I have been doing work in various bases and have been convincing myself of the fact that our base-10 number system is not special at all. Any other base ($2$, $16$, hell, even base $100$ could work if you can find enough characters to assign).  This got me thinking, would math work just as freely in non-integer bases?  Let me explain. A non-interger base would behave as integer bases do, with each digit being multiplied by a subsequent power of the base.     Example: $154$ in base-$\sqrt 2$ would represent $1\cdot (\sqrt{2})^2+5\cdot (\sqrt{2})^1+4\cdot (\sqrt{2})^0$.    This leads to some interesting questions about which digits to use, however.  In base-$10$, there are only $10$ digits. This idea translates easily to other integer bases, but the idea of a number of digits falls apart in non-integer bases, in fact, the idea of which digits to use seems vague too (why should $1$, $4$ and $5$, all integers, be digits in a base-$\sqrt 2$ system.)  However, the idea of only $10$ digits in base-$10$ is somewhat unnecessary. As weird as it is, you can use any digit (even non-integer digits) in base $10$.     Example: $(2)( \frac {1} {2})(50)(-3)_{10}=2\cdot 10^{3}+\frac {1} {2} \cdot 10^{2} + 50 \cdot 10^{1} + -3 \cdot 10^{0}=2547_{10}$   Some of you, myself included, may be cringing at the notation I've used for non-integer and non-positive digits, but when has  ever stopped any of us?  I think the question quickly becomes whether we could, not whether we should. Obviously, we shouldn't, but again, the purists among us live life without worrying about math's "shoulds".  Basically my question is if any of you have seen any work, serious or not, on this type of base system. I would love to hear what different fields of math would be changed in different ways. Obviously certain numbers could be represented in easy ways (we could calculate ALL the digits of $\pi$ in base-$\pi$).  From first thought, the idea of primes seems difficult to grasp, but I'm sure many other things are build natually from integer bases, if not from a base-$10$ system.
This is a question that keeps bothering me: much of mathematics was created by extending rules to new domains e.g. negative numbers were created by consistently extending arithmetic beyond positive integers. However, with number bases it seems impossible to do this. Number bases are integers, but could they be rational numbers? I've had previous discussions about this, see . One problem can be seen with this example: in base 2.5, 0.22 is greater than one. That's not what we want! What, if anything, can be done about this?  EDIT: To put it another way - if every positional number is less than any digit to its left, why doesn't this apply to fractional base positional numbers?
Problem on Sum of Digits of a Number I was given this problem to Prove if $Q(n)=Q(2n)$ implies $9∣n$. Where $Q(n)$ be the sum of digits of $n$. I tried to use the fact that sum of digits is congruent to the number modulo $9$ but I had no luck. Any help is appreciated. How to measure time taken by a function to execute How can I create an Excel spreadsheet with C# without requiring Excel to be installed on the machine that's running the code? When we measure the speed of a moving element we do it with the help of a reference frame. Now if we need to measure the speed of time, is it possible? Does time actually have a speed? Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$? Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated. What are Ctools plugin (content type, access, etc), and how does one create them? What are Ctools plugins, and how does one create them? While I basically understand what a determinant is, I wonder how this idea was developed?  What was the principal idea behind its origination?  I would like to know this so that I can have a better conceptualization of the determinant. How to apply a regex on a column fetched from DB I am using IBM DB2.  So, I am retrieving a column named SSN from a table. I have to apply the following regex on that column:  ([1-57-8][0-9]{2} |0([1-9][0-9] |[0-9][1-9]) |6([0-57-9][0-9] |[0-9][0-57-9])) ([1-9][0-9]|[0-9][1-9])([1-9]\d{3}|\d[1-9]\d{2}|\d{2}[1-9]\d|\d{3}[1-9])   Someone suggested to me that I have to write a function/procedure in the query to validate that the SSN picked from the db follows this regex else it should not be stored in the result set of the query.  I don't know how to write a function or a procedure in SQL, can you please help me out here? Sometimes, others just inform us about things without us asking them, other times they do not do that even if we ask them. I want a verb for when others do not give us information — whether because they do not like to or they just don't give it, but we cannot directly ask them to — but we actively extract the information from them sometimes without them being aware of it.   So the word I am looking for shouldn't be only for cases where we coerce or force others to give us information, and it shouldn't be only for cases where others do not like to give us information.  Example:     I needed that information. But I couldn't ask her for it, or most   probably she would not give to me even if I asked her. So, I ....   the information I wanted.   It could be a phrasal verb and so not just a single word, but I couldn't devise a  blank suitable for all possibilities!   Two verbs I've already had but I think are not suitable:  to give someone the third degree   To prize something from/out of somebody natbib: combining chicago style and sorting by appearance This is a  about how to handle email sent from your server being misclassified as spam.  