https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 12 Aug 2026 05:52:31 -0400https://pop.com.sg/when-is-it-allowed-to-take-apart-a-limit-multiplication-of-limits.htmlhttps://pop.com.sg/when-is-it-allowed-to-take-apart-a-limit-multiplication-of-limits.htmlWhen is it allowed to "take apart" a limit? Here's an example to show what I mean: $\displaystyle\lim_{n\to\infty}\frac{\frac 1 ne^{\frac 1n}(e-1)}{e^{\frac 1 n}-1}=e-1$ since we can "take apart"......Wed, 12 Aug 2026 09:46:09 -0400https://pop.com.sg/what-do-you-get-when-you-take-the-square-root-of-a-negative-imaginary-.htmlhttps://pop.com.sg/what-do-you-get-when-you-take-the-square-root-of-a-negative-imaginary-.htmlSimply, what do I get when I take the square root of negative imaginary number? But, it cant be an imaginary number since the answer to the $2^{\mathrm{nd}}$ power must equal the original negative ...Wed, 12 Aug 2026 09:41:40 -0400https://pop.com.sg/prove-that-sin-x-cos-x-sqrt2-sin-left-x-frac-pi4-right.htmlhttps://pop.com.sg/prove-that-sin-x-cos-x-sqrt2-sin-left-x-frac-pi4-right.html$$\sqrt2\sin\left(x+\frac\pi4\right)=\sqrt2\left(\sin(x)\cos\left(\frac\pi4\right)+\cos(x)\sin\left(\frac\pi4\right)\right)=\sin(x)+\cos(x)$$ Could you solve it from opposite? $$\sin(x)+\cos (x)=\s......Wed, 12 Aug 2026 09:37:11 -0400https://pop.com.sg/why-we-use-multiplication-rule-in-an-independent-event-in-probability.htmlhttps://pop.com.sg/why-we-use-multiplication-rule-in-an-independent-event-in-probability.htmlI know the independent concept and multiplication rules. Example Flip Coin 5 times flip, how many times success of Head? Ans : $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac......Wed, 12 Aug 2026 09:21:54 -0400https://pop.com.sg/limit-of-the-ratio-of-consecutive-fibonacci-numbers-duplicate.htmlhttps://pop.com.sg/limit-of-the-ratio-of-consecutive-fibonacci-numbers-duplicate.htmlI have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...Wed, 12 Aug 2026 09:11:01 -0400https://pop.com.sg/definition-of-continuity.htmlhttps://pop.com.sg/definition-of-continuity.htmlIt has been a year or so I took my course of real analysis, still could not understand these two definitions of continuity-(These two definitions are given as chapter 9 and 10 in the classroom reso......Wed, 12 Aug 2026 09:05:49 -0400https://pop.com.sg/what-is-the-meaning-of-evaluating-the-divergence-at-a-point.htmlhttps://pop.com.sg/what-is-the-meaning-of-evaluating-the-divergence-at-a-point.htmlReading this first, Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative ...Wed, 12 Aug 2026 09:02:27 -0400https://pop.com.sg/create-unique-number-from-2-numbers.htmlhttps://pop.com.sg/create-unique-number-from-2-numbers.htmlis there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result mu......Wed, 12 Aug 2026 09:01:44 -0400https://pop.com.sg/intuitive-proof-of-multivariable-changing-of-variables-formula-jacobia.htmlhttps://pop.com.sg/intuitive-proof-of-multivariable-changing-of-variables-formula-jacobia.htmlWhat is an intuitive proof of the multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory? I think that textbooks overcomplicate the proof. If possible, use ...Wed, 12 Aug 2026 09:00:25 -0400https://pop.com.sg/understanding-e-and-e-to-the-power-of-imaginary-number.htmlhttps://pop.com.sg/understanding-e-and-e-to-the-power-of-imaginary-number.htmlHow did the value of $e$ come from compound interest equation. What does the value of $e$ really mean... Capacitors and inductors charge and discharge exponentially, radioactive elements decay ...Wed, 12 Aug 2026 08:54:14 -0400https://pop.com.sg/suppose-that-a-and-b-are-positive-integers-such-that-a-2-3b-2-show-tha.htmlhttps://pop.com.sg/suppose-that-a-and-b-are-positive-integers-such-that-a-2-3b-2-show-tha.html...Use these and the Well-Ordering Principle to prove that no such $a$ and $b$ exist. From this it follows that $\sqrt{3} \notin \mathbb{Q}$ I have no idea where well-ordering principle comes in for ...Wed, 12 Aug 2026 08:46:55 -0400https://pop.com.sg/find-the-infinite-sum-of-the-series-sum-n-1-infty-frac-1-n-2-1.htmlhttps://pop.com.sg/find-the-infinite-sum-of-the-series-sum-n-1-infty-frac-1-n-2-1.htmlThis is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea......Wed, 12 Aug 2026 08:44:39 -0400https://pop.com.sg/why-the-complex-number-6-sin-240-circ-i-cos-240-circ-lies-in-first-qua.htmlhttps://pop.com.sg/why-the-complex-number-6-sin-240-circ-i-cos-240-circ-lies-in-first-qua.htmlproblem : find the quadrant in which $6(\sin(240^{\circ}) + i \cos(240^{\circ}))$ lies and find the principal argument then rewrite the complex number my attempt : $6(\sin(240^{\circ}) + i \co......Wed, 12 Aug 2026 08:12:35 -0400https://pop.com.sg/second-derivative-of-a-polar-coordinate.htmlhttps://pop.com.sg/second-derivative-of-a-polar-coordinate.htmlhttps://snag.gy/buiJve.jpg I'm not sure what the issue is, but I've calculated this several times and have been unable to produce the correct answer. I have followed the procedures. I went to cal......Wed, 12 Aug 2026 08:09:52 -0400https://pop.com.sg/prove-n-is-o-n-log-n-closed.htmlhttps://pop.com.sg/prove-n-is-o-n-log-n-closed.htmlIn order to prove that $n$ is $O(n\log n)$, as per my understanding if we have to say $f(n)$ is $O(g(n))$ then $\lim\limits_{n \to \infty} \frac{f(n)}{g(n)}= C$ Then in that case when I am taking......Wed, 12 Aug 2026 07:44:49 -0400https://pop.com.sg/how-to-intuitively-understand-eigenvalue-and-eigenvector.htmlhttps://pop.com.sg/how-to-intuitively-understand-eigenvalue-and-eigenvector.htmlI'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman. Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to unde......Wed, 12 Aug 2026 07:41:51 -0400https://pop.com.sg/to-convert-rotation-vector-to-rotation-matrix-the-rotation-vector-must.htmlhttps://pop.com.sg/to-convert-rotation-vector-to-rotation-matrix-the-rotation-vector-must.htmlI have a following formula to convert rotation vector($K\in \Bbb R^3$) to rotation matrix ($R \in SO(3)$) where $I$ is an identity matrix and K could be uniquely acquired from the conversion of an ......Wed, 12 Aug 2026 07:41:25 -0400https://pop.com.sg/what-does-it-mean-to-be-affinely-independent-and-why-is-it-important-t.htmlhttps://pop.com.sg/what-does-it-mean-to-be-affinely-independent-and-why-is-it-important-t.htmlI was studying linear optimization and i saw the term Affine independence. I came across this http://www.cis.upenn.edu/~cis610/geombchap2.pdf while trying to get a better understanding of the topic. ...Wed, 12 Aug 2026 07:38:35 -0400https://pop.com.sg/2d-rubik-039-s-cube.htmlhttps://pop.com.sg/2d-rubik-039-s-cube.htmlThere is a $3\times3$ matrix filled by numbers 1~9 that might look like this $$\begin{bmatrix}3 & 8 & 2 \\ 4 & 1 & 6 \\ 7 & 5 & 9\end{bmatrix}$$ All its rows and columns c......Wed, 12 Aug 2026 07:37:58 -0400https://pop.com.sg/solve-x-2-5x-3-x-2-x-in-mathbb-r.htmlhttps://pop.com.sg/solve-x-2-5x-3-x-2-x-in-mathbb-r.htmlSolve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ $x^2......Wed, 12 Aug 2026 07:31:33 -0400https://pop.com.sg/rate-of-change-of-cylindrical-roll-039-s-volume-as-it-unrolls.htmlhttps://pop.com.sg/rate-of-change-of-cylindrical-roll-039-s-volume-as-it-unrolls.htmlThis is purely a "for-fun" question. I was minding my own business in the washroom this morning when I began to unroll some toilet paper from the roll, and in typical Breaking Bad fashion (sorry if......Wed, 12 Aug 2026 07:22:31 -0400https://pop.com.sg/what-did-tao-prove-in-the-paper-almost-all-orbits-of-the-collatz-map-a.htmlhttps://pop.com.sg/what-did-tao-prove-in-the-paper-almost-all-orbits-of-the-collatz-map-a.htmlLast year, Terence Tao published a paper entitled "Almost All Orbits of the Collatz Map Attain Almost Bounded Values" (via arXiv). In layman's terms, could somebody please explain what this ...Wed, 12 Aug 2026 07:17:53 -0400https://pop.com.sg/equation-for-line-parallel-to-z-axis-and-intersects-x-axis-at-x-k-y-0-.htmlhttps://pop.com.sg/equation-for-line-parallel-to-z-axis-and-intersects-x-axis-at-x-k-y-0-.htmlHow would I pick $a,b,c$ to create a line that is parallel to $z$-axis and intersects $x$-axis at point $x=k, y=0, z=0$? $$ ax+by+cz=d $$...Wed, 12 Aug 2026 07:12:32 -0400https://pop.com.sg/derivative-method-of-proving-of-sqrt-x-ln-x.htmlhttps://pop.com.sg/derivative-method-of-proving-of-sqrt-x-ln-x.htmlI have a question about a proof I read, linked here, a proof that $\sqrt{x}>\ln x$. The second answerer considers the derivative of the function,$f\left(x\right)=\sqrt{x}-\ln x$. Now the deriv......Wed, 12 Aug 2026 07:09:47 -0400https://pop.com.sg/why-is-the-maximum-rayleigh-quotient-equal-to-the-maximum-eigenvalue.htmlhttps://pop.com.sg/why-is-the-maximum-rayleigh-quotient-equal-to-the-maximum-eigenvalue.html(Note: I'm only interested in real-valued matrices here, so I'm using "transpose" and "symmetric" instead of the more general "transjugate" and "Hermitian" in the hope that it will simplify the pro......Wed, 12 Aug 2026 06:57:15 -0400https://pop.com.sg/roll-1-die-and-square-or-roll-2-dice-and-multiply-which-has-higher-mea.htmlhttps://pop.com.sg/roll-1-die-and-square-or-roll-2-dice-and-multiply-which-has-higher-mea.htmlI am wondering if there is a fast way to tell which expected value is higher: "Roll 1 die and take the square of the number that comes up or roll 2 dice and multiply them". Thank you!...Wed, 12 Aug 2026 06:46:51 -0400https://pop.com.sg/circular-arrangements-at-a-round-table.htmlhttps://pop.com.sg/circular-arrangements-at-a-round-table.htmlQ: If six people, designated as $A,B,...,F,$ are seated about a round table how many different circular arrangements are possible, if arrangements are considered the same when one can be obtained f......Wed, 12 Aug 2026 06:46:31 -0400https://pop.com.sg/find-the-value-of-k-if-the-roots-of-an-equation-differ-by-2.htmlhttps://pop.com.sg/find-the-value-of-k-if-the-roots-of-an-equation-differ-by-2.htmlI need some help, Im trying to solve the below but to no avail, would appreciate your guidance :) Find the value of $k$ if the roots of $$3x^2+5x-k=0,$$ differ by two....Wed, 12 Aug 2026 06:43:57 -0400https://pop.com.sg/sum-of-2-consecutive-integers-is-x-then-their-product-will-be-closed.htmlhttps://pop.com.sg/sum-of-2-consecutive-integers-is-x-then-their-product-will-be-closed.htmlAccording to a source, the answer is $x^2-\frac{1}{4}$ Please explain...Wed, 12 Aug 2026 06:43:26 -0400https://pop.com.sg/integral-from-zero-to-zero.htmlhttps://pop.com.sg/integral-from-zero-to-zero.htmlit seems obvious that this integral is zero and so is the limit but what theorem we are using here? I see it's connected to Riemann sums with an interval=zero Right ? The function $\mathrm{f}$ is ...Wed, 12 Aug 2026 06:42:15 -0400https://pop.com.sg/is-there-a-way-to-solve-for-an-unknown-in-a-factorial.htmlhttps://pop.com.sg/is-there-a-way-to-solve-for-an-unknown-in-a-factorial.htmlI don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1. $n! = \text {a really big number}$ Ex. $n! = 9999999$ Is there a way to ...Wed, 12 Aug 2026 06:41:52 -0400https://pop.com.sg/at-what-rate-is-the-angle-theta-changing-when-10-ft-of-rope-is-out.htmlhttps://pop.com.sg/at-what-rate-is-the-angle-theta-changing-when-10-ft-of-rope-is-out.htmlA dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft. above the bow. The rope is hauled in a rate of 2 ft/s. At what rate is the angle $\theta$ changing when 10 ......Wed, 12 Aug 2026 06:31:28 -0400https://pop.com.sg/how-to-demonstrate-derivability.htmlhttps://pop.com.sg/how-to-demonstrate-derivability.htmlIf given a function $f$ that I am supposed to demonstrate is derivable, is it enough for me to try to compute its derivative $f'$ and if the result is continuous, can I then state that the orig......Wed, 12 Aug 2026 06:23:22 -0400https://pop.com.sg/creating-a-bijection-to-check-if-graphs-are-isomorphic.htmlhttps://pop.com.sg/creating-a-bijection-to-check-if-graphs-are-isomorphic.htmlTo prove that two graphs are isomorphic I was taught to first consider the bijection between the two graphs. I was never taught however the rules when coming up with the bijection. Is my only rule......Wed, 12 Aug 2026 06:20:45 -0400https://pop.com.sg/how-to-find-critical-points-of-multidimensional-function.htmlhttps://pop.com.sg/how-to-find-critical-points-of-multidimensional-function.htmlI have this function: $$ f(x,y) = x^2 - 2xy+ 4y^3$$ I calculated the gradient without problems: $$\nabla f(x,y) = \left(2x-2y , -2x + 12y^3\right)^T$$ This is where kind of got stuck, I know f......Wed, 12 Aug 2026 06:16:48 -0400https://pop.com.sg/find-the-number-of-ordered-pairs-of-positive-integers-x-y-that-satisfy.htmlhttps://pop.com.sg/find-the-number-of-ordered-pairs-of-positive-integers-x-y-that-satisfy.htmlafter some simplification, we get (2004y)/(y-2004)=x and by looking at the prime factorization of 2004 we observe there are only 12 ways in which 2004 can be expressed as the product of two intege......Wed, 12 Aug 2026 06:07:12 -0400https://pop.com.sg/prove-that-a-set-is-countable.htmlhttps://pop.com.sg/prove-that-a-set-is-countable.htmlShow using a proper theorem that the set {2, 3, 4, 8, 9, 16, 27, 32, 64, 81, … } is a countable set. Im lost, this is for school, but there is a huge language barrier between students and professo......Wed, 12 Aug 2026 06:07:03 -0400https://pop.com.sg/finding-the-points-of-intersection-in-polar-equations.htmlhttps://pop.com.sg/finding-the-points-of-intersection-in-polar-equations.htmlI must find the area of the region that lies