https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 04 Sep 2026 02:28:06 -0400https://pop.com.sg/what-does-within-the-same-order-of-magnitude-convey.htmlhttps://pop.com.sg/what-does-within-the-same-order-of-magnitude-convey.htmlThis question originates from a quandary about the meaning of the statement that two values are within the same order of magnitude. I wonder whether there is an established usage, of (rather more ...Fri, 04 Sep 2026 06:10:17 -0400https://pop.com.sg/are-all-sets-ordered-sets.htmlhttps://pop.com.sg/are-all-sets-ordered-sets.htmlThe definition of an ordered set according to W. Rudin in his book, Principles of Mathematical Analysis is: An ordered set is a set S in which an order is defined He also defined order in his boo......Fri, 04 Sep 2026 06:08:56 -0400https://pop.com.sg/finding-formula-for-partial-sum-of-polynomial-terms.htmlhttps://pop.com.sg/finding-formula-for-partial-sum-of-polynomial-terms.htmlFor example, we know that the following is true (and can be easily derived): $\sum\limits_{x=1}^{n}x = \frac{1}{2}n(n+1)$ But, what if we wanted to find the sum of a series like this: $\sum\limi......Fri, 04 Sep 2026 06:08:18 -0400https://pop.com.sg/what-do-small-and-large-mean-for-sets-and-categories.htmlhttps://pop.com.sg/what-do-small-and-large-mean-for-sets-and-categories.htmlCategories for the Working Mathematicians says In addition to the metacategory of all sets ~ which is not a set ~ we want an actual category $Set$, the category of all small sets. We shall ...Fri, 04 Sep 2026 06:00:00 -0400https://pop.com.sg/why-do-we-assume-in-the-principle-of-strong-complete-induction.htmlhttps://pop.com.sg/why-do-we-assume-in-the-principle-of-strong-complete-induction.htmlI get induction. (i) show that the statement holds when n=1 (or some basis) (ii) show that the statement holds for a general subsequent case, $n+1$. By PMI the statement holds for all cases. Do......Fri, 04 Sep 2026 05:57:48 -0400https://pop.com.sg/vieta-039-s-formula-failed.htmlhttps://pop.com.sg/vieta-039-s-formula-failed.htmlFind the value of $p$ if $p$ and $q$ are the roots of the equation, $x^2+px+q=0, \ \ \{x,p,q\}\in\ \mathbb{R}$ By using vieta's formula for sum and product of roots, $\begin{cases} p+q=-p \......Fri, 04 Sep 2026 05:54:25 -0400https://pop.com.sg/what-does-mean-in-math.htmlhttps://pop.com.sg/what-does-mean-in-math.htmlIn a Linear Algebra textbook I am reading, the following is stated: $b\notin \operatorname{span}(A \cup \{a\})\setminus \operatorname{span}(A)$. It does so without explaining what "$\setminus$" mea......Fri, 04 Sep 2026 05:45:19 -0400https://pop.com.sg/can-we-compute-the-area-of-a-quadrilateral-with-one-right-angle-when-w.htmlhttps://pop.com.sg/can-we-compute-the-area-of-a-quadrilateral-with-one-right-angle-when-w.htmlI took an IQ test for fun recently, but I take issue with the answer to one of the questions. Here's the question: My issue is that the explanation assumes angle DC is a right angle. Given that ...Fri, 04 Sep 2026 05:39:26 -0400https://pop.com.sg/a-demonstration-of-lagrange-039-s-form-for-the-remainder-of-a-taylor-s.htmlhttps://pop.com.sg/a-demonstration-of-lagrange-039-s-form-for-the-remainder-of-a-taylor-s.htmlThe following argument for Lagrange's Form for the Remainder of a Taylor polynomial is a typical one in analysis books. Even in the case of finding the remainder when the Taylor polynomial is a li......Fri, 04 Sep 2026 05:38:49 -0400https://pop.com.sg/how-does-sinc-interpolation-work.htmlhttps://pop.com.sg/how-does-sinc-interpolation-work.htmlEvery now and then I come across mention of sinc interpolation. Trying to read up on it, I have yet to get what it's about. I have done basic DSP work, have programmed stuff using FFT (using just a ...Fri, 04 Sep 2026 05:25:31 -0400https://pop.com.sg/microeconomics-calculating-tax-revenue-and-tax-incidence.htmlhttps://pop.com.sg/microeconomics-calculating-tax-revenue-and-tax-incidence.htmlAustralian Government has imposed a tax on Beer. Assume that the tax on Beer is $20 per unit (a unit is a carton of drinks) Assume the demand and supply functions for cartons of Beers per week are:......Fri, 04 Sep 2026 05:22:46 -0400https://pop.com.sg/why-does-the-graph-not-show-the-vertical-asymptote.htmlhttps://pop.com.sg/why-does-the-graph-not-show-the-vertical-asymptote.htmlI have a rational function in which the denominator is equal to zero when $x = -1.4142135$. So: $f(x)=\dfrac{1}{x + 1.4142135}$ Then the vertical asymptote at $x = -1.4142135$ is pretty clear. ......Fri, 04 Sep 2026 05:13:12 -0400https://pop.com.sg/simple-proof-of-krein-smulian.htmlhttps://pop.com.sg/simple-proof-of-krein-smulian.htmlI have a question concerning a simple and short proof of Krein-Shmulian, the proof can be found here https://people.math.ethz.ch/~jteichma/slides_ftap.pdf on page 35/36. Here the proof: Let $X$ be a ...Fri, 04 Sep 2026 05:04:37 -0400https://pop.com.sg/if-f-039-2-0-and-f-039-039-2-4-what-can-you-say-about-f.htmlhttps://pop.com.sg/if-f-039-2-0-and-f-039-039-2-4-what-can-you-say-about-f.htmlI was doing very well in Calculus up until this point. I realize that concavity and $f'$ and $f''$ require one to really visualize what is happening with a function, but can someone please help me......Fri, 04 Sep 2026 05:04:14 -0400https://pop.com.sg/why-characteristic-zero-and-not-infinite-characteristic.htmlhttps://pop.com.sg/why-characteristic-zero-and-not-infinite-characteristic.htmlThe characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we de......Fri, 04 Sep 2026 04:57:20 -0400https://pop.com.sg/notation-for-compact-sets.htmlhttps://pop.com.sg/notation-for-compact-sets.htmlIs there some accepted notation for a set that is compact? E.g. I am currently writing "... [blah] is true if for every compact set $A \subset \mathbb{R}^n$ and ...". I could simplify my writing if ...Fri, 04 Sep 2026 04:44:39 -0400https://pop.com.sg/how-many-3-digit-positive-integers-are-odd-and-do-not-contain-digit-5.htmlhttps://pop.com.sg/how-many-3-digit-positive-integers-are-odd-and-do-not-contain-digit-5.htmlIt is a GRE question. And it has been answered here. But I still want to ask it again, just to know why I am wrong. The correct is 288. My idea is, first I get the total number of 3-digit integer......Fri, 04 Sep 2026 04:29:59 -0400https://pop.com.sg/a-specific-example-of-the-gns-construction.htmlhttps://pop.com.sg/a-specific-example-of-the-gns-construction.htmlIn an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X......Fri, 04 Sep 2026 04:27:43 -0400https://pop.com.sg/the-transitive-closure-of-a-relation-r-on-set-a-whose-relation-matrix.htmlhttps://pop.com.sg/the-transitive-closure-of-a-relation-r-on-set-a-whose-relation-matrix.htmlThe transitive closure of a relation $R$ on set $A$ whose relation matrix My attempt: If I assumed node $(1, 2, 3)$, then given is: $R=\{(1, 2), (2, 3), (3, 1)\}$ The transitive closer would be......Fri, 04 Sep 2026 04:25:00 -0400https://pop.com.sg/proving-1-frac-1-4-frac-1-9-cdots-frac-1-n-2-leq-2-frac-1-n-for-all-n-.htmlhttps://pop.com.sg/proving-1-frac-1-4-frac-1-9-cdots-frac-1-n-2-leq-2-frac-1-n-for-all-n-.htmlQuestion: Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case. Attempt at soluti......Fri, 04 Sep 2026 04:23:26 -0400https://pop.com.sg/how-to-solve-int-c-im-z-dz-where-c-is-the-unit-circle.htmlhttps://pop.com.sg/how-to-solve-int-c-im-z-dz-where-c-is-the-unit-circle.htmlI want to solve $\int_C Im(z) dz$ where $C$ is the unit circle. The first thing is to parameterize the curve, $C$: $$ z(t)=e^{it} $$ where $0\leq t \leq 2\pi$ Then $$ \frac {dz}{dt} = ie^{it} $$ ......Fri, 04 Sep 2026 04:23:02 -0400https://pop.com.sg/isometries-and-orthogonal-matrices.htmlhttps://pop.com.sg/isometries-and-orthogonal-matrices.htmlI know how to show that multiplying by an orthogonal matrix preserves the angle and distance between two vectors. I have seen everywhere that Orthogonal matrices are kind of related to rotations and ...Fri, 04 Sep 2026 04:19:00 -0400https://pop.com.sg/if-type-theories-are-all-logics.htmlhttps://pop.com.sg/if-type-theories-are-all-logics.htmlSo it sounds like Higher Order Logic (HOL) and Type Theory are equivalent. Then there is Intuitionistic Logic and Intuitionistic Type Theory, but I'm not sure of the connection there. I am just ...Fri, 04 Sep 2026 04:19:00 -0400https://pop.com.sg/universal-factoring-method-or-list-of-methods-trinomials.htmlhttps://pop.com.sg/universal-factoring-method-or-list-of-methods-trinomials.htmlI am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it seems to exceedingly hard to find anything ...Fri, 04 Sep 2026 04:11:49 -0400https://pop.com.sg/for-which-angles-we-know-the-sin-value-algebraically-exact.htmlhttps://pop.com.sg/for-which-angles-we-know-the-sin-value-algebraically-exact.htmlFor example: $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$ $\sin(30^\circ) = \frac{1}{2}$ $\sin(45^\circ) = \frac{1}{\sqrt{2}}$ $\s......Fri, 04 Sep 2026 04:07:09 -0400https://pop.com.sg/finding-linear-dependence-relation.htmlhttps://pop.com.sg/finding-linear-dependence-relation.htmlFind a linear dependence relation between the following vectors: x1 = (1, -1, 2) x2 = (-3, 2, 1) x3= (1, 2, -3) x4= (2, 3, 1) I've already created a matrix and reduced and I know how to tell w......Fri, 04 Sep 2026 04:06:42 -0400https://pop.com.sg/how-to-simplify-sin-x-y-cos-y-cos-x-y-sin-y.htmlhttps://pop.com.sg/how-to-simplify-sin-x-y-cos-y-cos-x-y-sin-y.htmlthe question How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$ my steps I tried to use trig identities on the $\sin(x-y)$ and $\cos(x-y)$ and tried to distribute the others in but it didn't wo......Fri, 04 Sep 2026 03:58:46 -0400https://pop.com.sg/use-rational-zero-theorem-to-find-real-zeros-of-2x-3-3x-2-x-1.htmlhttps://pop.com.sg/use-rational-zero-theorem-to-find-real-zeros-of-2x-3-3x-2-x-1.htmlI am to use the rational zero theorem to find the real zeros of $2x^3-3x^2-x+1$. The provided answers are $\frac{1}{2}$, $\frac{1+\sqrt{5}}{2}$ and $\frac{1-\sqrt{5}}{2}$. I was able to get $\frac{......Fri, 04 Sep 2026 03:53:55 -0400https://pop.com.sg/is-there-such-an-inequality-between-product-and-integral-for-functions.htmlhttps://pop.com.sg/is-there-such-an-inequality-between-product-and-integral-for-functions.htmlGiven a measure space $(\Omega, \mathbb{F},\mu)$ and any two measurable real-valued functions $f,g$ defined on $\Omega$, I was wondering if there is an inequality like $$ \int_\Omega |f*g| d\mu \l......Fri, 04 Sep 2026 03:42:03 -0400https://pop.com.sg/finding-the-standard-deviation-of-a-probability-distribution.htmlhttps://pop.com.sg/finding-the-standard-deviation-of-a-probability-distribution.htmlHere is the question: The time, to the nearest whole minute, that a city bus takes to go from one end of its route to the other has the probability distribution shown. As sometimes happens with ...Fri, 04 Sep 2026 03:41:02 -0400https://pop.com.sg/unordered-sampling-with-replacement.htmlhttps://pop.com.sg/unordered-sampling-with-replacement.htmlI have difficulties with understanding of how do we count this. Why can't we just divide number of ways of ordered sample with replacement $n^k$ by $k!$, i.e. to get rid of permutations since we c......Fri, 04 Sep 2026 03:38:59 -0400https://pop.com.sg/how-to-find-the-area-of-a-triangle-with-lengths-of-heights.htmlhttps://pop.com.sg/how-to-find-the-area-of-a-triangle-with-lengths-of-heights.htmlGiven the lengths of 3 heights in a triangle, I need to find its area....Fri, 04 Sep 2026 03:37:07 -0400https://pop.com.sg/change-the-cartesian-integral-into-an-equivalent-polar-integral.htmlhttps://pop.com.sg/change-the-cartesian-integral-into-an-equivalent-polar-integral.htmlI started learning double integrals. I am trying to solve one problem but its not going well :). $$ \int_{-1}^{1} \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} \ln\bigl(x^2+y^2+1\bigr)\, dx\,dy $$ then I wro......Fri, 04 Sep 2026 03:33:02 -0400https://pop.com.sg/switching-iterated-integrals-from-dxdy-to-dydx.htmlhttps://pop.com.sg/switching-iterated-integrals-from-dxdy-to-dydx.htmlI am having issues switching an iterated integral from dxdy to dydx: Switch the integral $\int_0^2 \int_y^{2y}6xy$ dx dy to a dy dx integral in the form of $\int_0^2 \int_{...}^{...}6xydydx$ + $\...Fri, 04 Sep 2026 03:28:30 -0400https://pop.com.sg/how-to-solve-differential-equation-y-039-2y.htmlhttps://pop.com.sg/how-to-solve-differential-equation-y-039-2y.htmlHow to solve differential equation $y'=2y$ when $y(7)=6$ ? The answer is $6e^{2(x-7)}$, but i don't get how come it is $(x-7)$? I know that it is from formula of $y(x)=Ce^{kx}$, where $C=6$ and $......Fri, 04 Sep 2026 03:26:13 -0400https://pop.com.sg/amazing-isomorphisms-closed.htmlhttps://pop.com.sg/amazing-isomorphisms-closed.htmlJust as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether......Fri, 04 Sep 2026 03:17:53 -0400https://pop.com.sg/what-is-the-integral-of-x-ln-x.htmlhttps://pop.com.sg/what-is-the-integral-of-x-ln-x.htmlI need calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank you !...Fri, 04 Sep 2026 03:16:08 -0400https://pop.com.sg/why-do-we-not-measure-angles-like-this.htmlhttps://pop.com.sg/why-do-we-not-measure-angles-like-this.htmlAn