Does loss of life count as being dealt damage?
What are the differences between combat damage, direct damage, and loss of life?
Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable) This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers.  The source of confusion is this: if we abstract completely from the properties and operations on numbers and just think of them as strings of digits, it appears to be possible to list all the real numbers between $0$ and $1$ and all positive integers having the same digits in the same (or reverse) order, for example:  $$0.1 \to 1$$  $$0.01 \to 10$$  $$0.0101 \to 1010$$  $$0.110101 \to 101011$$  I use binary expansion for simplicity.  Since irrational numbers have infinite expansions, we would have to consider some kind of 'infinite natural numbers', like:  $$101110001\dots$$  or  $$\dots101110001$$  I think the main (and only) reason why such numbers don't exist is this requirement:   We need the sets of real and natural numbers to be ordered. We need to be able to do arithmetics on real and natural numbers.   Obviously, if we consider any of two types of 'infinite natural numbers' and the usual absolute value, we find that it's impossible to order them, or do arithmetics with them.  The only way is to change the norm (to $p-$adic norm for example).  While with the real numbers it's still possible to order them, and do arithmetics with them when they have infinite amount of digits in their expansion (or at least in principle).     Is my reasoning correct? Are the two requirements (ordering and arithmetics) enough to explain why natural numbers with infinite amount of digits do not exist, while real numbers do?   This confusion is not restricted to me alone, see  for example $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$ Prove that $\langle u,v\rangle=0\Longleftrightarrow \|u\|\leq \|u+av\|$.  So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints would be greatly appreciated.  Please no answers, I just want a small nudge to get me in the right direction, I've been stumped on this for a little bit.  Here's what I have so far:  ($\Longleftarrow$)  Assume $\|u\| \leq \|u+av\|$  $\|u+av\|^2=\langle u+av,u+av\rangle=\langle u,u\rangle+ \langle u,av\rangle+\langle av,u\rangle+\langle v,v\rangle=\|u\|^2+\bar{a} \langle u,v\rangle+a\langle v,u\rangle+\|v\|^2=\|u\|^2+ \bar{a}\bar{\langle v,u\rangle}+a\langle v,u\rangle+\|v\|^2$  Thoughts/ideas? After most recent kernel update (4.15.0-44-generic) my tablet freezes whenever plugged into docking station I performed the most recent kernel update on my Dell 5290 2-in-1 earlier today.  But moments after I plug it into its docking station, the screen flickers, blanks and freezes.  The only item left on the screen is the mouse pointer, and it is stuck.   If I leave the tablet not plugged in, it runs just fine.  It seems to be the docking station that triggers the freeze.  That said, I can log into the tablet via ssh even with the screen frozen.  So the system is still running.  But if I try to reboot the tablet via the reboot command, nothing appears to happen. sudo reboot -f will reboot the system, while sudo reboot will not.  Any ideas where I should start looking to identify this problem? checking if a number is a prime I was reading Wikipedia, and it was given that "all primes are of the form 6k ± 1"  (other than 2 and 3), where k = 1,2,3,4,...  Is this statement correct? If yes, can we use this to test if a given number is a prime number? For instance, we can say that 41 is a prime number, since there exists an integer K (K >= 1), such that 6k - 1 = 41 ==> k = 7.  I am confused why this test cannot be used to test if a number is prime?  Thanks, Sekhar Proof that $n^2 < 2^n$ How do I prove the following statement by induction?  $$n^2 \lt 2^n$$  $P(n)$ is the statement $n^2 \lt 2^n$  Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$  (since $k$ is any integer, I assume I have to prove this for positive and negative integers)  So let base case be $P(1)$, and I have to prove $P(n)$ for $n \ge 1$ and $n \lt 1$  $P(1)$ is $1^2 \lt 2^1$ which is clearly true.  Induction hypothesis: $k^2 \lt 2^k$  Inductive step $(k+1)^2 \lt 2^{(k+1)}$  $(k+1)^2 = k^2 + 2k + 1$  $k^2 + 2k + 1 \lt 2^k + 2k + 1$ by inductive hypothesis  Not sure how to proceed. Is my previous intuition that I have to prove this for $n \ge 1$ and $n \lt 1$ correct? When an option expires does that constitute a buy or sell A similar question recently asked about stock sales and option buys triggering wash sale rules.  If an option expires, is that considered a buy/sale ?  What I mean is: let's assume I have a call option with a target date of August 16. Now I sell the underlying stock somewhere before that (after July 16th). Does that trigger wash sale rules (if the option expires in the money, exercised by default) ?  Is an option expiring out of the money considered a sale ? How does editing work? Blender Freestyle : Vertical lines On Blender , a picture is shown with two Suzannes made of vertical/horizontal lines.    Click to enlarge  How to reproduce this effect with Blender Freestyle? Using modifiers? Custom shaders? Tsunami dampening mechanisms Sort array of objects by string property value Is it possible to install Ubuntu through network? I would like to install Ubuntu  for almost 120 users. But it's very difficult in using USB and drives.  Any suggestions? Answer appears automatically converted as a comment I posted an answer (pointing to another question already asked) but when I hit submit, I saw my answer appear as a comment to the question instead.   I tried it a couple of times and still the same result. The answer gets eaten up and is transformed into a comment.   Is this a bug? Find the set of $x>0$ such that the series $\sum\limits_n x^{\ln{n}}$ converges If $x&gt;0$, find the set of all values of $x$ such that series is convergent$$\sum_{n=1}^{\infty} x^{\ln{n}}$$   My attempt:- I used Ratio test for finding the set of all values of $x$ such that series is convergent.   $$\lim_{x\to\infty}\frac{x^{\ln{n+1}}}{x^{\ln{n}}}$$  $$=\lim_{x\to\infty}x^{\ln\frac{n+1}{n}}$$ This quantity must be less than one for getting a convergent series, I am not able to judge. Can you please help me to find the interval of convergence? Do you "gain" your 1st level? Let $R$ be a ring such that $x^4=x$ for every $x\in R$. Is this ring commutative? $$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime.  I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits: $-0.26959$?  I have looked at the "questions that may already have your answer" and some of the "similar questions," but they involve somewhat different formulas. How do I get the source code for eglibc in a format that will allow stepping through the source with gdb?    I have tried using apt-get source libc6-dev and that works for getting the source code, but I think that since the pre-processor directives were not applied that the line numbers that gdb has does not match with the source I downloaded. This makes it less useful when trying to step through the source and look at how functions in libc are working. Doubling the tails in two-sample permutation test throwing exceptions out of a destructor Most people say never throw an exception out of a destructor - doing so results in undefined behavior. Stroustrup makes the point that "the vector destructor explicitly invokes the destructor for every element. This implies that if an element destructor throws, the vector destruction fails... There is really no good way to protect against exceptions thrown from destructors, so the library makes no guarantees if an element destructor throws" (from Appendix E3.2).   seems to say otherwise - that throwing destructors are more or less okay.  So my question is this - if throwing from a destructor results in undefined behavior, how do you handle errors that occur during a destructor?  If an error occurs during a cleanup operation, do you just ignore it? If it is an error that can potentially be handled up the stack but not right in the destructor, doesn't it make sense to throw an exception out of the destructor?  Obviously these kinds of errors are rare, but possible. Using `M` instead of `k` when rep is >= 1M on https://chat.stackexchange.com/users I read on  ()    Shouldn't it be 1.03m or (if we really follow the conventions of the metric system) 1.03M instead?