# Activity for r~~‭

Type On... Excerpt Status Date
Edit Post #284742 Post edited:
1 day ago
Edit Post #285072 Initial revision 6 days ago
Answer A: What is the parallel for edges?
I would still use the term ‘edge’ for that case—as in, ‘These three arcs are the edges of a circular triangle.’ If necessary to prevent confusion, I might specify ‘curved edge’ or something similar. But while ‘edge’ in geometry often implies a strict line segment, in graph theory it simply denotes a ...
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6 days ago
Comment Post #285015 ^ This. Honestly, actually following this advice on all of your intuition questions would improve most of them substantially. But note that these are not examples of following this advice: ‘I don't understand.’ ‘How can this be?’ ‘This is sortilege!’ Instead: * Connect the thing you're askin...
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9 days ago
Comment Post #284997 All the above is true even in the absurdly unlikely case (perhaps you don't realize just how idiosyncratic your writing style is) that you've reposted someone else's closed question verbatim. You still know it's a question that doesn't meet SE's standards, and so you've decided to try it out here ins...
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9 days ago
Comment Post #284997 I don't think you understand. This question isn't closed because it's a repost from SE. This question is closed because it's a bad question. It would be a bad question regardless of whether it was original or a repost. However, the fact that it's a repost of a closed question means that not only have...
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9 days ago
Comment Post #284996 I recommend editing your question to be that, if that's what it is. The thing about a minimal border around two points is still imprecise; if that's an important part of the question, explain a bit more, and otherwise remove it.
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10 days ago
Comment Post #284997 of calling for a proof-by-picture by linking to a question the premise of which is that proofs-by-picture are often misleading and that students should pay more attention to stuff like algebra. Hopefully this is you poking a little fun at yourself, and not... a less charitable interpretation.
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11 days ago
Comment Post #284997 This was closed Somewhere Else (https://math.stackexchange.com/questions/4314855/without-trial-and-error-averageaveragea-b-c-vs-averagea-averageb-c), and should be closed here for the same reason. You got good feedback on that post, ignored it, and submitted exactly the same question here without try...
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11 days ago
Comment Post #284996 I believe there's a lovely question in here somewhere but it hasn't emerged yet. Some questions for you: What do you mean by ‘dot’? Do you mean the same thing as ‘point’, or are you using a different word intentionally to signify something else? Is a ‘minimal border around ... two dots’ something oth...
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11 days ago
Edit Post #284909 Post edited:
19 days ago
Edit Post #284909 Post edited:
20 days ago
Edit Post #284907 Post edited:
Retagging
21 days ago
Suggested Edit Post #284907 Suggested edit:
Retagging
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Edit Post #284909 Initial revision 21 days ago
Answer A: Third kind of infinite
By Cantor's diagonal argument, there can be no bijection between the power set of any set and the set itself. This is true for finite and infinite sets. So taking the power set ($\mathcal P$) of a set of some infinite cardinality always gives you a set with a strictly larger infinite cardinality. ...
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21 days ago
Edit Post #284742 Post edited:
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Edit Post #284742 Initial revision about 1 month ago
Answer A: What does upper indices represent?
A textbook homework problem might ask, ‘Speedy the snail creeps along at a steady pace of 60 cm per minute. How far does Speedy travel each second?’ The correct answer is, of course, one centimeter. Hopefully your teacher would also accept 10 mm or 0.01 m as equally valid answers; after all, they all...
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Edit Post #284741 Initial revision about 1 month ago
Answer A: Can the bijection for the Lost Boarding Pass Probability Problem, be formulated or pictured?
The bijection is just to swap the people sitting in the first and last seats. I feel like a ‘formula’ is more machinery than this simple concept is worth, but here, if this helps: let $P$ be the set of permutations on $\\{0\ldots n - 1\\}$, where for $p \in P$, $p(s)$ is the number of the passenge...
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Comment Post #284723 Any expression that yields the result of addition seems like it would be pretty much just addition. But, for example, perhaps you'd be happy with something like $\ln{e^a e^b}$? If not, maybe you could be clearer about what you would and would not accept.
