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I've read a number of the helpful Q&amp;As on photons that mention the mass/mass-less issue. Do I understand correctly that the idea of mass-less (a rest mass of 0) may be just a convention to make the equations work?  From a layperson's view, it's difficult to understand how a particle of light (photon) can be mass-less. A physical object (everyday world-large or quantum-small) must have a mass. Yet, if my understanding is correct, the mass of a moving object/particle increases in proportion to its speed/velocity...so that at the speed of light, its mass would be infinite. A photon travels at the speed of light, but it obviously doesn't have infinite mass, right? Can someone formulate a practical explanation that can be understood by middle-school to high school kids? Much thanks for the help.    Wow--your answers to my original Q below clear up much of my confusion. I now have the daunting task of going over these nuggets and working up an equation-less (hopefully) explanation of the mass-less photon for non-physicist types.  Yes, from a layperson's view, it does seem remarkable that an existing piece of matter-- which has to be made of physical substance--could have zero mass at rest (though a photon is never at rest). It would be almost understandable if a piece of matter made of nothing had zero mass, but that  seems to be an oxymoron, and "nothing" would equate to nonexistent, right?  In case you might find it interesting: I'm working on a writing project that posits we inhabit a universe that consists of matter (physical stuff) only, and that the NON-physical (aka supernatural) does not (and cannot) exist. For instance, if a purported supernatural phenomenon is found to actually exist, then by definition, its existence is proof that it is mundane/natural. All it would take to disprove this premise is reliable proof that ONE supernatural event has occurred. Despite thousands of such claims, that's never yet happened.  Who else better than physicists to confirm my premise? However, I do wish the TV physicists would explain the terms they throw about, some of which mislead/confuse their lay viewers. Case in point: "The universe is made up of matter and energy" (without properly defining the term "energy" as a property of matter).  The result is that laypersons are left with the impression that energy must therefore be something apart from or independent of matter (ie, nonphysical). Their use of the term "pure energy" without specifying exactly what that means adds to the confusion. (Thanks to your replies on this forum, I now understand that "pure energy" refers to photon particles.) However, "psychics" and other charlatans take advantage of such confusion by hijacking terms like energy (as in "psychic energy"), frequencies, vibrations, etc to give perceived scientific legitimacy to their claims that a supernatural spirit world, etc., exists. As you may realize, the majority of people in the US (per 2009 Harris Poll) and around the world believe in the existence of nonphysical/supernatural stuff such as ghosts and spirits.  My purpose is to give laypersons the information they need to distinguish what's real from what's not.  Thanks so much for help...And, PLEASE, add any further comments you think might be helpful/insightful to better inform laypersons. Get the size of the screen, current web page and browser window Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$ I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some questions from previous exams. One of the questions is the following:           Prove that $\displaystyle{\sum_{n = 0}^{2013} a_n z^n \neq 0}$ if the coefficients  satisfy $a_0 &gt; a_1 &gt; \dots &gt; a_{2013} &gt; 0$ and $|z| \leq 1$.      I tried to approach the problem by using the reverse triangle inequality to try to isolate the leading term, which I thought should be the constant coefficient, as follows:  \begin{array} . \left|  \sum_{n = 0}^{2013} a_n z^n \right| &amp;= \left| a_0 +  \sum_{n = 1}^{2013} a_n z^n \right| \\ &amp;\geq |a_0| - \left|  \sum_{n = 1}^{2013} a_n z^n \right|\\ &amp;\geq |a_0| -  \sum_{n = 1}^{2013} |a_n z^n|\\ &amp;\geq a_0 -  \sum_{n = 1}^{2013} a_n |z^n|\\ &amp;\geq a_0 -  \sum_{n = 1}^{2013} a_0 |z^n|\\ &amp;= a_0\left( 1  -  \sum_{n = 1}^{2013} |z^n| \right) \end{array}  Then for $|z| &lt; 1$, we have that this last expression is equal to  $$ a_0\left( 1  -  |z| \frac{|z|^{2013} - 1}{|z| - 1} \right) $$  and my hope was to prove that this was always positive, nevertheless, after failing to prove it, I plotted the expression inside the parenthesis as a function of $|z|$, and found out that it changes sign in the interval $(0, 1)$.  Question  Can my approach somehow be made to work? And if not, how can I prove that the sum $\displaystyle{\sum_{n = 0}^{2013} a_n z^n \neq 0}$ under the given conditions?  Thank you very much for any help. If $p$ is a prime such that $p≡1\pmod 7$ then $p ≡1\pmod{14}$ Prove or disprove the following:  1)  if $p$ is odd prime such that $p≡1\pmod 7$ then $p ≡1\pmod{14}$  2)  if $p$ is odd prime such that $p≡4 \pmod 7$ then $p ≡4\pmod{14}$  I think that both statement are true and have a similar proof  1) $p≡1\pmod 7$ so $p≡8\pmod 7$ then $7|p-8$ so $p-8= 7 k$ where $k$ is an integer so $p=8+7k$, so $p ≡1\pmod {14}$  It is that a correct proof? Thank you. What is the function represented by the power series $$ \sum_{k=2}^\infty\frac{x^k}{k(k-1)}\quad? $$  It looks like $\dfrac1{1-x}$ but I don't know. When ever I need to kill a background process I do ps -e | grep &lt;process_name&gt;  Which prints something like this 1766 ? 00:00:13 conky , Then I use the process ID to kill it like so kill 1766 .   