https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 28 Aug 2026 12:56:23 -0400https://pop.com.sg/a-little-question-about-clairaut-039-s-theorem.htmlhttps://pop.com.sg/a-little-question-about-clairaut-039-s-theorem.htmlClairaut's Theorem Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume that $f$ is defined on $U$ and its partial ...Fri, 28 Aug 2026 16:35:45 -0400https://pop.com.sg/what-is-a-jordan-cell.htmlhttps://pop.com.sg/what-is-a-jordan-cell.htmlGoogle has been surprisingly unhelpful for me. A homework problem from my algebra class asks me to Calculate p(A) where A is a Jordan cell and p is a polynomial. I'd like to attempt the problem ......Fri, 28 Aug 2026 16:32:04 -0400https://pop.com.sg/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlhttps://pop.com.sg/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlThe question is: If $\tan(\theta) = -\frac{8}{15}$, and $\sin(\theta) > 0$, then find $\cos(\theta)$. What I did was draw a triangle on the unit circle with sides $8, 15$ and therefore hypot......Fri, 28 Aug 2026 16:30:13 -0400https://pop.com.sg/finding-x-y-z-of-a-triangle.htmlhttps://pop.com.sg/finding-x-y-z-of-a-triangle.htmlMy younger brother was asked to do this problem and was given the problem exactly as the problem was given with no additional information. Basically, we are trying to find the values of x,y,and z ......Fri, 28 Aug 2026 16:28:19 -0400https://pop.com.sg/how-do-you-factor-x-3-8-0.htmlhttps://pop.com.sg/how-do-you-factor-x-3-8-0.htmlI factored $x^3 - 8 = 0$ and I only got $x = 2$, but the answer said it's $x=2$ and $x=-1 \pm \sqrt{3}i$? How do you get $x=-1 \pm\sqrt{3}i$?...Fri, 28 Aug 2026 16:25:39 -0400https://pop.com.sg/how-can-the-surd-sqrt-2-sqrt-3-be-expressed.htmlhttps://pop.com.sg/how-can-the-surd-sqrt-2-sqrt-3-be-expressed.htmlI was wondering how $\sqrt{2-\sqrt{3}}$ could be expressed in terms of $\frac{\sqrt{3}-1}{\sqrt{2}}$. I did try to solve both the expressions separately but none of them seemed to match. I would ...Fri, 28 Aug 2026 16:19:51 -0400https://pop.com.sg/integrate-frac-x-ln-x-sqrt-x-2-1-sqrt-x-2-1-closed.htmlhttps://pop.com.sg/integrate-frac-x-ln-x-sqrt-x-2-1-sqrt-x-2-1-closed.html$$\int \frac{x\ln\left(x+\sqrt{x^{2}+1}\right)}{\sqrt{x^{2}+1}}dx$$ I honestly have been struggling with this integral for 2 hours straight and none of my methods (including substitution and integ......Fri, 28 Aug 2026 16:19:48 -0400https://pop.com.sg/what-039-s-wrong-with-my-solution-for-equation-x-1-2-3x-1-2-10x-3-2.htmlhttps://pop.com.sg/what-039-s-wrong-with-my-solution-for-equation-x-1-2-3x-1-2-10x-3-2.htmlThe solution given in the website: Here's my solution: What am I doing wrong?...Fri, 28 Aug 2026 16:19:39 -0400https://pop.com.sg/how-to-solve-equations-when-logarithm-is-the-exponent.htmlhttps://pop.com.sg/how-to-solve-equations-when-logarithm-is-the-exponent.htmlHere is an example problem where the logarithm is expressed as an exponent. Please help me understand this concept its not properly covered in my textbook. $$ 3^{\log_3 (2k)} = 9 $$...Fri, 28 Aug 2026 16:01:27 -0400https://pop.com.sg/taking-the-area-of-isosceles-cross-sections-of-an-ellipse.htmlhttps://pop.com.sg/taking-the-area-of-isosceles-cross-sections-of-an-ellipse.htmlThe base of S is an elliptical region with boundary curve $25x^2 + 4y^2 = 100$. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. I have gotten ......Fri, 28 Aug 2026 15:49:18 -0400https://pop.com.sg/prove-that-arcsin-3-5-pi-is-irrational.htmlhttps://pop.com.sg/prove-that-arcsin-3-5-pi-is-irrational.htmlI was asked to prove that on the unit circle, the rational points are dense. My idea is to first show that the point (3/5,4/5) corresponds to an irrational angle (more precisely, its ratio with res......Fri, 28 Aug 2026 15:34:35 -0400https://pop.com.sg/how-can-i-use-qr-in-matlab-to-compute-lq-decomposition.htmlhttps://pop.com.sg/how-can-i-use-qr-in-matlab-to-compute-lq-decomposition.htmlI want to compute this: $$\begin{bmatrix} U\\ Y \end{bmatrix} = \begin{bmatrix} L_{11} &0 \\ L_{21} & L_{22} \end{bmatrix}\begin{bmatrix} Q_1\\ Q_2 \end{bmatrix}$$ Is this matlab command ...Fri, 28 Aug 2026 15:23:55 -0400https://pop.com.sg/is-there-anything-special-about-intersection-points-of-vesica-pisces-c.htmlhttps://pop.com.sg/is-there-anything-special-about-intersection-points-of-vesica-pisces-c.htmlTake a triangle $\triangle ABC$ On each pair of vertices of the triangle, place a pair of