For additional information you may find these similar questions helpful:                    Sometimes I want to send newsletters to my customers. The problem is, that some of the emails get caught as spam messages. Mostly by Outlook at the client (even in my own Outlook 2007).  Now I want to know what should be done to create "good" emails. I know about reverse lookup etc., but (for example), what about a unsubscribe link with an unique ID?  Does that increase a spam rating? Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers. I think i have proved that any number of the form $2^n$ can't be written as the sum of k consecutive positive integers, but i would be grateful if i could have some clarification as to whether this is a correct and valid proof or not.  Let $g=x+(x+1)+(x+2)+(x+3)+...+(x+(k-1))$, hence g is the sum of k consecutive positive integers.  $g=kx+\sum_{d=1}^{k-1}d$  Hence, $g=kx+\frac{k(k-1)}{2}$  Rearranging for x we find, $x=\frac{g}{k}-\frac{(k-1)}{2}$  So, for certain values of $k$, we can be sure that a positive integer value of $x$ can be found, and hence $g$ can be written as the sum of $k$ consecutive positive integers.There are two cases for $k$ to considor:  Case 1)  $k$ is odd. If $x$ exists, it must be the case $k$ | $g$, as for odd values of $k$, $\frac{(k-1)}{2}$ will alwyas be an integer, and hence $x$ will.  Case 2)  $k$ is even. $k$ does not divide $g$, but $\frac{g}{k}$ must be a multiple of $\frac{1}{2}$, and hence equals $\frac{h}{2}$, for some odd $h$.  Letting $j=2^n$, assume that we can write $j$ as the sum of $k$ consecutive positive integers.  If $k$ is odd, then if a $k$ exists, it must divide $j$, by Case 1. But, $j$ has no odd factors as it is a power of two, hence $k$ could not be odd.  If $k$ is even, by Case 2 it mut be the case $k$ does not divide $j$. Also, $\frac{j}{k}=\frac{h}{2}$, for some odd $h$.  But, this implies $2j$ has an odd factor, $h$. but, as $2j$ is a power of $2$, it can't have odd factors. So, $k$ can't be odd.  Hence, $k$ is neither odd or even, and this is a contradiction to our assumption. Hence no $x$ exists, and hence and number of the form $2^n$ can't be written as the sum of $k$ consecutive positive integers.  Any feedback on this proof would be appreciated. Could a ship warp into Borg cube and destroy it? During the Wolf-359 incident, 39 starships were destroyed, and 11,000 people were killed or assimilated.  Why didn’t one of the smaller ships warp into that one Borg cube? Doesn’t warping into it destroy anything?  Even if it just warps through without touching, one could program the navicom to self destruct while the ship is in the middle of Borg cube. Existence of $f'\in F$ such that $f^{[G:F]}=(f')^n$ for finite extension $F\subset G$ Let $F$ be a field and $u\in F$. Further, let $F\subset G$ be a finite extension and $v\in G$ such that $v^n=u$ for $n&gt;0$.     How would I show that there exists a $\tilde{u}\in F$ such that $u^{[G:F]}=(\tilde{u})^n$?   My first thought was to use the norm map $N_{G/F}(x)=\Pi_{\sigma\in X(G/F)}\sigma(x)$, where $X(G/F)$ is the fundamental set for $F\subset G$. However, I wouldn't know how to continue on this path. You can post a link to an android market application by useing a link that looks like:     market://search?q=pname:package    While browsing the market, is it possible to find an app's package name? Or is there a convenient way to get or share a link to a market app? Is there any way to post a link to a market app that normal (non-android) browsers can make use of? MySQL returns only one row I am writing a WP plugin. I have created a button for the user to click in thw WP dashboard. I need to run a function when a user clicks the button, shown with a red arrow. The code snippet for the button is:&lt;p&gt;&lt;button class="button button-primary"&gt;Update Media Titles and ALT Text&lt;/button&gt;&lt;/p&gt;. I have already created the function in my class like so:  public function kh_update_media_seo() {      //update media files title and alt tags here }   I can handle the code that goes in the function alone, I only need help making the button in the WP dashboard fire off this specific function when clicked.  Pardon me if this sounds dump or straight forward. It's my first time doing this.  My plugin is a one file plugin if that helps. Nested Virtualization What I do: I want to write a kernel module that does stuff to the KVM data structures. But I don't want to run it on the real machine, so I run one guest, inside it I run another two guests, and want to test my module.  The problem: VirtualBox (which now runs the outer guest), doesn't support nested virtualization, and when I cat /proc/cpuinfo | grep vmx I get no output.  What can I use that has nested virtualization?  You may look at my  to get an idea of what I want to do. What do I know, when deciding whether to cast Counterspell? I'm not entirely clear on how Counterspell works, when trying to counter higher-level spells: the CS description only talks about disrupting a creature you can see, within 60 feet, casting a spell.   However, when trying to counter a higher level spell:   Do I know what spell is being cast?  Do I at least know (somehow intuit, based on VSM components) the level of the spell? Or can I tell the spell/level of the spell by sensing a disturbance in The Weave? (I.e., if the spell is being cast using Subtle Spell, can I still tell what's going on?)   The point of the question is to determine how fruitful CS can be in combat--if I know what the other guy/gal is casting (or at least the level), I can decide whether I want to spend my L8 slot to counter his/her L8 slot. But if I have no idea, and I end up using a L8 slot to counter a L3 spell... seems a bit painful.  Hope this made sense! Batch file to create Folder based on Date and Copy Files from One Location to Another Location?