angle is a simple geometric figure that is obtained by taking a ray and rotating it about a point to form another ray, both of which have their starting point in common. The initial rays is call......Fri, 04 Sep 2026 03:12:46 -0400https://pop.com.sg/discrete-mathematics-relations-question.htmlhttps://pop.com.sg/discrete-mathematics-relations-question.htmlif r2 is in the set of N*N ( natural numbers) with (X,y) in the subset of r2, if and only if x+y=0 is it reflexive? is it symmetric? is it anti symmetric? is it Transitive? i said it is reflect......Fri, 04 Sep 2026 03:05:49 -0400https://pop.com.sg/find-the-maximum-radius-of-a-incircle.htmlhttps://pop.com.sg/find-the-maximum-radius-of-a-incircle.htmlI was soving geometry problem. And found this question , i tried to solve it .But i am unable to get the answer. The hypotenuse of a right triangle has length of $5$ cm. Determine its maximum pos......Fri, 04 Sep 2026 03:03:41 -0400https://pop.com.sg/how-to-prove-that-if-the-determinant-of-the-matrix-is-zero-then-at-lea.htmlhttps://pop.com.sg/how-to-prove-that-if-the-determinant-of-the-matrix-is-zero-then-at-lea.htmlFor a matrix $A$, if $\det(A)=0,$ prove and provide an example that at least one eigenvalue must be zero. At first, I tried using the identity that the product of eigenvalues is the determinant of......Fri, 04 Sep 2026 02:59:14 -0400https://pop.com.sg/simplifying-cos-arcsin-x-closed.htmlhttps://pop.com.sg/simplifying-cos-arcsin-x-closed.htmlI remember reading somewhere that we can simplify $\cos(\arcsin x)$ and $\sin(\arccos x)$ in terms of a polynomial by making the substitution $m=\arcsin x$ or $m=\arccos x$ (respectively), then ...Fri, 04 Sep 2026 02:55:21 -0400https://pop.com.sg/getting-a-mixed-probability-distribution-function-from-a-mixed-cumulat.htmlhttps://pop.com.sg/getting-a-mixed-probability-distribution-function-from-a-mixed-cumulat.htmlI am having trouble understanding how to get a complete mixed distribution function from a mixed CDF. The question is from my actuarial exam p manual. Namely, the discrete portion is confusing me. ......Fri, 04 Sep 2026 02:52:49 -0400https://pop.com.sg/integrate-over-t-if-dr-dt-sqrt-1-r-2.htmlhttps://pop.com.sg/integrate-over-t-if-dr-dt-sqrt-1-r-2.htmlI'm stuck on this step trying to solve a math puzzle. How do I integrate this?...Fri, 04 Sep 2026 02:50:30 -0400https://pop.com.sg/proof-that-the-runge-phenomenon-occurs.htmlhttps://pop.com.sg/proof-that-the-runge-phenomenon-occurs.htmlIs there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?...Fri, 04 Sep 2026 02:45:58 -0400https://pop.com.sg/fluid-force-on-semicircular-gate-partially-submerged.htmlhttps://pop.com.sg/fluid-force-on-semicircular-gate-partially-submerged.htmlWhat is the best way to approach this problem? Does the problem change since the gate is only partially submerged?...Fri, 04 Sep 2026 02:45:47 -0400https://pop.com.sg/figuring-out-the-percentage-of-a-in-relation-to-b.htmlhttps://pop.com.sg/figuring-out-the-percentage-of-a-in-relation-to-b.htmlI have two numbers 13.49 and 7.8 I want to know how to figure out how much 7.8 is of 13.49 in percentage e.g. 7.8 is 80% of 13.49 What is the best way to do this?...Fri, 04 Sep 2026 02:40:30 -0400https://pop.com.sg/divide-circle-into-9-pieces-of-equal-area.htmlhttps://pop.com.sg/divide-circle-into-9-pieces-of-equal-area.htmlI'd like to divide a unit circle disk into nine parts of equal area, using circle arcs as delimiting lines. The whole setup should be symmetric under the symmetry group of the square, i.e. 4 mirro......Fri, 04 Sep 2026 02:36:35 -0400https://pop.com.sg/sufficient-conditions-for-the-spectral-decomposition.htmlhttps://pop.com.sg/sufficient-conditions-for-the-spectral-decomposition.htmlI know that is possible to apply the spectral decomposition (diagonalization) to a matrix when the sum of the dimensions of its eigenspaces is equal to the size of the matrix. The spectral decompo......Fri, 04 Sep 2026 02:24:30 -0400https://pop.com.sg/solve-sin-3-x-cos-3-x-1.htmlhttps://pop.com.sg/solve-sin-3-x-cos-3-x-1.htmlSolve for $x\\ \sin^{3}x+\cos^{3}x=1$ $\sin^{3}x+\cos^{3}x=1\\(\sin x+\cos x)(\sin^{2}x-\sin x\cdot\cos x+\cos^{2}x)=1\\(\sin x+\cos x)(1-\sin x\cdot\cos x)=1$ What should I do next?...Fri, 04 Sep 2026 02:16:25 -0400https://pop.com.sg/limit-evaluation-conjugate-method-further-algebraic-manipulation.htmlhttps://pop.com.sg/limit-evaluation-conjugate-method-further-algebraic-manipulation.html$$\lim_\limits{x\to 