https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 18 Aug 2026 12:52:46 -0400https://pop.com.sg/calculate-average-wind-direction.htmlhttps://pop.com.sg/calculate-average-wind-direction.htmlWhat is the best way to average wind direction? I have found many conflicting suggestions elsewhere. Best one I saw is to average SINs of all angles in radians and take inverse SIN of the result. ......Tue, 18 Aug 2026 16:51:40 -0400https://pop.com.sg/finding-final-velocity-when-initial-velocity-distance-travelled-and-ac.htmlhttps://pop.com.sg/finding-final-velocity-when-initial-velocity-distance-travelled-and-ac.htmlI've been struggling with this, I need to find the final velocity of a car after it travels a certain distance when I know it's initial velocity and acceleration are known. Any help?...Tue, 18 Aug 2026 16:45:33 -0400https://pop.com.sg/trapezium-midsegment-theorem.htmlhttps://pop.com.sg/trapezium-midsegment-theorem.htmlA question from an ACT Math test: Please teach me a way to get the answer quickly, I must've missed some properties of trapezoid cause I can only think of making similar triangles to find the answ......Tue, 18 Aug 2026 16:35:48 -0400https://pop.com.sg/what-does-round-off-error-in-double-arithmetic-mean.htmlhttps://pop.com.sg/what-does-round-off-error-in-double-arithmetic-mean.htmlLet's say I have a function $f$ and furthermore let there also be a recursion formula for $f$. So I can evaluate $f(x)$ directly and I can evaluate $f(x)$ using the recursion formula. Now, in the ...Tue, 18 Aug 2026 16:32:15 -0400https://pop.com.sg/given-a-number-039-n-039-find-how-many-how-many-numbers-are-there-betw.htmlhttps://pop.com.sg/given-a-number-039-n-039-find-how-many-how-many-numbers-are-there-betw.htmlYou are given a number $N\le 10^{18}$. You need to find out how many numbers there exist in between $1$ to $N$, which doesn't contain the digit $'X'$ in it . Say $N = 5, X=4$ The answer is $4$. ......Tue, 18 Aug 2026 16:25:05 -0400https://pop.com.sg/given-that-the-customer-has-filed-a-claim-what-is-the-probability-that.htmlhttps://pop.com.sg/given-that-the-customer-has-filed-a-claim-what-is-the-probability-that.htmlAn automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 70% of the customers are good risks, 20% are medium risks, and 10% are......Tue, 18 Aug 2026 16:22:15 -0400https://pop.com.sg/what-does-3-3-mean.htmlhttps://pop.com.sg/what-does-3-3-mean.htmlI was taking a test and this was a question. It does not mean $3^3$, nor is it a typo. The superscript was before the $3$ and I have no idea what it means. I tried researching but couldn't find an ......Tue, 18 Aug 2026 16:20:43 -0400https://pop.com.sg/which-of-the-following-expressions-completes-the-identity-1-sec-2-x.htmlhttps://pop.com.sg/which-of-the-following-expressions-completes-the-identity-1-sec-2-x.htmlI simplified $1 + \sec^2 x$ to $1 + 1/\cos^2 x$ then $2/\cos^2 x$ but I am stuck here....Tue, 18 Aug 2026 16:15:22 -0400https://pop.com.sg/gale-stewart-theorem-open-games-are-determined-implies-closed-games-ar.htmlhttps://pop.com.sg/gale-stewart-theorem-open-games-are-determined-implies-closed-games-ar.htmlA Gale-Stewart game $G(A)$ is played on a set $A\subseteq\mathbb N^\mathbb N$. In this game, players p0 and p1 alternately pick a natural number, forming a sequence $\alpha:=\alpha_0\alpha_1\alpha_2\...Tue, 18 Aug 2026 16:11:28 -0400https://pop.com.sg/how-to-factor-a-four-term-polynomial-without-grouping.htmlhttps://pop.com.sg/how-to-factor-a-four-term-polynomial-without-grouping.html$$2x^3 + 9x^2 +7x -6$$ This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?...Tue, 18 Aug 2026 16:08:51 -0400https://pop.com.sg/rotate-graph-of-function-by-180.htmlhttps://pop.com.sg/rotate-graph-of-function-by-180.htmlsuppose that we have graph of function $$f(x)=1+x\cos(x)$$ and we should rotate it by $180$ degree,question is what is a function which describe new graph?answer is $$f(x)=x\cos(x)-1$$but i ca......Tue, 18 Aug 2026 16:08:12 -0400https://pop.com.sg/what-is-the-geodesic-between-a-point-and-a-line-geodesic-between-two-p.htmlhttps://pop.com.sg/what-is-the-geodesic-between-a-point-and-a-line-geodesic-between-two-p.htmlI found a similiar question that also asks for the distance from a point to a line but works on a sphere. Now I'm trying to figure out the length of the geodesic line d between the given geodesic......Tue, 18 Aug 2026 16:03:32 -0400https://pop.com.sg/find-the-least-positive-value-of-alpha-for-the-following-trignometric-.htmlhttps://pop.com.sg/find-the-least-positive-value-of-alpha-for-the-following-trignometric-.htmlGiven question is : $$\sin(5\alpha + \theta) = \cos(\theta - 3\alpha)$$ We are to find the least positive value of $\alpha$ for which above equation holds. The way I did is as, $$\sin5\alpha \cos......Tue, 18 Aug 2026 15:59:49 -0400https://pop.com.sg/how-to-calculate-the-area-of-a-parabolic-dish.htmlhttps://pop.com.sg/how-to-calculate-the-area-of-a-parabolic-dish.htmlI feel like this should be really easy, but I'm not sure if I'm doing it correctly so I'm going to give it a go here, and if I'm not very good at maths (I'm not) then you can hopefully correct me! ...Tue, 18 Aug 2026 15:54:20 -0400https://pop.com.sg/high-dimensional-optimization-algorithm.htmlhttps://pop.com.sg/high-dimensional-optimization-algorithm.htmlI have an optimization problem that at first sounds quite textbook. I have a convex objective function in $D$-dimensional space that is twice differentiable everywhere and has no local optima. ...Tue, 18 Aug 2026 15:50:30 -0400https://pop.com.sg/find-a-formula-for-the-area-of-the-triangle-with-vertices-in-polar-coo.htmlhttps://pop.com.sg/find-a-formula-for-the-area-of-the-triangle-with-vertices-in-polar-coo.htmlI don't know where to start with this homework problem. If anyone could help me start, or provide the solution, that'd be greatly appreciated....Tue, 18 Aug 2026 15:47:54 -0400https://pop.com.sg/defining-a-plane-using-only-2-points-and-an-axis.htmlhttps://pop.com.sg/defining-a-plane-using-only-2-points-and-an-axis.htmlSuppose you have a $3$-dimensional space in which there are 2 points ($A$ and $B$) defined (non identical). Now, you can define a line that goes through them but you cannot define a unique plane, b......Tue, 18 Aug 2026 15:43:06 -0400https://pop.com.sg/for-how-many-natural-numbers-a-is-the-expression-sqrt-frac-a-64-a-64-a.htmlhttps://pop.com.sg/for-how-many-natural-numbers-a-is-the-expression-sqrt-frac-a-64-a-64-a.htmlFor how many natural numbers $a$ is the expression $\sqrt{\frac{a+64}{a-64}}$ also a natural number? I cannot seem to do this without using trial and error....Tue, 18 Aug 2026 15:39:21 -0400https://pop.com.sg/how-to-find-the-length-of-diagonal-of-a-rhombus.htmlhttps://pop.com.sg/how-to-find-the-length-of-diagonal-of-a-rhombus.htmlsuppose,only the length of the side of a rhombus is given then how can I find the length of the diagonals?(without measuring the angle) Is there any equation where the side is related to the diagonal ...Tue, 18 Aug 2026 15:36:28 -0400https://pop.com.sg/how-do-i-solve-this-square-root-problem.htmlhttps://pop.com.sg/how-do-i-solve-this-square-root-problem.htmlI need to solve the following problem: $$\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}=\,?$$...Tue, 18 Aug 2026 15:31:40 -0400https://pop.com.sg/extending-the-p-adic-valuation.htmlhttps://pop.com.sg/extending-the-p-adic-valuation.htmlGiven a prime $p$, the $p$-adic valuation on the field $\mathbb{Q}$ is the map $\nu:\mathbb{Q}^*\to\mathbb{Z}$ given by $\nu(p^ka/b)=k$, where $a,b$ are prime to $p$. I want to consider extension......Tue, 18 Aug 2026 15:26:55 -0400https://pop.com.sg/how-to-find-the-n-th-term-of-the-series-1-22-333-4444-55555.htmlhttps://pop.com.sg/how-to-find-the-n-th-term-of-the-series-1-22-333-4444-55555.htmlHow to find the $n$-th term of the series $1, 22, 333, 4444, 55555..........?$ Here the $i^{\text{th}}$ term is the concatenation of $i$ for $i$ times. for ef $13$ the term = $13$ concatenated $13$......Tue, 18 Aug 2026 15:22:57 -0400https://pop.com.sg/on-when-and-when-not-to-divide-by-tan-theta.htmlhttps://pop.com.sg/on-when-and-when-not-to-divide-by-tan-theta.htmlThe textbook I'm using has this problem involving trig equations which goes $\tan(x) \sin^2(x)=2 \tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $\tan(x)$ ......Tue, 18 Aug 2026 15:22:28 -0400https://pop.com.sg/having-a-unique-solution-for-a-rectangular-matrix.htmlhttps://pop.com.sg/having-a-unique-solution-for-a-rectangular-matrix.htmlSuppose we are given a $4\times3$ matrix $A$ and a column $b$ such that $Ax=b$ always has a unique solution. What is the reduced row echelon form of $A$? I think that since the solution is unique,......Tue, 18 Aug 2026 15:18:55 -0400https://pop.com.sg/how-or-why-are-implicit-functions-actually-functions.htmlhttps://pop.com.sg/how-or-why-are-implicit-functions-actually-functions.htmlI'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function. I didn't quite get it at that time so I decided to ...Tue, 18 Aug 2026 15:13:54 -0400https://pop.com.sg/jacobian-calculation-with-homographic-coordinates.htmlhttps://pop.com.sg/jacobian-calculation-with-homographic-coordinates.htmlI'm trying to work my way through an image processing paper (Creating Full View Panoramic Image Mosaics and Environment Maps) and am stuck on how the Jacobian is calculated, especially terms ($J_{1......Tue, 18 Aug 2026 15:03:33 -0400https://pop.com.sg/third-order-differential-equations.htmlhttps://pop.com.sg/third-order-differential-equations.htmlI am having trouble solving the third order differential equation $y'''+y'=0$ It was given to me in a quiz (which I got wrong) with boundary conditions $y(0) = 0$ $y'(0)=2$ $y(\pi)=6$ I know t......Tue, 18 Aug 2026 15:03:15 -0400https://pop.com.sg/why-do-we-need-prime-ideals-in-the-spectrum-of-a-ring.htmlhttps://pop.com.sg/why-do-we-need-prime-ideals-in-the-spectrum-of-a-ring.htmlI'm reading Atiyah Macdonald, where they introduce in the exercises of chapter one a topological space $\operatorname{Spec}(A)$ associated to a ring $A$, which is defined as $\operatorname{Spec}(A) \...Tue, 18 Aug 2026 15:01:00 -0400https://pop.com.sg/cauchy-039-s-formula-for-repeated-integrals-proof-by-induction.htmlhttps://pop.com.sg/cauchy-039-s-formula-for-repeated-integrals-proof-by-induction.htmlI was trying to follow along with the proof on Wikipedia for Cauchy's formula for repeated integrals and I'm stuck on the last step. How do you go from $$\int_a^x \frac {d}{dy} \left( \int_a^y (y-t......Tue, 18 Aug 2026 14:59:31 -0400https://pop.com.sg/is-there-a-way-to-reverse-factorials-duplicate.htmlhttps://pop.com.sg/is-there-a-way-to-reverse-factorials-duplicate.htmlIs there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do factorials and factorial roots (for lack of a better term)? Here is an example: 5!......Tue, 18 Aug 2026 14:53:48 -0400https://pop.com.sg/given-a-point-p-outside-equilateral-delta-abc-but-inside-angle-abc-if-.htmlhttps://pop.com.sg/given-a-point-p-outside-equilateral-delta-abc-but-inside-angle-abc-if-.htmlGiven a point $P$ outside equilateral $\Delta ABC$ but inside $\angle ABC$, if the distance between $P$ to $BC,CA,AB$ are $h_1,h_2,h_3$ respectively, where $h_1 - h_2 + h_3 = 6$, find $[\Delta ABC]......Tue, 18 Aug 2026 14:50:35 -0400https://pop.com.sg/show-that-sin45-sin15-sin75.htmlhttps://pop.com.sg/show-that-sin45-sin15-sin75.htmlSteps I took: 1) Finding the value of the left hand side $$\sin45=\sin\frac { 90 }{ 2 } =\sqrt { \frac { 1-\cos90 }{ 2 } } =\sqrt { \frac { 1 }{ 2 } } =\frac { \sqrt { 2 } }{ 2 } $$ $$\sin15=......Tue, 18 Aug 2026 14:29:04 -0400https://pop.com.sg/how-do-we-write-the-derivation-of-distance-formula-of-two-points-from-.htmlhttps://pop.com.sg/how-do-we-write-the-derivation-of-distance-formula-of-two-points-from-.htmlI learnt the derivation of the distance formula of two points in first quadrant I.e., $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ where it is easy to find the legs of the hypotenuse (distance betwee......Tue, 18 Aug 2026 14:24:09 -0400https://pop.com.sg/asymptotic-and-exact-confidence-interval.htmlhttps://pop.com.sg/asymptotic-and-exact-confidence-interval.htmlAssume we toss a thumbtack 300 times. After every time, we note $1$ if it lands point up or $0$ if it lands point down. In summary, we get 124 times $1$. So we know that the number of the rounds w......Tue, 18 Aug 2026 14:17:41 -0400https://pop.com.sg/how-to-calculate-distance-run-on-athletics-track.htmlhttps://pop.com.sg/how-to-calculate-distance-run-on-athletics-track.htmlfirst time poster and definitely no maths expert. I am trying to solve a basic problem using an athletics track. The total distance around a standard athletics track is 400m: If you run in the first ...Tue, 18 Aug 2026 14:17:04 -0400https://pop.com.sg/how-to-calculate-a-weighted-gpa.htmlhttps://pop.com.sg/how-to-calculate-a-weighted-gpa.htmlHow would I be able to calculate my overall average grade (or grade point average) for these courses? Course Course Credit (weight) Course Grade Biology 101 45 86 Psychology 210 30 74 Chemistry......Tue, 18 Aug 2026 14:05:11 -0400https://pop.com.sg/set-of-algebraic-integers-is-closed-under-addition-and-multiplication.htmlhttps://pop.com.sg/set-of-algebraic-integers-is-closed-under-addition-and-multiplication.htmlIf $\alpha$ and $\beta$ are algebraic integers, then show $\alpha + \beta$ and $\alpha \times \beta$ are both algebraic integers. I know that an algebraic integer is a root of some monic polynomial ...Tue, 18 Aug 2026 14:05:06 -0400https://pop.com.sg/what-is-the-ratio-of-the-area-of-a-circle-to-its-radius-closed.htmlhttps://pop.com.sg/what-is-the-ratio-of-the-area-of-a-circle-to-its-radius-closed.htmlI solved this. $$πr^2/r=π r$$ In my book, they said it can not be determined. Am I right or wrong?...Tue, 18 Aug 2026 14:03:11 -0400https://pop.com.sg/how-were-the-sine-cosine-and-tangent-tables-originally-calculated.htmlhttps://pop.com.sg/how-were-the-sine-cosine-and-tangent-tables-originally-calculated.htmlAs I understand it... ahem... the (cosine, sine) vector was calculated for (30 degrees, PI/6), (45 degrees, PI/4) and (60 degrees, PI/3) angles etcetera, however, I would like know the original ...Tue, 18 Aug 2026 14:00:57 -0400https://pop.com.sg/learning-how-to-flip-equations.htmlhttps://pop.com.sg/learning-how-to-flip-equations.htmlI took Algebra and Geometry in high school, never thought I'd use them, then became a programmer. I guess I was wrong. To date, I have the hardest time taking equations and "flipping them," ie: ...Tue, 18 Aug 2026 13:56:04 -0400https://pop.com.sg/calculating-running-time-in-seconds-of-algorithms-of-a-given-complexit.htmlhttps://pop.com.sg/calculating-running-time-in-seconds-of-algorithms-of-a-given-complexit.htmlI've tried to find answers on this but a lot of the questions seem focused on finding out the time complexity in Big O notation, I want to find the actual time. I was wondering how to find the run......Tue, 18 Aug 2026 13:51:51 -0400https://pop.com.sg/difference-between-continuity-and-uniform-continuity.htmlhttps://pop.com.sg/difference-between-continuity-and-uniform-continuity.htmlI understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book ...Tue, 18 Aug 2026 13:50:49 -0400https://pop.com.sg/is-the-present-king-of-france-is-bald-studied-by-maths.htmlhttps://pop.com.sg/is-the-present-king-of-france-is-bald-studied-by-maths.htmlIntuitively, "The present King of France is bald." is false. But Bertrand Russell said it would mean that "The present King of France is not bald.", which seems to be false. This apparently leads......Tue, 18 Aug 2026 13:50:30 -0400https://pop.com.sg/symbol-of-a-partition.htmlhttps://pop.com.sg/symbol-of-a-partition.htmlI have a series of subsets that form a set however each subset is actually a partition. Currently I use the conventional subset symbol when writing the appropriate notation and then note the subse......Tue, 18 Aug 2026 13:50:01 -0400https://pop.com.sg/buchberger-039-s-algorithm.htmlhttps://pop.com.sg/buchberger-039-s-algorithm.htmlI am trying to calculate a Gröbner basis for $I=\langle \mathcal{B}\rangle$, where $\mathcal{B}=\{f=x_3-x_1^5, g=x_2-x_1^3\}$, with respect both lexicographic and graded reverse lexicographic order......Tue, 18 Aug 2026 13:43:28 -0400https://pop.com.sg/min-and-max-of-a-product.htmlhttps://pop.com.sg/min-and-max-of-a-product.htmlLet $x_i\in X_i\subset \mathbb{R}$ so that each $X_i$ is a compact set with no isolated points for all $1\leq i\leq n$. Let: $a_i\in X_i$ so that $|x_i|\leq|a_i|$ for all $x_i\in X_i$. $b_i = \max\...Tue, 18 Aug 2026 13:35:40 -0400https://pop.com.sg/a-proportionality-puzzle-if-half-of-5-is-3-then-what-039-s-one-third-o.htmlhttps://pop.com.sg/a-proportionality-puzzle-if-half-of-5-is-3-then-what-039-s-one-third-o.htmlMy professor gave us this problem. In a foreign country, half of 5 is 3. Based on that same proportion, what's one-third of 10? I removed my try because it's wrong....Tue, 18 Aug 2026 13:34:08 -0400https://pop.com.sg/probability-of-of-an-event-happening-at-least-once-in-a-sequence-of-in.htmlhttps://pop.com.sg/probability-of-of-an-event-happening-at-least-once-in-a-sequence-of-in.htmlI want to find the probability of flipping heads at least once if you flip a coin two times. The possible outcomes (we don't care about the order) are (each equally likely) TT, TH, HT, HH. Three o......Tue, 18 Aug 2026 13:31:53 -0400https://pop.com.sg/integration-over-a-triangle.htmlhttps://pop.com.sg/integration-over-a-triangle.htmlLet $\Delta$ be a triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$ in ${\bf R}^2$. I want to compute $$I=\int\limits_\Delta x^2\mathrm{e}^{y^2}\;\mathrm{d}A.