Assume that $\frac{x}{0}$ (and therefore $\frac{0}{0}$) is undefined and $\mathbb{Q}$ is our universum (when I tried to do it for $\mathbb{R}$ things just got way too complicated; if anyone knows a generalisation akin to mine — and possibly more correct — please add it to your answer).  We write  $$x^0 = x^{1} \cdot x^{-1} = \frac{x}{x} = 1; \quad x \neq 0.$$  We know that  $$0^x = 0 \; \text{ because } \; \underbrace{0 \cdot 0 \cdot \dots \cdot 0}_{x} = 0 \; \text{ for } \; x \in \mathbb{N}, $$  and  $$0^{a/b} = \sqrt[b]{0^a} = \sqrt[b]{0} = 0 \; \text{ for } \; a,b \in \mathbb{N} \; \text{ or } \; a,b \in\mathbb{Z}^-$$.  Then we consider the case  $$x^y \cdot x^0 = x^{y + 0} = x^y,$$  and plug for $x = 0$.  $$0^y \cdot 0^0 = 0^y = 0 \cdot 0^0 = 0 \; \text{ for } y \in \mathbb{Q}^+,$$  and now we have a scenario $0x = 0$ (where $0^0$ is our $x$) and we can't divide by zero (and write the "solution" $x = \frac{0}{0}$). Because we know that $0 \cdot z = 0$ for every $z \in \mathbb{Q}$ the solution set is $\mathcal{S} = \mathbb{Q}$.  So I've heard that asserting $0^0 = 1$ is why binomial theorem (don't ask me what that is, I've just heard of it, I'm a high school student currently) even works in the first place. So the way I see it, the assertion of $0^0 =1$, $0^0 = 0$ or any other equivalence $0^0 = z; \; z \in \mathbb{Q}$ is valid because $0z = 0; \; \forall z \in\mathbb{Q}$ holds. So we essentially select whatever value "fits" for us in a certain context, right?  Please correct any of my errors and outline the whole "being able to decide what the value is" thing. It will help me to understand mathematical concepts much deeper.  P.S.: I've blindly assumed the equivalence $\sqrt[b]{0}; \; b \in \mathbb{N}$, but I have no idea how to prove it. If you included the proof of this statement, that'd be a nice bonus. Thanks!
Could someone provide me with a good explanation of why $0^0=1$?  My train of thought:  $x&gt;0$  $0^x=0^{x-0}=0^x/0^0$, so  $0^0=0^x/0^x=\,?$  Possible answers:   $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined   PS. I've read the  on mathforum.org, but it isn't clear to me.
How to solve differential equations of the form $f'(x) = f(x + a)$ What could one do to find analytic solutions for $f'(x) = f(x + a)$ for various values of $a$?  I know that $c_1\sin(x + c_2)$ is solution when $a = \frac{1}{2}\pi$, and of course $c_1e^x$ when $a = 0$.  For instance, is there a function satisfying $f'(x) = f(x + 1)$?  What about negative or imaginary $a$?  Is there possibly a generalization? How do volume control headphones work? A recent question over on  got me to wondering how volume control headphones work.  I don't mean volume control which works by , but the kind of volume control that signals to the device to increase or decrease the signal output.  For instance, according to the , the volume controls are only for apple devices:     Universal Compatibility and Microphone Communication   LP comes with 2 cables for universal compatibility with all modern mobile and audio devices. The 3-Button remote microphone cable is designed for the latest Apple devices, including iPhone®,iPad®, iPod®, and Macbook series. The long audio-only cable and 1/4" adapter is universally compatible with all audio devices and professional equipment.   However this sort of device apparently works with some android phones and the question implies that it works with some computer sound cards on Windows machines too, but not being an Apple buyer (and only recently getting my first smartphone) I haven't seen this sort of thing before. I've done a quick google search, but can't find anything on standards for this technology.   How does this work?   Is it something as simple as shorting left/right/mic channels? If so, could using headphones of this type with equipment that doesn't support this technology, damage that equipment? Is it a serial signal, for instance sent over the mic channel? Does it just require a headphone  connector or a headphone  +microphone TRRS connector?   Does this technique have a name?  Incidentally, is someone could answer this, I can probably look up the rest of the information myself. *8') Is it as standard as it appears, or is everyone just following Apple? Is this something that is patented?   If so, who holds the patent and do they charge a license fee for people to use this technology? If not, is there an open standard for it?    There is some excellent information in  to the question  but it doesn't answer the specifics of how this type of volume (etc.) control works, since it now seems to be rather more common and standard. Is storing passwords hashed with the email more secure than without? Windows using too much RAM, how to diagnose resource hog I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. Find the asymptotic tight bound in  $$ T(n) = 4T\left(\frac{n}{2}\right) + n^{2}\log n. $$  where $ \log n= \log _{2}n $ and $T(1) = 1$.  I should solve this using all three common methods: iteration, master theorem and substitution.  It is highly likely that this kind of recurrence equation will be in my test in two days. Thank you very much in advance! How can I break an align environment for a paragraph? I would like to have two displayed equations, broken by a short paragraph, but with aligned equal signs (as if using the align environment).  Is there some way to "break" the align environment temporarily? If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$? Catch Regex too complicated when splitting a string in salesforce I have created a function wherein I split a string from the csv filecsvAsString.split('\n') . It is working when the string is not so big. But when I try to split a very large string, an error will occur      regex too complicated   what I want to do is to catch if an error occur during the split process. How will I do it ? I've tried to use try and catch but the problem still occur. Is there any other possible solution? Brightness changes while "Adaptive Brightness" is disabled in Windows 8.1/10 with Intel HD Graphics The brightness changes with the colors displayed on the screen on my Lenovo Thinkpads with Intel HD Graphics. Light colors make the screen brighter, dark make it darker. Same behavior in Windows 8.1 and 10.  