Those are neither 1.44MB nor   1.44MiB, they are 1.44 kilokibibytes. That's 1440kiB or 1'474'560 bytes. –   Third Let $a&gt;b&gt;0$ and let $n \in N$ satisfy $n \geq 2$.  Prove that $$a^{1/n}-b^{1/n}&lt;(a-b)^{1/n}.$$  If we let $a=x$ and $b=x-1$, then we need to show that $f(x):=x^{1/n}-(x-1)^{1/n}$ is decreasing for $1\leq x$ (which means that $f'(x)\leq 0$ and $f$ is differentiable), and evaluate $f$ at $1$ and $a/b$. Does equality of characteristic polynomials guarantee equivalence of matrices? I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and then the question.  True or False:   let $A,B \in \mathbb{Q}^{n \times n }$  Suppose $xI-A$ and $xI-B$ are equivalent then $\det(xI-A) = \det(xI-B)$.     I wrote in my notes this is true because any two matrices are equivalent in $\mathbb{Q[x]}^{n \times n}$ if and only if they have the same invariant factors.  Since the characteristic polynomial is a product of the invariant factors it follows that $\det(xI-A) = \det(xI-B)$.        True or False:   let $A,B \in \mathbb{Q}^{n \times n }$. If $\det(xI-A) = \det(xI-B)$ (in $\mathbb{Q}[x]$) then $xI-A$ and $xI-B$ are equivalent  in $\mathbb{Q}[x]^{n \times n}$   I think this is false because of the fact that equality of determinants is not strong enough to guarantee all invariant factors are equal but I do not have an example. Determining relationship (e.g. number of removes) with cousin? How do I figure out if someone is a first cousin, second cousin, cousin once-removed, etc? Suggested edits in top-bar put flags over the winterbash snowflake Suggested edits in the top bar put the flag count right over the winterbash snowflake. Looks pretty ugly if you ask me... The camera in Far Cry 3 is useful for tagging enemies from a distance but also includes the ability to take pictures.  This does not seem to do anything useful when used on animals or enemies.  Is there a purpose? Increasing the size of the memory: not all files are built successfully I am trying to compile a large file which contains a tikz picture with more data points than tex can handle.  Following the questions  and  I set to increase the capacity of tex by changing the file texmf.cnf in my path /usr/local/texlive/2012. I included the line  main_memory = 10000000   and on terminal I run  sudo fmtutil-sys --all   The problem is that this memory is not enough, and numbers bigger than that don't finish the fmtutil run properly. I get the error   fmtutil: Error! Not all formats have been built successfully.   To give a better idea of where the problem is coming from, if I only include in the file texmf.cnf the lines  main_memory.latex = 15000000 main_memory.pdflatex = 15000000   then these are the files which are not properly built:  This is a summary of all `failed' messages: `pdftex -ini  -jobname=latex -progname=latex -translate-file=cp227.tcx *latex.ini' failed `pdftex -ini  -jobname=pdflatex -progname=pdflatex -translate-file=cp227.tcx *pdflatex.ini' failed   How can I increase the size of the memory beyond 10000000? Am I doing anything wrong?  (BTW, I am running texlive 2012 on a mac) What is the type of string literals in C and C++? How can I represent an 'Enum' in Python? I'm mainly a C# developer, but I'm currently working on a project in Python.  How can I represent the equivalent of an Enum in Python? Evaluating a mathematical expression in a string C# Speech Recognition - Is this what the user said? 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I used to download 1/3 arcsecond elevation geotiffs from extract.cr.usgs.gov using web services.  But a few days ago, that stopped working, probably because USGS reorganized their site.  I tried emailing USGS and haven't heard back yet.    I could, of course, manually download states or degree tiles using the seamless server, but I want to do it programmatically so that users of my trip planner can get elevation data for their region without human intervention.  And SRTM isn't high enough resolution for bicycle/wheelchair trip planning.  Any ideas? How do I enable Scroll Lock? Why is Modus Ponens valid? I am having trouble understanding what defines Entailment operator. On Mathoverflow I posted this question on what I perceive to be .  Consider: :     P therefore Q  P            Therefore, Q   My question is could we have arbitrarily defined it such as:     P therefore Q  P            Therefore, P   However, if an advanced civilization receives information in the form of following nested structures/containers or packets, how will they interpret Modus Ponens if in an alien land Modus Ponens is not defined?    Therefore, it raises the validity of MP. How do we define the implication or entailment operator? Because if we use an argument that relies on implication or entailment operator ipso facto, then it is trivial that logic is circular.  Is there a flaw in my rationale?