$$ This is what I've done so far: Not......Tue, 18 Aug 2026 13:31:13 -0400https://pop.com.sg/find-the-nth-power-of-a-matrix.htmlhttps://pop.com.sg/find-the-nth-power-of-a-matrix.htmlLet matrix A is $$A=\left (\begin{array}{rrr} 1&0&0 \\ 1&1&0\\ 0&0&1 \end{array}\right)$$ How can I find the 30 th power of A.. Is diagonalization possible? I found the......Tue, 18 Aug 2026 13:20:38 -0400https://pop.com.sg/dodecahedron-proof-on-number-of-vertices-edges-and-faces.htmlhttps://pop.com.sg/dodecahedron-proof-on-number-of-vertices-edges-and-faces.htmlFind, with a proof, the number of vertices, edges, and faces of a dodecahedron. Its is clear that their are $20$ vertices, $30$ edges, and $12$ faces. I am not sure how to prove this though....Tue, 18 Aug 2026 13:08:55 -0400https://pop.com.sg/can-the-poisson-distribution-be-used-to-find-the-expected-value-of-tim.htmlhttps://pop.com.sg/can-the-poisson-distribution-be-used-to-find-the-expected-value-of-tim.htmlMy understanding of the Poisson Distribution is that its PMF $P(x=k) = \dfrac {\lambda^k e^{-\lambda}} {k!}$ refers to the probability of finding k events given an expected arrival expectancy $\lam......Tue, 18 Aug 2026 13:08:41 -0400https://pop.com.sg/matrix-representations-of-parabola.htmlhttps://pop.com.sg/matrix-representations-of-parabola.htmlContinuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...Tue, 18 Aug 2026 12:59:22 -0400https://pop.com.sg/how-to-evaluate-the-following-integral.htmlhttps://pop.com.sg/how-to-evaluate-the-following-integral.html$$P(S)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(a^2+c^2+ \lambda ~ b^2)} \delta[S-\sqrt{4b^2+(a-c)^2}] ~da~db~bc.$$ where $\lambda$ is a constant. How to Evaluate......Tue, 18 Aug 2026 12:54:35 -0400https://pop.com.sg/what-is-the-equation-of-a-3d-cone-with-generalised-tilt.htmlhttps://pop.com.sg/what-is-the-equation-of-a-3d-cone-with-generalised-tilt.htmlWhat is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that......Tue, 18 Aug 2026 12:49:06 -0400https://pop.com.sg/the-definition-of-ln-x.htmlhttps://pop.com.sg/the-definition-of-ln-x.htmlWhen I taught my student the logarithm, he asked me about the historical definition of $\ln(x)$. The first definition I found is that $$\ln(x)=\int_{1}^{x}{ \frac{dt}{t} } $$ Defined as the loga......Tue, 18 Aug 2026 12:43:58 -0400https://pop.com.sg/how-many-different-arrangements-are-possible-with-3-digits-numbers-fro.htmlhttps://pop.com.sg/how-many-different-arrangements-are-possible-with-3-digits-numbers-fro.htmlif there are numbers 1,2,3,4,5,6,7,8,9 how many different three digit values we can make such that the digits are in ascending order? since there are numbers from 1-9, I understand that the first ......Tue, 18 Aug 2026 12:43:38 -0400https://pop.com.sg/get-the-inverse-of-ratio-what-is-the-mathematical-term.htmlhttps://pop.com.sg/get-the-inverse-of-ratio-what-is-the-mathematical-term.htmlI need to create kind of the inversed ratio. What i mean: Let's say there are the numbers 1,1,2 the ratio is 0.25,0.25,0.5. The sum of the ratio is of course 1. But i want the "inversed" ratio so ......Tue, 18 Aug 2026 12:34:58 -0400https://pop.com.sg/operation-that-is-commutative-but-not-associative-duplicate.htmlhttps://pop.com.sg/operation-that-is-commutative-but-not-associative-duplicate.htmlDoes there exist an operation that is commutative but not associative? do you know what is operation?...Tue, 18 Aug 2026 12:32:32 -0400https://pop.com.sg/how-to-calculate-the-surface-area-of-parametric-surface.htmlhttps://pop.com.sg/how-to-calculate-the-surface-area-of-parametric-surface.htmlSuppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2\leq 1$. My homework is forcing me to use the parameterization $$\textbf{r}_1(s,t)= <......Tue, 18 Aug 2026 12:25:53 -0400https://pop.com.sg/rational-solutions-for-x-2-y-2-3.htmlhttps://pop.com.sg/rational-solutions-for-x-2-y-2-3.htmlAre there any rational numbers $x$ and $y$ such that $x^2+y^2=3$. I think there are no rational solutions, but I haven't been able to prove it....Tue, 18 Aug 2026 12:23:35 -0400https://pop.com.sg/why-is-sphere-non-euclidean-space.htmlhttps://pop.com.sg/why-is-sphere-non-euclidean-space.htmlThere are 5 axioms that define euclidean space and I believe that all hold also for a sphere. The definition of axioms from wikipedia: "To draw a straight line from any point to any