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Edit Post #284557 Post edited:
Typo
Comment Post #284561 Your source might be using [Einstein notation](https://en.wikipedia.org/wiki/Einstein_summation), but there are other notations that give different meanings to upper and lower indices. I think you need to include more of the surrounding context to know for sure—a raised index doesn't universally mean...
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Suggested Edit Post #284557 Suggested edit:
Typo
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Comment Post #284551 Cheers, I added a paragraph that should clear that up.
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Edit Post #284551 Post edited:
Clarify ‘abstract vector’ and tighten up some math expressions
Edit Post #284551 Post edited:
2 months ago
Edit Post #284551 Initial revision 2 months ago
Answer A: transpose matrix and general matrix is completely messed up
I suspect what's tripping you up here is the difference between applying a linear transformation to a vector versus applying a change of basis. Both involve matrix multiplication, but the nature of the transformation is different. First, some words about vectors. There are two ways to introduce st...
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2 months ago
Edit Post #284284 Post edited:
3 months ago
Suggested Edit Post #284284 Suggested edit:

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Edit Post #284281 Post edited:
3 months ago
Edit Post #284281 Post edited:
3 months ago
Edit Post #284281 Initial revision 3 months ago
Answer A: Is Pythagorean theorem really valid in higher dimensional space?
One generalization of the Pythagorean theorem to three dimensions is de Gua's theorem: > if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces ...
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3 months ago
Edit Post #284122 Initial revision 3 months ago
Answer A: What to do when there's lot of function of time? (For integration) Should I consider them as constant?
No, if you're integrating with respect to time, you can't treat functions of time as constants. Remember that, by the fundamental theorem of calculus, the result of your integration must be a function that, when differentiated, yields your original integrand. So if you incorrectly concluded that \...
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3 months ago
Comment Post #284051 There are many good questions about intuition and motivation, yes. I'm talking about a particular slim subset of those questions, where the next step in a proof is known, the reason that the step works is known, and the asker is only interested in, it would seem, the cognitive process that produced t...
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3 months ago
Comment Post #283900 @#53398, I agree that there are some ways that writing a proof is more like designing a bridge than writing a novel (and some ways that it is like neither). And yes, there are skeletons that get reused and are good to know. The sorts of questions I'm objecting to are not questions like, ‘What types o...
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3 months ago
Comment Post #283930 I'm not against (what I think is) the spirit of this answer, but I think it's out of place here. I'm not talking about questions where the asker doesn't know what the next step is; I'm talking about questions where the asker has been shown the next step, possibly understands how the step works, but w...
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3 months ago
Edit Post #283949 Post edited:
3 months ago
Edit Post #283949 Initial revision 3 months ago
Answer A: What're the orders for equation expressing?
Another aspect of operation precedence which Derek's otherwise excellent answer doesn't cover is operator associativity. Implicit in the PEMDAS convention is the idea that the arithmetic operators are left-associative, meaning that they should be grouped from left to right when a subexpression con...
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3 months ago
Edit Post #283899 Post edited:
3 months ago
Comment Post #283899 @DNB, I have given you specific feedback on each of these points on questions where they are applicable. At some point I stopped trying, since you didn't seem to be trying to incorporate the feedback into your future questions and I have better things to do with my time. But regardless, this question...
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3 months ago
Edit Post #283899 Post edited:
Demote the importance of the other issues with these particular posts
3 months ago
Comment Post #283930 Is this an answer to the question? It seems mostly off-topic (although I might have made the question too distracting by mentioning some other problems that questions can have).
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3 months ago
Comment Post #283886 If you know both children are girls, then the information that the winter child is a girl is redundant; you know that it's a girl regardless of whether the event is described as ‘at least one winter girl’ or ‘at least one winter child’.
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3 months ago
Edit Post #283900 Initial revision 3 months ago
Answer A: Is ‘How would you know to do the next step?’ always a bad question?
