https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 29 Aug 2026 17:44:06 -0400https://pop.com.sg/rearranging-formulas-involving-fractions.htmlhttps://pop.com.sg/rearranging-formulas-involving-fractions.htmlI'm currently studying some basic algebra (Not sure what the equivalent is for the US etc) in school currently. I'm having trouble with rearranging certain formulas. I attempted to try and solve ......Sat, 29 Aug 2026 21:31:24 -0400https://pop.com.sg/obtaining-the-frac-1-2-pi-factor-in-the-fourier-transform.htmlhttps://pop.com.sg/obtaining-the-frac-1-2-pi-factor-in-the-fourier-transform.htmlThis MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ ra......Sat, 29 Aug 2026 21:20:36 -0400https://pop.com.sg/centroid-of-a-region.htmlhttps://pop.com.sg/centroid-of-a-region.html$$y = x^3, x + y = 2, y = 0$$ I am suppose to find the centroid bounded by those curves. I have no idea how to do this, it isn't really explained well in my book and the places I have looked onlin......Sat, 29 Aug 2026 21:19:47 -0400https://pop.com.sg/proving-monotonic-sequence-theorem.htmlhttps://pop.com.sg/proving-monotonic-sequence-theorem.htmlA sequence $b_n$ is decreasing and bounded. Prove it it convergent. Proof: Since $b_n$ is bounded, $b_n > L$ where L is the greatest lower bound as per the completeness Axiom. Consider some ......Sat, 29 Aug 2026 21:06:08 -0400https://pop.com.sg/why-is-the-graph-of-x-y-2-and-y-sqrt-x-not-the-same.htmlhttps://pop.com.sg/why-is-the-graph-of-x-y-2-and-y-sqrt-x-not-the-same.htmlSo if you take $x=y^2$ and get the sqrt of both sides you get $y=\sqrt{x}$ so they are the same right? But when you graph them, $y=\sqrt{x}$ only shows the positive $y$ values because you can't sqr......Sat, 29 Aug 2026 21:05:44 -0400https://pop.com.sg/obscuring-squares-of-rubik-039-s-cube.htmlhttps://pop.com.sg/obscuring-squares-of-rubik-039-s-cube.htmlThis is a combinatorial question related to Rubik's cube $3\times3\times3$ (and, in the end, its generalizations $n\times n\times n$). I assume that the readers are familiar with this puzzle. Let's ...Sat, 29 Aug 2026 21:05:28 -0400https://pop.com.sg/writing-out-the-number-of-zeros-from-1-1-000-000.htmlhttps://pop.com.sg/writing-out-the-number-of-zeros-from-1-1-000-000.htmlIf you had to write out all the numbers between 1 and 1,000,000. How many zeros would you have to write in total. I was given that the answer is 488,895. Can any one show the clearest way possible......Sat, 29 Aug 2026 21:02:34 -0400https://pop.com.sg/in-set-theory-what-does-it-mean-for-a-variable-to-have-a-bar-symbol-ab.htmlhttps://pop.com.sg/in-set-theory-what-does-it-mean-for-a-variable-to-have-a-bar-symbol-ab.htmlPlease see the image below. What does the bar symbol mean in this context?...Sat, 29 Aug 2026 20:58:52 -0400https://pop.com.sg/how-to-simplify-abs-x-x.htmlhttps://pop.com.sg/how-to-simplify-abs-x-x.htmlI've been trying to find a way to simplify $\frac{|x|}{x}$ if $x$ is real and $\neq{0}$. The two possible outcomes to this are $\pm{1}$ but I believe there is one required answer. I've noticed that......Sat, 29 Aug 2026 20:57:22 -0400https://pop.com.sg/integrate-int-x-sin-2-x-dx.htmlhttps://pop.com.sg/integrate-int-x-sin-2-x-dx.htmlIntegrate $\int x\sin^2 (x) dx$ My attempt: $$=\int x\sin^2 (x) dx\\ =x^2\sin^2 (x) - \int 2\sin (x)\cos (x)x^2 dx\\ =x^2\sin^2 (x) - \int \sin (2x) x^2 dx.$$...Sat, 29 Aug 2026 20:54:02 -0400https://pop.com.sg/3-d-geometry-three-vertices-of-a-gm-abcd-is-3-1-2-1-2-4-1-1-2-find-the.htmlhttps://pop.com.sg/3-d-geometry-three-vertices-of-a-gm-abcd-is-3-1-2-1-2-4-1-1-2-find-the.htmlThe question is Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2). Find the coordinate of the fourth vertex. To get the answer I tried the distance formula, equated ......Sat, 29 Aug 2026 20:45:25 -0400https://pop.com.sg/calculate-the-vector-normal-to-the-plane-by-given-points.htmlhttps://pop.com.sg/calculate-the-vector-normal-to-the-plane-by-given-points.htmlHow can one calculate the vector normal to the plane that is determined by given