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}$$ I have tried evaluating the above limit by multiplying either both the conjugate of numerator and the denominator with no avail in exitin......Fri, 04 Sep 2026 02:11:54 -0400https://pop.com.sg/10-bit-2-039-s-complement-of-positive-numbers.htmlhttps://pop.com.sg/10-bit-2-039-s-complement-of-positive-numbers.htmlWhen doing this for a negative number, like -213 for example, I successfully found the answer by doing $(2^9 + -327)$ => $(2^9 - 213)$ taking the result of that, and using the division by 2 method, ...Fri, 04 Sep 2026 02:09:18 -0400https://pop.com.sg/why-does-the-sequence-frac-1-n-diverge.htmlhttps://pop.com.sg/why-does-the-sequence-frac-1-n-diverge.htmlMy math teacher said that it is not a Cauchy sequence thus it isn't convergent but divergent. It is bounded and its limit exists ($=0$). Now i am really confused. I thought that things started to get ...Fri, 04 Sep 2026 02:08:04 -0400https://pop.com.sg/the-circumference-of-the-circle-is-c-what-is-the-area-of-circle-in-ter.htmlhttps://pop.com.sg/the-circumference-of-the-circle-is-c-what-is-the-area-of-circle-in-ter.htmlThe circumference of the circle is $C$, what is the area of circle in terms of $C$? a). $\dfrac {C^2}{4\pi }$ b). $2\pi C$ c). $\dfrac {4}{3} \pi C^2$ d). $2\pi C^2$ My Attempt: $$\textrm {...Fri, 04 Sep 2026 02:05:54 -0400https://pop.com.sg/derivative-with-multiplication-and-division.htmlhttps://pop.com.sg/derivative-with-multiplication-and-division.htmlSo I have the following homework. I don't want the answer, only point me in the right direction please. Thanks. I'm stuck in the product rule. Do I apply the product rule twice or just one time af......Fri, 04 Sep 2026 02:01:50 -0400https://pop.com.sg/express-this-limit-as-a-definite-integral-no-interval-given-lim-limits.htmlhttps://pop.com.sg/express-this-limit-as-a-definite-integral-no-interval-given-lim-limits.htmlI am having trouble trying to convert a limit to a definite integral. I am unsure about how to go about this. I have already tried googling this but can not find anything that is comprehensive enou......Fri, 04 Sep 2026 02:00:21 -0400https://pop.com.sg/what-does-this-039-l-039-and-upside-down-039-l-039-symbol-mean.htmlhttps://pop.com.sg/what-does-this-039-l-039-and-upside-down-039-l-039-symbol-mean.htmlI need help finding out what the following symbols are called and what they do. I searched up math symbols but couldn't find them anywhere near there. $$\lceil{-3.14}\rceil=$$ $$\lfloor{-3.14}\rf......Fri, 04 Sep 2026 01:58:18 -0400https://pop.com.sg/write-the-exact-answer-using-base-10-logarithms.htmlhttps://pop.com.sg/write-the-exact-answer-using-base-10-logarithms.htmlSolve for x. $11^{-8x} = 18^{-x+10}$ Write the exact answer using base-10 logarithms. Every online tutorial I've looked up does not have a step by step process explaining how they got it....Fri, 04 Sep 2026 01:51:42 -0400https://pop.com.sg/differential-equations-ivp-application.htmlhttps://pop.com.sg/differential-equations-ivp-application.htmlI am given $$y'=0.05y-800$$ I am asked to: (a) Find all constant solutions of the differential equation. (b) Suppose $y = M$ is your constant solution from (a). Plot two solutions of the differen......Fri, 04 Sep 2026 01:47:59 -0400https://pop.com.sg/simplify-sqrt-4-frac-162x-6-16x-4-is-frac-3-sqrt-4-2x-2-2.htmlhttps://pop.com.sg/simplify-sqrt-4-frac-162x-6-16x-4-is-frac-3-sqrt-4-2x-2-2.html(I've been posting a lot today and yesterday, not sure if too many posts are frowned upon or not. I am studying and making sincere efforts to solve on my own and only post here as a last resort) I'm ...Fri, 04 Sep 2026 01:36:13 -0400https://pop.com.sg/proof-of-drinker-paradox-duplicate.htmlhttps://pop.com.sg/proof-of-drinker-paradox-duplicate.htmlI searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...Fri, 04 Sep 2026 01:34:31 -0400https://pop.com.sg/fair-5-sided-die.htmlhttps://pop.com.sg/fair-5-sided-die.htmlCommonly used polyhedral dice all have sides of the same shape. However: is this strictly required? Take a triangular prism. A triangular prism with small height compared to the base edge would al......Fri, 04 Sep 2026 01:33:09 -0400https://pop.com.sg/compute-the-double-sum-sum-n-m-0-n-neq-m-frac-1-n-left-m-2-n-2-right-f.htmlhttps://pop.com.sg/compute-the-double-sum-sum-n-m-0-n-neq-m-frac-1-n-left-m-2-n-2-right-f.htmlI