point." Defi......Tue, 18 Aug 2026 12:13:25 -0400https://pop.com.sg/size-of-eigenvalues-and-diagonal-dominance.htmlhttps://pop.com.sg/size-of-eigenvalues-and-diagonal-dominance.htmlIf $X$ is a diagonally dominant matrix and if $Y$ is a diagonal matrix with the diagonal elements of $X$, then how can one show that $ Y^{-1}(Y - X)$ has eigenvalues whose magnitudes are all strictly ...Tue, 18 Aug 2026 12:03:01 -0400https://pop.com.sg/finding-roots-of-z-3-8.htmlhttps://pop.com.sg/finding-roots-of-z-3-8.htmlI am having trouble finding the cubed roots of $8$ as a complex number. $$z^3 = 8+0i$$ $$z^3 = r^3 e^{3\theta i}=8e^{2i\pi k},\quad k\in \Bbb Z$$ $$\implies r=2,3\theta = 2\pi k\implies \theta = ......Tue, 18 Aug 2026 11:56:00 -0400https://pop.com.sg/when-to-stop-checking-if-a-transition-matrix-is-regular.htmlhttps://pop.com.sg/when-to-stop-checking-if-a-transition-matrix-is-regular.htmlThe definition that I have of a Transition Matrix for a Markov Chain is: A transition matrix is regular if some power of it is positive. Doesn't this mean though that in theory, you could keep ...Tue, 18 Aug 2026 11:53:03 -0400https://pop.com.sg/how-to-show-that-if-f-2-x-is-uniformly-continous-function-then-f-is-un.htmlhttps://pop.com.sg/how-to-show-that-if-f-2-x-is-uniformly-continous-function-then-f-is-un.html$f:R\to [0,\infty)$ is function such that $f^2(x)$ is uniformly continuous on R then I have to show that f is uniformly continuous ? My attempt : $|f^2(x)-f^2(y)|<\epsilon$ for $|x-y|<\del......Tue, 18 Aug 2026 11:45:35 -0400https://pop.com.sg/what-is-the-limit-of-lim-x-1-x-1-x-when-x-approaches-infinity-without-.htmlhttps://pop.com.sg/what-is-the-limit-of-lim-x-1-x-1-x-when-x-approaches-infinity-without-.htmlthis is my first post. I've been struggling whit this limit for too long (without using l'Hôpital's rule): $$\lim_{x\to {\infty}} \left(\frac{x-1}{x+1}\right)^x$$ My answer is $\frac1e$, but the ...Tue, 18 Aug 2026 11:35:18 -0400https://pop.com.sg/how-to-find-net-torque-closed.htmlhttps://pop.com.sg/how-to-find-net-torque-closed.htmlFor the meter stick in Figure 10-4, what is the magnitude of the net torque due to both forces $F_1$ and $F_2$ about an axis perpendicular to the page through point $A$? Is it clockwise or ...Tue, 18 Aug 2026 11:31:13 -0400https://pop.com.sg/proving-lim-x-n-a-n-as-x-goes-to-a-from-definition.htmlhttps://pop.com.sg/proving-lim-x-n-a-n-as-x-goes-to-a-from-definition.htmlUsing the $\varepsilon-\delta$ definition show that $\displaystyle \lim_{x \to a} x^n = a^n$ What I've done? Pretty much stuck on the step where one usually constructs delta: $\displaystyle |x^......Tue, 18 Aug 2026 11:28:56 -0400https://pop.com.sg/proof-that-there-are-infinitely-many-primes-congruent-to-1-mod-3.htmlhttps://pop.com.sg/proof-that-there-are-infinitely-many-primes-congruent-to-1-mod-3.htmlNeed a quick check on this proof. Show that there are infinitely many primes congruent to 1 mod 3. Proof: Suppose by contradiction finitely many primes $p≡1\pmod{3}$ say $p_1,...,p_k$. Define $m......Tue, 18 Aug 2026 11:28:22 -0400https://pop.com.sg/notation-convention-for-integer-in-a-certain-range.htmlhttps://pop.com.sg/notation-convention-for-integer-in-a-certain-range.htmlI am wondering what notation I should use when writing that some variable is an integer within some range. What is the most common way to do this? Here are some ideas I have but I'm not sure what the ...Tue, 18 Aug 2026 11:12:06 -0400https://pop.com.sg/calculating-the-divergence-of-a-tensor.htmlhttps://pop.com.sg/calculating-the-divergence-of-a-tensor.htmlI am working through a fluid dynamics paper and came across this equation: $$ \frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}=\nabla\cdot T - \frac{1}{\rho}\nabla \phi\tag1$$ where ......Tue, 18 Aug 2026 11:09:36 -0400https://pop.com.sg/find-the-derivative-of-y-2-x-as-a-function-of-y.htmlhttps://pop.com.sg/find-the-derivative-of-y-2-x-as-a-function-of-y.htmlFind the derivative of $$y^2=x$$ as a function of y. i have found for the function of x, it will be $$\pm\frac{1}{2\sqrt x}$$ however for the function of y will be $$\frac{dy}{dx}=2y$$ ? it loo......Tue, 18 Aug 2026 11:06:19 -0400https://pop.com.sg/where-does-b-2a-come-from.htmlhttps://pop.com.sg/where-does-b-2a-come-from.htmlIve came across the idea to find the minimum/maximum of a parabola , I keep running into the formula $-b/2a$ i wanted to know where does this actually