points? For example, given three points $P_1(5,0,0)$, $P_2(0,0,5)$ and $P_3(10,0,5)$, calculate the vector normal t......Sat, 29 Aug 2026 20:41:00 -0400https://pop.com.sg/prove-the-limit-is-sqrt-e.htmlhttps://pop.com.sg/prove-the-limit-is-sqrt-e.htmlHow do you show $$\lim\limits_{k \rightarrow \infty} \frac{\left(2+\frac{1}{k}\right)^k}{2^k}=\sqrt{e}$$ I know that $$\lim\limits_{k \to \infty} \left(1+\frac{1}{k}\right)^k=e$$ but I don't kno......Sat, 29 Aug 2026 20:39:06 -0400https://pop.com.sg/how-to-lower-upper-bound-n-using-1-x-leq-e-x.htmlhttps://pop.com.sg/how-to-lower-upper-bound-n-using-1-x-leq-e-x.htmlI need to prove for all positive integer $n$ $$ e\left(\frac{n}{e}\right)^n\leq n!\leq en\left(\frac{n}{e}\right)^n, $$ using the hint $1+x\leq e^x$ for all $x\in \mathbb{R}$. I did this: The h......Sat, 29 Aug 2026 20:27:06 -0400https://pop.com.sg/what-is-the-relationship-between-the-trigonometric-secant-and-the-geom.htmlhttps://pop.com.sg/what-is-the-relationship-between-the-trigonometric-secant-and-the-geom.htmlWhat is the difference between the geometric secant(the line that cuts two points of a curve) of a curve, and the trigonometric secant(=1/cosinex) ? If they are the same, can you explain how they a......Sat, 29 Aug 2026 20:22:10 -0400https://pop.com.sg/why-can-039-t-my-graphing-calculator-find-the-rref-of-the-transpose-of.htmlhttps://pop.com.sg/why-can-039-t-my-graphing-calculator-find-the-rref-of-the-transpose-of.htmlI know this is somewhat of an odd question, but I am having trouble with my TI-84 calculator and I don't know why. I'm trying to find the RREF of the transpose of a $4\times6$ matrix; for some reas......Sat, 29 Aug 2026 20:19:33 -0400https://pop.com.sg/one-to-one-correspondence-between-infinite-sets.htmlhttps://pop.com.sg/one-to-one-correspondence-between-infinite-sets.htmlWhat does it mean to have one-to-one correspondence between infinite sets. How would you solve them?...Sat, 29 Aug 2026 20:13:15 -0400https://pop.com.sg/area-enclosed-by-a-circle-and-leminscate.htmlhttps://pop.com.sg/area-enclosed-by-a-circle-and-leminscate.htmlFind the area enclosed by a circle $r=4\sin\theta$ and out of $r^2=8\cos 2\theta$ I have tried the following integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\int_{\sqrt{8\cos2\theta}}^{4\sin\theta}d......Sat, 29 Aug 2026 20:09:18 -0400https://pop.com.sg/int-sin-6-x-cos-x-dx.htmlhttps://pop.com.sg/int-sin-6-x-cos-x-dx.html$$\int\sin^6(x)\cos(x)\,dx.$$ Apparently... $$\int\sin^6(x)\cos(x)\,dx= \int\sin^6(x)\,d\sin(x)x = \frac{1}{7}\sin^7(x)+c.$$ Can somebody explain the second equality? Shouldn't there be a $dx$ a......Sat, 29 Aug 2026 20:05:46 -0400https://pop.com.sg/rational-equations-what-are-excluded-values.htmlhttps://pop.com.sg/rational-equations-what-are-excluded-values.htmlI have an example rational equation in a textbook which I was able to solve: $$\dfrac{3}{x-6} = \dfrac{5}{x}$$ Least common denominator is : $x(x-6)$ so: $$\frac{x(x-6)3}{x-6} = \dfrac{x(x-6)5}{x}......Sat, 29 Aug 2026 20:00:37 -0400https://pop.com.sg/why-the-derivative-of-inverse-secant-has-an-absolute-value.htmlhttps://pop.com.sg/why-the-derivative-of-inverse-secant-has-an-absolute-value.html$y=\operatorname{arcsec}x$ can be defined in two ways. The first restricts the domain of $\sec y$ to $[0,\pi], y\neq\frac{\pi}{2}$. So the range of $y$ goes between $[0,\frac{\pi}{2})\cup(\frac{\pi......Sat, 29 Aug 2026 20:00:07 -0400https://pop.com.sg/difficult-conversion-from-polar-equation-to-rectangular-equation.htmlhttps://pop.com.sg/difficult-conversion-from-polar-equation-to-rectangular-equation.htmlHow do we convert this into rectangular equation? $r=5\theta$...Sat, 29 Aug 2026 19:58:21 -0400https://pop.com.sg/volume-of-a-tent-with-canvas-stretched-from-a-circular-base-to-a-verti.htmlhttps://pop.com.sg/volume-of-a-tent-with-canvas-stretched-from-a-circular-base-to-a-verti.htmlFrom George Simmons' Calculus With Analytical Geometry, page 229 question 7: A tent consists of canvas stretched from a circular base of radius $a$ to