am trying to compute the following double sum $$\boxed{\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)}$$ I proceeded as following $$\sum_{n=1}^{\infty}\sum_{m......Fri, 04 Sep 2026 01:25:06 -0400https://pop.com.sg/definition-of-good-reduction.htmlhttps://pop.com.sg/definition-of-good-reduction.htmlLet $C/\mathbb{Q}$ be a curve. I am trying to understand the difference between (1) There is a model $\mathcal{C}/\mathbb{Z}$ of $C$ such that the fiber $\mathcal{C}_p$ is regular and (2) ......Fri, 04 Sep 2026 01:24:26 -0400https://pop.com.sg/symbol-for-the-set-of-integers-from-1-to-n-duplicate.htmlhttps://pop.com.sg/symbol-for-the-set-of-integers-from-1-to-n-duplicate.htmlIs there a special symbol for the set: $$ \{1, 2, 3, \dots, n\}$$, or $$\{x| 1\leq x\leq n, n \in \mathbb{Z} \} $$?...Fri, 04 Sep 2026 01:14:06 -0400https://pop.com.sg/an-open-ball-is-an-open-set.htmlhttps://pop.com.sg/an-open-ball-is-an-open-set.htmlProve that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open. My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an $r_1\in\mathbb{R^......Fri, 04 Sep 2026 01:12:47 -0400https://pop.com.sg/finding-plane-equation-given-two-lines.htmlhttps://pop.com.sg/finding-plane-equation-given-two-lines.htmlDetermine an equation of the plane containing the lines $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-5}{6}$; $r=\lt1,-1,5\gt+t\lt1,1,-3\gt$. I calculated the cross product between the directional vector......Fri, 04 Sep 2026 01:09:26 -0400https://pop.com.sg/could-someone-explain-me-what-is-a-clopen-set.htmlhttps://pop.com.sg/could-someone-explain-me-what-is-a-clopen-set.htmlI came across this term while studying for a test and got curious as to what this actually is. could someone please the concept in a relatively easy language?...Fri, 04 Sep 2026 01:00:20 -0400https://pop.com.sg/what-is-the-point-of-non-commutative-probability.htmlhttps://pop.com.sg/what-is-the-point-of-non-commutative-probability.htmlI read this article by Terry Tao about non-commutative probability. I honestly don't really get it. Apparently, we can generalize Kolmogorov's system of probability theory, in such a way that it i......Fri, 04 Sep 2026 01:00:13 -0400https://pop.com.sg/solving-cos4-theta-cos3-theta-to-find-the-roots-of-8x-3-4x-2-4x-1-0.htmlhttps://pop.com.sg/solving-cos4-theta-cos3-theta-to-find-the-roots-of-8x-3-4x-2-4x-1-0.htmlFind all the solution to the equation $\cos 4\theta=\cos 3\theta$. Hence or otherwise, show that the roots of the equation $$8x^3+4x^2-4x-1=0$$ are $$\cos \frac{2\pi}{7},\cos \frac{4\pi}{7},\cos \f......Fri, 04 Sep 2026 00:59:49 -0400https://pop.com.sg/what-are-the-common-abbreviation-for-minimum-in-equations.htmlhttps://pop.com.sg/what-are-the-common-abbreviation-for-minimum-in-equations.htmlI'm searching for some symbol representing minimum that is commonly used in math equations....Fri, 04 Sep 2026 00:59:48 -0400https://pop.com.sg/is-the-laplacian-a-vector-or-a-scalar.htmlhttps://pop.com.sg/is-the-laplacian-a-vector-or-a-scalar.htmlNeed to prove $\operatorname{div}(\nabla u)=\nabla ^2 u$ where $u=g(x,y,z)$ The RHS is the Lapacian which we were told is a vector. But $\nabla u=(g_x,g_y,g_z)$ and the divergence of that is $g_{x......Fri, 04 Sep 2026 00:56:13 -0400https://pop.com.sg/how-to-check-if-5-is-a-square-mod-701-without-a-calculator.htmlhttps://pop.com.sg/how-to-check-if-5-is-a-square-mod-701-without-a-calculator.htmlThe Legendre symbol tells us to calculate $5^{350} \mod 701$, but this question was on an exam where no calculators are allowed, so I wasn't able to do this question. How can you find if $5$ is a s......Fri, 04 Sep 2026 00:55:34 -0400https://pop.com.sg/difference-between-polynomial-functions-and-polynomials-and-why-these-.htmlhttps://pop.com.sg/difference-between-polynomial-functions-and-polynomials-and-why-these-.htmlHere's an excerpt from abstract algebra book that I'm reading and my question is given later: The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Giv......Fri, 04 Sep 2026 00:50:39 -0400https://pop.com.sg/what-is-the-function-fsolve-in-python-doing-mathematically.htmlhttps://pop.com.sg/what-is-the-function-fsolve-in-python-doing-mathematically.htmlIn the Python documentation for fsolve it says "Return the roots of the (non-linear) equations defined by func(x) = 0 given a starting estimate" f(x, *args). I wondered if anyone