have faced below problem while writing query, not sure why it is happening Please find the below example   SELECT 21000.000000/22000.000000 * 22000.000000   sql engine result is 20999.999999999999990000000  But if you do the same through calculator, we will be getting 21000. Why is it like this? and how to achieve the calculator result with out rounding off
Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen?
How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$? Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for the middle term. I got the limit as $n\to \infty$ of the function inside using Riemann sum but what I cannot get are the lower and upper bound. Ideas to improve website loading speed? Are there any ways in which I can improve the loading speed of a website?  I know that Google is really pushing loading speeds and with the popularity of mobile websites it's now imperative that sites load quickly. Show a matrix in detail Below it is a very useful command to create a matrix. But it has a short point to me since it shows Parentheses not Square brackets. My question it how should I change this to show square brackets? I appreciate your answer. Thank you so much.  \[ K =   \underbracket{\begin{pmatrix}    K_1 &amp; 0      &amp; 0    \\    0   &amp; \ddots &amp;    0   \\    0   &amp; 0      &amp; K_n    \\    \end{pmatrix}}_{d\times M} \] I would like to add my bibliography to my presentation in IEEE format. What should I add to my preamble and code.  \documentclass{beamer} \setbeamertemplate{footline}[page number]  \usepackage{graphicx} \usepackage{booktabs} \begin{document} \section{Reference} \begin{frame} \frametitle{Reference}  \end{frame}   I'm using the same bibTex which I use to make the IEEE paper. I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement? Bottom frame of listing appears on top of the next page I've been using Windows 10 for a couple of month now (joined the Insider Preview).  Now my hard drive started ticking so I've bought a new SSD and I want to move the windows installation to the new hard drive, which has been connected to the computer (unpartitioned). I don't mind and I even prefer reinstalling it over again.   Since I have the free version of Windows 10 pro, can I still reinstall it anew? Does the windows 10 installation offer disk management and partition like the older Windows installations used to? How should I reinstall Winodws 10 x64 the easiest but still remain on free mode? Batch fit images into ratio with blurred background I need to resize a lot of images into a 4:3 ratio with the desired effect.  Is there a way to do this using photoshop actions? (the background can be scaled or two copies moved to the sides, either way works)     image credits: How would I import/export Blender's preferences? I constantly stay up to date with each new release of Blender, but this means my preferences are wiped clean each time. How do I export/import these preferences to avoid loosing my carefully crafted set up each time? Resize image in Python without losing EXIF data I need to resize jpg images with Python without losing the original image's EXIF data (metadata about date taken, camera model etc.). All google searches about python and images point to the PIL library which I'm currently using, but doesn't seem to be able to retain the metadata. The code I have so far (using PIL) is this:  img = Image.open('foo.jpg') width,height = 800,600 if img.size[0] &lt; img.size[1]:     width,height = height,width  resized_img = img.resize((width, height), Image.ANTIALIAS) # best down-sizing filter resized_img.save('foo-resized.jpg')   Any ideas? Or other libraries that I could be using? Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$  Can you tell me if my proof is right:  $\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Z} / n \mathbb{Z}$ are both finite free $\mathbb{Z}$-modules with the basis consisting of one single element $\{ 1 \}$. So $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z})$ has the basis $\{ 1 \otimes 1 \}$.  Therefore, any element in $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z})$  is of the form $(ab) 1 \otimes 1$ and any element in $\mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ is of the form $k 1 = k$ where $k \in \{ 0, \dots , \gcd(n,m) \}$.  I would like to construct an isomorphism that maps $ab$ to some $k$. Let this map be $ab (1 \otimes 1) \mapsto ab \bmod \gcd(n,m)$.  This is a homomorphism between modules: it maps $0$ to $0$ because it maps the empty sum to the empty sum. It also fulfills $f(a + b) = f(a) + f(b)$ because there is only one element, $a = 1$.   It is surjective. So all I need to show is that it is injective. But that is clear too because if $ab \equiv 0 \bmod \gcd(m,n)$  then both $a \equiv 0 \bmod n$ and $b \equiv 0 \bmod m$ so the kernel is trivial.  