come from ? I know that the parabola has to be ...Tue, 18 Aug 2026 11:03:44 -0400https://pop.com.sg/how-can-you-prove-that-1-5-9-cdots-4n-3-2n-2-n-without-using-induction.htmlhttps://pop.com.sg/how-can-you-prove-that-1-5-9-cdots-4n-3-2n-2-n-without-using-induction.htmlUsing mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result wi......Tue, 18 Aug 2026 10:50:29 -0400https://pop.com.sg/how-to-simplify-a-sum-of-exponential-equation.htmlhttps://pop.com.sg/how-to-simplify-a-sum-of-exponential-equation.htmlSuppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ......Tue, 18 Aug 2026 10:49:36 -0400https://pop.com.sg/prove-that-if-f-n-in-o-g-n-then-g-n-in-omega-f-n.htmlhttps://pop.com.sg/prove-that-if-f-n-in-o-g-n-then-g-n-in-omega-f-n.htmlProve that if $f(n) \in O(g(n))$ then $g(n) \in \Omega(f(n))$. So I know that $$O(g(n)) \Rightarrow f(n) \leq c\cdot g(n)$$ and that $$\Omega(g(n)) \Rightarrow f(n) \geq c\cdot g(n)$$ but how ......Tue, 18 Aug 2026 10:47:54 -0400https://pop.com.sg/what-is-the-practical-application-of-factorials.htmlhttps://pop.com.sg/what-is-the-practical-application-of-factorials.htmlI'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer. I understand that a factorial of n items......Tue, 18 Aug 2026 10:33:18 -0400https://pop.com.sg/natural-deduction-proof-problem-with-existential-elimination.htmlhttps://pop.com.sg/natural-deduction-proof-problem-with-existential-elimination.htmlI have problems with the following proof: $$ \forall x \exists y (Rxy \land Py), \forall x \neg Rxx \vdash \neg \forall x \forall y (Px \implies (Py \implies x=y)) $$ The problem is with the applic......Tue, 18 Aug 2026 10:30:16 -0400https://pop.com.sg/25-chance-of-happening-but-has-to-happen-4-times-in-a-row-what-is-the-.htmlhttps://pop.com.sg/25-chance-of-happening-but-has-to-happen-4-times-in-a-row-what-is-the-.htmlSay you need to win 4 times in a row at a 25% success chance for each event, how do I calculate the chance of winning? Overall, how does this calculation goes? given that 1st is a win, what is the ......Tue, 18 Aug 2026 10:16:05 -0400https://pop.com.sg/difference-between-state-space-and-sample-space.htmlhttps://pop.com.sg/difference-between-state-space-and-sample-space.htmlAren't they both the same? According to me they are just different terminologies used for different types of systems. What I understand is that sample space is used when we talk about static system......Tue, 18 Aug 2026 10:08:55 -0400https://pop.com.sg/which-value-goes-on-which-axis-in-plot-a-against-b.htmlhttps://pop.com.sg/which-value-goes-on-which-axis-in-plot-a-against-b.htmlWhen the question asks to plot $a$ against $b$, which value should go on which axis?...Tue, 18 Aug 2026 10:07:49 -0400https://pop.com.sg/can-039-t-understand-a-recursive-definition-of-concatenation-of-two-st.htmlhttps://pop.com.sg/can-039-t-understand-a-recursive-definition-of-concatenation-of-two-st.htmlI'm reading Rosen's Discrete Mathematics and its applications(6ed), but I can't understand a recursive definition about concatenation of two strings: Two strings can be combined via the operatio......Tue, 18 Aug 2026 09:44:44 -0400https://pop.com.sg/probability-of-getting-a-triple-when-rolling-12-fair-12-sided-dice-onc.htmlhttps://pop.com.sg/probability-of-getting-a-triple-when-rolling-12-fair-12-sided-dice-onc.htmlThere are $12^{12}$ = 8,916,100,448,256 total possible outcomes. I decided to approach it by calculating the probability of rolling the $12$ dice WITHOUT getting a triple and subtract from $1$. I f......Tue, 18 Aug 2026 09:27:04 -0400https://pop.com.sg/what-is-x-times-sqrt-x.htmlhttps://pop.com.sg/what-is-x-times-sqrt-x.htmlDoes $x \times \sqrt x = x$? I thought it was correct because sqrt is the opposite of multiplying by a number, so I figured by multiplying by a number it would balance out and be that number norm......Tue, 18 Aug 2026 09:26:15 -0400https://pop.com.sg/representing-ln-x-as-a-power-series-centered-at-2-without-computing-an.htmlhttps://pop.com.sg/representing-ln-x-as-a-power-series-centered-at-2-without-computing-an.htmlI am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old ser......Tue, 18 Aug 2026 09:11:34 -0400https://pop.com.sg/what-is-the-number-of-numbers.htmlhttps://pop.com.sg/what-is-the-number-of-numbers.htmlFor a positive integer $n≥2$, let $f(n)$ denote the smallest prime divisor of $n$. Find the number of integers $n$, $2≤n≤1000$, such