vesica pisces circles with the properties that each circle has the same radius as the other circle of th......Fri, 28 Aug 2026 15:09:15 -0400https://pop.com.sg/how-to-find-the-dimensions-of-a-rectangle-given-a-the-area-and-the-dia.htmlhttps://pop.com.sg/how-to-find-the-dimensions-of-a-rectangle-given-a-the-area-and-the-dia.htmlI need to write concise formulas of these two:$\space \space \space$ $(1)$ finding the dimensions of a rectangle given the Area and the diagonal $\space \space \space$$(2)$ finding the dimensions o......Fri, 28 Aug 2026 15:01:12 -0400https://pop.com.sg/how-to-find-the-length-of-one-of-the-sides-of-a-triangle-given-the-are.htmlhttps://pop.com.sg/how-to-find-the-length-of-one-of-the-sides-of-a-triangle-given-the-are.htmlThe triangles are drawn to scale. The first triangle has side lengths of 17, 17, 16 while the second triangle has side lengths of 17,17,$k$. The triangles have the same area. Find the value of $k$ ...Fri, 28 Aug 2026 15:00:27 -0400https://pop.com.sg/what-039-s-the-intuition-behind-definition-of-chaotic-function.htmlhttps://pop.com.sg/what-039-s-the-intuition-behind-definition-of-chaotic-function.htmlI read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos......Fri, 28 Aug 2026 14:54:26 -0400https://pop.com.sg/normalizer-of-the-normalizer-of-the-sylow-p-subgroup.htmlhttps://pop.com.sg/normalizer-of-the-normalizer-of-the-sylow-p-subgroup.htmlIf $P$ is a Sylow $p$-subgroup of $G$, how do I prove that normalizer of the normalizer $P$ is same as the normalizer of $P$ ?...Fri, 28 Aug 2026 14:53:47 -0400https://pop.com.sg/can-a-5th-degree-equation-with-repeated-roots-be-solved-by-radicials.htmlhttps://pop.com.sg/can-a-5th-degree-equation-with-repeated-roots-be-solved-by-radicials.htmlI realized that generally a 5th degree equation cannot be solved by radicals, but what if we have a 5th degree equation with repeated roots, how about its solvability by radicals? Intuitively, it w......Fri, 28 Aug 2026 14:50:29 -0400https://pop.com.sg/how-to-draw-a-phase-portrait-of-a-two-dimensional-ode.htmlhttps://pop.com.sg/how-to-draw-a-phase-portrait-of-a-two-dimensional-ode.htmlIf we're given: $$\dot{x}=-x+y$$ $$\dot{y}=xy-1$$ How do I draw a phase portrait of this system? I don't understand which direction the arrows are supposed to point. This is what I got so far: I ...Fri, 28 Aug 2026 14:46:14 -0400https://pop.com.sg/finding-the-largest-singular-value-easily.htmlhttps://pop.com.sg/finding-the-largest-singular-value-easily.htmlI'm only interested in finding the largest singular value of an $m\times n$ real matrix $A$. I don't need the corresponding (left and right) singular vectors. Is there a way to do so without perf......Fri, 28 Aug 2026 14:43:04 -0400https://pop.com.sg/classification-of-pde-into-linear-nonlinear.htmlhttps://pop.com.sg/classification-of-pde-into-linear-nonlinear.htmlConsider the following second order PDEs \begin{align} u_{t} + v_{t} + x^{4}u_{xx} + v_{yy} &=0\,\, (1)\\ u_{t} + v_{t} + xu_{xx} + v_{yy} + x^{2}v_{y}&=0\,\,(2) \end{align} I want to ...Fri, 28 Aug 2026 14:41:03 -0400https://pop.com.sg/finding-the-order-of-an-element-in-the-group-of-integers-modulo-n-with.htmlhttps://pop.com.sg/finding-the-order-of-an-element-in-the-group-of-integers-modulo-n-with.htmlI'm a bit new to abstract algebra and while learning it, I've come across a somewhat tedious problem. I'm curious as to whether or not there is some modular congruency trick/number theory that let......Fri, 28 Aug 2026 14:33:08 -0400https://pop.com.sg/convert-angle-radians-to-a-heading-vector.htmlhttps://pop.com.sg/convert-angle-radians-to-a-heading-vector.htmlI have been looking everywhere trying to find out how to convert an angle in radians (expressed as -Pi to Pi) to a heading vector. The only [x,y] answer I have found is, [cos(angle), sin(angle)] , ...Fri, 28 Aug 2026 14:24:35 -0400https://pop.com.sg/user-aretor-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-aretor-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 28 Aug 2026 14:23:41 -0400https://pop.com.sg/proving-the-so-called-well-ordering-principle.htmlhttps://pop.com.sg/proving-the-so-called-well-ordering-principle.htmlIs there anything wrong with the following proof? Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing......Fri, 28 Aug 2026 14:18:20 -0400https://pop.com.sg/sample-standard-deviation-given-population-standard-deviation.htmlhttps://pop.com.sg/sample-standard-deviation-given-population-standard-deviation.htmlHow do you find the sample standard deviation when given the population standard deviation? What formula do you use? If you can make up an example that would be great....Fri, 28 Aug 2026 14:08:47 -0400https://pop.com.sg/why-are-3d-transformation-matrices-4-times-4-instead-of-3-times-3.htmlhttps://pop.com.sg/why-are-3d-transformation-matrices-4-times-4-instead-of-3-times-3.htmlBackground: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually h......Fri, 28 Aug 2026 14:07:52 -0400https://pop.com.sg/let-t-be-the-centroid-of-a-triangle-abc-and-let-p-be-the-midpoint-of-t.htmlhttps://pop.com.sg/let-t-be-the-centroid-of-a-triangle-abc-and-let-p-be-the-midpoint-of-t.htmlQuestion : Let $T$ be the centroid of a triangle $ABC$, and let $P$ be the midpoint of the side $\overline{AC}$. The line containing the point $T$ parallel to the line $BC$ intersects the side $\ov......Fri, 28 Aug 2026 14:00:53 -0400https://pop.com.sg/how-to-express-logical-equivalence-arrow-using-pierce-039-s-arrow.htmlhttps://pop.com.sg/how-to-express-logical-equivalence-arrow-using-pierce-039-s-arrow.htmlI am really confused about this and not sure how to show $P\iff Q$ with the ↓ arrow and only the ↓ arrow. I understand that $P \iff Q$ is $P\implies Q$ and $Q\implies P$. I also know that $P\impl......Fri, 28 Aug 2026 13:45:44 -0400https://pop.com.sg/why-are-the-trig-functions-defined-by-the-counterclockwise-path-of-a-c.htmlhttps://pop.com.sg/why-are-the-trig-functions-defined-by-the-counterclockwise-path-of-a-c.htmlMy understanding is that $\cos$ is defined by the value of $x$ as you trace the graph of a circle counterclockwise, starting at the point $(1, 0)$. Similarly, $\sin$ traces the $y$ value. I underst......Fri, 28 Aug 2026 13:32:13 -0400https://pop.com.sg/find-y-function-of-cos-y-x.htmlhttps://pop.com.sg/find-y-function-of-cos-y-x.htmlLet's say that there is no limit to $x$, then I believe $y = \arccos(x) + \pi n $ because $\cos(y) = x$ for $x + \pi n , n\in \mathbb{Z}$. However, in the last step of the following calculation, it......Fri, 28 Aug 2026 13:28:50 -0400https://pop.com.sg/prove-series-frac-1-x-ln-x-diverges-duplicate.htmlhttps://pop.com.sg/prove-series-frac-1-x-ln-x-diverges-duplicate.htmlProve series $$\sum\limits_{n=2}^{\infty}\frac{1}{n\ln n}$$ doesn't converge without integral test. I really don't know any hint for $\ln n$. So I understand I have to show $$\frac{1}{n\ln n} > ......Fri, 28 Aug 2026 13:22:32 -0400https://pop.com.sg/lagrange-multipliers-with-multiple-constraints.htmlhttps://pop.com.sg/lagrange-multipliers-with-multiple-constraints.htmlI want to maximize $f(x,y,z)=x+y+z$, according to the constraints $g_1(x,y,z)=x^2-y^2-1=0$ and $g_2(x,y,z)=2x+z-1=0$ . So I get $5$ equations using lagrange multipliers solving $\nabla f = \lambda......Fri, 28 Aug 2026 13:15:05 -0400https://pop.com.sg/optimization-related-rate-searchlight.htmlhttps://pop.com.sg/optimization-related-rate-searchlight.htmlI can't seem to figure out how to set up this problem. I'm a bit confused on the relationship between the variables here; I've seen a similar problem involving a searchlight but the angle of $\thet......Fri, 28 Aug 2026 13:13:11 -0400https://pop.com.sg/center-of-mass-of-a-circle.htmlhttps://pop.com.sg/center-of-mass-of-a-circle.htmlHow would one find the center of mass of a circle? The center of mass of a rod is given by: $$\frac{1}{M}\int^{L}_{0}\rho x dx$$ So, for a sphere, it would be an area integral, such as: $$\frac{......Fri, 28 Aug 2026 13:11:14 -0400https://pop.com.sg/normalizing-eigenvectors-from-pauli-matrices.htmlhttps://pop.com.sg/normalizing-eigenvectors-from-pauli-matrices.htmlFor this example of a Pauli matrix, \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} I found that one of its eigenvectors (for $\lambda = 1$) is \begin{bmatrix} -i \\ 1 \end{bmat......Fri, 28 Aug 2026 13:00:04 -0400https://pop.com.sg/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.htmlhttps://pop.com.sg/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.html(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this su......Fri, 28 Aug 2026 12:59:08 -0400https://pop.com.sg/is-the-division-property-of-equality-just-a-special-case-of-the-multip.htmlhttps://pop.com.sg/is-the-division-property-of-equality-just-a-special-case-of-the-multip.htmlMy textbook clearly states after the lesson on Transforming Equations: Addition and Subtraction "Notice