a vertical semicircular rod fastened to the ...Sat, 29 Aug 2026 19:34:17 -0400https://pop.com.sg/is-it-possible-to-change-the-pole-and-or-the-polar-axis-in-a-polar-coo.htmlhttps://pop.com.sg/is-it-possible-to-change-the-pole-and-or-the-polar-axis-in-a-polar-coo.htmlCiting Wikipedia's article on polar coordinates... In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance f......Sat, 29 Aug 2026 19:31:42 -0400https://pop.com.sg/is-polynomial-in-general-the-same-as-polynomial-function.htmlhttps://pop.com.sg/is-polynomial-in-general-the-same-as-polynomial-function.htmlThe algebra text book says, a polynomial in one variable over $\mathbb{R}$ is given by, $$f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$$ Where $x$ is an unknown quantity which commutes ......Sat, 29 Aug 2026 19:27:23 -0400https://pop.com.sg/does-the-fact-that-a-17-i-2-imply-that-the-matrix-a-must-be-i-2.htmlhttps://pop.com.sg/does-the-fact-that-a-17-i-2-imply-that-the-matrix-a-must-be-i-2.htmlDoes the fact that $A^{17} = I_2$ imply that the matrix $A$ must be $I_2$? Since the question does not specify whether the entries of $A$ are allowed to be complex valued functions, I said that IF......Sat, 29 Aug 2026 19:23:25 -0400https://pop.com.sg/convert-vector-into-diagonal-matrix.htmlhttps://pop.com.sg/convert-vector-into-diagonal-matrix.htmlGiven a vector $[x_1,x_2,x_3, \dots, x_n]^T$, is it possible to obtain a diagonal matrix, $ \left[\begin{array}{c c c c c} x_1 & 0 & 0 & \dots & 0\\ 0 & x_2 & 0 & \dots......Sat, 29 Aug 2026 19:22:18 -0400https://pop.com.sg/c-alculating-flux-using-the-divergence-theorem-when-the-divergence-is-.htmlhttps://pop.com.sg/c-alculating-flux-using-the-divergence-theorem-when-the-divergence-is-.htmlI calculated the divergence of my vector field $\langle x^2 + y^2, y^2 + z^2, 1 − 2xz − 2yz\rangle$ to be $0$. The flux is meant to be over the unit hemisphere. If I do use the divergence theorem, ......Sat, 29 Aug 2026 19:10:45 -0400https://pop.com.sg/is-this-the-general-solution-to-this-2x4-matrix.htmlhttps://pop.com.sg/is-this-the-general-solution-to-this-2x4-matrix.htmlI am a bit in doubt if I am doing this correct, I am row reducing and trying to get as close to a reduced echelon form, but is that even what I want? I am not sure if I am seeking the right answers......Sat, 29 Aug 2026 19:08:24 -0400https://pop.com.sg/cot-theta-8-9-cos-theta-0-why-does-the-answer-have-a-positive-sin-and-.htmlhttps://pop.com.sg/cot-theta-8-9-cos-theta-0-why-does-the-answer-have-a-positive-sin-and-.htmlQuestion 50 Mathway Link: https://www.mathway.com/problem/NTcyNjQ2OTM4OA My Work: I understand why it's in quadrant 4 but I don't understand why I got a negative sin and a positive cos. Is there ...Sat, 29 Aug 2026 19:07:56 -0400https://pop.com.sg/rotating-a-plane-in-3d-space.htmlhttps://pop.com.sg/rotating-a-plane-in-3d-space.htmlI have a plane defined by 3 points. I want to rotate that plane so that it will be possible to change the orientation of the plane in any dimensions. For example, what I want to do is converting th......Sat, 29 Aug 2026 19:06:06 -0400https://pop.com.sg/area-of-triangle-when-2-sides-and-no-angle-known.htmlhttps://pop.com.sg/area-of-triangle-when-2-sides-and-no-angle-known.htmlIt is quite possible this question has no answer -- that is, the area cannot be determined from the information given. It's a question I've created myself as I study for the GRE. No trigonometry is......Sat, 29 Aug 2026 19:05:44 -0400https://pop.com.sg/proving-the-trichotomy-of-order-for-the-natural-numbers.htmlhttps://pop.com.sg/proving-the-trichotomy-of-order-for-the-natural-numbers.htmlIs my proof correct? The trichotomy of order for natural numbers states: Let $a,b$ be natural numbers. Then exactly one of the following statements is true: I. $a < b$ II. $a = b$ III. $a &......Sat, 29 Aug 2026 19:04:10 -0400https://pop.com.sg/subtracting-30-percent-of-which-number-gives-2000.htmlhttps://pop.com.sg/subtracting-30-percent-of-which-number-gives-2000.htmlSubtracting $30$ percent of which number