I believe that yes, such questions are intrinsically bad. There are two ways to attempt to answer such a question. First, the answerer could try to get into the head of the author of the particular proof being examined, and figure out how they made the necessary leap. If the proof is well written,...
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3 months ago
Edit Post #283899 Initial revision 3 months ago
Question Is ‘How would you know to do the next step?’ always a bad question?
We have a user who keeps posting questions of the form, ‘How would you [tortured synonym for “know”] to [do the next step in a proof]?’ Leaving aside the various other reasons that these posts are bad^[I don't want those other issues to be a distraction from the central question here. I'm only mentio...
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3 months ago
Edit Post #283877 Initial revision 3 months ago
Answer A: Getting backward of partial differentiation's chain rule
The difference between a partial derivative and a total derivative is that the partial derivative measures the change in a function when only one of its arguments varies, while the total derivative measures the change in a function when all of its arguments vary. The $\frac{\partial z}{\partial x}$ n...
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3 months ago
Edit Post #283707 Post edited:
4 months ago
Edit Post #283707 Initial revision 4 months ago
Answer A: Find limits of integration in polar coordinates
Regardless of the coordinate system, the principle of finding the limits of integration is the same: find the minimum and maximum value of the independent coordinate that enclose the region of interest. In polar coordinates, for these two particular curves, the region of interest is best described...
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4 months ago
Comment Post #283593 $f(x) = \frac{1}{x-1}$ is considered continuous. It's not defined at $x = 1$, so it has a disconnected domain, but it is continuous at all values where it is defined.
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4 months ago
Comment Post #283385 This is not a question about math.
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4 months ago
Edit Post #283173 Initial revision 4 months ago
Answer A: How can "information about the birth season" bring "at least one is a girl" closer to "a specific one is a girl"?
‘More specific information’, here, is informally referring to the value of the (positive) likelihood ratio: the ratio between how likely it is to get that information if a proposition is true, and how likely it is to get that information if the proposition is false. (I say ‘informally’ because, as yo...
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4 months ago
Edit Post #282600 Post edited:
4 months ago
Comment Post #282615 Nope. As written, this is a complete answer to your original question. You asked a new question here in the comments.
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4 months ago
Comment Post #282892 Have you tried to understand why? Take some example values of $s$, $c$, and $n$, and see what happens to the value of that fraction as $n$ increases.
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4 months ago
Edit Post #282892 Initial revision 4 months ago
Answer A: How does $\lim\limits_{n \to \infty} \dfrac{p[s + (1 - s)c^n]}{p[ s + (1 - s)c^n] + (1 - p)[s + (1 - s)w^n]} = p?$
> So if $n \to \infty$, then all terms above containing $c^n \to 0$. This is false. It's true that as $n \to \infty$, $c^n \to 0$ (for $0 < c < 1$). But a term like $\frac{s}{c^n}$ will diverge instead of converging to $0$. You haven't done yourself any favors by dividing everything by $c^n$...
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4 months ago
Comment Post #282615 An algebraic expression is made exclusively of numbers, variables, and algebraic operations. $\ldots$ is none of these. This really isn't any more complicated than that. Doing algebra with informal notations can get you into trouble if you don't keep in mind what the informal notations mean. This is ...
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4 months ago
Comment Post #282873 You moved your coloring back one paragraph. Look at the paragraph that is now above your red phrase. That is the explanation. And you need to explain why you think 2 and 3 are true; I can't read your mind.
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4 months ago
Comment Post #282873 The answer to 1 was given just two paragraphs earlier. 2 and 3 are not true and not implied by anything in the solution.
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4 months ago
Edit Post #282780 Post edited:
Think you missed a term in your last equation
5 months ago
Suggested Edit Post #282780 Suggested edit:
Think you missed a term in your last equation
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Comment Post #282737 Is this a homework assignment? What have you tried? Where are you stuck? We can help but we aren't going to do the assignment for you.
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5 months ago
Edit Post #282712 Initial revision 5 months ago
Answer A: How to derive some trigonometric formulas?