that the subtraction property of equality is just a special case of the addition property, s......Fri, 28 Aug 2026 12:55:53 -0400https://pop.com.sg/showing-a-matrix-is-not-diagonalizable.htmlhttps://pop.com.sg/showing-a-matrix-is-not-diagonalizable.htmlFor eigenvalue 1, the eigenspace I got was $span[ 1,0,0]$, and for eigenvalue 4, the eigenspace I got was $span[ 1,-3,9]$. Do they look right? The reason the matrix is not diagonalizable is becaus......Fri, 28 Aug 2026 12:55:45 -0400https://pop.com.sg/integral-of-arcsin-x-x-2.htmlhttps://pop.com.sg/integral-of-arcsin-x-x-2.htmlI did the integration by parts and got this expression, but then I am stuck on how to take it further. I tried substituting u=1-x^2, but then I had to do a partial fraction decomposition (which I d......Fri, 28 Aug 2026 12:54:00 -0400https://pop.com.sg/complex-conjugation-continuous-function.htmlhttps://pop.com.sg/complex-conjugation-continuous-function.htmlHow can I show that complex conjugation is a continuous function? I tried looking at open sets $U$ and then the preimage. Can assume preimage is not empty so that if $z$ is in preimage then $f(z) \......Fri, 28 Aug 2026 12:51:42 -0400https://pop.com.sg/how-to-calculate-the-integral-of-exponential-functions.htmlhttps://pop.com.sg/how-to-calculate-the-integral-of-exponential-functions.htmlHaving an integral like $\int_{2}^{10}{\frac{x}{\ln x}}dx$ How does this function turns to an exponential integral of the form: $ \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt.\,$ For ...Fri, 28 Aug 2026 12:48:56 -0400https://pop.com.sg/example-of-function-of-bounded-variation.htmlhttps://pop.com.sg/example-of-function-of-bounded-variation.htmlI want to show that $f(x) = x \sin(\frac{1}{\sqrt{x}})$ , with $f(0)=0$ on $[0,1]$ is of bounded variation. To do that I Don't want to use of showing this is continuously differentiable hence, BV. ......Fri, 28 Aug 2026 12:39:15 -0400https://pop.com.sg/given-this-transformation-matrix-how-do-i-decompose-it-into-translatio.htmlhttps://pop.com.sg/given-this-transformation-matrix-how-do-i-decompose-it-into-translatio.htmlI have this problem from my Graphics course. Given this transformation matrix: $$\begin{pmatrix} -2 &-1& 2\\ -2 &1& -1\\ 0 &0& 1\\ \end{pmatrix}$$ I need to extract ...Fri, 28 Aug 2026 12:36:14 -0400https://pop.com.sg/power-series-to-evaluate-a-definite-integral-of-f-x-frac1-1-x-7.htmlhttps://pop.com.sg/power-series-to-evaluate-a-definite-integral-of-f-x-frac1-1-x-7.htmlFind a power series expansion of the function $f(x)=\dfrac1{1+x^7}$ about $x=0$. Use it to evaluate $\int_0^1 \dfrac1{1+x^7} dx$. Your answer should be expressed in the form of numerical series (a ......Fri, 28 Aug 2026 12:34:32 -0400https://pop.com.sg/calculating-values-for-cubic-bezier-curve.htmlhttps://pop.com.sg/calculating-values-for-cubic-bezier-curve.htmlI'm trying to use cubic bezier curves for some non-linear animations in my iOS app. Let's say I'm animating position of some element on the screen. I'm using this curve from cubic-bezier.com for ...Fri, 28 Aug 2026 12:29:39 -0400https://pop.com.sg/find-s-where-s-sqrt-3-5-2-sqrt-13-sqrt-3-5-2-sqrt-13-why-am-i-getting-.htmlhttps://pop.com.sg/find-s-where-s-sqrt-3-5-2-sqrt-13-sqrt-3-5-2-sqrt-13-why-am-i-getting-.html$\large S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$ Multiplying by conjugate: $\large S=\dfrac {-3}{\sqrt[3] {5+2 \sqrt {13}}-\sqrt[3]{5-2 \sqrt {13}}}$ From the original: $\large S-......Fri, 28 Aug 2026 12:20:20 -0400https://pop.com.sg/how-to-show-if-a-complex-function-is-analytic.htmlhttps://pop.com.sg/how-to-show-if-a-complex-function-is-analytic.htmlJust began the study of complex analysis. Let $$ f(x,y) = x^2 - y^2 + 2 i xy - x - iy. $$ I need to determine if this function is analytic. This means I have to show the partials satisfy the Cauchy-...Fri, 28 Aug 2026 12:18:21 -0400https://pop.com.sg/understanding-vector-projection.htmlhttps://pop.com.sg/understanding-vector-projection.htmlI'm learning about vector projection. I understand how to perform it, but I still can't understand what it actually means and what it gives me. Here is a common definition: Vector projection of a ...Fri, 28 Aug 2026 12:14:31 -0400https://pop.com.sg/integral-of-1-x-2.htmlhttps://pop.com.sg/integral-of-1-x-2.htmlI know that the integral of $1/x^2$ is $(-1/x +C)$ Can we split $(1/x^2)$ as $(1/x)(1/x)$ and say that the integral of $(1/x^2)$ is $\ln x\cdot\ln x$? PS: I am