https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 01 Sep 2026 17:56:18 -0400https://pop.com.sg/evaluate-int-cos-cos-x-dx.htmlhttps://pop.com.sg/evaluate-int-cos-cos-x-dx.htmlEvaluate $\int \cos(\cos x)~dx$ I tried to use chain rule but failed. Can anyone help me please?...Tue, 01 Sep 2026 21:51:11 -0400https://pop.com.sg/how-would-you-find-and-classify-all-of-the-critical-points-of-a-functi.htmlhttps://pop.com.sg/how-would-you-find-and-classify-all-of-the-critical-points-of-a-functi.htmlSuppose you had the function $f(x,y)=x^{3}-y^{3}+9xy$. The critical points of this function would be the values of $(x,y)$, such that $f_{x}=f_{y}=0$. We have $f_{x}=3x^{2}+9y$ and $f_{y}=-3y^{2}+......Tue, 01 Sep 2026 21:37:12 -0400https://pop.com.sg/what-is-the-equation-of-a-straight-line.htmlhttps://pop.com.sg/what-is-the-equation-of-a-straight-line.htmlI'm reading a book named "Geometry, A comprehensive course" by Dan Pedoe at the moment. But I don't understand when they talk about the equation of a straight line... Here is what is written: The ...Tue, 01 Sep 2026 21:25:51 -0400https://pop.com.sg/use-pohlig-hellman-to-solve-discrete-log.htmlhttps://pop.com.sg/use-pohlig-hellman-to-solve-discrete-log.htmlWe have $$7^x = 166 \pmod{433}$$ I need to find $x$ using the Pohlig-Hellman algorithm....Tue, 01 Sep 2026 21:20:05 -0400https://pop.com.sg/solving-given-initial-value-problem-using-the-method-of-laplace-transf.htmlhttps://pop.com.sg/solving-given-initial-value-problem-using-the-method-of-laplace-transf.htmlI have the problem asking me to solve the initial value problem using the method of Laplace Transforms given $$ y''+y=f(t); \ \ \ \ \ y(0) = 0, y'(0)=1 $$ where $$ f(t)=\begin{cases} 0,& 0<......Tue, 01 Sep 2026 21:16:31 -0400https://pop.com.sg/slant-asymptote-and-limit-problem.htmlhttps://pop.com.sg/slant-asymptote-and-limit-problem.htmlThe graph $y = f(x)$ has a slant asymptote along the line $y = mx + b$ (with $m \not= 0$) if $$ \displaystyle\lim_{x \rightarrow \infty} |f(x) - (mx+b)| = 0. $$ Describe algebraically the condition......Tue, 01 Sep 2026 21:13:45 -0400https://pop.com.sg/how-many-positive-integers-n-satisfy-n-p-n-s-n.htmlhttps://pop.com.sg/how-many-positive-integers-n-satisfy-n-p-n-s-n.htmlLet $P(n)$ denote the product of digits of $n$ and let $S(n)$ denote the sum of digits of $n$. Then how many positive integers $n$ satisfy $$ n = P(n) + S(n) $$ I think I solved it, but I need......Tue, 01 Sep 2026 21:12:35 -0400https://pop.com.sg/is-x-5-2x-4-irreducible-in-mathbb-q-x.htmlhttps://pop.com.sg/is-x-5-2x-4-irreducible-in-mathbb-q-x.htmlI was asked this question on my exam but I didn't know how to solve it. Eisenstein criterion can't be applied, rational roots theorem shows that it doesn't have a linear factor because possible roo......Tue, 01 Sep 2026 21:11:05 -0400https://pop.com.sg/how-can-we-prove-that-the-field-of-complex-number-is-not-isomorphic-to.htmlhttps://pop.com.sg/how-can-we-prove-that-the-field-of-complex-number-is-not-isomorphic-to.htmlI want to prove this using contradiction, supposing that there is a ring isomorphism between the two and then finding a contradiction....Tue, 01 Sep 2026 21:07:39 -0400https://pop.com.sg/laplace-transform-of-cos-at-t.htmlhttps://pop.com.sg/laplace-transform-of-cos-at-t.htmlIf someone could help me solve for $$\mathcal{L}\left\{\frac{\cos(at)}{t}\right\}$$ it would be great. Step-by-step I have so far: $$\begin{align}\int_0^\infty \frac{\cos(at)\space e^{-st}}{t}\sp......Tue, 01 Sep 2026 20:53:48 -0400https://pop.com.sg/questions-tagged-improper-integrals.htmlhttps://pop.com.sg/questions-tagged-improper-integrals.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 20:52:49 -0400https://pop.com.sg/is-the-figure-the-circumference-of-a-unit-circle.htmlhttps://pop.com.sg/is-the-figure-the-circumference-of-a-unit-circle.htmlA friend of mine taught me the following question. I've never heard such a strange and interesting question! Qustion: Supposing that a figure $S$, which is constituted by points, satisfies the ...Tue, 01 Sep 2026 20:50:29 -0400https://pop.com.sg/inconsistencies-in-the-definition-of-derivative-of-a-polynomial-over-a.htmlhttps://pop.com.sg/inconsistencies-in-the-definition-of-derivative-of-a-polynomial-over-a.htmlA problem I came across defines a particular differentiation operator $D$ over the set of polynomials $\{P\}$ over a field $F$ with "the normal formula; that is $D(\sum_{i=0}^n a_nx^i) = \sum_{i=1}^n ...Tue, 01 Sep 2026 20:49:09 -0400https://pop.com.sg/how-do-i-calculate-a-weighted-average-from-two-averages.htmlhttps://pop.com.sg/how-do-i-calculate-a-weighted-average-from-two-averages.htmlI have two sets of averaged data I am working with, which account for a score and the average amount of users that achieved this. For example: Average Score $4$, Total Number of participants (wh......Tue, 01 Sep 2026 20:47:15 -0400https://pop.com.sg/what-does-asymptotically-proportional-mean.htmlhttps://pop.com.sg/what-does-asymptotically-proportional-mean.htmlIn an article about Metcalfe's law, I've read that $n(n-1)/2$ is asymptotically proportional to $n^2$. What does this mean? PS: I did find asymptotically optimal but I'm not sure if they mean the......Tue, 01 Sep 2026 20:39:34 -0400https://pop.com.sg/infinite-product-of-1-1-2n-terms.htmlhttps://pop.com.sg/infinite-product-of-1-1-2n-terms.htmlWhat is the value of the infinite product $$ \frac{1}{2} \frac{3}{4} \frac{5}{6}\cdots = \prod_{n=1}^{\infty} \frac{2n-1}{2n} = \prod_{n=1}^{\infty} \left(1-\frac{1}{2n}\right) = \lim_{n\to\infty}......Tue, 01 Sep 2026 20:38:15 -0400https://pop.com.sg/vector-field-using-green-039-s-theorem-other-integration.htmlhttps://pop.com.sg/vector-field-using-green-039-s-theorem-other-integration.htmlSo am I supposed to be using green's theorem for the first question, but where I'm confused is that there are three variables if I do, dx dy dz (I haven't learn how to use green's theorem for 3 ...Tue, 01 Sep 2026 20:37:10 -0400https://pop.com.sg/proving-that-an-irrational-number-exists-near-every-rational-number-du.htmlhttps://pop.com.sg/proving-that-an-irrational-number-exists-near-every-rational-number-du.htmlShow that arbitrarly close to any rational number there is a real (non-rational) number. In other words, show that to each real $\varepsilon>0$ and each rational $r\in\mathbb Q$ there exists $x\in\...Tue, 01 Sep 2026 20:30:24 -0400https://pop.com.sg/locally-vs-globally-lipschitz-confusion.htmlhttps://pop.com.sg/locally-vs-globally-lipschitz-confusion.htmlIs there any difference in a function being locally Lipschitz on $\mathbb{R^n}$ and being globally Lipschitz?...Tue, 01 Sep 2026 20:30:19 -0400https://pop.com.sg/how-do-the-floor-and-ceiling-functions-work-on-negative-numbers.htmlhttps://pop.com.sg/how-do-the-floor-and-ceiling-functions-work-on-negative-numbers.htmlIt's clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take $\,-0.8\,$ to $\,-1,\,$ then do you ta......Tue, 01 Sep 2026 20:24:02 -0400https://pop.com.sg/technique-to-draw-relative-velocities-vectors-and-diagrams.htmlhttps://pop.com.sg/technique-to-draw-relative-velocities-vectors-and-diagrams.htmlHow do you start drawing a diagram in a problem with relative velocities? I always see my coursemates and my lecturer drawing them in a second, as soon as they read the text, I asked them how they ......Tue, 01 Sep 2026 20:23:04 -0400https://pop.com.sg/why-does-tan-1-sqrt-x-2-1-sec-1-x.htmlhttps://pop.com.sg/why-does-tan-1-sqrt-x-2-1-sec-1-x.htmlWhy does $\tan^{-1}(\sqrt{x^2-1})=\sec^{-1}(x)$ How do you derive $\tan^{-1}(\sqrt{x^2-1})$ from $\sec^{-1}(x)$ ? (By $^{-1}$ I mean the inverse function)...Tue, 01 Sep 2026 20:21:26 -0400https://pop.com.sg/is-being-a-right-triangle-both-necessary-and-sufficient-for-the-pythag.htmlhttps://pop.com.sg/is-being-a-right-triangle-both-necessary-and-sufficient-for-the-pythag.htmlI recently encountered a Stack Overflow question (since closed) in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. ...Tue, 01 Sep 2026 20:20:22 -0400https://pop.com.sg/area-between-two-curves-with-absolute-value.htmlhttps://pop.com.sg/area-between-two-curves-with-absolute-value.htmlI have the two equations: y = 1 - 2x^2 y = abs(x) I solved the two equations using the absolute value for: abs(x) and -abs(x) Using these positive and negative absolute values I solved for the ...Tue, 01 Sep 2026 20:16:24 -0400https://pop.com.sg/idempotent-elements-of-z-4-times-z-8.htmlhttps://pop.com.sg/idempotent-elements-of-z-4-times-z-8.htmlI am trying to find idempotent elements of the direct product $Z_4 \times Z_8$. By the definition of direct product we have: $$Z_4 \times Z_8=\left\{(0,0),(0,1),.....