knew the mathemati......Fri, 04 Sep 2026 00:45:29 -0400https://pop.com.sg/constructibility-of-the-17-gon.htmlhttps://pop.com.sg/constructibility-of-the-17-gon.htmlComment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore. Assume we are using this set of axioms $A$ for plane euclidean geo......Fri, 04 Sep 2026 00:43:17 -0400https://pop.com.sg/cosine-rule-to-find-a-side-of-a-triangle.htmlhttps://pop.com.sg/cosine-rule-to-find-a-side-of-a-triangle.htmlI would like to ask you for help for the following problem: The sides of a $\triangle ABC$ are $BC=2,AB=5,AC=\sqrt{22}$ and point $D$ lies on $AC$, such that $BD=3$. Find $CD$. I have just studied......Fri, 04 Sep 2026 00:29:28 -0400https://pop.com.sg/drawing-a-phase-portrait-given-eigenvectors.htmlhttps://pop.com.sg/drawing-a-phase-portrait-given-eigenvectors.htmlI am a bit confused on how the author here drew the phase portraits in the following picture. The second eigenvalue is larger than the first. For large and positive t’s this means that the solu......Fri, 04 Sep 2026 00:25:50 -0400https://pop.com.sg/what-is-the-definition-of-2-5-2-5-factorial.htmlhttps://pop.com.sg/what-is-the-definition-of-2-5-2-5-factorial.htmlI was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater ......Fri, 04 Sep 2026 00:22:06 -0400https://pop.com.sg/grandi-039-s-series-tends-to-1-2-but-why-is-this-considered-a-valid-su.htmlhttps://pop.com.sg/grandi-039-s-series-tends-to-1-2-but-why-is-this-considered-a-valid-su.htmlGrandi's series, $$1+1-1+1-1+1-1+...$$ can be expressed as the below: $$\sum_{n=0}^\infty(-1)^n$$ Two valid sums that make sense to me are $1$, and $0$, depending on how you approach the series......Fri, 04 Sep 2026 00:14:39 -0400https://pop.com.sg/how-can-i-show-that-a-function-is-non-decreasing.htmlhttps://pop.com.sg/how-can-i-show-that-a-function-is-non-decreasing.htmlI have the following function: $$F(x)=\begin{cases} \dfrac{x}{x+1},& \text{if }x\ge 0,\\ 0,& \text{otherwise}\end{cases}$$ I want to prove that it is right-continuous and non-decreasing. ......Fri, 04 Sep 2026 00:14:12 -0400https://pop.com.sg/questions-tagged-factoring.htmlhttps://pop.com.sg/questions-tagged-factoring.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 04 Sep 2026 00:12:16 -0400https://pop.com.sg/how-can-i-simply-prove-that-the-pearson-correlation-coefficient-is-bet.htmlhttps://pop.com.sg/how-can-i-simply-prove-that-the-pearson-correlation-coefficient-is-bet.htmlFor building a recommendation system, I also use the Pearson correlation coefficient. This is the definition: $r(x, y)=\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n (x_i-\bar{x......Fri, 04 Sep 2026 00:07:51 -0400https://pop.com.sg/optimisation-in-two-variables-second-order-conditions.htmlhttps://pop.com.sg/optimisation-in-two-variables-second-order-conditions.htmlI have a problem understanding second-order conditions at a critical point in finding critical points of two-variable functions. Let's consider $f(x,y)$. The first-order conditions are $\frac{\pa......Fri, 04 Sep 2026 00:01:53 -0400https://pop.com.sg/why-is-it-that-0-5-0-5-equals-sin-pi-4.htmlhttps://pop.com.sg/why-is-it-that-0-5-0-5-equals-sin-pi-4.htmlIf you put $0.5^{0.5}$ into a calculator you will see that: $$0.5^{0.5}\approx0.707106781187$$ And, if you also put $\sin(\frac{\pi}{4})$ into that same calculator you will get: $$\sin(\frac{\pi}......Thu, 03 Sep 2026 23:56:40 -0400https://pop.com.sg/counting-the-number-of-real-roots-of-a-polynomial.htmlhttps://pop.com.sg/counting-the-number-of-real-roots-of-a-polynomial.htmlI am interested in solving problems which involve finding the number of real roots of any polynomial. Suppose I take a function $$f(x)=x^6+x^5+x^4+x^3+x^2+x+1$$ This does not have any real roots b......Thu, 03 Sep 2026 23:49:41 -0400https://pop.com.sg/what-is-the-modulus-of-a-matrix.htmlhttps://pop.com.sg/what-is-the-modulus-of-a-matrix.htmlI came across the following: "Let f be continuously differentiable with Lipschitz gradient, i.e., $ ||\nabla f(\mathbf{x}) - \nabla f(\mathbf{y})|| \leq L||(\mathbf{x} - \mathbf{y})|| $ where L ......Thu, 03 Sep 2026 23:48:58 -0400https://pop.com.sg/is-upper-hemi-continuity-and-upper-semi-continuity-the-same-thing-for-.htmlhttps://pop.com.sg/is-upper-hemi-continuity-and-upper-semi-continuity-the-same-thing-for-.htmlDespite of difference in their name, they seem to be the same definition. Some texts use the