https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 31 Aug 2026 02:47:03 -0400https://pop.com.sg/what-is-the-difference-between-completeness-and-soundness-in-first-ord.htmlhttps://pop.com.sg/what-is-the-difference-between-completeness-and-soundness-in-first-ord.htmlCompleteness is defined as if given $\Sigma\models\Phi$ then $\Sigma\vdash\Phi$. Meaning if for every truth placement $Z$ in $\Sigma$ we would get $T$, then $\Phi$ also would get $T$. If the previous ...Mon, 31 Aug 2026 06:23:14 -0400https://pop.com.sg/composition-method-of-generating-random-variables.htmlhttps://pop.com.sg/composition-method-of-generating-random-variables.htmlI'm a student learning about methods for generating random variables. But I'm having a hard time finding examples or youtube tutorials on composition method and algorithm of generating random vari......Mon, 31 Aug 2026 06:11:15 -0400https://pop.com.sg/solving-a-hard-algebraic-equation.htmlhttps://pop.com.sg/solving-a-hard-algebraic-equation.htmlThe question is: Rectangular floor mats have an area of $x^2 + 2x - 15 \text{ cm}^2$. The length is $x + 5\text{ cm}$, and the width is $100\text{ cm}$. How do I find out the value of $x$, and......Mon, 31 Aug 2026 05:57:46 -0400https://pop.com.sg/geometric-series-how-to-find-r-and-n-given-the-last-term-the-first-ter.htmlhttps://pop.com.sg/geometric-series-how-to-find-r-and-n-given-the-last-term-the-first-ter.html$S_n=76$, $a=36$, last term is $16$ given that this is a geometric series how to find $r$ and number of terms I have tried to solve simultaneously but failed ....Mon, 31 Aug 2026 05:56:48 -0400https://pop.com.sg/how-to-find-perpendicular-vector-to-another-vector.htmlhttps://pop.com.sg/how-to-find-perpendicular-vector-to-another-vector.htmlHow do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\m......Mon, 31 Aug 2026 05:55:30 -0400https://pop.com.sg/proof-of-the-derivative-of-ln-x.htmlhttps://pop.com.sg/proof-of-the-derivative-of-ln-x.htmlI'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + ......Mon, 31 Aug 2026 05:54:35 -0400https://pop.com.sg/equivalence-between-relational-algebra-statements.htmlhttps://pop.com.sg/equivalence-between-relational-algebra-statements.htmlThis is a practice question for a relational algebra question which I don't understand. Consider two relations $R(A, B)$ and $S(B,C)$. Which of the following relational algebra expressions is not ...Mon, 31 Aug 2026 05:51:55 -0400https://pop.com.sg/shouldn-039-t-we-check-for-conditionally-convergent-in-ratio-test-done.htmlhttps://pop.com.sg/shouldn-039-t-we-check-for-conditionally-convergent-in-ratio-test-done.html(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute val......Mon, 31 Aug 2026 05:49:33 -0400https://pop.com.sg/difference-between-topologically-complete-space-and-complete-metric-sp.htmlhttps://pop.com.sg/difference-between-topologically-complete-space-and-complete-metric-sp.htmlDefinition: Topological space $(X,\tau)$ is called topologically complete if there is metric $d$ on $X$ which induces the topology $\tau$ of $X$ and $(X,d)$ is complete metric space. Also the foll......Mon, 31 Aug 2026 05:47:43 -0400https://pop.com.sg/if-x-and-y-are-rational-then-is-x-y-also-rational.htmlhttps://pop.com.sg/if-x-and-y-are-rational-then-is-x-y-also-rational.htmlI can think of the counter example $x = 2$ and $y = 1/2$ but how would a proof to disprove this look like?...Mon, 31 Aug 2026 05:37:46 -0400https://pop.com.sg/what-do-the-negative-and-positive-signs-next-to-the-limit-mean.htmlhttps://pop.com.sg/what-do-the-negative-and-positive-signs-next-to-the-limit-mean.htmlI’m sorry for this awkward question, but I don’t know how else I should explain it, I tried to do some research online, but couldn’t find anything. For example, the notation looks like: $$\lim_{x \......Mon, 31 Aug 2026 05:31:47 -0400https://pop.com.sg/how-to-calculate-z-score-without-standard-deviation.htmlhttps://pop.com.sg/how-to-calculate-z-score-without-standard-deviation.htmlI have been scratching my head about this for ages. I am studying intro stats for a Psychology Masters, and I can't get my head round this example problem: Example S.6.1 from https://newonlinecour......Mon, 31 Aug 2026 05:23:02 -0400https://pop.com.sg/system-in-a-6x6-matrix-of-ordinary-differential-equations.htmlhttps://pop.com.sg/system-in-a-6x6-matrix-of-ordinary-differential-equations.htmlOne