I am having a hard time understanding why the division algorithm works during base conversion even after reading some of the answers given on this site.  Obviously I am missing a formal introduction to these topics hence the difficulty. So I am asking if introductory books on number theory like - "A Friendly Introduction to Number theory" or "A Gentle Introduction to Number Theory" would solve this problem?
I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this method works, and I just can't seem to see anything, although I know what type of intuition I am trying to make out.   So, my question is why we can use the division algorithm to convert between the decimal bases and any other number bases?   An example from the decimal system to binary would be appreciated.
Can you combine multiple lists with LINQ? Say I have two lists:  var list1 = new int[] {1, 2, 3}; var list2 = new string[] {"a", "b", "c"};   Is it possible to write a LINQ statement that will generate the following list:  var result = new []{      new {i = 1, s = "a"},     new {i = 1, s = "b"},     new {i = 1, s = "c"},     new {i = 2, s = "a"},     new {i = 2, s = "b"},     new {i = 2, s = "c"},     new {i = 3, s = "a"},     new {i = 3, s = "b"},     new {i = 3, s = "c"} };   ?  Edit: I forgot to mention I didn't want it in query syntax. Anyway, based on preetsangha's answer I've got the following:  var result = list1.SelectMany(i =&gt;  list2.Select(s =&gt; new {i = i, s = s})); How to create svg symbols that have modifiable fill color, stroke color and stroke width? I want to create some svg symbols for Qgis using Inkscape, those symbols must have attributes as described in the question. For the last two days, I have been experimenting as instructed by  and mimicked some symbols came bundled with qgis that have desired attributes without any success.  Finally I experimented with the simplest form of symbol: I created an svg that contains only a circle in Inkscape and tried to modify it.  The original file () have this line:  &lt;path    sodipodi:type="arc"    style="fill:#ffffff;fill-opacity:1;stroke:#000000;stroke-width:1.01988637;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none"   I modified it into:  &lt;path    sodipodi:type="arc"    style="fill:param(fill) #ffffff;fill-opacity:1;stroke:param(outline) #000000;stroke-width:param(stroke-width) 1.01988637;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none"   Saved as  and selected it as point symbol, but Qgis couldn't change all three parameters at the same time.  What did I do wrong or should do differently?  My system: Qgis 1.8.0 on Ubuntu 12.04 64 bit, Inkscape 0.48, Gedit 3.4.1 text editor. How do I get transparency to work with the Principled BSDF shader? How do I get the Cycles Principled BSDF shader to support a base color image such as a 32bit png with alpha or a tga with an alpha channel, and get the transparency to work. I don't see anywhere on the Principled BSDF shader that handles transparency.  Without success, I have tried:   Properties Editor>Render tab>Film rollout>Transparent Properties Editor>Material tab>Settings rollout>Viewport Alpha (changing to other than Opaque). Adding a Transparent BSDF and Diffuse BSDF after the base color image node and using a Mix Shader to combine them. How to enable sleep/standby on Windows Server 2012 Hyper V How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$ Beginner Question: How to close/split panels? Why are hand-built wheels considered to be better than machine-built wheels? Why are hand-built wheels considered to be better than machine-built wheels? I would think that a machine-built wheel would be better and cheaper, since machines generally do a more precise job than humans. Is there any way to import a .json model into Blender? How to Select Columns in Editors (Atom,Notepad++, Kate, VIM, Sublime, Textpad,etc) and IDEs (NetBeans, IntelliJ IDEA, Eclipse, Visual Studio, etc) I'm currently working on the following problem for my computer theory class. It goes as follows:     Let $A$ and $B$ be regular expressions. Show then that $A^*B$ is the solution   of $X = AX + B$   Am I supposed to let $X = A^*B$? If so then   $A^*B = A(A^*B) + B$  I can understand that  $A(A^*B)$ could be reduced to $A^*B$ but what happens with the "${}+ B$" part of the expression? I've read up on Kleene algebra and found an axiom that seems relevant in which      $b + ax ≤ x → a^∗b ≤ x$   but I'm not actually sure if it applies. Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems? Is there a bijection of the natural numbers which swaps $\frac{1}{n}$-summable subsets with $\frac{1}{\sqrt{n}}$-summable subsets? Let me start with a precise statement of the question.  For a subset $A\subseteq \mathbb{N}$ and a series of positive real numbers $\sum_{n=0}^\infty a_n$, I'll use the notation $\sum_A a_n$ as a shorthand for $\sum_{n\in A} a_n$.  Is there a bijection $f:\mathbb{N}\rightarrow \mathbb{N}$ with the property that for every $A\subseteq \mathbb{N}$, $\sum_A \frac{1}{n}$ converges iff $\sum_{f(A)} \frac{1}{\sqrt{n}}$ converges?  Background: Fix a series of positive terms $\sum_{n=0}^\infty a_n$.  Given a subset $A\subseteq \mathbb{N}$, call it $a_n$-small if $\sum_A a_n$ converges.  The following proposition is easy to prove:  Proposition:  The set $\{A\subseteq \mathbb{N}: A\text{ is } a_n\text{-small}\}\cup \{\mathbb{N}\}$ is a topology of closed sets on $\mathbb{N}$.  I will use the notation $(\mathbb{N},a_n)$ to refer to this topology.  Then one can ask how topological properties of $(\mathbb{N},a_n)$ are related to series properties of $\sum a_n$.  