name "hemi-continuity", others use the other name; none of those notes clarify the difference between hemi-...Thu, 03 Sep 2026 23:47:46 -0400https://pop.com.sg/confused-with-natural-logarithms.htmlhttps://pop.com.sg/confused-with-natural-logarithms.htmlHow can we solve the following natural logarithms? I'm confused with this stuff: $\ln(x+1) - \ln x = \ln 3$ $\ln(x+1) + \ln x = \ln 2$...Thu, 03 Sep 2026 23:38:17 -0400https://pop.com.sg/spectrum-of-adjacency-matrix-of-complete-graph.htmlhttps://pop.com.sg/spectrum-of-adjacency-matrix-of-complete-graph.htmlFooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of $K_5$. A = [0 1 1 1 1; 1 0 1 1 1; 1 1 0 1 1; ......Thu, 03 Sep 2026 23:28:20 -0400https://pop.com.sg/what-is-the-taylor-expansion-of-x-2-closed.htmlhttps://pop.com.sg/what-is-the-taylor-expansion-of-x-2-closed.htmlWhat is the Taylor Expansion of the expression $x^2$ ? Is it possible to express this as a general expression evaluated at any point? Edit I have been asked to explain why this question is differ......Thu, 03 Sep 2026 23:22:07 -0400https://pop.com.sg/weird-integral-symbol-mathrel-int.htmlhttps://pop.com.sg/weird-integral-symbol-mathrel-int.htmlWhat does this integral sign mean ($\int$ with line going through the middle)? $$ \mathrel{\int\!\!\!\!\!\!-} $$ (It had something to do with the Beckenbach-Radó Theorem)...Thu, 03 Sep 2026 23:18:49 -0400https://pop.com.sg/formal-symbol-for-the-integer-division-operation.htmlhttps://pop.com.sg/formal-symbol-for-the-integer-division-operation.htmlThe integer division is a common and useful operation in Computer Science. It comes up in many domains, as in the manipulation of matrices and grids. Is there any formal symbol for this operation?......Thu, 03 Sep 2026 23:13:11 -0400https://pop.com.sg/if-an-ideal-contains-the-multiplicative-identity-then-it-is-the-whole-.htmlhttps://pop.com.sg/if-an-ideal-contains-the-multiplicative-identity-then-it-is-the-whole-.htmlI have to prove that if $I\subseteq R$ is an ideal, and $1\in I$, then $I=R\,$. So I know $I\subseteq R$ is an ideal if $a,b\in I$ implies $a+b\in I$, and if $a\in I$, $r\in R$, then $ra\in I$. I'm ...Thu, 03 Sep 2026 23:12:13 -0400https://pop.com.sg/taylor-series-of-frac-1-1-x-2.htmlhttps://pop.com.sg/taylor-series-of-frac-1-1-x-2.htmlI have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to constru......Thu, 03 Sep 2026 23:07:18 -0400https://pop.com.sg/does-the-order-of-operations-matter-with-just-addition-and-subtraction.htmlhttps://pop.com.sg/does-the-order-of-operations-matter-with-just-addition-and-subtraction.htmlHad a debate on whether you could do addition/subtraction in any order you want. Specifically, for the following: $9 - 4 + 3$ We both agree that the answer is 8. I argue that, by giving addition a ...Thu, 03 Sep 2026 22:59:55 -0400https://pop.com.sg/help-with-finding-the-directional-derivative-of-a-f-x-y-z-xy-xz-yz-at-.htmlhttps://pop.com.sg/help-with-finding-the-directional-derivative-of-a-f-x-y-z-xy-xz-yz-at-.htmlI need to find the directional derivative of $f(x,y,z)=xy+xz+yz$ at $P(1,2,3)$ in the direction of $\overrightarrow{v}=\langle 2,1,-1 \rangle$ I think I started this incorrectly and would greatly ...Thu, 03 Sep 2026 22:57:32 -0400https://pop.com.sg/team-balancing-score-normalisation-for-gamification.htmlhttps://pop.com.sg/team-balancing-score-normalisation-for-gamification.htmlThis is the first time I post in Mathematics as I'm normally in programing. But for this question, I think I can get better help here. I'm trying to create a gamification system where different tea......Thu, 03 Sep 2026 22:48:18 -0400https://pop.com.sg/convective-derivative-vs-divergence-of-velocity.htmlhttps://pop.com.sg/convective-derivative-vs-divergence-of-velocity.htmlWhat is the physical significance or difference between Convective derivative : $\vec{v} \cdot \nabla $ and the Divergence of velocity $\nabla \cdot \vec{v}$? I have understood the convective deriv......Thu, 03 Sep 2026 22:47:48 -0400https://pop.com.sg/a-ladder-leans-against-a-wall-but-the-problem-asks-how-long-is-the-lad.htmlhttps://pop.com.sg/a-ladder-leans-against-a-wall-but-the-problem-asks-how-long-is-the-lad.htmlhere's the problem: A ladder leans against a wall with its base 10 ft from it. When the ladder is pulled 3 ft further from the wall, the upper end slides down 7 ft. How long is the ladder? Here's w......Thu, 03 Sep 2026 22:31:13 -0400