Can't use algebra, you say? Nonsense! All you need is Euler's formula $$e^{i\theta} = \cos\theta + i\sin\theta$$ and these identities just fall out of the algebra. For example: \begin{align} \sin (\alpha + \beta) &= \frac1{2i} \left(e^{i(\alpha + \beta)} - e^{-i(\alpha + \beta)}\right)\\\\ ...
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5 months ago
Edit Post #282665 Initial revision 5 months ago
Answer A: $\sum_{k=0}^{n} \binom{n}{k}=2^{n} \overset{?}{\iff} \sum_{k=0}^{n} \binom{2n+1}{k}=2^{2n}$
The green result can be used to prove the red one, yes. Start with the fact that $\binom{m}{k} = \binom{m}{m - k}$. If this isn't obvious to you, you can derive it either by looking at the usual formula for the binomial, or by considering that choosing the members of a subset is equivalent to choo...
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5 months ago
Edit Post #282654 Initial revision 5 months ago
Answer A: How does the change of variable $\color{red}{r↦n−r}$ transmogrify $\sum\limits_{r=0}^n2^{n-r}\binom{n+r}{n}=2^{2n}$ into $\sum\limits_{k=0}^{n} \frac{\binom{2n-k}{n}}{2^{2n-k}}=1$?
$n$ is free in the expression $\sum{r=0}^n 2^{n-r}\binom{n+r}{n}$. The only variable not free in that expression is $r$. What $\sum{r=0}^n$ means is to treat what comes next as a function of $r$, and compute the sum of that function evaluated at $0$, $1$, and so on up to $n$. If $n$ is used inside th...
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5 months ago
Comment Post #282652 The variance argument is only relevant if you draw from the same urn multiple times, which isn't part of the original problem. With one draw, the author is correct that you should be indifferent. Your argument is interesting additional information, but I think it's misleading to present it as a count...
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5 months ago
Edit Post #282638 Post edited:
5 months ago
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5 months ago
Edit Post #282638 Post edited:
5 months ago
Edit Post #282638 Initial revision 5 months ago
Answer A: A formal-logic formula for decimal to binary conversion
If you have a non-negative integer $n$, and the base $b$ representation of $n$ is a sequence of digits, numbered from the right as $dk\dots d2d1d0$, then $di = \left\lfloor \frac{n}{b^i} \right\rfloor \bmod b$. For example, decimal 10 is binary 1010 because:  d3 = \left\lfloor \frac{10}{2^3}...
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5 months ago
Comment Post #282615 $\ldots$ doesn't work like that; you can't simply substitute a variable into that kind of informal, descriptive expression without thinking about what the notation as a whole represents. It's not an algebraic expression, so the rules of algebra don't apply. $\ldots$ is a signal to look at the pattern...
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5 months ago
Comment Post #282623 I think the feedback given in the other thread is better. Citing edition and page number doesn't make the question any clearer; it just makes it possible for someone to do detective work to fill in details that the asker should have provided in the first place. An image of the page doesn't lead to a ...
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5 months ago
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5 months ago
Edit Post #282625 Initial revision 5 months ago
Answer A: Questions with a quote/screenshot and a request to explain
My approach is downvote and, if it's a first infraction, comment explaining the issue with the question. Repeat offenders get downvotes and, perhaps if this one-guy's-opinion calcifies into actual community policy, flags. I'd like us to start doing a better job of holding the line against low-effo...
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5 months ago
Comment Post #282616 An understandable mistake, but one you perhaps could have caught yourself if you had thought about your own interpretation a little more. $k$ doesn't make sense as the number of ways to choose $k$ team members, any more than $$$^n_k$$$ makes sense as the number of ways to choose one of them to be cap...
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5 months ago
Comment Post #282611 We've had words before about the appropriateness of screenshots in questions. In this case, the quoted text doesn't seem to be relevant to your question. The question is about the screenshotted text, which is completely different, and from an online source which I seem to be able to copy from just fi...
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5 months ago
Comment Post #282615 I don't even know what the middle expression in that line is supposed to mean. You start at $n$, and then descend to $n - 1$, and by the end somehow... wrap around to $n + 2$, $n + 1$? What's happening in between, where the ellipsis is? How did you arrive at that as what $LHS\|_{k=1}$ should represen...