inside the first curve and outside the second curve for: $ r^2 = 8cos2\theta, r = 2$ How do I find $a$ and $b$? My solution manual tells me that the c......Wed, 12 Aug 2026 06:05:45 -0400https://pop.com.sg/calculating-flux-across-s-a-graph-of-function-g-x-y-over-unit-disc-in-.htmlhttps://pop.com.sg/calculating-flux-across-s-a-graph-of-function-g-x-y-over-unit-disc-in-.htmlLet $~F = \langle x,y,z \rangle~$ be a vector field on $ R^3$ . Let $S$ be the graph of $ g(x,y)=1- y^2 + x^2 $ over the unit disc $D$ in $ R^2$ and let $S$ be equipped with the upward orientation. ...Wed, 12 Aug 2026 05:54:52 -0400https://pop.com.sg/proving-absolute-value-inequalities.htmlhttps://pop.com.sg/proving-absolute-value-inequalities.htmlI need help proving the last two cases for the following inequality: $\bigl|\lvert x\rvert-\lvert y\rvert\bigr| \le \lvert x-y\rvert$. Case 1: $x > 0$ and $y > 0$: the inequality simplif......Wed, 12 Aug 2026 05:49:36 -0400https://pop.com.sg/why-combining-two-quadratic-equations-of-a-circle-and-a-parabola-creat.htmlhttps://pop.com.sg/why-combining-two-quadratic-equations-of-a-circle-and-a-parabola-creat.htmlWe have a parabola and a circle with the following equations and their graph placed at the end of my question. Parabola: $y^2 = 4x -4$ Circle: $(x-2)^2 + y^2 = 9$ My goal was to calculate their ...Wed, 12 Aug 2026 05:39:50 -0400https://pop.com.sg/is-it-true-that-if-a-b-c-d-then-ad-bc.htmlhttps://pop.com.sg/is-it-true-that-if-a-b-c-d-then-ad-bc.htmlIs it true that, for all positive $A$, $B$, $C$, $D$, if $A-B>C-D$ then $AD>BC$ ? I'm just wondering about this, and I tested it with a bunch of integers and haven't found a counterexample y......Wed, 12 Aug 2026 05:39:49 -0400https://pop.com.sg/complex-numbers-converting-to-polar-form.htmlhttps://pop.com.sg/complex-numbers-converting-to-polar-form.htmlI understand the basics of converting to polar form but I have just come across a question that I haven't seen before. Usually the complex number is expressed as $z=a+bi$, but this time the complex ...Wed, 12 Aug 2026 05:29:40 -0400https://pop.com.sg/value-of-sin-pi-18-by-nested-radicals.htmlhttps://pop.com.sg/value-of-sin-pi-18-by-nested-radicals.htmlI recently asked a question about how to find value of $\sin(\pi/18)$ and I understand that there is no expression for $\sin(\pi/18)$ that uses the ordinary arithmetical operations. but I know that......Wed, 12 Aug 2026 05:25:11 -0400https://pop.com.sg/solving-the-ode-x-2-1-y-039-039-2xy-039-2y-x-2-1-2.htmlhttps://pop.com.sg/solving-the-ode-x-2-1-y-039-039-2xy-039-2y-x-2-1-2.htmlI want to solve this ODE: $$(x^2 - 1)y'' - 2xy' + 2y = (x^2 - 1)^2.$$ I found out that $y_1 = x$ and $y_2 = x^2+1$ are solutions of the associated homogeneous equation, $x$ by inspecting, an......Wed, 12 Aug 2026 05:21:16 -0400https://pop.com.sg/show-that-for-all-real-numbers-a-and-b-ab-le-1-2-a-2-b-2-duplicate.htmlhttps://pop.com.sg/show-that-for-all-real-numbers-a-and-b-ab-le-1-2-a-2-b-2-duplicate.htmlso as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b\in \mathbb R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ......Wed, 12 Aug 2026 05:15:22 -0400https://pop.com.sg/using-square-area-finding-quadrilateral-area.htmlhttps://pop.com.sg/using-square-area-finding-quadrilateral-area.htmlArea of square ABCD is 169 and that of square EFGD is 49. Find area of quadrilateral FBCG I am stuck and just thinking which way can be helpful for me finding this area of quadrilateral FBCG. Inst......Wed, 12 Aug 2026 05:11:23 -0400https://pop.com.sg/difference-between-factorization-theorem-and-fischer-neymann-theorem-f.htmlhttps://pop.com.sg/difference-between-factorization-theorem-and-fischer-neymann-theorem-f.htmlDifference between Factorization theorem and Fischer-Neymann theorem for t to be sufficient estimator of $\theta$ Factorization theorem says for parameter t to be sufficient statistic of $\theta$,......Wed, 12 Aug 2026 05:09:15 -0400https://pop.com.sg/where-039-s-the-error-in-this-2-1-fake-proof-duplicate.htmlhttps://pop.com.sg/where-039-s-the-error-in-this-2-1-fake-proof-duplicate.htmlI'm reading Spivak's Calculus: 2 What's wrong with the following "proof"? Let $x=y$. Then $$x^2=xy\tag{1}$$ $$x^2-y^2=xy-y^2\tag{2}$$ $$(x+y)(x-y)=y(x-y)\tag{3}$$ $$x+y=y\tag{4}$$ $$2y=y......Wed, 12 Aug 2026 04:57:55 -0400https://pop.com.sg/total-injective-relations-are-always-functions.htmlhttps://pop.com.sg/total-injective-relations-are-always-functions.htmlI was watching the following video, and around 6:12, he