new to high school. And this is my fi......Fri, 28 Aug 2026 12:02:29 -0400https://pop.com.sg/is-there-any-formula-to-find-the-nth-element-in-a-sequence-where-commo.htmlhttps://pop.com.sg/is-there-any-formula-to-find-the-nth-element-in-a-sequence-where-commo.htmlTo explain my question, here is an example. Below is an AP: 2, 6, 10, 14....n Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is ...Fri, 28 Aug 2026 11:56:56 -0400https://pop.com.sg/is-there-a-straight-line-solution-for-this-separable-differential-equa.htmlhttps://pop.com.sg/is-there-a-straight-line-solution-for-this-separable-differential-equa.htmlThis is the question: Solve $$x^2 \frac{\text{d}y}{\text{d}x} = y^2 - 3xy -5x^2 .$$ Are there any straight line solutions? If so, state them. For the case $y(1)=0$, state the solution ...Fri, 28 Aug 2026 11:56:03 -0400https://pop.com.sg/what-is-the-difference-between-probability-density-function-and-probab.htmlhttps://pop.com.sg/what-is-the-difference-between-probability-density-function-and-probab.htmlWhats the difference between probability density function and probability distribution function?...Fri, 28 Aug 2026 11:48:02 -0400https://pop.com.sg/set-intersection-the-same-as-logical-conjunction.htmlhttps://pop.com.sg/set-intersection-the-same-as-logical-conjunction.htmlI have been exploring more about set theory beyond my textbook and I have ran into something I couldn't explain. Can you use logical conjunction/disjunction on sets and are they the same as union/...Fri, 28 Aug 2026 11:45:14 -0400https://pop.com.sg/mathematical-notation-for-distinct-numbers.htmlhttps://pop.com.sg/mathematical-notation-for-distinct-numbers.htmlHow can I represent a set of distinct numbers in shorthand mathematical notation? In particular I am trying to write that there are 4 distinct integers $a,b,c,d$. For three numbers, I could say: ......Fri, 28 Aug 2026 11:17:31 -0400https://pop.com.sg/equations-for-cubic-regression.htmlhttps://pop.com.sg/equations-for-cubic-regression.htmlSo, I'm making a simple program for drawing graphs, and I'm looking at making some simple best-fit curves using some basic regression analysis. I've happily got linear and quadratic regression work......Fri, 28 Aug 2026 11:09:24 -0400https://pop.com.sg/depressed-cubic.htmlhttps://pop.com.sg/depressed-cubic.htmlI have this string of questions, but when I look up depressed cubic, I don't quite understand. I'm not asking for all of these questions to be answered explicitly, but for some explanation on depre......Fri, 28 Aug 2026 11:05:09 -0400https://pop.com.sg/equation-for-curves-of-high-density-in-bifurcation-diagram.htmlhttps://pop.com.sg/equation-for-curves-of-high-density-in-bifurcation-diagram.htmlIs there an equation for the high density curves in the chaotic regions of the bifurcation diagram for the logistic map? I'm talking about the sinusoidal-looking dark curves in the following picture. ...Fri, 28 Aug 2026 10:55:43 -0400https://pop.com.sg/how-to-compute-confidence-interval-for-variance-with-unknown-mean-from.htmlhttps://pop.com.sg/how-to-compute-confidence-interval-for-variance-with-unknown-mean-from.htmlWhen mean is known, we note that $\frac{\bar{(X-a)^2}n}{\sigma ^2}$ has a Chi-squared distribution with $n$ degrees of freedom. However, what to do when $a$ is unknown? I can't just substitute sample ...Fri, 28 Aug 2026 10:50:25 -0400https://pop.com.sg/is-there-a-symbol-for-antiparallel.htmlhttps://pop.com.sg/is-there-a-symbol-for-antiparallel.htmlI've been doing some work where I've needed to talk about vectors that are parallel and those that are antiparallel, parallel to the negative of the other vector. Is there a symbol for this? I ......Fri, 28 Aug 2026 10:24:28 -0400https://pop.com.sg/divergence-of-series-with-n-factorial.htmlhttps://pop.com.sg/divergence-of-series-with-n-factorial.htmlI need to find the convergence or divergence of $$\sum_{n=1}^\infty \frac{n!}{3^n}$$ The first thing I did was to take the nth root. So I got: $$\lim_{n\rightarrow\infty}\big(\frac{n!