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5 months ago
Edit Post #282616 Initial revision 5 months ago
Answer A: Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"?
$$$^nk$$$ is the number of ways to choose $k$ team members from $n$ players. $k$ is the number of ways to choose one captain from $k$ team members. (It isn't the number of ways to choose a captain from the original group of people, as you suggested; that would be $n$.) $k$$^nk$$$ is the product...
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5 months ago
Edit Post #282615 Initial revision 5 months ago
Answer A: If k = 1, why $n(n-1) \dots \color{red}{(n-k+1)} = n$?
You started on the right track with $n(n - 1)\ldots(n - [k - 3])(n - [k - 2])(n - [k - 1])$. What you should have noticed is that there are $k$ terms in that product. The first term subtracts 0 from $n$, the second term subtracts 1, and so on, with the $i$th term subtracting $i - 1$, all the way u...
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5 months ago
Comment Post #282611 You screenshotted a website? Is this an elaborate troll or something?
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5 months ago
Comment Post #282609 In a spree of low-quality questions, this one stands out as exceptional.
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5 months ago
Suggested Edit Post #282600 Suggested edit:

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Edit Post #282027 Initial revision 6 months ago
Answer A: Why mustn't the proportion of smokers among married people be the same as the proportion of smokers in the whole population?
The key word is exactly. If I flip a fair coin, I expect about half of my flips to be heads. If I flip it twice, exactly one head is quite likely. If I flip it twenty times, exactly ten heads is not that unusual, but still a little lucky maybe. If I flip that coin one million times and get exactly 50...
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6 months ago
Comment Post #282015 Completely off topic, but: if you've been taught that, in English, choosing a more obscure word always demonstrates mastery of the language, you've been taught incorrectly. Inappropriate use of obscure words looks far sillier than common words used correctly. (If you are fortunate enough even to find...
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6 months ago
Comment Post #281724 Does your kid know about 30-60-90 triangles? Knowing that the short side is half the length of the hypotenuse would be enough.
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7 months ago
Edit Post #281638 Initial revision 7 months ago
Answer A: difference between quotient rule and product rule
If you work out deriving the quotient rule yourself using the exact trick you're highlighting, you can see that the quotient rule is nothing more than the product rule and the chain rule used together. If you rearrange a quotient to use the product rule, then you'll very likely be using the chain rul...
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7 months ago
Comment Post #280716 Never claimed otherwise! In my first paragraph, I specifically referenced ‘multiplication of square matrices’ as an example of ring multiplication. Cross products are an example of my third case, things that don't conform rigorously to ring multiplication or category products, but which borrow the na...
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8 months ago
Comment Post #280842 _Counterexamples in Analysis_ (Gelbaum and Olmsted), p. 126, offers $\vec{F}(x, y, z) = (x^2 + y^2 + z^2)^{-\frac{3}{2}}(x\vec{i} + y\vec{j} + z\vec{k})$, defined everywhere but the origin, as a counterexample. This suggests that a simply connected domain is not sufficient—IIRC it needs to be 2-conne...
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10 months ago
Comment Post #280857 I would say, in addition to ‘is also typed into the post’, that the text should be understandable without the image—so, in particular, having the main question of the post refer to a circled part of a screenshot can still be a problem, even if that circled part appears elsewhere in the post (because ...
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10 months ago
Edit Post #280854 Initial revision 10 months ago
Answer A: How can I visualize this Compound Interest Chart with indefinite integrals?
Neither of those charts presents those integrals; the first illustrates annually compounded interest, and the second compares a few different compounding schedules to continuous compounding, but not in a way that highlights the integrals in the text explanation. If you wanted an illustration of th...
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10 months ago
Comment Post #280847 In financial math, the term ‘point’ is used precisely to indicate what you call ‘absolute increase’. The difference is that everyone in the field will know what you mean if you say ’basis point’ or ‘percentage point’, and people will ask for clarification if you say ‘absolute increase’. Your wording ...