that $f(n)=5$. All the numbers I could think of were $5,25,35,55......Tue, 18 Aug 2026 09:07:29 -0400https://pop.com.sg/prove-the-trigonometric-identities-cot-x-cot-2x-csc-2x.htmlhttps://pop.com.sg/prove-the-trigonometric-identities-cot-x-cot-2x-csc-2x.htmlProve $\cot(x) - \cot(2x) =\csc(2x)$. I start to solve from LHS, and change all the terms into $\sin$ and $\cos$, but I could not prove it into $\csc(2x)$....Tue, 18 Aug 2026 09:03:45 -0400https://pop.com.sg/express-the-following-complex-numbers-in-the-form-of-a-bi.htmlhttps://pop.com.sg/express-the-following-complex-numbers-in-the-form-of-a-bi.htmlI'm struggling with the following exercises: I tried to use the reasoning as follows: $$(a+bi)^n=(re^{\theta i})^n=r^ne^{\theta in}=r^n(\cos(\theta n)+i\sin(\theta n))$$ So for the first one I d......Tue, 18 Aug 2026 09:00:07 -0400https://pop.com.sg/the-difference-of-two-consecutive-perfect-squares-is-always-odd.htmlhttps://pop.com.sg/the-difference-of-two-consecutive-perfect-squares-is-always-odd.htmlI am working on another homework assignment about proofs. The question is: Prove or find counterexample: the difference of two consecutive perfect squares is odd? There is no counterexample corre......Tue, 18 Aug 2026 08:35:54 -0400https://pop.com.sg/i-ab-be-invertible-leftrightarrow-i-ba-is-invertible-duplicate.htmlhttps://pop.com.sg/i-ab-be-invertible-leftrightarrow-i-ba-is-invertible-duplicate.htmlassume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance...Tue, 18 Aug 2026 08:34:19 -0400https://pop.com.sg/do-the-taylor-series-of-sin-x-and-cos-x-depend-on-the-identity-sin-2-x.htmlhttps://pop.com.sg/do-the-taylor-series-of-sin-x-and-cos-x-depend-on-the-identity-sin-2-x.htmlI had this crazy idea trying to prove the Pythagorean trigonometric identity;$$\sin^2x+\cos^2x=1$$by squaring the infinite Taylor series of $\sin x$ and $\cos x$. But it came out quite beautiful, ...Tue, 18 Aug 2026 08:30:23 -0400https://pop.com.sg/help-with-proof-of-injection-and-surjection.htmlhttps://pop.com.sg/help-with-proof-of-injection-and-surjection.htmlFor the record, I am sorry, I haven't yet learnt how to use LaTeX I have a function $f(x) = 2x^3 - 1$ My proof of injection is as follows: $f$ is one to one for all $x_1,x_2$ element $X$, if $f(x......Tue, 18 Aug 2026 08:14:51 -0400https://pop.com.sg/relationship-between-primes-and-fibonacci-sequence.htmlhttps://pop.com.sg/relationship-between-primes-and-fibonacci-sequence.htmlI recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connect......Tue, 18 Aug 2026 07:58:37 -0400https://pop.com.sg/integral-definition-of-e.htmlhttps://pop.com.sg/integral-definition-of-e.htmlI know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...Tue, 18 Aug 2026 07:54:49 -0400https://pop.com.sg/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlhttps://pop.com.sg/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlProblem : How many zeros are in all natural numbers from $1$ to $2019$ ? My attempt : From $1$ to $100$ we have $11$ zeros Now from $101$ to $110$ we have $10$ zeros So from $20$ zeros ...Tue, 18 Aug 2026 07:43:23 -0400https://pop.com.sg/optimizing-svd-factors.htmlhttps://pop.com.sg/optimizing-svd-factors.htmlProblem Description My aim is to find the SVD decomposition of the minimizer of $F:\mathbb{R}^{m\times n}\to\mathbb{R}$ $$ U_*\Sigma_* V^\top_* = A_*\qquad \text{where} \qquad A_* = \arg\min_A F(A)......Tue, 18 Aug 2026 07:37:53 -0400https://pop.com.sg/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlhttps://pop.com.sg/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlIf $A$ triangle's side length are $25,25$ and $30$ and $B$ triangle's length are $25,25$ and $40$ what is the ratio between the two areas of the triangles? #math...Tue, 18 Aug 2026 07:36:12 -0400https://pop.com.sg/i-have-a-doubt-in-roman-numericals.htmlhttps://pop.com.sg/i-have-a-doubt-in-roman-numericals.htmlThe below question, I am looking at the 80, They translated it as LXXX, but I can also write it as XXC right?...Tue, 18 Aug 2026 07:20:21 -0400https://pop.com.sg/how-do-inverse-functions-exist-for-exponential-functions.htmlhttps://pop.com.sg/how-do-inverse-functions-exist-for-exponential-functions.htmlI know that they exist for exponential functions (we currently have them in class), but to me it doesn't seem "reasonable" when I look at the definition of what an inverse function is. The inverse is ...Tue, 18 Aug 2026 07:14:30 -0400