}{3^n}\big)^\......Fri, 28 Aug 2026 10:12:48 -0400https://pop.com.sg/probably-an-easy-question-e-2-ln-3-without-a-calculator.htmlhttps://pop.com.sg/probably-an-easy-question-e-2-ln-3-without-a-calculator.htmlHow to evaluate $e^{2\ln(3)}$ without using a calculator? I've tried playing with $\ln$ but couldn't figure out...Fri, 28 Aug 2026 10:11:04 -0400https://pop.com.sg/angle-between-curves-at-point-of-intersection.htmlhttps://pop.com.sg/angle-between-curves-at-point-of-intersection.htmlQuestion reads- Find the acute angle between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $y=x^......Fri, 28 Aug 2026 10:08:10 -0400https://pop.com.sg/arc-length-definition.htmlhttps://pop.com.sg/arc-length-definition.htmlI've always felt that the notion of 'area under the curve' is more precise than 'length of a curve', because of the Riemann integral and the reasoning used there: First, we assume the quantity we ......Fri, 28 Aug 2026 10:07:37 -0400https://pop.com.sg/how-does-the-focus-directrix-definition-of-a-conic-section-apply-to-a-.htmlhttps://pop.com.sg/how-does-the-focus-directrix-definition-of-a-conic-section-apply-to-a-.htmlOne of the definitions of a conic section is that the conic section is a locus of points whose distance to the focus $F$ is a constant multiple of distance between them and the directrix $D$, i.e. ......Fri, 28 Aug 2026 10:06:06 -0400https://pop.com.sg/how-to-solve-x-2-3-4.htmlhttps://pop.com.sg/how-to-solve-x-2-3-4.htmlOK I know this sounds pretty stupid, but I am stuck on solving $x^{{2}/{3}}=4$. I rewrote it to $\sqrt[3]{x^2}=4$, but I don't know what to do next. Would the radical go away if I took the $\sqrt[3......Fri, 28 Aug 2026 10:06:05 -0400https://pop.com.sg/simple-theorems-that-are-instances-of-deep-mathematics.htmlhttps://pop.com.sg/simple-theorems-that-are-instances-of-deep-mathematics.htmlSo, this question asks about how useful computational tricks are to mathematics research, and several people's response was "well, computational tricks are often super cool theorems in disguise.&...Fri, 28 Aug 2026 10:04:56 -0400https://pop.com.sg/why-is-i-2-equal-to-1-duplicate.htmlhttps://pop.com.sg/why-is-i-2-equal-to-1-duplicate.htmlIn this KhanAcademy link at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why. Why is this so? What is the intuition behind it?...Fri, 28 Aug 2026 09:54:34 -0400https://pop.com.sg/block-matrix-multiplication.htmlhttps://pop.com.sg/block-matrix-multiplication.htmlIf $A,B$ are $2 \times 2$ matrices of real or complex numbers, then $$AB = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot \left[ \begin{array}{cc} b_{......Fri, 28 Aug 2026 09:54:13 -0400https://pop.com.sg/pick-5-box-play-pa-lottery-odds.htmlhttps://pop.com.sg/pick-5-box-play-pa-lottery-odds.htmlI was looking over the PA lottery odds and payout table for the Pick 5 "Quinto" game, specifically for the second row "Boxed - 5 in any order - 5 distinct digits" http://www.palottery.state.pa.us/......Fri, 28 Aug 2026 09:48:34 -0400https://pop.com.sg/how-long-does-the-ball-take-to-reach-half-of-its-terminal-velocity.htmlhttps://pop.com.sg/how-long-does-the-ball-take-to-reach-half-of-its-terminal-velocity.htmlA ball of mass m = 0.1kg falls from rest under the influence of gravity (on earth) in a medium that provides resistance that is proportional to its velocity. For a velocity of 0.2 m/s, the resistance ...Fri, 28 Aug 2026 09:44:02 -0400https://pop.com.sg/how-can-i-graph-the-derivative-of-1-4th-of-a-circle-or-a-semicircle-in.htmlhttps://pop.com.sg/how-can-i-graph-the-derivative-of-1-4th-of-a-circle-or-a-semicircle-in.htmlI'm having trouble with questions like these. In the first image, the original function is what is the two sharp lines and a semicircle in between. I understand how to find and graph the derivative......Fri, 28 Aug 2026 09:42:16 -0400https://pop.com.sg/sum-and-difference-of-tangents-in-degrees-duplicate.htmlhttps://pop.com.sg/sum-and-difference-of-tangents-in-degrees-duplicate.html$$\tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ$$ At first, I tried to break the tangent into multiple angles… $$\text{FAILED}$$ Then I proceeded by converting it into other trigonometric ...Fri, 28 Aug 2026 09:38:05 -0400https://pop.com.sg/adding-2-absolute-values-together-x-2-x-3-5-duplicate.htmlhttps://pop.com.sg/adding-2-absolute-values-together-x-2-x-3-5-duplicate.htmlI came across a very basic absolute value question $|x+2| + |x-3| =5.