must find general solution for $$y' = \left(\begin{matrix} 1&2&-1&-2&1&2\\ -1&-2&1&2&-1&-2\\ 2&4&-2&-4&2&4\\ -2&-4&2&4&......Mon, 31 Aug 2026 05:19:12 -0400https://pop.com.sg/cosets-and-quotient-groups-what-is-the-difference.htmlhttps://pop.com.sg/cosets-and-quotient-groups-what-is-the-difference.htmlSo I'm very new to group theory and I'm trying to understand the difference between a coset and a quotiënt group. To me it seems that a quotient group is always also a coset but not the other way a......Mon, 31 Aug 2026 04:56:13 -0400https://pop.com.sg/valuation-rings.htmlhttps://pop.com.sg/valuation-rings.htmlWhat's the spectrum of a valuation ring? How to describe morphisms from it to a scheme? Is it enough to set the image of generic point and of a maximal ideal and correspondent map of local rings?...Mon, 31 Aug 2026 04:53:44 -0400https://pop.com.sg/user-alvin-jin-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-alvin-jin-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 31 Aug 2026 04:52:01 -0400https://pop.com.sg/is-g-x-log-x-convex-function.htmlhttps://pop.com.sg/is-g-x-log-x-convex-function.htmlThe graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that $......Mon, 31 Aug 2026 04:38:25 -0400https://pop.com.sg/binomial-theorem-and-complex-numbers.htmlhttps://pop.com.sg/binomial-theorem-and-complex-numbers.htmlUse the Binomial Theorem to calculate $z^4$ where $z$ is the complex number $2 − i$. Simplify your answer using the fact that $i^2 = −1$. Your final answer should be in the form $a + bi$ where $a$ ......Mon, 31 Aug 2026 04:32:53 -0400https://pop.com.sg/derivative-of-ln-1-beta-x-1.htmlhttps://pop.com.sg/derivative-of-ln-1-beta-x-1.htmlHow do I differentiate the term $\ln((1+\beta)^x-1)$ with respect to $x$? Is it possible to do it this way: $$\frac{1}{(1+\beta)^x-1}$$ But i get stuck if i do the normal differentiation....Mon, 31 Aug 2026 04:26:51 -0400https://pop.com.sg/which-symbol-is-used-to-represent-an-isometric-isomorphism-between-two.htmlhttps://pop.com.sg/which-symbol-is-used-to-represent-an-isometric-isomorphism-between-two.htmlI've been looking for a symbol which could be used to represent an Isometric Isomorphism between two normed spaces. Reading many books or PDF's, I noticed that authors use different symbols for it ......Mon, 31 Aug 2026 04:25:48 -0400https://pop.com.sg/fractions-with-negative-numerators-and-denominators.htmlhttps://pop.com.sg/fractions-with-negative-numerators-and-denominators.htmlI've been having this doubt in my mind.. I really don't get the concept behind fractions with negative numerators and denominators being equal to there positive form. For eg.: $\frac{-2}{-3} = \fra......Mon, 31 Aug 2026 04:21:40 -0400https://pop.com.sg/the-definition-and-meaning-of-machine-epsilon-in-matlab.htmlhttps://pop.com.sg/the-definition-and-meaning-of-machine-epsilon-in-matlab.htmlI am taking a introductory course in numerical mathematics, using MATLAB and a numerical math text that refers to MATLAB often. In the text, the machine precision is defined as: The distance $\e......Mon, 31 Aug 2026 04:13:55 -0400https://pop.com.sg/what-is-a-countable-set.htmlhttps://pop.com.sg/what-is-a-countable-set.htmlWhat is a countable set? In Wikipedia we read this definition: In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbe......Mon, 31 Aug 2026 04:08:25 -0400https://pop.com.sg/what-is-the-order-of-the-symmetry-group-of-the-cube.htmlhttps://pop.com.sg/what-is-the-order-of-the-symmetry-group-of-the-cube.htmlI figured out that there are 24 rotational symmetries, shown below. Rotation fixing: faces = 9 diagonals = 8 edges = 6 identity = 1 total = 24 Now, I don't know why the symmetry group of the 3-cub......Mon, 31 Aug 2026 04:07:15 -0400https://pop.com.sg/how-to-calculate-the-number-of-possible-combinations-with-repetitions.htmlhttps://pop.com.sg/how-to-calculate-the-number-of-possible-combinations-with-repetitions.htmlI would like to know the difference between "permutations with repetition" and "ways to choose $k$ elements from a set of $n$ elements if repetitions are allowed". For instance: In a set $S$ of $k$ ...Mon, 31 Aug 2026 03:54:31 -0400https://pop.com.sg/is-zero-to-the-power-of-an-imaginary-number-still-zero.htmlhttps://pop.com.sg/is-zero-to-the-power-of-an-imaginary-number-still-zero.htmlI