(2,7),(3,7)\right\}$$ Now by the ...Tue, 01 Sep 2026 20:12:29 -0400https://pop.com.sg/why-is-decidable-included-in-turing-recognizable.htmlhttps://pop.com.sg/why-is-decidable-included-in-turing-recognizable.html(pic from "introduction to the theory of computation" - Michael Sipser) According to my understanding: Turing-recognizable languages are languages whice are accepted by a Turing machine; ...Tue, 01 Sep 2026 20:11:48 -0400https://pop.com.sg/what-is-ext-short-for-in-ext-functor-duplicate.htmlhttps://pop.com.sg/what-is-ext-short-for-in-ext-functor-duplicate.htmlStrangely, I've never heard Ext functors referred to by any other name, and so I'm not sure what "Ext" actually means. The only thing I can think of that "Ext" might stand for is "Exterior", which ......Tue, 01 Sep 2026 20:10:08 -0400https://pop.com.sg/find-all-possible-values-which-following-expression-can-take-sqrt-x-2-.htmlhttps://pop.com.sg/find-all-possible-values-which-following-expression-can-take-sqrt-x-2-.htmlGiven expression is: $\sqrt{x^2-7x+6}$. Now i first find values of x for which this is a valid question by putting guy inside square root equals to greater than 0. I get $x \in[-\infty,1]\cup[6,\......Tue, 01 Sep 2026 20:05:49 -0400https://pop.com.sg/fair-and-unfair-coin-probability.htmlhttps://pop.com.sg/fair-and-unfair-coin-probability.htmlI am stuck on this question. A coin with $P(H) = \frac{1}{2}$ is flipped $4$ times and then a coin with $P(H) = \frac{2}{3}$ is tossed twice. What is the probability that a total of $5$ heads o......Tue, 01 Sep 2026 19:54:49 -0400https://pop.com.sg/why-fourier-transform-uses-the-notation-x-jw-and-x-e-j-omega.htmlhttps://pop.com.sg/why-fourier-transform-uses-the-notation-x-jw-and-x-e-j-omega.htmlUsually books (e.g. Oppenheim) use the notation X(jw) for the continuous time Fourier Transform and X(e^j$\Omega$) for the discrete time Fourier transform. AFAIK the only parameter is w o $\Omega$, ...Tue, 01 Sep 2026 19:50:30 -0400https://pop.com.sg/normalizing-an-eigenvector-to-have-unit-length.htmlhttps://pop.com.sg/normalizing-an-eigenvector-to-have-unit-length.htmlI worked out a problem where one of the eigenvectors is $(1+ \sqrt{2}, 1)$. How do I normalize this vector? Should I use approximate values?...Tue, 01 Sep 2026 19:42:48 -0400https://pop.com.sg/minimum-possible-number-of-friendships.htmlhttps://pop.com.sg/minimum-possible-number-of-friendships.htmlHere is the problem: There are 2000 people on a social network. Each person sends 1000 friend requests. Two people are friends if they've sent a friend request to each other. What is the minimum ...Tue, 01 Sep 2026 19:40:12 -0400https://pop.com.sg/when-will-the-ball-hit-the-ground.htmlhttps://pop.com.sg/when-will-the-ball-hit-the-ground.htmlA girl is throwing a ball. If I have an equation $h= -4.9*t^2+20t+1.2$, where $t$ is number of second after the girl has thrown the ball, and $h$ is the height above ground. I want to find out w......Tue, 01 Sep 2026 19:30:00 -0400https://pop.com.sg/volume-and-lateral-surface-area-are-equal.htmlhttps://pop.com.sg/volume-and-lateral-surface-area-are-equal.htmlI need to express the radius $r$ of the right circular cone as a function of its height $h$ given that its volume equals to its lateral surface area. I know the two equations $\pi r \times \sqrt{......Tue, 01 Sep 2026 19:27:55 -0400https://pop.com.sg/permutations-vs-combinatorial-vs-factorials-vs-exponents.htmlhttps://pop.com.sg/permutations-vs-combinatorial-vs-factorials-vs-exponents.htmlI'm currently working on a probability course, and I am constantly having trouble figuring out when to use permutations vs. combinations vs. factorials vs. exponents in order to calculate sample si......Tue, 01 Sep 2026 19:27:54 -0400https://pop.com.sg/why-can-a-5x4-matrix-never-be-surjective.htmlhttps://pop.com.sg/why-can-a-5x4-matrix-never-be-surjective.htmlMy thought is that since a matrix can only be surjective if null(T)= n-m, no matrix can have a nullity of -1 therefore a 5x4 matrix cannot be surjective. Is this a valid proof?...Tue, 01 Sep 2026 19:27:49 -0400https://pop.com.sg/rgb-to-hsv-color-conversion-algorithm.htmlhttps://pop.com.sg/rgb-to-hsv-color-conversion-algorithm.htmlI'm a programmer looking to build an RGB to HSV color converter. I found an algorithm, but I have very little mathematical background and I'm not quite sure what's going on with it. A step-by-step ...Tue, 01 Sep 2026 19:26:59 -0400https://pop.com.sg/gaussian-elimination-with-4-variables.htmlhttps://pop.com.sg/gaussian-elimination-with-4-variables.htmlAfter