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10 months ago
Comment Post #280851 That won't do; that definition requires the limit to be a finite number, but you're working with an infinite limit. Also, downvoted because once again, you're using images of text instead of writing your question such that no images are required.
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10 months ago
Edit Post #280799 Post edited:
10 months ago
Suggested Edit Post #280799 Suggested edit:

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Comment Post #280717 The graph in the upper-right corner of the proof box is all you need. Imagine that's a graph of $f_1$. Once you have $f_1$, the details of $f$ and $g$ don't matter and graphing them is neither necessary nor useful, though you can always imagine what that would look like if you want.
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10 months ago
Edit Post #280717 Initial revision 10 months ago
Answer A: How can I generalize a picture for the Mean Value Theorem to the Generalized MVT?
Plotting $f$ and $g$ together is unlikely to lead to intuition about the GMVT; they're just two arbitrary curves, whereas the value from that diagram of the MVT comes from the fact that the yellow line is constructed to give intuition. The art of drawing intuitive figures is largely about figuring ou...
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10 months ago
Edit Post #280716 Initial revision 10 months ago
Answer A: Why should a non-commutative operation even be called "multiplication"?
Usually (not always, but true for the examples of multiplication of real numbers and multiplication of square matrices), a binary operation that is called some variant on ‘multiplication’ is the multiplicative operation of some ring). That's the closest thing to a rigorous definition of multiplicatio...
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10 months ago
Edit Post #280168 Post edited:
Fix TeX formatting
11 months ago
Comment Post #280206 I suspect $k$ is an idiosyncratic or just unfortunate choice of independent variable. It looks like the author is just saying that $\mathcal{M}$ is the identity function, like $f(x) = x$—at least, I don't see any reason to say anything more nuanced than that about $\mathcal{M}$, given what's discusse...
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11 months ago
Edit Post #280204 Post edited:
Fix TeX formatting
11 months ago
Edit Post #280206 Initial revision 11 months ago
Answer A: Given a triangle with squares on two sides, the line segments joining the centres of the squares to the midpoint of the third side are equal and perpendicular
Your calculations look correct to me, but I think this is perhaps meant to be seen geometrically (apologies if this is already obvious to you and you were looking for a non-geometric explanation). Let the vertices of the triangle be $x$, $y$, $z$, starting at the vertex shared by the two squares, ...
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11 months ago
Suggested Edit Post #280204 Suggested edit:
Fix TeX formatting
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Edit Post #280171 Initial revision 11 months ago
Answer A: How can I deduce which operation removes redundacies?
You have 8 people, and you have to pick 3 of them to get identical gifts. Even though the gifts are identical, let's give them labels: A, B, and C. Let's also number the people 1 to 8. The first step is to figure out how many ways there are to hand out gifts in order. You have 8 choices for gift A...
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11 months ago
Suggested Edit Post #280168 Suggested edit:
Fix TeX formatting
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Edit Post #280120 Initial revision 12 months ago
Answer A: Product of empty set of elements vs. product over empty indexing set — is there any difference?
tl;dr: Having one logical choice for a notation that is consistent with the rest of one's definitions is not the same as defining that notation to mean that choice. Until one has done so, the notation has no meaning (in a strict reading of a math text). In this excerpt, Lang is defining, presumabl...
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12 months ago
Edit Post #280065 Post edited:
12 months ago
Edit Post #280065 Initial revision 12 months ago
Answer A: Why always rationalize a denominator?
Mathematically, absolutely nothing is wrong with fractions that don't have the smallest positive integer denominator that they could have, as far as I know. Pedagogically, the policy does help with there being a unique answer that can be checked quickly, when grading multiple homework assignments ...
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12 months ago
Comment Post #278141 You might be interested in [Troll](http://hjemmesider.diku.dk/~torbenm/Troll/), a language and interpreter for expressing complex dice probabilities. The case you're interested in would be sum largest k nDs in the Troll language (with suitable constants for $k$, $n$, and $s$). Perhaps one of the pa...
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