$ Initially, I thought the answer was $x=-2$ and $x=3$ because I let each absolute values be either positive and negative and ......Fri, 28 Aug 2026 09:36:55 -0400https://pop.com.sg/rectangular-box-optimization-problem.htmlhttps://pop.com.sg/rectangular-box-optimization-problem.htmlA rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs \$0.31 per square foot, the material for the sides costs $0.05 per square foot, and the materia......Fri, 28 Aug 2026 09:27:34 -0400https://pop.com.sg/create-an-isosceles-triangle-from-only-equilateral-triangles-possible-.htmlhttps://pop.com.sg/create-an-isosceles-triangle-from-only-equilateral-triangles-possible-.htmlMy friend recently asked me a question. And I think I have a nice solution but wanted to share this question here to see how you approach this problem and hopefully with a solid proof. The question......Fri, 28 Aug 2026 09:26:15 -0400https://pop.com.sg/expected-waiting-time-for-bus.htmlhttps://pop.com.sg/expected-waiting-time-for-bus.htmlSuppose that the inter-arrival time between consecutive buses is 15 minutes with probability 0.5 and 30 minutes with probability 0.5. You just arrived at a bus stop. What is your expected waiting t......Fri, 28 Aug 2026 09:19:59 -0400https://pop.com.sg/conditions-for-the-mean-value-theorem.htmlhttps://pop.com.sg/conditions-for-the-mean-value-theorem.htmlthe mean value theorem which most of us know starts with the conditions that $f$ is continuous on the closed interval $[a,b]$ and differentiable on the opened interval $(a,b)$, then there exists a ......Fri, 28 Aug 2026 09:13:34 -0400https://pop.com.sg/verification-of-proof-for-when-a-number-is-divisible-by-4.htmlhttps://pop.com.sg/verification-of-proof-for-when-a-number-is-divisible-by-4.htmlI have never taken a number theory course and so am only going off of the first few chapters in an introductory number theory book. The divisibility property I wish to prove is the following: Def......Fri, 28 Aug 2026 09:09:57 -0400https://pop.com.sg/root-test-applied-to-power-series.htmlhttps://pop.com.sg/root-test-applied-to-power-series.htmlWhen I apply the root test to power series I know that $$R= \frac{1}{\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_n|}}}$$ So if I get $1$ as the result of the limit the radius of convergence of the p......Fri, 28 Aug 2026 08:57:29 -0400https://pop.com.sg/subtract-binary-numbers-with-1-s-complement.htmlhttps://pop.com.sg/subtract-binary-numbers-with-1-s-complement.htmlI'm trying to figure out how to subtract two binary numbers with a complementary one or two, When I need to address carrier and when not? Do I need to solve the problem in decimal numbers? And ...Fri, 28 Aug 2026 08:56:09 -0400https://pop.com.sg/product-of-banded-matrices.htmlhttps://pop.com.sg/product-of-banded-matrices.htmlHow can one show that the product of two banded matrices is a banded matrix with upper and lower bandwidths equal to the sum of the upper and lower bandwidths (respectively) of the multiplicands ? ......Fri, 28 Aug 2026 08:52:25 -0400https://pop.com.sg/norm-of-multiplication-and-multiplication-of-norms.htmlhttps://pop.com.sg/norm-of-multiplication-and-multiplication-of-norms.htmlIt is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$. Is the following true for all $p \in \mathbb{R}$: $$\|u \cdot v \|_p \le \|u \|_p \cdo......Fri, 28 Aug 2026 08:51:24 -0400https://pop.com.sg/difference-between-magnitude-of-gradient-vs-directional-derivative-of-.htmlhttps://pop.com.sg/difference-between-magnitude-of-gradient-vs-directional-derivative-of-.htmlI've read that the directional derivative is the rate of change of a function $f$ in a given direction $\mathbf{v}$, given as $\nabla f\cdot \mathbf{v}$. I've also read (perhaps incorrectly) that the ...Fri, 28 Aug 2026 08:50:31 -0400https://pop.com.sg/how-does-the-second-derivative-imply-concavity.htmlhttps://pop.com.sg/how-does-the-second-derivative-imply-concavity.htmlconsider this equation, $$y=x(400-x)$$the second derivative of this equation is $$y''=-2$$ As far as I know, a negative sign in the second derivative indicates the curve will concave down. As it is a ...Fri, 28 Aug 2026 08:49:33 -0400https://pop.com.sg/optimal-strategy-for-deal-or-no-deal.htmlhttps://pop.com.sg/optimal-strategy-for-deal-or-no-deal.htmlWhen I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy t......Fri, 28 Aug 2026 08:46:23 -0400https://pop.com.sg/finding-the-exact-value-of-tan-pi-5.htmlhttps://pop.com.sg/finding-the-exact-value-of-tan-pi-5.htmlHi, I realise there has been a question already asked regarding this particular exact value, but this question requires for it to be done under different conditions, which is the part I require hel......Fri, 28 Aug 2026 08:33:23 -0400https://pop.com.sg/puzzle-123456789-100-with-three-operations.htmlhttps://pop.com.sg/puzzle-123456789-100-with-three-operations.htmlGiven the