What is the formal definition of fraction?  Everyone says something different.  Thank you for your answers.
Is $\frac{1}{\sqrt{2}}$ a fraction or not?   I just came across a Grade 8 student's class work where the teacher has given the above as an example to differentiate fractions from rational numbers. I think this is wrong fundamentally, as $\sqrt2$ is irrational while all fractions would be rational. Please advise. Thanks!
Clear 301-redirect cache in Firefox Is it possible to clear only 301-redirects from Firefox's cache? CTRL+F5 doesn't do the job since you just reload the target without caching but do not clear the cached redirect.  An option to disable caching of those redirects completely would be fine, too - but some way (possibly through an extension) to remove those cached redirects would be better.  Note: I'm not looking for answers that disable all caching (including CSS, JS, etc.) I want to use symbols from wasysym and marvosym packages:  \usepackage{wasysym} \usepackage{marvosym}   but it gives an errors as both packages have the command of \CheckedBox. I do not want to use this command, but since both packages have it, it makes confusion for latex. How can I ignore one command to have both packages? Do tag edits count for Archaeologist? Change elements on different positions within a list Why are there $1 / 1.602176634 \times 10^{-19}$ electrons in a coulomb? Why that exact number of electrons in one coulomb? who decided it? there is nothing wrong with the number, it just seems slightly messy. Why didn't the scientific community just settle on an easier number, such as $1\times10^{-19}$ for example? Python does not find packages installed without Anaconda Entry to the US with an expired Nexus Is it possible to enter the US through the Nexus lane while my Nexus card is being renewed? And how? Start TOC from chapter 1 and also page numbering Why does $1+2+3+\cdots = -\frac{1}{12}$? My final goal is to generate Impulse Response Functions in R.  I have variables that are non stationary when I set k = 5 in a Unit Root test, and they are cointegrated which to my understanding prompts the use of the VECM, from which the Vec2Var argument is used to then generate IRFs. However, my response functions from this methodology do not decay over time and mostly do not revert to the zero line.  Furthermore, I noticed that when I input first differenced variables into the VECM as opposed to level data above, response function do revert to the zero line. Therefore my first question is: 1) is it appropriate to use differenced variables in a VECM model?  Secondly, as an alternative I am considering restricting the number of lags used in the Unit Root tests so that the variables are not all non-stationary; in this case 2) would the use of VECM be void and the VAR with first differences be a more appropriate model? Again, using differenced data is giving me better response functions that revert to zero in the long run. However, 3) is it ok to use the VAR for stationary/non-stationary data in levels where these variables are still cointegrated?  Also, I am using growth rate variables, 4) should I still long transform all variables and use logs in all tests?  Thank you for the help! How to check for an active Internet connection on iOS or macOS? Add Appendix with correct page to ToC What's the canonical way to check for type in Python? What is the best way to check whether a given object is of a given type? How about checking whether the object inherits from a given type?  Let's say I have an object o. How do I check whether it's a str? No such file or directory but I can see it! I'm trying to run a python script, on a headless Raspberry PI using winSCP and get the following error message:  Command '"./areadetect_movie_21.py"' failed with return code 127 and error message /usr/bin/env: python : No such file or directory.   When I try and run from terminal, I get:  : No such file or directory.   I try a similar python script, in the same directory, with the same python shebang, the same permissions and using the same user pi, and it works.  I also do a ls and I can see the file, so I don't know why it will not run. What is "summoning sickness" and how does it affect my creatures? Note: this is intended to be a canonical question about summoning sickness.  "Summoning sickness", broadly, is the rule that stops you from attacking or paying {T} costs with a creature that you just played. But how exactly does it work? How does it interact with other rules and effects?  For example, if I'm trying to figure out how a creature is affected:   Does the status matter (tapped/untapped, face down/face up, etc.)? What if it came into play, but I didn't cast it? What if it's a token? What if it was on the battlefield when my turn started, but it wasn't a creature? What if it stops being a creature? What if it's a creature and also another card type? What if it's a double-faced card that just transformed? What if it changes control? What if it's a copy of another creature? Which of the creature's activated abilities can I use? Are any of its other abilities affected? Can I tap the creature to pay a cost that is not a {T} cost?   (These examples are primarily taken from questions about summoning sickness that have already been asked on the site) When is $\sin(x) = \cos(x)$? How do I solve the following equation?  $$\cos(x) - \sin(x) = 0$$   I need to find the minimum and maximum of this function:   $$f(x) = \frac{\tan(x)}{(1+\tan(x)^2}$$  I differentiated it, and in order to find the stationary points I need to put the numerator equal to zero. But I can't find a way to solve this trigonometric equation. How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$? Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise? Every planar graph has a vertex of degree at most 5. Find the general solution of $xy''-(1+x)y'+y=x^2$ knowing that the homogeneous equation has the following solution: $e^{ax}$, where $a$ is a parameter you have to find.  I have found that $a=1$ or $a=1/x$. Is time dilation an illusion? It is said that we can verify time dilation by flying a very accurate clock on a fast jet or spaceship and prove that it registers less time than the clocks on earth. However, the clocks on earth would be moving relative to the clock on the spaceship, and since time always dilates and never goes faster regardless of the direction of relative motion,  the clocks on earth should register less time than the clock on the spaceship.  Is this true? Whenever there is a fast-moving object such as a rocket do all clocks on earth really become slow?  If the rocket with the clock landed after moving at relativistic speed, would its clock and the earth's clock again show the same time since during its travel both appeared slow to each other?  Or is all this just an illusion, ie. the clocks just appear to be slow to each other but in actually run at normal speed, and neither is behind when the rocket actually lands?