leads to $2000$? $$x - (30 \text{ percent of }x) = 2000$$ for example: $$4000 - (50\text{ percent of }4000) = 2000$$...Sat, 29 Aug 2026 19:04:00 -0400https://pop.com.sg/the-relation-between-cardinality-l1-norm-and-l2-norm-of-a-vector.htmlhttps://pop.com.sg/the-relation-between-cardinality-l1-norm-and-l2-norm-of-a-vector.htmlFor every $u\in \mathbb{R}^n$, $\textbf{Card}(u)=q$ implies ${\lVert u \rVert}_1 \leq \sqrt{q} {\lVert u \rVert}_2$ where $\textbf{Card}(u)$ is the number of non-zero element (so the L0-norm). Why ...Sat, 29 Aug 2026 19:03:33 -0400https://pop.com.sg/surface-fitting-through-the-least-squares-039-general-regression-matri.htmlhttps://pop.com.sg/surface-fitting-through-the-least-squares-039-general-regression-matri.htmlThe last page of this paper attemps to fit a paraboloid to a set of given 3D points through the general linear regression matrix formula , my question here is to what extent is this approach valid?......Sat, 29 Aug 2026 18:55:38 -0400https://pop.com.sg/unable-to-get-coordinates-and-angles-on-ogee-curve.htmlhttps://pop.com.sg/unable-to-get-coordinates-and-angles-on-ogee-curve.htmlI am trying to do some CAD designing with an Ogee curve. My Ogee is slightly more general than the typical one (where R1 = R2 = 1/2 W). In my Ogee the opening width (W) does not have to equal 2 * ......Sat, 29 Aug 2026 18:55:17 -0400https://pop.com.sg/finite-differences-second-derivative-as-successive-application-of-the-.htmlhttps://pop.com.sg/finite-differences-second-derivative-as-successive-application-of-the-.htmlThe finite difference expressions for the first, second and higher derivatives in the first, second or higher order of accuracy can be easily derived from Taylor's expansions. But, numerically, the ...Sat, 29 Aug 2026 18:53:42 -0400https://pop.com.sg/how-to-find-a-local-maximum-and-local-minimum-of-a-function.htmlhttps://pop.com.sg/how-to-find-a-local-maximum-and-local-minimum-of-a-function.htmlMy question is related to how to find a local maximum and local minimum. As far I know, for the first we should find derivative of the function and set it to zero. For exmaple, suppose our func......Sat, 29 Aug 2026 18:49:06 -0400https://pop.com.sg/how-to-set-up-lagrangian-optimization-with-matrix-constrains.htmlhttps://pop.com.sg/how-to-set-up-lagrangian-optimization-with-matrix-constrains.htmlSuppose we have a function $f: \mathbb{R} \to \mathbb{R} $ which we want to optimize subject to some constraint $g(x) \le c$ where $g:\mathbb{R} \to \mathbb{R} $ What we do is that we can set up a ...Sat, 29 Aug 2026 18:45:29 -0400https://pop.com.sg/how-do-i-solve-x-1-1-x-is-it-possible.htmlhttps://pop.com.sg/how-do-i-solve-x-1-1-x-is-it-possible.htmlI'm trying to find the intersection of $y=x$ and $y=1.1^x$. I've tried logging both sides and doing every log law on it, but can't figure it out. I know these equations intersect twice thanks to de......Sat, 29 Aug 2026 18:45:24 -0400https://pop.com.sg/automatic-proof-checker.htmlhttps://pop.com.sg/automatic-proof-checker.htmlSorry for the noob question: does it exist an automatic proof checker? I mean, some kind of programming language that validates the steps of a proof. I don't speak of automatic proof finder, just......Sat, 29 Aug 2026 18:44:43 -0400https://pop.com.sg/what-does-it-mean-to-be-a-transitive-permutation.htmlhttps://pop.com.sg/what-does-it-mean-to-be-a-transitive-permutation.htmlLet $X$ be a finite set and let $G$ be a group. What is the meaning of $G$ being a "transitive permutation" on the set $X$?...Sat, 29 Aug 2026 18:31:06 -0400https://pop.com.sg/prove-that-lim-n-to-infty-h-n-n-0-h-n-is-the-n-th-harmonic-number-usin.htmlhttps://pop.com.sg/prove-that-lim-n-to-infty-h-n-n-0-h-n-is-the-n-th-harmonic-number-usin.htmlI can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that $\li......Sat, 29 Aug 2026 18:24:57 -0400https://pop.com.sg/graph-of-complicated-equation.htmlhttps://pop.com.sg/graph-of-complicated-equation.htmlGraph of the equation $(x+y) (x^2 + y^2 -1) = 0$ is just the line $y=-x$ and