sequence 123456789: You can insert three operations ($+$,$-$,$\times$,$/$) into this sequence to make the equation = 100. My question is: is there a way to solve this without brute forc......Fri, 28 Aug 2026 08:30:30 -0400https://pop.com.sg/actually-playable-games-based-on-graphs.htmlhttps://pop.com.sg/actually-playable-games-based-on-graphs.htmlIn computer science lessons, we have recently got the task to program something using graphs. Due to my enthusiasm for computer games, i would really prefer to implement a concept for a game. The ...Fri, 28 Aug 2026 08:29:01 -0400https://pop.com.sg/how-to-show-c-n-k-c-n-n-k.htmlhttps://pop.com.sg/how-to-show-c-n-k-c-n-n-k.htmlI am doing a question from the textbook "Calculus and Statistics" by Gemigani. If $n = 2m$ and $k= 1,2, \ldots ,m$ Prove that $C(n,k)= C(n,n-k)$ Ok so my approach begins with writing out the for......Fri, 28 Aug 2026 08:21:55 -0400https://pop.com.sg/find-a-second-degree-polynomial-that-goes-through-3-points.htmlhttps://pop.com.sg/find-a-second-degree-polynomial-that-goes-through-3-points.htmlI am having trouble calculating the quadratic curve $f(x)$ that goes through 3 points; for example a curve that goes through $A(1,3), B(-1,-5), and C(-2,12)$. I can only guess that the curve is upw......Fri, 28 Aug 2026 08:17:37 -0400https://pop.com.sg/why-is-there-no-online-record-of-pentagram-area-formula-closed.htmlhttps://pop.com.sg/why-is-there-no-online-record-of-pentagram-area-formula-closed.htmlI'm talking about the short version of the area of a pentagram inscribed in a circle when the radius is given and so Area $=1.123r^2$. I only got this formula from a book but I researched about it......Fri, 28 Aug 2026 08:14:30 -0400https://pop.com.sg/how-to-find-the-roots-of-x-4-1.htmlhttps://pop.com.sg/how-to-find-the-roots-of-x-4-1.htmlI'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a......Fri, 28 Aug 2026 08:14:18 -0400https://pop.com.sg/differential-geometry-and-transition-maps-as-communative-diagrams.htmlhttps://pop.com.sg/differential-geometry-and-transition-maps-as-communative-diagrams.htmlConsider the picture below. I know pictures are usually looked down upon on this site but I have no experience with TikZ. Anyway, I am learning about regular surfaces and transition maps in differ......Fri, 28 Aug 2026 08:10:19 -0400https://pop.com.sg/the-derivative-of-log-2-n-5.htmlhttps://pop.com.sg/the-derivative-of-log-2-n-5.htmlThe derivative of $ (\log_2 n)^5$? (log base 2) Hey everyone, I am not sure how to go about this question because I am not sure what to do with the power $5$ ( or any other power ) in the log? Shou......Fri, 28 Aug 2026 08:07:22 -0400https://pop.com.sg/finding-all-orthogonal-matrices-of-a-given-form.htmlhttps://pop.com.sg/finding-all-orthogonal-matrices-of-a-given-form.htmlI am given this problem. I need to find all $a,b,c,d \in \mathbb{R}$ for which the matrix $A$ is orthogonal. $$A:=\frac{1}{3} \begin{pmatrix} -1 & 2 & a \\ 2 & 2 & b \\ 2 & c ......Fri, 28 Aug 2026 08:03:27 -0400https://pop.com.sg/lusin-039-s-theorem-proof-verification-folland-chap-2-exercise-44.htmlhttps://pop.com.sg/lusin-039-s-theorem-proof-verification-folland-chap-2-exercise-44.htmlConisder Lusin's Theorem (Folland Chap 2 Ex 44): If $f: [a,b] \rightarrow \mathbb{C}$ is Lebesgue measurable and $\epsilon > 0$, there is a compact subset $E \in [a,b]$ s.t. $\mu(E^C) < \...Fri, 28 Aug 2026 08:00:38 -0400https://pop.com.sg/can-i-break-up-log-a-b.htmlhttps://pop.com.sg/can-i-break-up-log-a-b.htmlFor constants $a$ and $b$, I know that I can break up $\log(a/b)$ into $\log(a) - \log(b)$. Can I conveniently break up $\log(a - b)$ somehow into several terms?...Fri, 28 Aug 2026 07:52:45 -0400https://pop.com.sg/how-to-find-whether-equation-of-angle-bisector-represents-the-obtuse-o.htmlhttps://pop.com.sg/how-to-find-whether-equation-of-angle-bisector-represents-the-obtuse-o.htmlTwo lines: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are given. I know that the equation of its bisectors is ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\...Fri, 28 Aug 2026 07:48:41 -0400https://pop.com.sg/chance-on-throwing-a-six-with-6-dice.htmlhttps://pop.com.sg/chance-on-throwing-a-six-with-6-dice.htmlThe chance to throw a 6 with one die is 1/6 And 6 times 1/6 = 1 So, if I throw with 6 dice, the chance to throw at least 1 six should be 1. But when I throw 6 dice, I sometimes don't throw any 6......Fri, 28 Aug 2026 07:47:09 -0400