this step however, I end up with 2 simultanious equations but still with four unknowns, I'm not sure if my gaussian elimination is wrong or not but how will i find the exact values of x,y,z a......Tue, 01 Sep 2026 19:21:48 -0400https://pop.com.sg/why-do-we-say-dn-dt-kn.htmlhttps://pop.com.sg/why-do-we-say-dn-dt-kn.htmlSo, I just started doing differential equations and I'm trying to understand the basic idea of dN/dt = kN. I understand the whole process of solving a differential equation (the separation of varia......Tue, 01 Sep 2026 19:19:43 -0400https://pop.com.sg/when-is-a-graph-balanced-bipartite.htmlhttps://pop.com.sg/when-is-a-graph-balanced-bipartite.htmlI have a quick question: is there any sufficient condition (theorem, lemma, proposition,...) to show that a graph (vertices do not have the same degree) is balanced bipartite?...Tue, 01 Sep 2026 19:17:02 -0400https://pop.com.sg/confused-about-matrix-tracing.htmlhttps://pop.com.sg/confused-about-matrix-tracing.htmlI need help with this one question I found from my Linear Algebra textbook. While I was doing problems and checking my work from the back of my textbook, I found out one of the problems I did is wr......Tue, 01 Sep 2026 19:06:58 -0400https://pop.com.sg/how-to-find-the-inradius-of-a-triangle-with-given-side-lengths.htmlhttps://pop.com.sg/how-to-find-the-inradius-of-a-triangle-with-given-side-lengths.htmlI need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Thank you....Tue, 01 Sep 2026 19:02:53 -0400https://pop.com.sg/widest-part-of-a-sector.htmlhttps://pop.com.sg/widest-part-of-a-sector.htmlCoordinates of the center of a circle, radius, initial and final angles of a sector are given. How to find coordinates of the endpoints of a segment (A and B) connecting borders of sector in its wi......Tue, 01 Sep 2026 19:01:01 -0400https://pop.com.sg/how-to-find-vector-orthogonal-to-a-given-pair-of-vectors.htmlhttps://pop.com.sg/how-to-find-vector-orthogonal-to-a-given-pair-of-vectors.html(a) $[2,-3,5], [4,-5,1]$ Solution: $[2,-3,5] \times [4,-5,1] = [22,18,2]$ No idea how they got $ [22,18,2]$...Tue, 01 Sep 2026 18:54:30 -0400https://pop.com.sg/total-average-of-averages-not-same-as-the-average-of-total-values.htmlhttps://pop.com.sg/total-average-of-averages-not-same-as-the-average-of-total-values.htmlI am having a strange problem while computing the overall percentage. Let me demonstrate my problem using an example. Assume that I am receiving multiple batches of apple from the vendor and the nu......Tue, 01 Sep 2026 18:53:54 -0400https://pop.com.sg/how-to-find-coordinates-of-reflected-point.htmlhttps://pop.com.sg/how-to-find-coordinates-of-reflected-point.htmlHow can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If P is a reflection (image) of point (3, -3) in the line $2y = ......Tue, 01 Sep 2026 18:35:36 -0400https://pop.com.sg/importance-of-rolle-039-s-and-lagrange-039-s-theorem-in-daily-life-for.htmlhttps://pop.com.sg/importance-of-rolle-039-s-and-lagrange-039-s-theorem-in-daily-life-for.htmlI am school teacher and teaching math. I know the Rolle's theorem and Lagrange's theorem. I can solve the problems numerically then and there. However, I was completely failed to explain the signif......Tue, 01 Sep 2026 18:32:49 -0400https://pop.com.sg/inscribing-square-in-circle-in-just-seven-compass-and-straightedge-ste.htmlhttps://pop.com.sg/inscribing-square-in-circle-in-just-seven-compass-and-straightedge-ste.htmlProblem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the ...Tue, 01 Sep 2026 18:30:31 -0400https://pop.com.sg/conditional-probability-involving-sum-of-bernoulli-trials.htmlhttps://pop.com.sg/conditional-probability-involving-sum-of-bernoulli-trials.htmlI have $X_i (i=1,...,n)$ which has $\text{Bernoulli}(p)$ or $\text{binomial}(1,p)$ distribution and each $X_i$ is independent with $X_i=1$ with probability $p$ and $X_i=0$ with probability $(1-p)$. I ...Tue, 01 Sep 2026 18:27:36 -0400https://pop.com.sg/evaluating-a-surface-integral-using-symmetry.htmlhttps://pop.com.sg/evaluating-a-surface-integral-using-symmetry.htmlLet $S$ denote the surface of the cylinder $x^2+y^2=4,-2\le z\le2$ and consider the surface integral $$\int_{S}(z-x^2-y^2)\,dS$$ How can one use geometry and symmetry to evaluate the integral without ...Tue, 01 Sep 2026 18:24:10 -0400https://pop.com.sg/a-question-about-the-arctangent-addition-formula.htmlhttps://pop.com.sg/a-question-about-the-arctangent-addition-formula.htmlIn the arctangent formula, we have that: $$\arctan{u}+\arctan{v}=\arctan\left(\frac{u+v}{1-uv}\right)$$ however, only for $uv<1$. My question is: where does this condition come from? The situa......Tue, 01 Sep 2026 18:22:37 -0400https://pop.com.sg/how-to-derive-a-function-that-depends-on-another-function-evaluated-at.htmlhttps://pop.com.sg/how-to-derive-a-function-that-depends-on-another-function-evaluated-at.htmlI'm coming from physics, but I figured this is a mathematical question, so here I am. I have a scalar $f(\vec{x},\vec{y})$ that is a function of two different points $\vec{x}$ and $\vec{y}$ in spac......Tue, 01 Sep 2026 18:16:07 -0400https://pop.com.sg/calculating-sqrt-28-cdot-29-cdot-30-cdot-31-1.htmlhttps://pop.com.sg/calculating-sqrt-28-cdot-29-cdot-30-cdot-31-1.htmlIs it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work....Tue, 01 Sep 2026 18:09:47 -0400https://pop.com.sg/why-is-the-empty-set-a-subset-of-every-set-duplicate.htmlhttps://pop.com.sg/why-is-the-empty-set-a-subset-of-every-set-duplicate.htmlTake for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?...Tue, 01 Sep 2026 17:55:14 -0400https://pop.com.sg/is-it-valid-to-calculate-standard-deviation-for-n-2.htmlhttps://pop.com.sg/is-it-valid-to-calculate-standard-deviation-for-n-2.htmlMy colleagues and I are having discussion whether it's valid to calculate standard deviation for $n=2$ or not? I think it's valid since I can calculate based on the equation, but higher N will give......Tue, 01 Sep 2026 17:52:39 -0400https://pop.com.sg/how-to-integrate-cos-x-e-ix-dx.htmlhttps://pop.com.sg/how-to-integrate-cos-x-e-ix-dx.htmlHow to integrate $$\int \cos(t)e^{it}dt $$ I tried integrating by parts twice, but it doesn't work because an i shows up in the end and one gets $0=….$...Tue, 01 Sep 2026 17:49:06 -0400https://pop.com.sg/given-coordinates-of-beginning-and-end-of-two-intersecting-line-segmen.htmlhttps://pop.com.sg/given-coordinates-of-beginning-and-end-of-two-intersecting-line-segmen.htmlThere are two line segments. I know for sure they intersect (so I don't have to check it). For both line segment I know coordinates of its both ends. With what formula can I find coordinates of their ...Tue, 01 Sep 2026 17:43:38 -0400https://pop.com.sg/derivative-of-y-6xe-6x-e-6x.htmlhttps://pop.com.sg/derivative-of-y-6xe-6x-e-6x.htmlCan some one walk me through how to find the derivative of $y=6xe^{6x}-e^{6x}$? I know the answer should be $36e^{6x}x$. I know I am supposed to use the product and chain rule, I just don't ...Tue, 01 Sep 2026 17:39:14 -0400https://pop.com.sg/null-space-of-a-vector.htmlhttps://pop.com.sg/null-space-of-a-vector.htmlAssuming I have a vector $v_1 \in \mathbb{R}^n $ of dimension $n$. (edit what I mean is e.g. $v_1 \in \mathbb{R}^n $) I want to describe the set of "all the vectors that are orthogonal to $v_1$"......Tue, 01 Sep 2026 17:36:58 -0400https://pop.com.sg/how-to-convert-a-vector-to-a-unit-vector-if-its-magnitude-is-smaller-t.htmlhttps://pop.com.sg/how-to-convert-a-vector-to-a-unit-vector-if-its-magnitude-is-smaller-t.htmlIn my javascript project, I need to convert a vector to a unit vector. I know that I can divide its x and y components by its magnitude. But what if the vector's magnitude is smaller than one? How to ...Tue, 01 Sep 2026 17:34:46 -0400https://pop.com.sg/questions-tagged-conic-sections.htmlhttps://pop.com.sg/questions-tagged-conic-sections.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 17:29:27 -0400https://pop.com.sg/two-numbers-two-mathematicians-puzzle.htmlhttps://pop.com.sg/two-numbers-two-mathematicians-puzzle.htmlThis is one of the most beautiful and difficult puzzles I have encountered. I have talked to several people, but I still don't know the solution - I do however know that the solution exists. Someone ...Tue, 01 Sep 2026 17:28:25 -0400https://pop.com.sg/formula-for-calculating-the-total-interest-payable-over-the-life-of-a-.htmlhttps://pop.com.sg/formula-for-calculating-the-total-interest-payable-over-the-life-of-a-.htmlwondering if someone can help a non-mathematician out. I am looking for the formula for calculating the total interest payable over the life of a loan. Given that we know: P: Principal (amount) of ...Tue, 01 Sep 2026 17:25:57 -0400https://pop.com.sg/evaluate-integral-with-fractional-exponents.htmlhttps://pop.com.sg/evaluate-integral-with-fractional-exponents.htmlCan someone please help me integrating this? $$\displaystyle \int \frac{w\mathrm dw}{(5-3w)^{2/3}}$$ I tried substituting $5-3w = u$ and $-3\mathrm dw = \mathrm du$. So $w = (u-5)/3$ and then we ......Tue, 01 Sep 2026 17:24:30 -0400https://pop.com.sg/what-is-the-039-physical-039-explanation-of-a-division-by-a-fraction.htmlhttps://pop.com.sg/what-is-the-039-physical-039-explanation-of-a-division-by-a-fraction.htmlFor example, dividing by 2, means we cut something in two. But dividing by 0.5, can only be explained with multiplying something by 2. So, is there a "physical" explanation of dividing by 0.5? Is......Tue, 01 Sep 2026 17:22:43 -0400https://pop.com.sg/setting-lineard-equation-ode-into-standard-form-proving-existence-of-u.htmlhttps://pop.com.sg/setting-lineard-equation-ode-into-standard-form-proving-existence-of-u.htmlI'm working on proving existence and uniqueness of a local solution to ODE's of the Lienard variety $$y'' + f(y)\,y' + g(y) = 0$$ I'm trying to put this into standard $y' = f(t,y)$ system So I see ...Tue, 01 Sep 2026 17:22:05 -0400https://pop.com.sg/contradiction-proof-of-the-product-of-two-irrational-numbers.htmlhttps://pop.com.sg/contradiction-proof-of-the-product-of-two-irrational-numbers.htmlI am wondering what is wrong with my contradiction proof that "The product of two irrational numbers is irrational". I understand that there are examples where this is not true: $\sqrt{2} * \frac{1}{\...Tue, 01 Sep 2026 17:16:27 -0400https://pop.com.sg/recurrence-for-number-of-words-of-length-r-over-n-with-no-three-consec.htmlhttps://pop.com.sg/recurrence-for-number-of-words-of-length-r-over-n-with-no-three-consec.htmlQuestion Let $b_{n,r}$ be the number words of length $r$ over $[n]=\{1,2,\dotsc, n\}$ with no three consecutive letters the same. Show that $$ b_{n,r}=(n-1)(b_{n,r-1}+b_{n,r-2})\quad (r>2)......Tue, 01 Sep 2026 16:47:48 -0400https://pop.com.sg/showing-that-x-4-4x-3-11x-2-10x-9-is-irreducible-over-mathbb-q.htmlhttps://pop.com.sg/showing-that-x-4-4x-3-11x-2-10x-9-is-irreducible-over-mathbb-q.htmlLet $p(x) = x^4 + 4x^3 + 11x^2 + 10x + 9$. How to how that it is irreducible over $\mathbb{Q}$? The first thing I tried was Eisenstein, but that obviously doesn't work with the polynomial in that ......Tue, 01 Sep 2026 16:46:37 -0400https://pop.com.sg/volume-of-a-hyperbolic-cube-shaped-solid-in-euclidean-space.htmlhttps://pop.com.sg/volume-of-a-hyperbolic-cube-shaped-solid-in-euclidean-space.htmlI'm searching for a formula for the volume (and, if available, also other information like area, etc.) of a so-called "hyperbolic cube": I couldn't find anything in Wikipedia, and also MathWorld's "...Tue, 01 Sep 2026 16:20:59 -0400https://pop.com.sg/does-cos-pi-2-really-equal-zero.htmlhttps://pop.com.sg/does-cos-pi-2-really-equal-zero.htmlI was watching this video and at 5:30 $\cos \frac{\pi}{2}$ is substituted with $0$. Using my calculator the result I got was $-0.5$. Which one is correct?...Tue, 01 Sep 2026 16:14:47 -0400https://pop.com.sg/where-can-one-find-a-list-of-prime-numbers.htmlhttps://pop.com.sg/where-can-one-find-a-list-of-prime-numbers.htmlI am looking for the biggest list of precomputed prime numbers one can find and download. Where should I look?...Tue, 01 Sep 2026 16:10:30 -0400https://pop.com.sg/finding-the-adjoint-of-an-operator.htmlhttps://pop.com.sg/finding-the-adjoint-of-an-operator.htmlThis is from my homework, I'm totally lost as to how to proceed. Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $(Tf)(x) = \int^x_0 f(s) \ ds$ What is the adjoint of $T$? ......Tue, 01 Sep 2026 16:06:55 -0400https://pop.com.sg/since-is-a-subset-of-every-set-it-is-a-subset-of-itself-and-hence-dupl.htmlhttps://pop.com.sg/since-is-a-subset-of-every-set-it-is-a-subset-of-itself-and-hence-dupl.htmlI am bit confused with the concept of empty set here. Since {} is a subset of every set, it is a subset of itself? and hence {} = {{}}? Also, say A = {a}, but since {} is a subset of A, is it tru......Tue, 01 Sep 2026 15:59:24 -0400https://pop.com.sg/decomposition-of-a-biquadratic-polynomial.htmlhttps://pop.com.sg/decomposition-of-a-biquadratic-polynomial.htmlLet $f(t):=T^4+aT^2+b$ be a polynomial in a field $K[T]$ such that it can be decomposed in two irreducible factors of degree 2. My intuition then is screaming to me