the circle $x^2 + y^2 = 1$. Is it generally true that the graph of $f(x,y) \cdot g(x,y) = 0$ may be drawn as union of g......Sat, 29 Aug 2026 18:24:46 -0400https://pop.com.sg/derivative-of-the-square-root-of-x-closed.htmlhttps://pop.com.sg/derivative-of-the-square-root-of-x-closed.htmlGiven $y = \sqrt x$ and nothing more, using the formula of a limit $$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ (that is, f prime of x equals the limit of h approaching zero with the equation ((f......Sat, 29 Aug 2026 18:24:08 -0400https://pop.com.sg/is-there-any-intuition-behind-why-the-derivative-of-cos-x-is-equal-to-.htmlhttps://pop.com.sg/is-there-any-intuition-behind-why-the-derivative-of-cos-x-is-equal-to-.htmlwhy is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it ju......Sat, 29 Aug 2026 18:10:15 -0400https://pop.com.sg/can-an-equilateral-triangle-be-an-isosceles-triangle-too.htmlhttps://pop.com.sg/can-an-equilateral-triangle-be-an-isosceles-triangle-too.htmlI've looked in a math book that an isosceles triangle has at least two congruent sides. I also know that the words "at least" mean this symbol: $\ge$, which means "is greater than or equal to" or ......Sat, 29 Aug 2026 17:57:57 -0400https://pop.com.sg/cannot-understand-how-to-simplify-algebraic-fractions-with-opposite-ne.htmlhttps://pop.com.sg/cannot-understand-how-to-simplify-algebraic-fractions-with-opposite-ne.htmlI'm working on the Simplify Rational Expressions with Common Binomial Factors (https://www.khanacademy.org/math/algebra2/rational-expressions-equations-and-functions/simplify-rational-expressions/e/...Sat, 29 Aug 2026 17:51:53 -0400https://pop.com.sg/user-1123581321-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-1123581321-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 29 Aug 2026 17:50:55 -0400https://pop.com.sg/prove-continuity-of-piecewise-function-using-epsilon-delta.htmlhttps://pop.com.sg/prove-continuity-of-piecewise-function-using-epsilon-delta.htmlSuppose we have a function $\phi$ so that $$\phi (x)=\cases{f(x) & \text{ if } x\le 0\\ g(x)& \text{ if } x>0.}$$ where $f$ is continuous on $(-\infty,0]$ and $g$ is continuous o......Sat, 29 Aug 2026 17:50:49 -0400https://pop.com.sg/show-that-a-positive-operator-on-a-complex-hilbert-space-is-self-adjoi.htmlhttps://pop.com.sg/show-that-a-positive-operator-on-a-complex-hilbert-space-is-self-adjoi.htmlLet $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A : \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{......Sat, 29 Aug 2026 17:37:38 -0400https://pop.com.sg/what-is-the-correct-formula-for-the-washer-method.htmlhttps://pop.com.sg/what-is-the-correct-formula-for-the-washer-method.htmlAlmost everywhere I look the formula is: $$ \pi \int_b^a {\left(f(x)^2 - g(x)^2\right) dx} $$ where f(x) is the big function and g(x) is the smaller function. Though I've run into problems while ...Sat, 29 Aug 2026 17:34:19 -0400https://pop.com.sg/what-is-the-difference-between-a-kernel-and-a-function.htmlhttps://pop.com.sg/what-is-the-difference-between-a-kernel-and-a-function.htmlI have been looking around for this question, but all results I found only describe the definition and not the answer I seek. Is "kernel" basically a synonym of "function"? When should be the time......Sat, 29 Aug 2026 17:15:50 -0400https://pop.com.sg/duplication-formula-for-gamma-function.htmlhttps://pop.com.sg/duplication-formula-for-gamma-function.htmlUsing the Weierstrass definition for $\Gamma(x)$ and $\Gamma\Big(x + \frac12\Big)$, how can I prove the duplication formula? This is problem $10.7.3$ in the book Irresistible Integrals, by Boros an......Sat, 29 Aug 2026 17:11:51 -0400https://pop.com.sg/computing-the-value-of-zeta-6-and-zeta-8.htmlhttps://pop.com.sg/computing-the-value-of-zeta-6-and-zeta-8.htmlI know the closed form formula for calculating the values of Zeta function at even integers. I was able to derive it from the coefficients of the power series of $-\dfrac{\pi x}{2}\cot(\pi x)$. Now......Sat, 29 Aug 2026 17:08:01 -0400https://pop.com.sg/what-is-the-precise-definition-of-the-ambient-space.htmlhttps://pop.com.sg/what-is-the-precise-definition-of-the-ambient-space.htmlFor instance, what is the ambient space of a singleton $\{x\}$, where $x \in \mathbb{R}$? Can it be the singleton itself? $\mathbb{R}$? $\mathbb{R}^n$? or some arbitary set that happens to contain......Sat, 29 Aug 2026 17:00:10 -0400https://pop.com.sg/how-do-i-evaluate-the-integral-of-a-function-g-x-using-a-graph.htmlhttps://pop.com.sg/how-do-i-evaluate-the-integral-of-a-function-g-x-using-a-graph.htmlI attempted to solve this problem in the following manner: Part1: The area under the curve is defined by the integral below: $$ \int_0^{10} g(x)dx $$ and can be easily obtained by using the Are......Sat, 29 Aug 2026 16:59:24 -0400https://pop.com.sg/solve-initial-value-problem-ivp.htmlhttps://pop.com.sg/solve-initial-value-problem-ivp.htmlI have equations \begin{align} 7z'(x) &= x^2 + y(x)^2 + z(x)^2 \\ y''(x) &= -(y'(x) + 7y(x))\sin(z(x)) \end{align} where $$(y(0),y'(0),z(0))=(1.5,-2.6,0.5)$$ The task is to solve this pro......Sat, 29 Aug 2026 16:55:31 -0400https://pop.com.sg/constructive-intermediate-value-theorem-ivt.htmlhttps://pop.com.sg/constructive-intermediate-value-theorem-ivt.htmlI'm trying to learn a bit about intuitionistic/constructive mathematics, as I want to understand a little about topos theory and homotopy type theory (HoTT). I'm confused as to why the intermediate ...Sat, 29 Aug 2026 16:51:20 -0400https://pop.com.sg/how-to-simply-increase-probability-of-an-event.htmlhttps://pop.com.sg/how-to-simply-increase-probability-of-an-event.htmlLets say there is 40% chance that an event will happen. Than it is said that the probability of the event is increased by five times. I do realize that the resulting percentage is not 200% or even ......Sat, 29 Aug 2026 16:47:48 -0400https://pop.com.sg/is-it-possible-to-make-a-2d-net-of-a-spherical-truncated-icosahedron-a.htmlhttps://pop.com.sg/is-it-possible-to-make-a-2d-net-of-a-spherical-truncated-icosahedron-a.htmlI know that a Truncated Icosahedron is the underlying shape that makes a Football (Soccer ball). In real life, the 20 Hexagons and 12 Pentagons are cut out of fabric and a balloon inside is blown......Sat, 29 Aug 2026 16:47:23 -0400https://pop.com.sg/trigonometric-equation-involving-tangent-how-do-i-solve-it.htmlhttps://pop.com.sg/trigonometric-equation-involving-tangent-how-do-i-solve-it.htmlI would like to know how to solve this and most importantly make intuitive sense of it. Thanks in advance! Equation: $\tan(x) = -1.5$ (using a calculator with $\arctan(-1.5)$) My Results: $x = -0......Sat, 29 Aug 2026 16:42:54 -0400https://pop.com.sg/prove-2-boolean-expressions-are-equal-by-multiplying-out-and-simplifyi.htmlhttps://pop.com.sg/prove-2-boolean-expressions-are-equal-by-multiplying-out-and-simplifyi.htmlI was asked to prove that the following is TRUE by "multiplying it out" and simplifying: $(A+BC)(A+DE)=A+BCDE$ I am already familiar with the theorem $A+BC=(A+B)(A+C)$ which would be enough pr......Sat, 29 Aug 2026 16:41:54 -0400https://pop.com.sg/two-linearly-independent-vectors-perpendicular-to-vector-u.htmlhttps://pop.com.sg/two-linearly-independent-vectors-perpendicular-to-vector-u.htmlI'm having trouble with these types of questions. I have the following vector $u = (4, 7, -9)$ and it wants me to find 2 vectors that are perpendicular to this one. I know that $(4,7,-9)\cdot (x,y......Sat, 29 Aug 2026 16:38:50 -0400https://pop.com.sg/help-with-coding-class-exercise-find-the-half-of-a-number-without-divi.htmlhttps://pop.com.sg/help-with-coding-class-exercise-find-the-half-of-a-number-without-divi.htmlThe teacher explained more or less what we have to do. My question is about the maths part not the coding part. The exercise is: Create a function that given a positive integer will return the int......Sat, 29 Aug 2026 16:23:43 -0400https://pop.com.sg/the-derivative-of-constant-raised-to-the-power-of-n-which-is-raised-to.htmlhttps://pop.com.sg/the-derivative-of-constant-raised-to-the-power-of-n-which-is-raised-to.htmlI