For example, I can show:  Proposition:  The following are equivalent.  $\sum a_n$ converges.  $(\mathbb{N},a_n)$ is discrete.  $(\mathbb{N},a_n)$ is disconnected.  $(\mathbb{N},a_n)$ is Hausdorff    And there are other nice things.  For example, $(\mathbb{N},a_n)$ is compact iff $(\mathbb{N},a_n)$ is cofinite iff $\liminf a_n &gt; 0$. With this language, my question can be reformulated as...  Are $\left(\mathbb{N}, \frac{1}{n}\right)$ and $\left(\mathbb{N}, \frac{1}{\sqrt{n}}\right)$ homeomorphic?  I have made very little progress on this.  Of course, the identity function $i:(\mathbb{N}, \frac{1}{\sqrt{n}})\rightarrow (\mathbb{N}, \frac{1}{n})$ is a continuous bijection, but the inverse map is not continuous.  Also, if there is such a bijection, there there is such a bijection with agrees with $i$ on any preassigned finite set. Edit  I thought I add in a slightly suprising (to me, at least) example when things work out to be homeomorphic. Begin with a convergent series $\sum c_n$. and divergent series $\sum a_n$ with $\lim a_n = 0$.  Create a new series $b_n$ using all the terms of $c_n$ and $a_n$ (in whatever order you wish).  Then $(\mathbb{N}, a_n)$ and $(\mathbb{N}, b_n)$ are homeomorphic.  The idea is that since $\lim a_n = 0$, there is a convergent infinite subseries $\sum_A a_n$.  Then we use a bijection which with $A$ and $A\cup \{\text{indices of }c_n\}$ to &quot;squeeze&quot; the $c_n$ in without changing the topology.  Of course, I am glossing over many details, but I can include them if desired. What ways could you create a data extension that contains all the subscribers (SubscriberKeys) that are currently “in” a Journey (Journey Builder)?  The Journey I would like to do this for is a complex activity with multiple decision branches resulting in multiple possible journey durations. For example; if the customer didn’t open Email1 after 2 days, then send Email2; else Exit Journey. If the customer didn’t open Email2 after 2 days, then send Email3; else Exit Journey. Wait 2 more days, then End Journey. The minimum journey duration for the above example is 2 days, while the maximum is 6 days.  Note that I don’t want a list of subscribers that “entered” a Journey, but a list of the ones that are “currently in” a Journey and have not yet exited. Answer must account for non-standard Journey exits including Unsubscribes/Contact Deletes, etc. Android: How to draw route directions google maps API V2 from current location to destination How do you cast a List of supertypes to a List of subtypes? "I think him to be about 50" or "I think he is about 50"? Prove that if $a&gt;0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $.   Using the fact that $a \gt 0$, multiply by $a$ on both sides and get everything to one side we have; $a^{11}-a^{10}-a+1 \geq 0$. By factoring $(a^{10}-1)(a-1) \geq 0 $.  I am not sure how to proceed any further. Cardinality of a totally ordered union Let $(X_{\alpha})_{\alpha \in A}, (Y_{\alpha})_{\alpha \in A}$ be families of sets such that for $\{X_{\alpha} \mid \alpha \in A \}$ and $\{Y_{\alpha} \mid \alpha \in A \}$ are totally ordered by inclusion. We have $(X_{\alpha},Y_{\alpha}) \neq (X_{\beta},Y_{\beta})$ for $\alpha \neq \beta$. Suppose that $|X_{\alpha}| \leq |Y_{\alpha}|$ for any $\alpha \in A$. It is true that $|\bigcup_{\alpha} X_{\alpha}| \leq |\bigcup_{\alpha} Y_{\alpha}|$? Topological/Geometrical justification for $\text{CFT}_2$ being special It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n&gt;2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are from a much wider class of maps, holomorphic/antiholomorphic maps. I was wondering to know if there is any topological or geometrical description for this.   To show what I mean, consider this example: in $\mathbb{R}^n$ for $n&gt;2$ interchanging particles can only change the wave function to itself or its minus. It is related to the fundamental group of $\mathbb{R}^n-x_0$ ($x_0$ is a point in $\mathbb{R}^n$ and $\pi_1(\mathbb{R}^n-\{x_0\})=e$ for $n&gt;2$) but this is not true for $n=2$.   I want to know whether exists any topological invariant or just any geometrical explanation that is related to the fact that I mentioned about conformal maps on $\mathbb{R}^n$. RAM not clearing, alway in standby I'm having a problem with my ram on windows 7. The ram is clearing or programs are using and dumping it faster than it can clear. I have 8gb of ram I'm only using 4gb and the rest is in Standby. At best time I can get 100mb of free but its mostly at 15mb or less. I suspect a memory leak but I have no way to figure out what program it is. I have not installed any new hardware recently and newly installed programs have bin closed. Does any one have an idea of what is going on and how to fix it?   I'm sorry this is so little information but that is all I have as well.  Edit: I tried RamMap and found out the my "Mapped File" has 3,291,784k ram in standby how do I figure out what program and how do I limit it's ability to set ram in standby?  Edit: I keep getting a problem when I will have 0 free ram and my computer will stop working properly. It will become outrageously slow where it will take 3-4 minutes to start task manager so I can close some programmes. When I free up some ram by closing some programmes every thing runs normally. It does not matter which programs I closes only that I free up some ram. I know a lot of you say it's supposed to be like that and that's how windows works. But its obviously a problem and my computer has never worked that way neither has any of my other computers. I would like to fix this please help me.