just want to make sure that $0^i = 0$, but for some reason I couldn't find anything about this online. Is this true? --Background-- I'm trying to prove that some exponent is zero. I thought I'd ...Mon, 31 Aug 2026 03:50:57 -0400https://pop.com.sg/show-that-f-x-3x-2-2x-1-is-continuous-at-2.htmlhttps://pop.com.sg/show-that-f-x-3x-2-2x-1-is-continuous-at-2.htmlPlease check my proof Let$\delta ,\epsilon >0$ $$|x-2|<\delta \rightarrow |f(x)-f(2)<\epsilon $$ $$ \rightarrow |3x^{2}-2x+1-3x^{2}-2x+1|<\epsilon $$ $$ \...Mon, 31 Aug 2026 03:47:45 -0400https://pop.com.sg/solve-exponential-equation-with-exponent-variables.htmlhttps://pop.com.sg/solve-exponential-equation-with-exponent-variables.htmlOk, the question is: Solve the following for x: $2^x4^{x-1}=70$ I have just asked Wolfram Alpha, of course though, it supplies an answer without revealing its working. I started to try the pro......Mon, 31 Aug 2026 03:45:34 -0400https://pop.com.sg/how-to-calculate-norm-of-matrix-using-any-orthogonal-basis.htmlhttps://pop.com.sg/how-to-calculate-norm-of-matrix-using-any-orthogonal-basis.htmlShow that for $X \in \mathrm{M}_n(\mathbb{C})$ and any orthonormal basis $\{u_1, \ldots , u_n\}$ of $\mathbb{C}^n$, we have $$\|X\|^2=\sum_{j,k}^n|\langle u_j,Xu_k\rangle |^2.$$ My Attempt: I t......Mon, 31 Aug 2026 03:42:34 -0400https://pop.com.sg/can-a-sequence-be-called-convergent-divergent-if-it-has-finite-number-.htmlhttps://pop.com.sg/can-a-sequence-be-called-convergent-divergent-if-it-has-finite-number-.htmlNo explanation required for the question I guess....Mon, 31 Aug 2026 03:35:53 -0400https://pop.com.sg/is-there-a-good-way-to-compute-christoffel-symbols.htmlhttps://pop.com.sg/is-there-a-good-way-to-compute-christoffel-symbols.htmlLets say you have a Riemannian Manifold $(M,g)$, and you have some given chart where $g = g_{ij} dx_i dx_j$ and you wish to compute the Christoffel symbols for the Riemannian connection in this cha......Mon, 31 Aug 2026 03:35:28 -0400https://pop.com.sg/where-is-greens-theorem-used.htmlhttps://pop.com.sg/where-is-greens-theorem-used.htmlWhere is Greens theorem used? I think it's weird going from a vector field to calculating a volume on a scalar field, where do we use this kind of calculation?...Mon, 31 Aug 2026 03:31:39 -0400https://pop.com.sg/optimization-of-the-surface-area-of-a-open-rectangular-box-to-find-the.htmlhttps://pop.com.sg/optimization-of-the-surface-area-of-a-open-rectangular-box-to-find-the.htmlA rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...Mon, 31 Aug 2026 03:21:54 -0400https://pop.com.sg/total-distance-travelled-using-integrals.htmlhttps://pop.com.sg/total-distance-travelled-using-integrals.htmlFor time $t \geq 0$, the velocity of a particle moving along the $x$-axis is given by $\mathrm{v}\left(t\right) = \sin\left(t\right)$, where $t$ is measured in seconds and $\mathrm{v}\left(t\right)......Mon, 31 Aug 2026 03:14:46 -0400https://pop.com.sg/is-a-triangle-with-two-equal-angles-always-isosceles.htmlhttps://pop.com.sg/is-a-triangle-with-two-equal-angles-always-isosceles.htmlAn isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles......Mon, 31 Aug 2026 03:13:53 -0400https://pop.com.sg/p-norm-with-p-to-infinity-duplicate.htmlhttps://pop.com.sg/p-norm-with-p-to-infinity-duplicate.htmlI have to show that: for all vectors $v\in \Bbb R^n$: $\lim_{p\to \infty}||v||_p = \max_{1\le i \le n}|v_i|$ with the $||\cdot ||_p$ norm defined as $$ ||\cdot ||_p: (v_1, \dots ,v_n) \to (\sum^n_{......Mon, 31 Aug 2026 03:08:52 -0400https://pop.com.sg/how-to-integrate-sec-3-x-dx-duplicate.htmlhttps://pop.com.sg/how-to-integrate-sec-3-x-dx-duplicate.htmlPossible Duplicate: Indefinite integral of secant cubed How to integrate $\sec^3 x \, dx$? Can someone please give a method, I tried separating $\sec^3 x$ as $\sec x(\sec^2 x)$ then applying by-...Mon, 31 Aug 2026 02:59:22 -0400https://pop.com.sg/change-from-product-to-sum.htmlhttps://pop.com.sg/change-from-product-to-sum.htmlWe know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : $$\prod\limits_......Mon, 31 Aug 2026 02:57:51 -0400https://pop.com.sg/which-of-these-sentences-are-propositions-what-are-the-truth-values-of.htmlhttps://pop.com.sg/which-of-these-sentences-are-propositions-what-are-the-truth-values-of.htmlWhich