need a reminder of how to go about taking the derivative of $c^{n^x}$ where $x$ and $c$ are constants. An example would be $$\large 4^{n^2}$$ Thanks in advance....Sat, 29 Aug 2026 16:09:09 -0400https://pop.com.sg/abel-differential-equation-of-second-kind.htmlhttps://pop.com.sg/abel-differential-equation-of-second-kind.htmlConsider this Abel differential equation: $$4\left[ y(z)+ z^2\right] y'(z)-(1+6z)y(z)=0$$ Is there a way to separate variables? I just tried some change of variables, but with no good results....Sat, 29 Aug 2026 16:06:01 -0400https://pop.com.sg/proving-difference-of-two-powers-using-induction.htmlhttps://pop.com.sg/proving-difference-of-two-powers-using-induction.htmlProve, that $\forall n \in \mathbb{N}$ the following identity holds: $$a^n - b^n = (a-b)\sum_{k=0}^{n-1} a^k b^{n-k-1}.$$ For $n = 1$ we get $a^1 - b^1 = a - b$ on LHS, and on RHS we get $(a-b)\su......Sat, 29 Aug 2026 16:00:02 -0400https://pop.com.sg/maximum-value-of-function-on-unit-circle.htmlhttps://pop.com.sg/maximum-value-of-function-on-unit-circle.htmlFind the maximum value of the function $f(x,y) = (x+2y)^2$ on the unit circle $x^2+y^2=1$. I defined a new function $g(x)= x^2 +4x\sqrt{1-x^2} +4(1-x^2) $ and let its derivative equal to zero. So......Sat, 29 Aug 2026 15:56:52 -0400https://pop.com.sg/what-does-it-mean-to-call-something-naive-in-mathematics.htmlhttps://pop.com.sg/what-does-it-mean-to-call-something-naive-in-mathematics.htmlI apologize if this is a less-than appropriate place for the question, but I've seen this term come up repeatedly in textbooks and lectures without any knowledge of what it indicates. There is alwa......Sat, 29 Aug 2026 15:52:15 -0400https://pop.com.sg/archimedes-039-derivation-of-the-spherical-cap-area-formula.htmlhttps://pop.com.sg/archimedes-039-derivation-of-the-spherical-cap-area-formula.htmlArchimedes derived a formula for the area of a spherical cap. so Archimedes says that the curved surface area of a spherical cap is equal to the area of a circle with radius equal to the distance ...Sat, 29 Aug 2026 15:46:45 -0400https://pop.com.sg/find-the-perimeter-of-any-triangle-given-the-three-altitude-lengths.htmlhttps://pop.com.sg/find-the-perimeter-of-any-triangle-given-the-three-altitude-lengths.htmlThe altitude lengths are 12, 15 and 20. I would like a process rather than just a single solution....Sat, 29 Aug 2026 15:21:29 -0400https://pop.com.sg/user-akhil-mathew-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-akhil-mathew-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 29 Aug 2026 15:12:42 -0400https://pop.com.sg/how-to-find-the-eigenvalues-of-tridiagonal-toeplitz-matrix.htmlhttps://pop.com.sg/how-to-find-the-eigenvalues-of-tridiagonal-toeplitz-matrix.htmlAssume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & &\\ b & a & c &am......Sat, 29 Aug 2026 15:04:20 -0400https://pop.com.sg/parallel-surface.htmlhttps://pop.com.sg/parallel-surface.htmlFor a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$ How could I show the ...Sat, 29 Aug 2026 14:57:27 -0400https://pop.com.sg/expectation-of-hitting-time-for-simple-symmetric-random-walk.htmlhttps://pop.com.sg/expectation-of-hitting-time-for-simple-symmetric-random-walk.htmlAssume there is a simple symmetric random walk $$S_n=X_1+...+X_n,\quad S_0=0$$ where $\mathbb P(X_i=\pm 1)=\frac{1}{2}$. Define $T=\inf\{n:S_n=1\}$. How to compute $\mathbb E(T)$? My idea: if $\m......Sat, 29 Aug 2026 14:43:14 -0400https://pop.com.sg/right-english-wording-for-counterexamples-to-a-theorem.htmlhttps://pop.com.sg/right-english-wording-for-counterexamples-to-a-theorem.htmlThis question is about the right English wording. I give here what I call "counterexamples to Banach fixed-point theorem". What I do, is that I look to what happen if some hypothesis of the theore......Sat, 29 Aug 2026 14:34:06 -0400https://pop.com.sg/deriving-surface-area-of-a-sphere-from-the-circumference.htmlhttps://pop.com.sg/deriving-surface-area-of-a-sphere-from-the-circumference.htmlgiven