Why 4.7 is 4.6999999 in c
Is floating point math broken?
On the proof that the product of diagonal matrices is a diagonal matrix? Let $A [a_{i,j}]$  and $B =[b_{ij}]$   be two diagonal matrices in   $M_n(F)$. Let $i \ne j \in \left\{1, \ldots, n \right\}$. Using the   product rule, we obtain      $$(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$      We claim that $a_{ik}b_{kj} =0$ for all $k \in \left\{1, \ldots, n \right\}$ thus $(AB)_{ij}=0$ for all $i \ne j$ and $AB$ is diagonal. To    prove the claim, note that since $i \ne j$ , we have $i \ne k$ or $    k\ne j$. Thus either $a_{ik} = 0$ (since $A$ is diagonal) or $b_{kj} =  0$ (since $B$ is diagonal), thus in all cases $a_{ik}b_{kj}=0$ and the   claim is proved.   Where does the bit "we have $i \ne k$ or $k\ne j$" come from? I understand that since $i \ne j$, and $k$ and $j$ run over the same set of natural numbers, we also have $i \ne k$. But the "or $k \ne j$" baffles me. I understand that $k \ne j$ implies that we can have $i=k$. But that's from the other direction ($\Longleftarrow).$  Also, when it says $a_{ik}=0$ or $b_{kj}=0$ it's talking about when $i \ne k$ or $k \ne j$ right? Many thanks. I have this  problem with Ubuntu 16.04 LTS. All programs and all menus have black boarders and Terminal is displayed black:    The problem appeared after installing  tool and installing all Kali Tools. Digging in deeper to Javascript Remoting and I'm a bit confused.  The  starts off by showing how to set up everything and uses code like the following:  function getRemoteAccount() {     var accountName = document.getElementById('acctSearch').value;      // This remoting call will use the page's timeout value     Visualforce.remoting.Manager.invokeAction(         '{!$RemoteAction.AccountRemoter.getAccount}',         accountName,          handleResult     ); }   However, I've also seen examples where Visualforce.remoting.Manager.invokeAction is never called.  They end up looking something like the following:  function getText() {   var text = "";   JSRemotingBasicsController.doGetText(     function(result, event) {       // do something here     }   ); }   I'm wondering what is the point to using Visualforce.remoting.Manager.invokeAction when you can just reference the RemoteAction directly like the second example?  Is there something I'm missing?  Is there a time where the second example doesn't work? Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $. Why do Lambertian Reflectors Privilege the Observer? $G$ is non abelian simple group of order $<100$ then $G\cong A_5$ How to hide users from the GDM login screen? I have recently added several new users, that I need for qmail. Now they appear in the box in the login screen and clutter it, and I have to scroll to find my user. How can I hide those users from the login box? What is a simple way to create a binary relation symbol on top of another? What is a simple way to have \ll on top of \gg with equal size and vertically center aligned and have them work as a binary relation symbol like \lessgtr? The commands \underset and \overset cannot be used because they do not treat the two symbols equally, and they do not vertically center. Secure hash and salt for PHP passwords It is currently said that MD5 is partially unsafe. Taking this into consideration, I'd like to know which mechanism to use for password protection.  This question,   suggests that hashing multiple times may be a good idea, whereas  suggests using salt.  I'm using PHP. I want a safe and fast password encryption system. Hashing a password a million times may be safer, but also slower. How to achieve a good balance between speed and safety? Also, I'd prefer the result to have a constant number of characters.   The hashing mechanism must be available in PHP It must be safe It can use salt (in this case, are all salts equally good? Is there any way to generate good salts?)   Also, should I store two fields in the database (one using MD5 and another one using SHA, for example)? Would it make it safer or unsafer?  In case I wasn't clear enough, I want to know which hashing function(s) to use and how to pick a good salt in order to have a safe and fast password protection mechanism.  Related questions that don't quite cover my question: Is there an equivalent of computation of physical processes in nature? I was watching a waterfall in the Austrian Alps. There were thousands of water droplets falling down, splattering on the stones below. I thought - how does nature find out so quickly where each droplet of water should go? To find out what happens to a falling droplet of water, one can use the laws of motion and calculate the trajectory. To calculate, one needs some amount of time, some machine (the brain, computer) and some energy to feed the machine. Does it make sense to ask what is the equivalent of this computation in nature? How does nature find out so quickly how things should move? More generally, to find out how anything should happen? Where is the &quot;calculation&quot; in nature? There's no room or energy for a machine in the particles that make up things. Homeomorphism that maps given points to given points How many of Trelawney's prophecies have come true? How to prevent iTerm2 from creating overlaps and broken lines after pasting multiline chunks of text? If I paste multiline chunks of text into iTerm2, I always a get mess of output. It seems limited to only some programs running interactively (such as Bash, IRB - Python / VIM work flawlessly). Using Apple's native Terminal, I can paste whatever I want wherever I want - it works. For example, this sample text ():  echo Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec a diam lectus. echo Sed sit amet ipsum mauris. Maecenas congue ligula ac quam viverra nec consectetur ante hendrerit. echo Donec et mollis dolor. Praesent et diam eget libero egestas mattis sit amet vitae augue. echo Nam tincidunt congue enim, ut porta lorem lacinia consectetur. Donec ut libero sed arcu vehicula ultricies a non tortor. echo Lorem ipsum dolor sit amet, consectetur adipiscing elit. Aenean ut gravida lorem. Ut turpis felis, pulvinar a semper sed, adipiscing id dolor. echo Pellentesque auctor nisi id magna consequat sagittis. Curabitur dapibus enim sit amet elit pharetra tincidunt feugiat nisl imperdiet. echo Ut convallis libero in urna ultrices accumsan. Donec sed odio eros. Donec viverra mi quis quam pulvinar at malesuada arcu rhoncus. [snip]   pasted into IRB produces this in iTerm2:    while it produces this in native Terminal:    I am running the latest versions of OSX and iTerm2. What's going wrong here? Continuity of absolute value of a function Let $f(x)$ be a continuous function. Prove that $\left|f(x)\right|$ is also continuous.  Is it correct to say that, by the reverse triangle inequality, $\left|f(x)-f(c)\right| \geq \left|f(x)\right|-\left|f(c)\right|$ in all cases, so we will always have that for all $\left|x-c\right|&lt;\delta$ implies $\left|f(x)-f(c)\right|\leq \epsilon$?  I am not sure if my solution adequately uses the functional form, since I just used the equation and not my actual function. Please help with the proper solution! Chatroom owners can star their own messages I was able to star my own message by pinning it and unpinning it.  Pinning a message adds a star, even to your own messages, but when the pin is removed, the star remains. USB flash drive not working or is appearing as an empty disk drive, Disk Management reports "No Media" with 0 bytes size After a brief discussion with @MonicaCellio (In the comment) - It has come to my attention that perhaps I chose the wrong words to describe my original idea (In the edit history). When I wrote "social features" I was actually suggesting an alternative way to for users to share their knowledge using a new type of post and articles and not just by questions and answers.   The basics of my idea was to allow users to write articles on topics relevant to the community on the SE websites just as any answer posted is expected  to be too. Allowing users to write their own content (I.E posts/articles/etc..) will be granted as a privilege for advanced users so that not anyone could write stupid things, users that will write content must be established and well-familiar with the rules of SE.   I know anyone can start a blog, but it's require frequent maintenance and content updates, while writing on SE ensure that it addressed to a large, well-established community, with a similar fields of interest.  Posts writing will follow the same guidelines as asking and answering questions does, but it will provide an alternative for users that are mostly answering specific questions, as a new method to contribute to the community.   By writing, one may also reveal more about their personality, wit, way of thinking, ETC. That's when the "social" takes place - You'll also have "authors" and not just "users" you can remember by their catchy name, but people that you can remember by the field of interest they are writing content about.   They key points will be:    Users will be able write rich content with advanced WYSIWYG HTML editor, content that doesn't require a question to answer, but that is still relevant to the SE website it is writing on. Users will be able to Upvote/Downvote other users posts. Posts will be listed on their author profile, so it will not just show the activity, but additional layer of user generated content. Popular posts with high reputation could be associated to tags making the information of tags richer with user generated content Users will be able to include links in answers not only to other questions, but also to posts. User will be able to comment on post to allow the author to correct and improve it. It will improve the interaction and offline discussions (Which are prohibited when it come to long discussions on questions and answers). I believe that users will be happy to share their own tutorials, tips and tricks - Because they are already doing so when answering questions, but before the can start, they'll have to earn enough reputation to ensure that only high profile users will be able to write high profile content, and not just any rubbish. It will provide another type of method for voting and getting earning reputation Users will be able to include content in their CVs on SO Jobs.  Opportunity to earn some new badges.   So it's a Mega feature request and it require some detailed design work, but I do think that it will invite a new type of activity for users on SO (and the rests of the sites of-course). i would like to know how to prove that if $a^2+b^2=c^2$ then   a) $3|a$ or $3|b$  b) $5|a$ or $5|b$ or $5|c$  c) $4|a$ or $4|b$   Also that $a^2+b^2+c^2+1$ can't be divisible by 8  Thank you very much! Elementary argument for conservation laws from symmetries *without* using the Lagrangian formalism It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. BM01 2011/12 Question 6 Geometry Problem Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A,B$ and $C$ are $D, E$ and $F$ respectively. Prove that $DE +DF \le BC$ and determine the triangles for which equality holds. The altitude from A is the line through A which is perpendicular to BC. The foot of this altitude is the point D where it meets BC. The other altitudes are similarly defined.   Thanks in advance for any contributions.