of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2 + 3 = 5. d) ......Mon, 31 Aug 2026 02:53:07 -0400https://pop.com.sg/utilization-of-m-m-c-queue.htmlhttps://pop.com.sg/utilization-of-m-m-c-queue.htmlWhat is the utilization of m/m/c queue ? in some texts such as http://www.amazon.com/Queueing-Networks-Markov-Chains-Applications/dp/0471565253 said: individual server utilization $\rho =(\lambda)......Mon, 31 Aug 2026 02:49:20 -0400https://pop.com.sg/some-commutator-identities.htmlhttps://pop.com.sg/some-commutator-identities.htmlOn the way to study Lang's algebra, I cannot solve this problem. See page 69. Let G be a group and denote the commutator of x and y by & $[x,y]=xyx^{-1}y^{-1}$. I wanna prove that if $[x,y]=......Mon, 31 Aug 2026 02:45:19 -0400https://pop.com.sg/why-is-it-valid-to-multiply-both-sides-of-an-equation-by-its-complex-c.htmlhttps://pop.com.sg/why-is-it-valid-to-multiply-both-sides-of-an-equation-by-its-complex-c.htmlThis is the closest explanation I can find, although I still don't fully understand. We know what’s happening: division is subtracting an angle and shrinking the magnitude. By multiplying top ......Mon, 31 Aug 2026 02:44:13 -0400https://pop.com.sg/how-often-in-years-do-calendars-repeat-with-the-same-day-date-combinat.htmlhttps://pop.com.sg/how-often-in-years-do-calendars-repeat-with-the-same-day-date-combinat.htmlE.g. I'm using this formulas for calculating day of week (Julian calendar): \begin{align} a & = \left\lfloor\frac{14 - \text{month}}{12}\right\rfloor\\ y & = \text{year} + 4800 - a \\ m &a......Mon, 31 Aug 2026 02:25:18 -0400https://pop.com.sg/clarification-of-the-definition-of-tempered-growth.htmlhttps://pop.com.sg/clarification-of-the-definition-of-tempered-growth.htmlThis is a rather simple question, but I don't have many resources I've been able to consult in this regard (meaning that a Google search hasn't yielded any meaningful results). I just want to clari......Mon, 31 Aug 2026 02:16:58 -0400https://pop.com.sg/what-is-different-between-g-n-and-gn-n.htmlhttps://pop.com.sg/what-is-different-between-g-n-and-gn-n.htmlWhat is different between $G/N$ and $GN/N$? ($G$ denotes a group, and N does a normal subgroup of G) If you look at the element of each group above, I think $G/N$ contains element of form $gN$, ......Mon, 31 Aug 2026 02:10:56 -0400https://pop.com.sg/in-a-math-paper-what-is-a-remark.htmlhttps://pop.com.sg/in-a-math-paper-what-is-a-remark.htmlI sometimes see paragraphs labeled 'Remark.' However, papers that include remarks also include unlabeled explanatory paragraphs (i.e. all the other writing in the article) that seem to be remarks. ......Mon, 31 Aug 2026 02:06:33 -0400https://pop.com.sg/the-definition-of-distance-and-how-to-prove-the-ruler-postulate-in-euc.htmlhttps://pop.com.sg/the-definition-of-distance-and-how-to-prove-the-ruler-postulate-in-euc.htmlI have started to read some books about geometry. At the moment I've just started to read Hilbert's axioms and also some elementary books for highschool. From the basic perspective of an axiomatic ......Mon, 31 Aug 2026 02:04:52 -0400https://pop.com.sg/proof-for-symmetry-property-of-congruent-segments.htmlhttps://pop.com.sg/proof-for-symmetry-property-of-congruent-segments.htmlI am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$). One of the exercises i......Mon, 31 Aug 2026 01:55:40 -0400https://pop.com.sg/what-is-the-difference-between-o-2-n-and-2-o-n.htmlhttps://pop.com.sg/what-is-the-difference-between-o-2-n-and-2-o-n.htmlIn the context of complexity, I have seen both $O(2^{n})$ and $2^{O(n)}$ used. Whats the difference between the two?...Mon, 31 Aug 2026 01:44:39 -0400https://pop.com.sg/what-is-the-norm-of-a-complex-number.htmlhttps://pop.com.sg/what-is-the-norm-of-a-complex-number.htmlMany articles on the web state that the norm of $z = a +bi$ is \begin{align} \sqrt{z*z^*} &= \sqrt{(a + bi)(a - bi)}\\ &= \sqrt{a^2 - abi + abi - b^2i^2}\\ &= \sqrt{a^2 - b^2\sqrt{-1......Mon, 31 Aug 2026 01:41:47 -0400https://pop.com.sg/recurrence-tree-method-t-n-3-2n-3-1-why-is-the-solution-in-log-3-2-and.htmlhttps://pop.com.sg/recurrence-tree-method-t-n-3-2n-3-1-why-is-the-solution-in-log-3-2-and.htmlI feel this is more of a question in the logarithms, but I am confused why this tree reoccurrence $$T(n) = 3T(2n/3)+1$$ solution is derived from $\log_{3/2}$ rather than $\log_{2/3}$? My thinking......Mon, 31 Aug 2026 01:30:50 -0400https://pop.com.sg/how-does-an-exponent-work-when-it-039-s-less-than-one.htmlhttps://pop.com.sg/how-does-an-exponent-work-when-it-039-s-less-than-one.htmlI'm rather familiar with exponents, I know that $y^x = y_1 \cdot y_2 \cdot y_3 .... y_x$, but what if the exponent is less than one, how would that work? I put in my computer $25^{1/2}$ anyway, ...Mon, 31 Aug 2026 01:30:08 -0400https://pop.com.sg/sweepstakes-probability.htmlhttps://pop.com.sg/sweepstakes-probability.htmlIs there a way to determine the probability of winning a particular sweepstakes with only the following information: the estimated odds of winning are $1$ in $13,000$ contestants must be owners (...Mon, 31 Aug 2026 01:16:56 -0400https://pop.com.sg/can-a-circle-have-a-negative-radius.htmlhttps://pop.com.sg/can-a-circle-have-a-negative-radius.htmlI am working on a problem that asks if a given sphere intersects the zx-plane. The equation of the sphere is $(x-2)^2+(y+6)^2+(z-4)^2=5^2$ Can someone please explain to me how the sphere does not ...Mon, 31 Aug 2026 01:12:57 -0400https://pop.com.sg/infinity-divided-by-infinity-and-dirac-delta.htmlhttps://pop.com.sg/infinity-divided-by-infinity-and-dirac-delta.htmlA dirac delta produces something that's infinitely long, and it could also be seen as infinitely thin. Why do we define the surface of a dirac delta to be 1. If length$\times$width $= \infty \cdot......Mon, 31 Aug 2026 01:07:22 -0400https://pop.com.sg/necessary-condition-for-the-mean-value-theorem.htmlhttps://pop.com.sg/necessary-condition-for-the-mean-value-theorem.htmlGive an example which demonstrates that continuity is a necessary condition for the mean value theorem: I thought in this function: $$g(x) = \begin{cases} x + 1 & x < 1 \\[4pt] x - 1 &......Mon, 31 Aug 2026 00:54:49 -0400https://pop.com.sg/best-math-books-or-apps-for-adults-to-learn-math-from-the-beginning.htmlhttps://pop.com.sg/best-math-books-or-apps-for-adults-to-learn-math-from-the-beginning.htmlI lost a possible job because I didn't know how to multiply and subtract negative valued integers. I also don't know how fraction manipulation works. What reference books can I read that can help f......Mon, 31 Aug 2026 00:43:48 -0400https://pop.com.sg/how-to-prove-that-frac-d-dx-sin-x-cos-x.htmlhttps://pop.com.sg/how-to-prove-that-frac-d-dx-sin-x-cos-x.htmlI have to prove that $\dfrac{d}{dx}\sin(x)=\cos(x)$. I used the definition of a derivative: $$\dfrac{d}{dx}f(x)=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$$ $$\dfrac{d}{dx}\sin(x)=\lim\limits_{h\t......Mon, 31 Aug 2026 00:41:05 -0400https://pop.com.sg/why-does-the-base-b-not-matter-in-the-change-of-base-logarithm-equatio.htmlhttps://pop.com.sg/why-does-the-base-b-not-matter-in-the-change-of-base-logarithm-equatio.htmlI've recently started learning about logarithms and was just curious as to why the base "b" in the "change the base" formula doesn't matter, as in it doesn't matter what value y......Mon, 31 Aug 2026 00:39:26 -0400https://pop.com.sg/solving-2-sin-theta-cos-theta-sin-theta-0.htmlhttps://pop.com.sg/solving-2-sin-theta-cos-theta-sin-theta-0.htmlThe question is to solve the following question in the range $-\pi \le \theta \le \pi$ $$2\sin\theta\cos\theta + \sin\theta = 0$$ I missed the obvious sin factorisation so proceeded as below. I s......Mon, 31 Aug 2026 00:37:27 -0400https://pop.com.sg/how-to-calculate-q-expansion-coefficients-of-a-modular-form.htmlhttps://pop.com.sg/how-to-calculate-q-expansion-coefficients-of-a-modular-form.htmlCan you please explain for me how we can calculate the coefficients of $q$-expansion series of a modular form function $f$? I am really confused....Mon, 31 Aug 2026 00:35:24 -0400https://pop.com.sg/find-number-of-solutions-of-2x-2-5x-3-x-1-0.htmlhttps://pop.com.sg/find-number-of-solutions-of-2x-2-5x-3-x-1-0.htmlProblem: Find number of solutions of $|2x^2-5x+3|+x-1=0$ Solution: Case 1: When $2x^2-5x+3 \geq 0$ Then we get, $2x^2-5x+3+x-1=0$ x=1,1 Case 2: When $2x^2-5x+3 < 0$ Then we get, $-2x^2+5x-3+x-1=......Mon, 31 Aug 2026 00:33:10 -0400https://pop.com.sg/difference-between-mathbb-r-linear-and-mathbb-c-linear-maps.htmlhttps://pop.com.sg/difference-between-mathbb-r-linear-and-mathbb-c-linear-maps.htmlSuppose