that then we neccessarily have ......Tue, 01 Sep 2026 15:54:49 -0400https://pop.com.sg/a-general-formula-for-n-distinguishable-arrangements-of-length-ell-of-.htmlhttps://pop.com.sg/a-general-formula-for-n-distinguishable-arrangements-of-length-ell-of-.htmlIs there a general formula for the number of distinguishable arrangements of length $\ell$ of a sequence of size $s$, given the repeated elements? So, For example, would there be a general formula......Tue, 01 Sep 2026 15:49:37 -0400https://pop.com.sg/what-is-the-cardinality-of-powerset-of-a-union-set.htmlhttps://pop.com.sg/what-is-the-cardinality-of-powerset-of-a-union-set.htmlIs there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), a......Tue, 01 Sep 2026 15:46:40 -0400https://pop.com.sg/31-331-3331-33331-333331-3333331-33333331-are-prime.htmlhttps://pop.com.sg/31-331-3331-33331-333331-3333331-33333331-are-prime.html31,331,3331, 33331,333331,3333331,33333331 are prime. This law can continue it? Will there emerge a composite number? Without using a computer how to judge....Tue, 01 Sep 2026 15:40:09 -0400https://pop.com.sg/2015-amc-10b-problem-21.htmlhttps://pop.com.sg/2015-amc-10b-problem-21.htmlThe problem and solutions I've attempted to solve another AMC 10 problem, and the problem is basically like this: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of......Tue, 01 Sep 2026 15:36:11 -0400https://pop.com.sg/continuum-between-addition-multiplication-and-exponentiation.htmlhttps://pop.com.sg/continuum-between-addition-multiplication-and-exponentiation.htmlI noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we f......Tue, 01 Sep 2026 15:32:32 -0400https://pop.com.sg/when-is-991n-2-1-a-perfect-square.htmlhttps://pop.com.sg/when-is-991n-2-1-a-perfect-square.htmlWhat should be the value of $n$ so that the number obtained after adding $1$ to $991$ times its square is itself a perfect square? Can you please give me a few hints on this topic with a few speci......Tue, 01 Sep 2026 15:28:15 -0400https://pop.com.sg/number-of-edges-in-n-cube.htmlhttps://pop.com.sg/number-of-edges-in-n-cube.htmlIf the number of edges in an $n$-cube (also denoted as $Q_n$) is $k$, what is the number of edges in an $(n+1)$-cube (denoted as $Q_{n+1}$)? Express your answer in terms of $k$ and $n$. How can I ......Tue, 01 Sep 2026 15:18:48 -0400https://pop.com.sg/how-to-show-that-a-cauchy-sequence-of-sequences-converges.htmlhttps://pop.com.sg/how-to-show-that-a-cauchy-sequence-of-sequences-converges.htmlLet $c_0$ be the space of real-valued sequences $\{x_n\}$ which converge to zero, equipped with the metric $d(\{x_n\}, \{y_n\}) = sup_n |x_n − y_n|$. I want to show that the metric space $(c_0, d)$......Tue, 01 Sep 2026 15:09:48 -0400https://pop.com.sg/sequence-with-no-accumulation-point.htmlhttps://pop.com.sg/sequence-with-no-accumulation-point.htmlConstruct an example of a set of real numbers E that has no points of accumulation and yet has the property that for every ε > 0 there exist points x, y ∈ E so that 0 < |x − y| < ε. so i kno......Tue, 01 Sep 2026 15:05:04 -0400https://pop.com.sg/radical-and-radical-ideal.htmlhttps://pop.com.sg/radical-and-radical-ideal.htmlI am trying to get used to the terminology of commutative algebra. Let $R$ be a commutative ring with unity. Then I know radical $\sqrt{I} := \{ a\in R| a^n \in I, \textrm{for some $n \in \mathbb{Z......Tue, 01 Sep 2026 15:03:57 -0400https://pop.com.sg/in-percent-change-why-do-we-divide-the-difference-by-the-original-numb.htmlhttps://pop.com.sg/in-percent-change-why-do-we-divide-the-difference-by-the-original-numb.htmlIf $a = 90$ and $b = 100$, and we are calculating the percent increase going from $a$ to $b$, we calculate the actual numerical difference $b - a = 10$. But why divide the difference into the star......Tue, 01 Sep 2026 15:02:13 -0400https://pop.com.sg/laplace-transform-of-the-square-of-first-derivative.htmlhttps://pop.com.sg/laplace-transform-of-the-square-of-first-derivative.htmlI want to compute the laplace transformation of this equation: $$4 -0.1{\operatorname{d}\!x(t)\over\operatorname{d}\!t} - 0.01\left({\operatorname{d}\!x(t)\over\operatorname{d}\!t}\right)^2 = 0.9{\...Tue, 01 Sep 2026 15:01:59 -0400https://pop.com.sg/how-to-calculate-a-b-bmod-n-and-b-bmod-n-closed.htmlhttps://pop.com.sg/how-to-calculate-a-b-bmod-n-and-b-bmod-n-closed.htmlConsider the following expression: (a - b) mod N Which of the following is equivalent to the above expression? 