Many thanks for your help!! I remember watching this one show when I was a kid in the 80's/early 90's in Canada. It was a live-action show about these two kids and had some puppet/muppet-like characters, but it kind of had a dark/mysterious element to it. I can't remember too much about it except portions of some episodes, but the main characters were kids, and they had some mysterious artifact (kind of like a compass?) that took them to this other world where there were giants and other mysterious beings that we rarely saw. The two kids weren't the only one's going to this place either, they find out kids from all around the world that also had this same artifact (and they all fit together like a round puzzle?). I seem to recall that the giants in this other world were the ones who made the artifacts, and were dying/or in desperate need of the help of the kids.  A couple episodes I remember:   There was one where the leaves were falling, and the world was doomed to end as soon as the last leaf touched the ground, so they're trying to figure out how to prevent all the leaves from falling, but after a long and suspenseful episode (for a kid) the last leaf dramatically falls, but out of no where the main girl catches the leaf and saves the world (it was very exciting for my young self, and I remember being very impressed with the pretty girl for thinking to catch the leaf).     There was another episode where the kids meet these two old guys with long white hair (muppets) sullenly working in a mine/tunnel, they're in tattered old clothes working away with their pick axes or hammers and they are connected to each other at the beard. This long old white beard grows out of one of their faces right into the other's. The kids save the day by cutting the old guy's beard in the middle, freeing them from each other. They celebrate being free from their bonds and excitedly thank the kids, but after the kids leave they continue to work away, this time in much brighter spirits. Seemingly unbeknownst to them though, as they work, their beards slowly and subtly grow back together.   Anyone else remember this show? Rectifiability of a curve Let $f$ be a function defined on $[0,1]$ by   $$f(x) = { 0, \text{ if } x = 0} $$ $$f(x) = { x \sin \frac 1 x , \text{ if } 0 &lt; x \leq 1} $$  Prove that the curve $\{(x, f(x)) : x \in [0,1]\}$ is not rectifiable.   I'm not sure how to approach this. The general idea seems logical, we're proving that the length of the curve is infinite, but the method seems difficult to find. Say I have this file:  hello world hello world   This program  #!/bin/bash  for i in $(cat $1); do     echo "tester: $i" done   outputs  tester: hello tester: world tester: hello tester: world   I'd like to have the for iterate over each line individually ignoring whitespaces though, i.e. the last two lines should be replaced by  tester: hello world   Using quotes for i in "$(cat $1)"; results in i being assigned the whole file at once. What should I change? Is there a rule to know when to pronounce letter c as a "soft c" (s) or "hard c" (k)? How to extend WIFI from ethernet What is the best (or preferred method) of setting a url parameter on a visualforce page embedded on a standard layout page?    This is easy to do when creating a custom link or a button, but I don't get how to do this on page load via the Apex Controller for the page.   When using a custom button you can do something like /apex/MyPageName?parameter=value  Can I use something like this?   ApexPages.currentPage.getParameters().put('Id',myId) My PC was working yesterday without any problem. I shut it down and tried to open this morning. It prints out these messages on the black screen:  systemd-udevd[713]: failed to execute '/bin/systemctl' '/bin/systemctl start --no-block nvidia-persistenced.service': No such file or directory  systemd-udevd[852]: failed to execute '/bin/systemctl' '/bin/systemctl stop--no-block nvidia-persistenced': No such file or directory   It then continued with the login screen. I entered my password and pressed enter. It waited for a while and the login screen reappeared again.  How can I fix the problem? I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$:  \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2 \mathbb{Z} \end{align}  I'm aware that there are answers to this question on this website. However, almost all of the answers seem to use facts about modules and vector spaces defined over fields. This question is posed in section 7.6 in Dummit and Foote's Abstract Algebra. This section is part of the chapters which introduces the basics of ring theory (rings, homomorphisms, ideals, Chinese Remainder Theorem etc.).  I'm not even sure how to being answering this question. I know that if $R$ were an integral domain, then we would have that $R \cong \mathbb{Z} / 2 \mathbb{Z} $.   Please only give hints, considering that that the statement has to be proved using the material only included up till chapters 9 in the textbook. Hiding and redirecting extensions on my webpage