we have a $f\colon \mathbb{C} \to \mathbb{C} $. I am confused when people say $f$ is $\mathbb{R}$-linear map or $\mathbb{C}$-linear map. What is the difference between these two ?...Mon, 31 Aug 2026 00:21:18 -0400https://pop.com.sg/what-039-s-the-difference-between-a-logic-an-internal-logic-language-o.htmlhttps://pop.com.sg/what-039-s-the-difference-between-a-logic-an-internal-logic-language-o.htmlmaybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all......Mon, 31 Aug 2026 00:15:30 -0400https://pop.com.sg/euler-angles-and-gimbal-lock.htmlhttps://pop.com.sg/euler-angles-and-gimbal-lock.htmlCan someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...Mon, 31 Aug 2026 00:12:46 -0400https://pop.com.sg/when-is-the-intersection-of-a-cone-and-a-cylinder-is-planar.htmlhttps://pop.com.sg/when-is-the-intersection-of-a-cone-and-a-cylinder-is-planar.htmlI'm trying to understand better the link between the algebraic and geometric interpretation of the ellipse, and I've wondered about the intersection of a right circular cylinder and a right circula......Mon, 31 Aug 2026 00:04:31 -0400https://pop.com.sg/trace-of-the-cissoid-of-diocles.htmlhttps://pop.com.sg/trace-of-the-cissoid-of-diocles.htmlI want to show that you can parameterized the cissoid of Diocles using $$ \alpha(t) = \left(\frac{2at^2}{1+t^2}, \frac{2at^3}{1+t^2}\right),\quad\text{where }t=\tan\theta. $$ Here is an image of the ...Mon, 31 Aug 2026 00:04:07 -0400https://pop.com.sg/algorithms-for-symbolic-manipulation.htmlhttps://pop.com.sg/algorithms-for-symbolic-manipulation.htmlIf you take a look at WolframAlpha, or other computer algebraic system, you will find that it is able to do symbolic manipulation like real humans. For example, if you type in an integral, it can ......Sun, 30 Aug 2026 23:59:46 -0400https://pop.com.sg/what-does-the-integral-of-position-with-respect-to-time-mean.htmlhttps://pop.com.sg/what-does-the-integral-of-position-with-respect-to-time-mean.htmlThe integral of acceleration with respect to time is velocity. The integral of velocity with respect to time is position. What is the integral of position with respect to time, and what does it me......Sun, 30 Aug 2026 23:58:45 -0400https://pop.com.sg/equivalence-of-repeating-decimal.htmlhttps://pop.com.sg/equivalence-of-repeating-decimal.htmlGRE exam question asks what is greater $.\overline{717}$ or $.\overline{71}$ I believe both are equal, but GRE says that $.\overline{717}$ is greater. But why? If they repeat for infinity, is......Sun, 30 Aug 2026 23:47:52 -0400https://pop.com.sg/existance-of-multiple-of-n-with-only-0-and-1-as-it-039-s-digits-duplic.htmlhttps://pop.com.sg/existance-of-multiple-of-n-with-only-0-and-1-as-it-039-s-digits-duplic.htmlPossible Duplicate: Proof that a natural number multiplied by some integer results in a number with only one and zero as digits I read this somewhere recently: For any natural number $n$, there ex......Sun, 30 Aug 2026 23:46:21 -0400https://pop.com.sg/find-the-tangential-and-normal-components-of-the-acceleration-vector-f.htmlhttps://pop.com.sg/find-the-tangential-and-normal-components-of-the-acceleration-vector-f.htmlFind the tangential and normal components of the acceleration vector for the curve $$ \vec r (t) = 〈 2t,t^2 ,\ln t 〉$$ I found this problem in a notebook and I did a part, but I was stuck and I ......Sun, 30 Aug 2026 23:45:17 -0400https://pop.com.sg/find-x-intercept-of-a-logarithmic-function.htmlhttps://pop.com.sg/find-x-intercept-of-a-logarithmic-function.htmlFind the x-intercept of: $3-4\log _{10}\left(2x\right) $ My process: $4\log _{10}\left(2x\right)=3 $ $\log _{10}\left(2x\right)\:=\:\frac{3}{4}$ How would I find the x-intercept from here? $......Sun, 30 Aug 2026 23:38:00 -0400https://pop.com.sg/largest-subset-with-no-pair-summing-to-power-of-two.htmlhttps://pop.com.sg/largest-subset-with-no-pair-summing-to-power-of-two.htmlFor positive integer $n$, define the set $A_n=\{0,1,\ldots,n\}$. What is the size of the largest subset of $A_n$ such that the sum of any two (not necessarily distinct) elements in it is not a powe......Sun, 30 Aug 2026 23:35:33 -0400https://pop.com.sg/understanding-the-heine-cantor-theorem.htmlhttps://pop.com.sg/understanding-the-heine-cantor-theorem.htmlThe "Heine–Cantor theorem" states: If $f : M → N$ is a continuous function between two metric spaces, and $M$ is compact, then $f$ is uniformly continuous. I do not doubt its validity, of course, ......Sun, 30 Aug 2026 23:33:15 -0400https://pop.com.sg/boolean-algebra-finding-the-maxterm-expansion-of-f-a-b-c-d-a-b-d-a-039.htmlhttps://pop.com.sg/boolean-algebra-finding-the-maxterm-expansion-of-f-a-b-c-d-a-b-d-a-039.htmlI wanted to show the maxterm expansion of the following Boolean Function below $f(A,B,C,D) = (A + B + D)(A' + C)(C + D)$ What I know: From what I know, the domain is f(A, B, C, and D). No terms ar......Sun, 30 Aug 2026 23:30:07 -0400https://pop.com.sg/best-program-for-multiplying-many-multivariable-polynomials.htmlhttps://pop.com.sg/best-program-for-multiplying-many-multivariable-polynomials.htmlI want to make a table with two columns: The first column will consist of many(possibly hundreds) of polynomials in two variables. The second column will be a function applied to all of the polyn......Sun, 30 Aug 2026 23:10:47 -0400https://pop.com.sg/how-to-find-inverse-fourier-transform.htmlhttps://pop.com.sg/how-to-find-inverse-fourier-transform.htmlI have the function $$ \delta(f-2) $$ How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$. The correct answer is $$ e^{4\pi ......Sun, 30 Aug 2026 23:09:30 -0400https://pop.com.sg/derivative-on-endpoints.htmlhttps://pop.com.sg/derivative-on-endpoints.htmlwhat's the derivative of $f(x)= x^{2}$ ($x\geq 0$) when x=0? from my understand, it doesn't exist because even $\lim_{h \to 0^{+}}\frac{f(x+h)-f(x)}{h}$ is 0, but $\lim_{h \to 0^{-}}\frac{f(x+h)-f......Sun, 30 Aug 2026 23:06:45 -0400https://pop.com.sg/user-raymond-hettinger-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-raymond-hettinger-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 30 Aug 2026 23:04:43 -0400https://pop.com.sg/comparison-test-of-sum1-ln-n-n.htmlhttps://pop.com.sg/comparison-test-of-sum1-ln-n-n.htmlHere's what I came up with: For this problem, I'm required to use a comparison test to determine if $\Sigma1/ln(n)^n$ converges or diverges. By intuition, I am thinking that $\Sigma1/ln(n)^n$ conv......Sun, 30 Aug 2026 22:56:43 -0400https://pop.com.sg/is-x-not-actually-equal-to-e-ln-x.htmlhttps://pop.com.sg/is-x-not-actually-equal-to-e-ln-x.htmlYeah, this is a silly question, but I can't seem to convince myself that the graph $f(x)=x$ is really equal to the graph of $g(x)=e^{\ln(x)}$. Specifically, doesn't this fail on negative values of ......Sun, 30 Aug 2026 22:56:07 -0400https://pop.com.sg/why-is-frac-987654321-123456789-8-0000000729.htmlhttps://pop.com.sg/why-is-frac-987654321-123456789-8-0000000729.htmlMany years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill si......Sun, 30 Aug 2026 22:52:21 -0400https://pop.com.sg/units-in-exponent-e-g-solve-2-3-years.htmlhttps://pop.com.sg/units-in-exponent-e-g-solve-2-3-years.htmlWhat happens to units in an exponent? My math textbook just introduced the exponential equation: $$A_t = Pe^{rt}$$ I've always made it a point in solving math problems to include the units in ev......Sun, 30 Aug 2026 22:51:13 -0400https://pop.com.sg/formula-for-a-stadium-shape-2d-capsule.htmlhttps://pop.com.sg/formula-for-a-stadium-shape-2d-capsule.htmlThis shape (Known as a stadium, it's basically a 2D capsule.) I need a formula to draw this shape on a graph, preferably in that orientation (on its side). What I've tried: I've tried adapting......Sun, 30 Aug 2026 22:42:45 -0400https://pop.com.sg/checking-if-matrix-columns-are-linearly-independent.htmlhttps://pop.com.sg/checking-if-matrix-columns-are-linearly-independent.htmlaccording to the definition of linear dependency vectors $v_1,...,v_n$ are linearly independent $iff$ $c_1v_1$+$c_2v_2$+$...