Let $n \in \mathbb{N}$. Using the Euclidean algorithm, it is straightforward to see that every natural number can be written as  $$n = \sum_{j=0}^m \epsilon_j(n) 2^j $$  where $\epsilon_j(n) \in \{0,1\}$.   Is there an easy way to show that this way of writing the number is unique?
I am trying to show that how the binary expansion of a given positive integer is unique.  According to this link, , All I see is that I can recopy theorem 3-1's proof?  Is this polished enough of an argument. Thanks
Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? Can "doubt" sometimes mean "question"? I often see questions on Stack Exchange sites which I presume are written by non-native English speakers who use the word "doubt" in place of the word "question". Is this a case of misunderstanding the correct meaning or are people being taught that this is correct usage? What is inertia of a body? The definition of  is  "Inertia is the resistance offered by the body whenever its state of motion is changed."   What is inertia of a body? Is inertia actually a force exerted by the body?  If so, then why is there no application of inertia in numerical problems involving application of force on a body? What is the actual application of inertia in theory? Can we express a trigonometric function for the product of two angles as a function of trigonometric functions of its factors?  For example: Is there a formula for $\sin(xy)$ as a function of $\sin x$ and $\sin y$ or other trigonometric functions of $x$ and $y$. I need to run IE 10 in 32-bit mode on 64-bit Windows 7 Enterprise N.  When I run IE 10 from "C:\Program Files (x86)\Internet Explorer" I can see in task manager that I have only one process "iexplorer.exe" and no process "iexplorer.exe *32".  I have an application with an ActiveX that is supported only on 32-bit, and the application can't be loaded on 64-bit mode.  Is there a way I can force IE10 to run in 32-bit mode? \varepsilon vs. \epsilon "register" keyword in C? What does the register keyword do in C language?  I have read that it is used for optimizing but is not clearly defined in any standard.  Is it still relevant and if so, when would you use it? Allow rejected editors to be @replied to How can I design a tree graph with node shapes as this photo? What is the best way to remove accents (normalize) in a Python unicode string? I have a Unicode string in Python, and I would like to remove all the accents (diacritics). I found on the web an elegant way to do this (in Java):  convert the Unicode string to its long normalized form (with a separate character for letters and diacritics) remove all the characters whose Unicode type is &quot;diacritic&quot;.  Do I need to install a library such as pyICU or is this possible with just the Python standard library?  And what about python 3? Important note: I would like to avoid code with an explicit mapping from accented characters to their non-accented counterpart. If gravity isn't a force, then why do we learn in school that it is? I have studied some of Einstein's Theory of General Relativity, and I understand that it states that gravity isn't a force but rather the effects of objects curving space-time. If this is true, then why are we instructed in middle school that it is a force? I had to take ownership of the special folder C:\Program Files\WindowsApps to fix . The issue is now resolved, so I'd like to reset the permissions to the way they were before I took ownership of the folder.  I reverted NT SERVICE\TrustedInstaller to the owner and removed my account's permission entries:     But I still have full control of the folder and can browse the folder in Explorer without getting the usual warning that I would have to take ownership:     Is it possible to restore the default permissions of this folder? Gradient Boosting for Linear Regression - why does it not work? Why is my "Committed" memory so much higher than my actual RAM space? I have run two virus scans and neither of them have found anything so far (Malware Bytes and Avira Rescue). During gaming I will get Windows popups saying that my system is low of memory and recommends closing the game. This can also happen while browsing. My computer seems to become sluggish a lot and will hang up often, such as when clicking a link from a messenger program to open in my browser.   I am unsure what is causing this. My memory usage on the processes tab doesn't add up to such a high percentage of my actual memory.    As a side note, is my Disk Usage supposed to spike up to 100% sometimes? Research seems to say so. What does "All things X" mean? I've seen the expression in some company names; for example , .  Is the expression "All things X" a parody of some other popular phrase, or does it have its own meaning? Or is it just a misuse of grammar? Iterating through a Collection, avoiding ConcurrentModificationException when removing objects in a loop We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. Pokemon keep running away and Pokestops don't drop anything when spinning. What is going on? I can't do anything now. Can't catch Pokemon, can't get items. Does anyone have the same problem? How to fix slow wireless on machines with Intel wireless cards? I have looked through all the previous questions but turning power management off for the Wi-Fi card didn't work. This problem seems to be with the Lenovo Thinkpads. I switched to Ubuntu 11.10 from Windows 7 3 days ago so I'm new to the operating system. What's the difference between "Service" and "/etc/init.d/"? I am making the check for update script for my theme  I have 2 text files. First one is called "current.txt" and contains the current version. There is 4.1.1 string in that text file.  Second one is called "latest.txt" and contains the latest version. There is 4.2 string in this text file.  So here is the code  echo "Checking update"; x=$(cat ./current.txt) y=$(cat ./latest.txt) if [ "$x" -eq "$y" ] then        echo There is version $y update else        echo Version $x is the latest version fi   What it mean is if current.txt is NOT the same with latest.txt then it will say "there is version 4.2 update". If not, it will say "version 4.1.1 is the latest version"  But when I try to run it. I get this error  Checking update ./test.sh: line 4: [: 4.1.1: integer expression expected Version 4.1.1 is the latest version   So what am I doing wrong with this?