Is there any way I can simplify this ? Make it quicker ? reduce the amount of typing ? Here's a scenario:  Someone asked about "How to build a SQL query to get count and blah blah blah..."  His only tag is . I added the  and  tags to it.  Someone else answered with a nice SQL statement that has a "group by" statement which seems "logically inevitable".  I proceed to add a tag . I did that on the basis that someone who searches for  will be able to find this post useful, just as I did. My concern is, it wasn't described at all in the question, nor was it implied, unlike  and .  What is the community opinion?  The scenario above is a real one . And the one who answered it nicely is Jon Skeet. Finder search doesn't work properly after Yosemite update Ever since I upgraded to Yosemite, search no longer works properly in Finder. Spotlight works fine but Finder search just gives me a blank window with a spinning ball.  I've tried reindexing and I've also tried deleting Spotlight's .plist file. I restart my MacBook Pro after doing either. Finder search will work for one or two queries and stop again. I don't know what else I can do. Illustrator CC: Turn off align objects to pixel grid I have inadvertently created an illustration with the align new objects to pixel grid turned on (not sure why it was on, I did not know or ever turn it on) and want to now scale the image. When I turned it off, I scale and it is still affecting the illustration.  So I copy and pasted it into a new document (Making sure that the option was turned of when I created the new document), and it is still affecting the illustration.  How can I scale this object without it being affected by this option? Beginner Physics Resources? I'm interested in learning physics. I do realize that the subject is large and that it would be easier if I had a specific area of interest. However, I do not. I suppose I want to learn about the fundamentals of it all; the axioms that combine all physics fields. Or, in other words, a high school physics class.   Specifically, a book or series of videos would be helpful. I looked over MIT and unfortunately the material wasn't for me. I don't mean to be "picky" so I am not completely ruling out any resource just yet.   Thanks in advance. Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. How to differentiate a given inner product (CSIR DEC(2018) Let $A$ be a $n \times n$  invertible  matrix.Define a function $F:\mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ by $F(x,y)=\langle Ax,y\rangle$. Let $DF(x,y)$ denotes the derivative of $F$ at $(x,y)$ which is a linear transformation from $\mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R} $. Then      1) if $x\neq 0$,then  $DF(x,0)\neq 0$      2) if $y\neq 0$,then  $DF(0,y)\neq 0$      3)  if $(x,y)\neq (0,0)$,then $DF(x,y)\neq 0$      4) if $x=0 \; or \;y=0$,then $DF(x,y)= 0$   I don't even know how to start the solution and also how to take derivative in higher dimensions.  Please help!  Thank you. Did merging Black Holes in GW150914 give up entropy and information to the gravitational waves, since they lost 3 solar masses? Since the final Black Hole (BH) had 3 solar masses less of mass than the original binary BH, it seems the 2 BHs lost mass, and with it event surface area, entropy, and information. If that came from inside the BHs it seems that it would be a contradiction of what a BH is --'nothing escapes its event horizon'. Some may say that came from the rotational energy of the pair, but the new BH now seems to have less than the two original ones summed. So, did some of that escape the horizons? How to prove this result involving the quotient maps and connectedness? Given topological spaces $X$ and $Y$, where $Y$ is connected, let $p \colon X \to Y$ be a quotient map. If, for each point $y \in Y$, the set $p^{-1}(\{y\})$ is connected, then how to prove that $X$ is connected also?   By the map $p$ being a quotient map is meant the following:   The map $p$ is surjective, and, for any subset $V$ of $Y$, the set $V$ is open in $Y$ if and only if $p^{-1}(V)$ is open in $X$. If I am travelling on a car at around 60 km/h, and I shine a light, does that mean that the light is travelling faster than the speed of light? The title says it all.  If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?  So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?  How should I answer his question? I am reading Schwarz QFT and I reached the mass renormalization part. So he introduces, after renormalization, a physical mass, defined as the pole of the renormalized propagator, and a renormalized mass which can be the physical mass, but can also take other values, depending on the subtraction schemes used. Are these masses, other than the physical one, observable in any way experimentally, or are they just ways to do the math easier (using minimal subtraction instead of on-shell scheme, for example)?   Also, in the case of charge renormalization, the explanation was that due to the vacuum polarization, the closer you are to the charge, the more you see of it, so the charge of the particle increases with the momentum you are testing the particle with. However, I am not sure I understand, from a physical point of view, why do you need to renormalize the mass. Is this physical mass (defined as the pole of the renormalized propagator) the same no matter how close you get, or it also changes with energy? And if it changes, what is shielding it at big distances? Is there a way to delete all stack exchange accounts at the same time?  I also want to remove all traces of my posts on the site. Is that possible?   Will I have to delete all of my posts manually?  Can I use the stack exchange API for this? Restrictive vs Non-restrictive Relative Clause Show reference label when it is undefined windows bat file error