$+$c_nv_n≠0$. One can also do the gaussian elemination to get which colu......Sun, 30 Aug 2026 22:37:17 -0400https://pop.com.sg/area-in-an-equilateral-triangle-that-is-closer-to-the-centroid-than-to.htmlhttps://pop.com.sg/area-in-an-equilateral-triangle-that-is-closer-to-the-centroid-than-to.htmlWhat percent of the area of an equilateral triangle is closer to the centroid of the triangle than to any edge of the triangle?...Sun, 30 Aug 2026 22:36:54 -0400https://pop.com.sg/why-is-sqrt-12-2-sqrt-3.htmlhttps://pop.com.sg/why-is-sqrt-12-2-sqrt-3.htmlWhy $\sqrt {12} = 2 \sqrt 3$? It is obvious? If we considered the function $f(s) = s^2 $ it is injective on positive numbers so we obtain the conclusion. But in the same time it is an equality bet......Sun, 30 Aug 2026 22:32:59 -0400https://pop.com.sg/solution-of-a-given-legendre-equation.htmlhttps://pop.com.sg/solution-of-a-given-legendre-equation.htmlI was trying to solve following question : $$ (1-x^2)y''-2xy'+n(n+1)y=0 $$ has the $n^{th}$ degree polynomial solution $y_n(x)$ such that $y_n(1) =3$. We are given $\int_{-1}^{1} [y_n^2(x) + y_{......Sun, 30 Aug 2026 22:29:52 -0400https://pop.com.sg/how-to-be-good-at-proving-duplicate.htmlhttps://pop.com.sg/how-to-be-good-at-proving-duplicate.htmlI'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know ho......Sun, 30 Aug 2026 22:27:22 -0400https://pop.com.sg/partial-derivative-with-constant.htmlhttps://pop.com.sg/partial-derivative-with-constant.htmlGiven $$c = 0.03 + 0.08a$$ What is $\dfrac{\partial c}{\partial a}$? I'm guessing $\dfrac{\partial c}{\partial a} = 0.03 + 0.08 = 0.11$? The constant is confusing me. Thanks....Sun, 30 Aug 2026 22:15:31 -0400https://pop.com.sg/why-is-the-definition-of-the-absolute-value-x-1-the-way-it-is.htmlhttps://pop.com.sg/why-is-the-definition-of-the-absolute-value-x-1-the-way-it-is.htmlIn my notebook it is given that for the above function, we would have: $f(x) = {-(x+1), x<-1; (x+1), x\geq-1}$ What I don't get is why did we take $-1$ instead of $0$ as is the case for the fu......Sun, 30 Aug 2026 21:42:19 -0400https://pop.com.sg/equivalence-relation-showing-that-an-operation-is-well-defined.htmlhttps://pop.com.sg/equivalence-relation-showing-that-an-operation-is-well-defined.htmlLet $r \in \mathbb{N}$. Observe the equivalence relation $\sim \subseteq \mathbb{Z} \times \mathbb{Z}$ defined as $x \sim y :\Leftrightarrow (\exists k \in \mathbb{Z})(y=x+kr)$. Show that an operat......Sun, 30 Aug 2026 21:41:59 -0400https://pop.com.sg/the-derivative-of-the-electric-field-for-a-uniformly-charged-rod.htmlhttps://pop.com.sg/the-derivative-of-the-electric-field-for-a-uniformly-charged-rod.htmlThe formula for the electric field at a point due to a charge $Q$ (just considering the magnitude) at some distance $x$ away from the point is $E=\dfrac{k_eQ}{x^2}$ where $k_e$ is a constant equal to ...Sun, 30 Aug 2026 21:28:12 -0400https://pop.com.sg/showing-ha-h-if-and-only-if-a-belongs-to-h.htmlhttps://pop.com.sg/showing-ha-h-if-and-only-if-a-belongs-to-h.htmlLet be $H$ a subgroup. Show $Ha=H$ if and only if $a$ belongs to $H$. Here what I have understand of the proof: $a$ must belong to $Ha$ because $ea=a$ where $e$ is the identity in $H$. So the onl......Sun, 30 Aug 2026 21:24:55 -0400https://pop.com.sg/summing-the-digits-of-pi.htmlhttps://pop.com.sg/summing-the-digits-of-pi.htmlI find $\pi$ and other transcendental numbers quite fascinating. Lately I've been playing around with sequences and $\pi$ and found something relatively intriguing. Consider $\pi$ written in base......Sun, 30 Aug 2026 21:17:29 -0400https://pop.com.sg/what-is-2-32-1-in-decimal-notation.htmlhttps://pop.com.sg/what-is-2-32-1-in-decimal-notation.htmlI came across decimal notation, and surprisingly I have never heard of it. I have heard of scientific notation and just thought this was decimal notation. However, when I want to find a definition ......Sun, 30 Aug 2026 21:16:00 -0400https://pop.com.sg/definition-of-a-sheaf-making-sense-of.htmlhttps://pop.com.sg/definition-of-a-sheaf-making-sense-of.htmlI have started to read an english translation of Serre's FAC. Immediately a sheaf is defined. The more categorical definition given on wikipedia actually makes more sense to me, but I would like to ...Sun, 30 Aug 2026 21:15:52 -0400https://pop.com.sg/fourier-transform-of-a-radial-function.htmlhttps://pop.com.sg/fourier-transform-of-a-radial-function.htmlConsider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or......Sun, 30 Aug 2026 21:14:49 -0400https://pop.com.sg/proving-that-the-dot-product-is-distributive.htmlhttps://pop.com.sg/proving-that-the-dot-product-is-distributive.htmlI know that one can prove that the dot product, as defined "algebraically", is distributive. However, to show the algebraic formula for the dot product, one needs to use the distributive property i......Sun, 30 Aug 2026 21:03:51 -0400