says "Whenever there is a total injective relation, there is always a total injective function" as well. https://ocw.mit.edu/courses/electr......Wed, 12 Aug 2026 04:56:35 -0400https://pop.com.sg/is-there-a-way-to-solve-sin-x-x.htmlhttps://pop.com.sg/is-there-a-way-to-solve-sin-x-x.htmlNote: Question was originally to solve it algebraically, though I've decided to change it to analytically due to the comments and answers. When trying to solve $\sin(x)=x$, the obvious first solut......Wed, 12 Aug 2026 04:53:31 -0400https://pop.com.sg/euler-and-the-factorial-function.htmlhttps://pop.com.sg/euler-and-the-factorial-function.htmlI recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that "Euler observed that $\...Wed, 12 Aug 2026 04:51:05 -0400https://pop.com.sg/asymptote-of-a-line.htmlhttps://pop.com.sg/asymptote-of-a-line.htmlIf a rational function is defined as $f(x) = \frac{p(x)} {q(x)}$ for any two polynomials $p(x)$ and $q(x)$; given $p(x) = 2x+1$ (polynomial of degree 1) and $q(x) = 3$ (polynomial of degree zero), ......Wed, 12 Aug 2026 04:50:22 -0400https://pop.com.sg/properties-of-triangles-that-have-3-equal-120-degree-angles-around-an-.htmlhttps://pop.com.sg/properties-of-triangles-that-have-3-equal-120-degree-angles-around-an-.htmlWhere is the point in a triangle, that when connected to the three vertices forms equal angles of 120 degrees? Does this point exist in all triangles? For a triangle with side lengths of 2,4,5, this ...Wed, 12 Aug 2026 04:40:22 -0400https://pop.com.sg/projection-of-ellipsoid.htmlhttps://pop.com.sg/projection-of-ellipsoid.htmlFind the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in adva......Wed, 12 Aug 2026 04:23:59 -0400https://pop.com.sg/what-is-a-covering-set-of-a-sierpinski-number-what-does-it-do.htmlhttps://pop.com.sg/what-is-a-covering-set-of-a-sierpinski-number-what-does-it-do.htmlRecently a new prime number has been discovered, which eliminates one of the six remaining candidates for the smallest Sierpinski numbers. So I was reading the wikipedia article about the Sierpinski ...Wed, 12 Aug 2026 04:16:36 -0400https://pop.com.sg/when-given-the-dimensions-of-a-cylinder-a-x-b-x-c-what-does-each-lette.htmlhttps://pop.com.sg/when-given-the-dimensions-of-a-cylinder-a-x-b-x-c-what-does-each-lette.htmlWhen looking at fire extinguishers online, the dimensions are always listed in the above format (A x B x C). What does each letter represent? Example: 3" x 5.5" x 14.9" I assume the third (C) is ...Wed, 12 Aug 2026 04:12:59 -0400https://pop.com.sg/choose-a-coefficient-so-that-the-system-of-equations-has-exactly-one-s.htmlhttps://pop.com.sg/choose-a-coefficient-so-that-the-system-of-equations-has-exactly-one-s.htmlChoose the value of $a$ from $\{-4, 88\}$ so that the system of equations has exactly one solution: $$2x +20y+3z=1$$ $$2x +2y+41z=2$$ $$ax -22y-44z=-3$$ I tried solving the system for $a=-4$ and ......Wed, 12 Aug 2026 04:12:31 -0400https://pop.com.sg/why-does-trapezoidal-rule-have-potential-error-greater-than-midpoint.htmlhttps://pop.com.sg/why-does-trapezoidal-rule-have-potential-error-greater-than-midpoint.htmlI can approximate the area beneath a curve using the Midpoint and Trapezoidal methods, with errors such that: $Error_m \leq \frac{k(b-a)^3}{24n^2}$ and $Error_T \leq \frac{k(b-a)^3}{12n^2}$. Does......Wed, 12 Aug 2026 04:07:36 -0400https://pop.com.sg/a-place-to-learn-about-math-etymology.htmlhttps://pop.com.sg/a-place-to-learn-about-math-etymology.htmlI was recently wondering where the word `kernel' comes from in mathematics. I am sure the internet must know. I did manage to find http://www.pballew.net/etyindex.html#k which contains the ori......Wed, 12 Aug 2026 03:58:08 -0400https://pop.com.sg/integer-coefficients-of-cubic-equation-imply-integer-roots.htmlhttps://pop.com.sg/integer-coefficients-of-cubic-equation-imply-integer-roots.htmlProblem: Let $a,b,c$ be three integers for which the sum $ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$ is integer. Prove that each of the three numbers $ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac......Wed, 12 Aug 2026 03:55:33 -0400https://pop.com.sg/in-classical-geometry-why-is-a-line-considered-to-be-parallel-to-itsel.htmlhttps://pop.com.sg/in-classical-geometry-why-is-a-line-considered-to-be-parallel-to-itsel.htmlA definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is pr......Wed, 12 Aug 2026 03:54:20 -0400https://pop.com.sg/find-difference-between-a-positive-and-a-negative-number.htmlhttps://pop.com.sg/find-difference-between-a-positive-and-a-negative-number.htmlI