Show that $\sqrt n$ is irrational if $n$ is not a perfect square, using the method of infinite descent. I know how to prove this by doing a contradiction proof and using The Fundamental Theorem of Arithmetic, but now I'm asked to use infinite descent to prove it. Then the very next problem says &quot;Why does the method of the text fail to show that $\sqrt n$ is irrational if $n$ is a perfect square?&quot; I'm confused by this. Any hints or solutions are greatly appreciated. I was thinking of the standard argument, let $\sqrt n = {a\over b}$ where $gcd(a,b)=1$ and then through some algebra arrive at a common factor for both $a$ and $b$ which contradicts the fact that $gcd(a,b)=1$ and so we can apply this over and over again, but then I don't understand how the next problem says to explain why this method fails.
I found the following proof arguing for the irrationality of $\sqrt{2}$.     Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q &gt; 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::= (\sqrt{2} - 1)q$. Clearly $0 &lt; q' &lt; q$. But an easy computation shows that $(\sqrt{2} - 1)q'$ is a nonnegative integer, contradicting the minimality of $q$.   Clearly, this is a vague proof in that it expects the reader to work out the math involved on his own. I took that as an exercise and worked out the following:  Assuming $\sqrt{2}$ to be rational, let $$\sqrt{2} = \frac{p}{q}$$ where $p$ and $q$ are least possible integers.  Clearly then, $\sqrt{2} - 1$ would also be rational, $$\sqrt{2} - 1 = \frac{p-q}{q}$$ where both $(p-q)$ and $q$ are also integers.  Multiplying both sides by $q$, we get $$(\sqrt{2} - 1)q = (p-q) = q'$$ where $q'$ is an integer. Using, $2&gt;\sqrt{2} &gt; 1$ , we get $$1&gt;\sqrt{2}-1&gt;0$$ proving that $q'$ is a non-negative integer.  As stated in the proof, computing $(\sqrt{2}-1)q'$ we get, $$(\sqrt{2}-1)q' = \frac{(p-q)^2}{q}$$which is not necessarily an integer (in fact necessarily not an integer). However, the proof claims it to be.  Am I missing something or something is wrong with how I proceeded?  Here is a link to the textbook where I found this proof: , (p 32, problem 1.14).  P.S. This is my first post here and my first time with TeX. I'd also like to know if there is anything wrong with the way my question is presented.
Cannot get onto Apple Developer Forums Every time I attempt to log on to the apple developer forums it tells me I need to create a new user account (even though I already have one), and when I try to sign up it gives me an error:   Anyone else have this issue and/or know how to fix it? Thanks! Can low-gravity planets sustain a breathable atmosphere? If astronauts could deliver a large quantity of breathable air to somewhere with lower gravity, such as Earth's moon, would the air form an atmosphere, or would it float away and disappear? Is there a minimum amount of gravity necessary to trap a breathable atmosphere on a planet? Change to default start folder for Windows command prompt My new Windows XP work computer's command prompt automatically brings up "H:\&gt;" when I open it.  Is there a way for me to change that safely?  Specifically, to "C:\foo\bar\Something Else"?  I found the method of changing the Autorun value through Google, but it came with a warning that it "might affect the functionality of batch scripts."  Bonus note: I was originally getting at this issue in , but I didn't ask very directly.  The answer I ended up accepting was really good, so I wanted to leave it there. Elegant proofs that similar matrices have the same characteristic polynomial? Uncountable ordinals without power set axiom Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a set? Non-principal ideal in $K[x,y]$? How do I use superposition to solve a circuit? Yes, this is a pedagogical question. While answering another recent question, I wanted to refer the OP to concise instructions for using superposition to solve circuits. I found that all the easily found resources online were somewhat deficient. Typically they were unclear about what kinds of circuits superposition applies to, or about the actual method to apply the superposition theorem to a circuit problem. So,   What kinds of circuits can be solved by superposition?  How are different kinds of sources treated when solving by superposition?  What are the steps to solve a circuit using the superposition theorem? I took a look at my profile, where I stuck my flair. Here's the flair:    However, I don't have 28k rep. According to chat, I have 19.7k.  Note: I know about the combined flair - I've had this in my profile for at least a year. It had been showing me the same as in chat, now it isn't.   What is going on here?  Edit: I'm reporting a bug here, that the combined flair is not showing the correct number, as it has in the past. Bayes' Theorem Intuition I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote:     The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.   The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two? In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while now but to no avail and would be grateful if you could help me. Why can't I transform faces, edges, or vertices without transforming the whole object? My problem is that I sometimes lose the ability to independently translate or otherwise transform any parts of a single mesh. If I try, for example, to just translate several faces, edges, or vertices, then the entire mesh will stretch and displace itself in order to follow the translation. Or, alternatively, if I try to translate just one face then the whole mesh will move as if I'm translating in Object Mode.  I'm not really sure I know how to describe this problem properly. I don't even know how to reliably reproduce it. It seems to happen to any given mesh at random, and it doesn't stop happening to that mesh unless I delete and remake it.  I assume that I must be doing something wrong to cause this to happen. Does anyone have any ideas? Thanks. Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? Why is unity feedback used? Why is the unity feedback system used? This block diagram is for a 2 DOF ball balancing using a PID controller. Preserve blend mode effect when underneath layers are disabled? (Photoshop) My regex is matching too much. How do I make it stop? How to get the filename without the extension from a path in Python? I am working on a large C++ project in Visual Studio 2008, and there are a lot of files with unnecessary #include directives. Sometimes the #includes are just artifacts and everything will compile fine with them removed, and in other cases classes could be forward declared and the #include could be moved to the .cpp file.  Are there any good tools for detecting both of these cases? How to Query all opportunities ids from account I write a sample query to fetch account Ids from cases of a particular record type.Then i wrote another sample query parent to child relationship to fetch opportunity ids from accounts.  My sample code  List&lt;Case&gt; c=new List&lt;Case&gt;(); c=[Select Id,accountId from case where RecordTypeId = '01290000000sF3L']; List&lt;Id&gt; accountIds= new List&lt;Id&gt;(); List&lt;Account&gt; accounts=new List&lt;Account&gt;(); List&lt;Opportunity&gt; opps=new List&lt;Opportunity&gt;(); for(Case cl:c)  { accountIds.add(cl.accountId); } accounts=[Select Id,name,(select Id,name from opportunities) from account        where Id in:accountIds]; System.debug('Data in accounts '+accounts);   But my issue is how to get opportunity id from above Soql accounts means how to add those opportunity id's to the list.  for(Account ao:accounts) { //How to fetch Opportunity ids from above account list. } Why are floating point numbers inaccurate? I have installed Ubuntu on, but now I am trying to install NVIDIA CUDA Toolkit for the GPU. When I install it, it says that there is no driver found, so I have tried to install the driver, but I don't know why after installation, the system not coming up. Maybe it is because of the wrong version of the driver.   When I tried to detect the driver version using sudo lshw -C display , it showed this information:    But here it doesn't say that what is the product name of the driver, and also when I go Software &amp; Updates -> Additional Drivers, then it says NVIDIA Corporation: Unknown.    What should I do?