1) ((a mod N) + (-b mod N)) mod N 2) ((a mod N) - (b mod N)) mod N Also, how is (......Tue, 01 Sep 2026 14:59:43 -0400https://pop.com.sg/computing-volume-by-cross-section-method.htmlhttps://pop.com.sg/computing-volume-by-cross-section-method.htmlThe base of the solid below is the region in the $xy$-plane bounded by the $x$-axis,the graph of y = $\sqrt{x}$ and the line $ x = 3 $. Find the volume of the solid. Each cross-section of S ...Tue, 01 Sep 2026 14:56:17 -0400https://pop.com.sg/how-do-i-calculate-the-area-of-a-curved-irregular-plane-such-as-an-aer.htmlhttps://pop.com.sg/how-do-i-calculate-the-area-of-a-curved-irregular-plane-such-as-an-aer.htmlSo I need to calculate the wing area of an airbus a350 and all I have is this blueprints. Thought about searching functions that fit the wing from the top perspective and then integrate, but I dont......Tue, 01 Sep 2026 14:54:13 -0400https://pop.com.sg/how-long-will-it-take-to-learn-algebra-1-2-geometry-and-pre-calculus-c.htmlhttps://pop.com.sg/how-long-will-it-take-to-learn-algebra-1-2-geometry-and-pre-calculus-c.htmlI've previously learned algebra (wasn't the best, wasn't the worst) and have since forgotten most (basically all) of it. I'm currently homeschooled and I would like to go to college in two years. D......Tue, 01 Sep 2026 14:50:58 -0400https://pop.com.sg/how-the-jacobian-of-the-transformation-can-be-shown-to-not-depend-on-x.htmlhttps://pop.com.sg/how-the-jacobian-of-the-transformation-can-be-shown-to-not-depend-on-x.htmlTransform the random variables, $X_i$, $i=1,2,\ldots,n$ to $$ \begin{align} Y_1 & =\bar X \\ Y_2 & =X_2-\bar X \\ Y_3 & = X_3-\bar X \\ & {}\ \vdots \\ Y_n & =X_n-\bar X \end{......Tue, 01 Sep 2026 14:32:46 -0400https://pop.com.sg/probability-of-winning-twice-on-a-wheel-of-fortune.htmlhttps://pop.com.sg/probability-of-winning-twice-on-a-wheel-of-fortune.htmlA wheel of fortune is divided into 40 sectors, numbered from 1 to 40. Tickets are sold representing each sector. Tickets are \$1 each. All 40 tickets must be sold before the wheel can be spun. Only......Tue, 01 Sep 2026 14:30:29 -0400https://pop.com.sg/probability-question-with-black-and-white-marbles.htmlhttps://pop.com.sg/probability-question-with-black-and-white-marbles.htmlJust looking for some help with this question. The answer seems obvious but I am questioning myself as it seems a little too obvious. Max has a bag containing 2 black marbles and 2 white marbles.......Tue, 01 Sep 2026 14:25:39 -0400https://pop.com.sg/why-are-rings-called-rings.htmlhttps://pop.com.sg/why-are-rings-called-rings.htmlI've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks....Tue, 01 Sep 2026 14:22:22 -0400https://pop.com.sg/how-to-solve-a-quintic-polynomial-equation.htmlhttps://pop.com.sg/how-to-solve-a-quintic-polynomial-equation.htmlI know that not all quintics are solvable. But how do I identify the class of solvable ones?...Tue, 01 Sep 2026 14:22:00 -0400https://pop.com.sg/guess-the-value-of-the-limit.htmlhttps://pop.com.sg/guess-the-value-of-the-limit.htmlSimplify and guess the value of $\lim_{x\to 2}\frac {x^3-8}{x-2}$. I'm very confused on how to do this. Please help!...Tue, 01 Sep 2026 14:18:31 -0400https://pop.com.sg/what-does-the-phrase-except-possibly-mean.htmlhttps://pop.com.sg/what-does-the-phrase-except-possibly-mean.htmlThis term was used by my math professor when he was teaching us limits. This term also appears online like this "Let f be a function which is defined on some open interval containing $a$ except pos......Tue, 01 Sep 2026 14:11:41 -0400https://pop.com.sg/finding-angle-in-an-equilateral-triangular-pyramid.htmlhttps://pop.com.sg/finding-angle-in-an-equilateral-triangular-pyramid.htmlGiven an equilateral triangular pyramid (refer the below diagram) $\Delta ABC$ & $P$ is any point inside the triangle such that ${PA}^{2}={PB}^{2}+{PC}^{2}$, then $\angle BPC$ is - I am unable......Tue, 01 Sep 2026 14:00:36 -0400https://pop.com.sg/how-to-tell-if-3-connected-points-are-connected-clockwise-or-counter-c.htmlhttps://pop.com.sg/how-to-tell-if-3-connected-points-are-connected-clockwise-or-counter-c.htmlI have three points: p1(x1,y1) p2(x2,y2) and p3(x3,y3). I am connecting p1 to p2 to p3. how can I tell if the triangle was drawn clockwise or counter-clockwise? How can I generalize this for n po......Tue, 01 Sep 2026 13:49:49 -0400