The "adaptive brightness" feature in Windows is disabled. Remove automatically added keyboard inputs and prevent them from coming back (Windows 10) I have 2 languages installed on my computer, both with a single input method.  I have 2 keyboards: CES-CSQ and ENG-US.    However lately (maybe after the last Win10 update) I started to see 2 additional keyboards in my systray  - namely CES-CS and ENG-CSQ.    Is there there a way to remove those 2 input options (CES-CS, ENG-CSQ) which I do not see in the Control panel's Language options?  Solution described at  question does not apply to my problem because I do not see input options that I want remove in Control panel's Language options. Allow cancelling of stars from message Currently, the only way to cancel stars in a chat transcript is when the chat-message is visible in the chatroom's sidebar.    However, in a fast-moving, multi-star chatroom, the window of opportunity to cancel can be short, and rapid changes in starring can quickly make this impossible to do in a timely way.  We use stars as markers in moderator chat events (where the Community managers disseminate important information to the moderator teams) as well as for town halls during elections.  Feature request: Allow mods/room owners to cancel stars from the chat message menu box, including in the transcript. Does Java Connection.close rollback? An image alt="" text should contain information about that image in the context of the page, so for instance if you have an article about cats on skateboards and then a photograph of the cat on his skateboard you'd probably give it the alt text of alt="Fluffy pulling some mad tricks on his pimped-out board".          &ndash; Wireframes created with   However, what should happen when that image is a link to a page (such as the Wikipedia cat page)?  I don't think there is any point using the title attribute here because screenreaders just ignore that when an alt attribute exists - the only people who would see the title attribute are mouse users who hover over the image, so that excludes keyboard only users too. It's not really any benefit to accessibility to include a title in this situation (Overall the title attribute is a bit useless really).  So, what should the image ALT text say in this case?   states:     Provide text alternatives for any non-text content so that it can be changed into other forms people need, such as large print, braille, speech, symbols or simpler language.    Therefore giving it the alt text of cat article page on Wikipedia is incorrect because that doesn't describe the image, neither does skateboarding cat image that links to the wikipedia article because the alt text should refer to the image itself, not the link associated with it. "Whom" or "who" where the referent is both subject and object? I understand there has been so much on this topic but I am still confused. I get that if the person is the subject it is who and anything else is whom. However, I'm really struggling to work out this rule for the following sentence:     The scope of which his account of morality applies to is therefore restricted to those whom/who the ‘notion of justification to a being of that kind makes sense.’   Should it be who or whom? I can't work out the subject and object. Short story set in the near future where all wars are settled on the moon using automated machines? Plot Details/Summary  Another one where I have only sketchy memories of the tale, I fear.   The setting for this one is the near future (from the writer's perspective, which I think was the 1970s).  Man's reached a point where bloodshed and loss of life are no longer necessary to settle conflicts.  Instead, all disputes are settled through an annual combat held on the surface of the moon, using the best combat machines each nation can come up with.  The protagonist, as I recall, is an engineer for the Americans.  He recollects past conflicts, and the circumstances for this years fracas.  Much of the combat is told from his observers' perspective.  The actual conflict is a rather clever affair, with a description of each team's designs and all the tricks and gee-whiz things they built into them.  The protagonist's team manages to prevail, when they managed to use an attack as a feint, with the real attack being a swarm of microscopic machines (they did not use the term nanites) that destroyed the opposing machine from the inside.  Timeframe/Other Details  I think this one dates from the 1970s.  I don't think it more recent, given my recollections of when I read it, and also the technology descriptions they used in it.  There is an outside chance that the combat is not between rival nations, but rival corporations.  That would change the premise of "man evolving beyond bloodshed", of course.  I think it's nations fighting, though.  I am almost 100% certain this was in a sci-fi magazine, not an anthology. Since when did kidnapping come to include adults too? As per , the word 'kidnap' originated to denote nabbing away of a child. When and how did kidnap come to denote nabbing of adults?  Update: Just found a link to a  that mentions kidnapping of souls. Now not sure how correct the above link is. Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.  For the P $\implies$ Q part, I know you can get a surjection $Y\to X$ by mapping $y$ to $x$ if $y=f(x)$ for some $x\in X$ and mapping $y$ to some arbitrary $\alpha\in X$ if $y\in Y\setminus f(X)$. But I don't know about the Q $\implies$ P part.  Could someone give an elementary proof of the theorem? Run command on another(new) terminal window Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$. I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.  If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that $\overrightarrow{AB}=(5,2)=\overrightarrow{CD}$. Similarly $\overrightarrow{AC}=\overrightarrow{BD}$. So I see that these points indeed form a parallelogram.  It is assignment from linear algebra class. I wasn't sure if I had to like use a matrix or something. Sometimes a question I ask will get answers that don't help me, or maybe it doesn't receive any answers at all.  How about making a notification when somebody links to my question?   It could be something like this:     New information regarding your question might be available at {link}   It might help me getting an answer from related questions that weren't there when I first asked the question.