am trying to work out the percentage difference between two numbers (one of which is negative), NOT PERCENTAGE CHANGE. I have done quite a bit of reading, but there doesn't appear to be a definit......Wed, 12 Aug 2026 03:47:31 -0400https://pop.com.sg/prove-the-median-of-a-uniform-distributions-is-frac-1-2-a-b.htmlhttps://pop.com.sg/prove-the-median-of-a-uniform-distributions-is-frac-1-2-a-b.htmlLet X~U(a,b) with a and b in the real line, such that b>a with X's pdf given by $$f_X(x)=\frac{1}{b-a}\mathbb{1}(a<x< b)$$ Show the median of X's distribution is given by $$m=\frac{1}{2}(b+a)......Wed, 12 Aug 2026 03:33:43 -0400https://pop.com.sg/getting-a-hermite-polynomial-expansion-of-gaussian-with-given-variance.htmlhttps://pop.com.sg/getting-a-hermite-polynomial-expansion-of-gaussian-with-given-variance.htmlI am trying to find an expansion of centered Gaussian - $\frac{1}{\sqrt{2\pi}\sigma}\exp({-\frac{x^2}{2\sigma^2})}$ in terms of Hermite polynomials. Namely to calculate $a_n=\frac{1}{\sqrt{2\pi}\......Wed, 12 Aug 2026 03:32:26 -0400https://pop.com.sg/parallelogram-constructed-through-medians.htmlhttps://pop.com.sg/parallelogram-constructed-through-medians.htmlBdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ......Wed, 12 Aug 2026 03:06:59 -0400https://pop.com.sg/what-is-the-practical-difference-between-a-differential-and-a-derivati.htmlhttps://pop.com.sg/what-is-the-practical-difference-between-a-differential-and-a-derivati.htmlI ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a functio......Wed, 12 Aug 2026 03:06:25 -0400https://pop.com.sg/sum-of-perfect-cubes-formula-proof-without-induction-duplicate.htmlhttps://pop.com.sg/sum-of-perfect-cubes-formula-proof-without-induction-duplicate.htmlis there any proof for the sum of cubes except induction supposition? there are some proofs using induction in below page Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using ...Wed, 12 Aug 2026 03:03:40 -0400https://pop.com.sg/integral-of-d-2y-dx-2-dy.htmlhttps://pop.com.sg/integral-of-d-2y-dx-2-dy.htmlI saw this as a step in a calculation and it was confusing. The left cancels down to: $$\int \frac{d^2y}{dx^2}dy$$ But surely the answer is $\frac{d^2y}{dx^2}y$; instead, $\frac {d^2}{dx^2}$ is being ...Wed, 12 Aug 2026 03:03:29 -0400https://pop.com.sg/union-and-difference-of-convex-set.htmlhttps://pop.com.sg/union-and-difference-of-convex-set.htmlsuppose X,Y are two convex sets x1, x2 in X and y1, y2 in Y defn of X and Y being convex: tx1+(1-t)x2 in X ty1+(1-t)y2 in Y it is clear that: 1) X+Y is convex. 2) X intersection Y is convex 3) ...Wed, 12 Aug 2026 02:37:20 -0400https://pop.com.sg/integrate-e-2x-sin-x.htmlhttps://pop.com.sg/integrate-e-2x-sin-x.htmlFirst of all, sorry for my poor English. Can I integrate the following using integration by parts, because I've tried a lot and it didn't work. (And sorry if it's too easy for you, guys! I'm such a......Wed, 12 Aug 2026 02:31:59 -0400https://pop.com.sg/how-to-prove-sum-frac-1-n-1-1-n-is-divergent.htmlhttps://pop.com.sg/how-to-prove-sum-frac-1-n-1-1-n-is-divergent.htmlWe know that $\sum \frac{1}{n^p}$ is convergent for $p>1$. However the series $\sum {\frac{1}{n^{1+1/n}}}$ is apparently divergent since $1+1/n$ tends to 1 as $n$ tends to infinity. But how to p......Wed, 12 Aug 2026 02:25:59 -0400https://pop.com.sg/complex-analysis-trigonometric-conjugate.htmlhttps://pop.com.sg/complex-analysis-trigonometric-conjugate.htmlI'm having trouble with this assignment: Show that $ \overline {\cos(z)} = \cos(\bar z)$, where $\bar z$ represents conjugate of $z$. I know that the two are equal, but how to mathematically show......Wed, 12 Aug 2026 02:25:08 -0400https://pop.com.sg/graph-of-negative-exponential-function.htmlhttps://pop.com.sg/graph-of-negative-exponential-function.htmlIs it possible to graph a function $Y = (-2)^x$ where $x \geq 1$ It is written in my text that exponential functions are only defined for positive bases, but why not negative bases having restricti......Wed, 12 Aug 2026 02:20:16 -0400https://pop.com.sg/limit-approaching-infinity-of-sine-function.htmlhttps://pop.com.sg/limit-approaching-infinity-of-sine-function.htmlI'd like to ask a question which I have been reflecting on for some time now. What is the limit of: $f(x) = \sin(x)$ as $x$ tends to infinity? As we know, the function has a definite value for each ...Wed, 12 Aug 2026 02:17:33 -0400https://pop.com.sg/what-039-s-an-isomorphism.htmlhttps://pop.com.sg/what-039-s-an-isomorphism.htmlI'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces, the hoeomorphism of topological spaces, and have some feeling for the ...Wed, 12 Aug 2026 02:13:33 -0400https://pop.com.sg/why-is-the-chain-rule-applied-to-derivatives-of-trigonometric-function.htmlhttps://pop.com.sg/why-is-the-chain-rule-applied-to-derivatives-of-trigonometric-function.htmlI'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives?......Wed, 12 Aug 2026 02:10:18 -0400https://pop.com.sg/reflexive-graph-meaning-of-the-reflection.htmlhttps://pop.com.sg/reflexive-graph-meaning-of-the-reflection.htmlHere on the page 1 there is a definition of reflexive graph. I need an intuition how it works the morphism $e:X_0\to X_1.$ What is it and to what edge in $X_1$ it sends a vertex from $X_0$?...Wed, 12 Aug 2026 01:57:46 -0400https://pop.com.sg/leibniz-convergence-test.htmlhttps://pop.com.sg/leibniz-convergence-test.htmlI want to prove if $\sum\limits_{n=2}^\infty (-1)^n \frac{1}{n!}$ is convergent or not. I can prove this by the $ratio \ test$. Isn't this series an alternating series with $a_n = (-1)^n$ and $b_......Wed, 12 Aug 2026 01:56:17 -0400https://pop.com.sg/hadamard-039-s-inequality-proof.htmlhttps://pop.com.sg/hadamard-039-s-inequality-proof.htmlI have the following inequality to prove. With $A \in M_n(R)$ show that: $$ (\det(A))^2 \leq \prod_{i=1}^n\left( \sum_{k=1}^n A_{k,i}^2\right) $$ What I already have: I found out that: $$G(v_1,......Wed, 12 Aug 2026 01:55:24 -0400https://pop.com.sg/elliptical-version-of-pythagoras-theorem.htmlhttps://pop.com.sg/elliptical-version-of-pythagoras-theorem.htmlConsider any right triangle $\triangle ABC$. We focus on one side, $AC$, and we take the midpoint $E$ of this side. Then, we draw the circle with center in $E$ and passing by $A,C$. If we take the ...Wed, 12 Aug 2026 01:52:00 -0400https://pop.com.sg/continuous-mapping-on-a-compact-metric-space-is-uniformly-continuous.htmlhttps://pop.com.sg/continuous-mapping-on-a-compact-metric-space-is-uniformly-continuous.htmlI am struggling with this question: Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks fo......Wed, 12 Aug 2026 01:49:58 -0400https://pop.com.sg/how-do-we-find-the-value-of-x-where-the-tangent-line-to-f-x-is-horizon.htmlhttps://pop.com.sg/how-do-we-find-the-value-of-x-where-the-tangent-line-to-f-x-is-horizon.htmlConsider the function $f(x)=9x^2 +7x$ The derivative is $f'(x)=18x+7$ The slope of the tangent to the graph of $f(x)$ at $x=2$ is $43$ The equation for the tangent line at $x=2$ is $y=43x-36$ I fo......Wed, 12 Aug 2026 01:19:24 -0400https://pop.com.sg/uniform-convergence-problem-of-sequences-of-functions.htmlhttps://pop.com.sg/uniform-convergence-problem-of-sequences-of-functions.htmlI have been trying to solve this problem for hours... At this point, I believe that there is a mistake in the problem, but I assume that I am in the one who is wrong. May you help me? Here is the ...Wed, 12 Aug 2026 01:16:01 -0400https://pop.com.sg/algorithm-of-tree-partition.htmlhttps://pop.com.sg/algorithm-of-tree-partition.htmlI am looking for algorithm which allows to do following: Let $T$ is some tree, where each vertex $V_i$ has its cost $c_i$. I am looking for algorithm which allows to split $T$ (by removing two ed......Wed, 12 Aug 2026 01:09:29 -0400https://pop.com.sg/why-is-the-distance-between-two-circles-spheres-that-don-039-t-interse.htmlhttps://pop.com.sg/why-is-the-distance-between-two-circles-spheres-that-don-039-t-interse.htmlFrom GRE 0568 From MathematicsGRE.Com: I'm guessing the idea applies to circles also? Is there a way to prove this besides the following non-elegant way? Form a line between centers $C_1$ and $C......Wed, 12 Aug 2026 01:09:25 -0400https://pop.com.sg/why-are-the-two-probabilities-in-the-monty-hall-problem-dependent-on-e.htmlhttps://pop.com.sg/why-are-the-two-probabilities-in-the-monty-hall-problem-dependent-on-e.htmlLike most people, one of the first things I did after ringing in the new year was get into a discussion about the Monty Hall problem. Past discussions typically amounted to the other person saying,......Wed, 12 Aug 2026 00:50:25 -0400https://pop.com.sg/get-rid-of-the-square-roots-of-the-denominator-dfrac-1-sqrt-7-2-sqrt-5.htmlhttps://pop.com.sg/get-rid-of-the-square-roots-of-the-denominator-dfrac-1-sqrt-7-2-sqrt-5.htmlHow to get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$? I squared the whole denominator, but that didn't help. Also I searched for a propriety or identity ......Wed, 12 Aug 2026 00:47:41 -0400https://pop.com.sg/rudin-039-s-principles-of-mathematical-analysis-or-apostol-039-s-mathe.htmlhttps://pop.com.sg/rudin-039-s-principles-of-mathematical-analysis-or-apostol-039-s-mathe.htmlI am in high school and have no access to a professor or anyone. I previously used Calculus Volume I by Tom Apostol and Spivak's Calculus (for the differential calculus bit). I can choose between ...Wed, 12 Aug 2026 00:46:57 -0400https://pop.com.sg/diagonally-dominant-matrix-by-rows-and-or-by-columns.htmlhttps://pop.com.sg/diagonally-dominant-matrix-by-rows-and-or-by-columns.htmlConsider a binary square matrix ${\bf B} \in \{0,1\}^{N \times N}$, with $b_{i,i} = 0$ (this can be, for example, the adjacency matrix of a directed graph). Let's define $r_i = \sum_{j=1}^N b_{i,j......Wed, 12 Aug 2026 00:43:05 -0400https://pop.com.sg/what-does-p-to-q-mean-in-propositional-logic.htmlhttps://pop.com.sg/what-does-p-to-q-mean-in-propositional-logic.htmlI have a question that is focussed on a more language oriented side of propositional logic. In such a way that I have no clue what the question means. I want to know what the "-q to -p" part means......Wed, 12 Aug 2026 00:38:04 -0400https://pop.com.sg/find-all-irreducible-monic-polynomials-in-mathbb-z-2-x-with-degree-equ.htmlhttps://pop.com.sg/find-all-irreducible-monic-polynomials-in-mathbb-z-2-x-with-degree-equ.htmlFind all irreducible monic polynomials in $\mathbb{Z}/(2)[x]$ with degree equal or less than $5$. This is what I tried: It's evident that $x,x+1$ are irreducible. Then, use these to find all redu......Wed, 12 Aug 2026 00:18:18 -0400https://pop.com.sg/integrating-the-area-of-a-quarter-circle.htmlhttps://pop.com.sg/integrating-the-area-of-a-quarter-circle.html$A=\int_0^r\sqrt{r^2-x^2}\,dx$ is the area of a quarter-circle. Show why, with a graph and thin rectangles. Calculate this integral by substituting $x=r\sin\theta$ and $\,dx = r\cos\theta\,d\theta$......Wed, 12 Aug 2026 00:16:12 -0400https://pop.com.sg/integrate-int-frac-1-sqrt-x-2-1-dx.htmlhttps://pop.com.sg/integrate-int-frac-1-sqrt-x-2-1-dx.htmlI am solving an ODE $$y^2 \, dx + \left(x\sqrt{y^2 - x^2} - xy\right)dy=0$$ I let $y=xu$ and did some algebra, and I ended up with this $$\ln(x)+C=\int \frac{1}{\sqrt{u^2 - 1}} \, du - \int \fra......Wed, 12 Aug 2026 00:15:16 -0400https://pop.com.sg/compounded-interest-differential-equation.htmlhttps://pop.com.sg/compounded-interest-differential-equation.htmlYou borrow $8000 to buy a car. The lender charges an annual rate of 10% compounded continuously. You make payments of k dollars per year continuously. A) write a differential equation describing the ...Wed, 12 Aug 2026 00:13:53 -0400https://pop.com.sg/proof-by-induction-involving-factorials.htmlhttps://pop.com.sg/proof-by-induction-involving-factorials.htmlMy "factorial" abilities are a slightly rusty and although I know of a few simplifications such as: $(n+1)\,n! = (n+1)!$, I'm stuck I have to prove by induction that: $$\sum_{i=1}^n\frac{i-1}{i!}......Wed, 12 Aug 2026 00:11:42 -0400https://pop.com.sg/how-is-the-dot-product-of-two-cartesian-unit-vectors-equal-to-the-kroe.htmlhttps://pop.com.sg/how-is-the-dot-product-of-two-cartesian-unit-vectors-equal-to-the-kroe.htmlIn my lectures, the relationship $\underline{\hat{e_k}} \cdot \underline{\hat{ e_j}}$ is given to be equal to $\delta_{kj}$ - how is this so? From my understanding this is equal to $ \left( \begin{...Wed, 12 Aug 2026 00:01:28 -0400https://pop.com.sg/example-of-non-compact-set-for-rationals.htmlhttps://pop.com.sg/example-of-non-compact-set-for-rationals.htmlthere is an example from lectures which i do not understand. given a closed set [0,1] , for the below set K, how is there not a finite subcover on K? why is the set K=[0,1] $\bigcap \mathbb{Q}$ ......Tue, 11 Aug 2026 23:48:43 -0400https://pop.com.sg/evaluating-int-0-infty-frac-arctan-x-2-ln-1-x-4-x-x-8-x-4-1-mathrm-dx.htmlhttps://pop.com.sg/evaluating-int-0-infty-frac-arctan-x-2-ln-1-x-4-x-x-8-x-4-1-mathrm-dx.htmlSo, my friend has challenged me to solve the following integral $\displaystyle \tag*{} I=\int_0^\infty \frac{\arctan (x^2) \ln(1+x^4)}{x(x^8+x^4+1)} \mathrm dx$ I started by doing $x^2=t$, so $\...Tue, 11 Aug 2026 23:44:16 -0400https://pop.com.sg/unequal-circles-within-circle-with-least-possible-radius.htmlhttps://pop.com.sg/unequal-circles-within-circle-with-least-possible-radius.htmlIt is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to ...Tue, 11 Aug 2026 23:43:12 -0400