the circumference of a circle, which is 2πr, how many times do I have to add it to itself to cover a whole surface of a sphere and deriving 4πr^2?...Sat, 29 Aug 2026 14:32:29 -0400https://pop.com.sg/best-strategies-for-039-squid-game-039-episode-6-game-4-marble-game-pl.htmlhttps://pop.com.sg/best-strategies-for-039-squid-game-039-episode-6-game-4-marble-game-pl.htmlTwo players each get $n=10$ marbles. Every alternating turn, one player (say, the player whose turn it is) hides an amount of own marbles in fist, and, the other player must guess if hidden amount ......Sat, 29 Aug 2026 14:29:13 -0400https://pop.com.sg/multiplying-two-inequalities.htmlhttps://pop.com.sg/multiplying-two-inequalities.htmlSuppose we have two inequalities $$a\leq x\leq b\tag{1}$$ $$c\leq y\leq d\tag{2},$$ where $a,b,c,d>0$. Then can I conclude that $$ac\leq xy\leq bd\quad ?$$ My attempt: Since $a,b,c,d>0$ and $\...Sat, 29 Aug 2026 14:16:22 -0400https://pop.com.sg/find-the-remainder-of-a-number-with-a-large-exponent-duplicate.htmlhttps://pop.com.sg/find-the-remainder-of-a-number-with-a-large-exponent-duplicate.htmlI have to find the remainder of $10^{115}$ divided by 7. I was following the way my book did it in an example but then I got confused. So far I have, $\overline{10}^{115}$=$\overline{10}^{7*73+4}......Sat, 29 Aug 2026 14:13:51 -0400https://pop.com.sg/how-to-represent-the-ceiling-function-using-mathematical-notation.htmlhttps://pop.com.sg/how-to-represent-the-ceiling-function-using-mathematical-notation.htmlHow to represent the ceiling function using mathematical notation? I need an equation to input in a program because it doesn't except the ceiling function so it has to be some sort of mathematical ...Sat, 29 Aug 2026 14:11:06 -0400https://pop.com.sg/slopes-of-inverse-functions.htmlhttps://pop.com.sg/slopes-of-inverse-functions.htmlI have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ......Sat, 29 Aug 2026 14:07:53 -0400https://pop.com.sg/generate-integer-solutions-from-a-hyperbolic-equation.htmlhttps://pop.com.sg/generate-integer-solutions-from-a-hyperbolic-equation.htmlI want to generate any integer solution from the equation $\sqrt{8r^2+1} = n$. I know there are integer solutions to this, but I have no idea how to approach this. (When working with whole numbers)......Sat, 29 Aug 2026 14:07:43 -0400https://pop.com.sg/how-to-find-the-angle-of-a-rectangle-vertex.htmlhttps://pop.com.sg/how-to-find-the-angle-of-a-rectangle-vertex.htmlbefore I go further I want you to know that I'm developing a collision detection system in a programming language (Javascript). I'm not used to math terms (it was like 10 years ago when I was in ......Sat, 29 Aug 2026 14:01:11 -0400https://pop.com.sg/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-p.htmlhttps://pop.com.sg/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-p.htmlIt’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an a......Sat, 29 Aug 2026 13:39:39 -0400https://pop.com.sg/analysis-determine-radius-and-interval-of-convergence-for-these-power-.htmlhttps://pop.com.sg/analysis-determine-radius-and-interval-of-convergence-for-these-power-.htmlI needed to determine the radius and interval of convergence for each of the following; $\mathbb{One:}$ $$\sum_{n=0}^{\infty}\frac{3^nx^n}{n!}$$ $\mathbb{Two:}$ $$\sum_{n=0}^{\infty}\frac{x^n}{(1......Sat, 29 Aug 2026 13:24:29 -0400https://pop.com.sg/how-to-find-the-length-of-the-line-segment-of-this-following-triangle.htmlhttps://pop.com.sg/how-to-find-the-length-of-the-line-segment-of-this-following-triangle.htmlGiven: right triangle ABC with C as a right angle. Line DE is formed, such that ED is perpendicular to AB. D at AB and E at CB. If AC=AD= 8 cm and CE=DE, what is the length of BE? My thinking(sin......Sat, 29 Aug 2026 13:18:28 -0400https://pop.com.sg/example-of-a-quantifier-elimination-procedure-for-a-simple-but-nontriv.htmlhttps://pop.com.sg/example-of-a-quantifier-elimination-procedure-for-a-simple-but-nontriv.htmlIs there a simple-but-nontrivial example of a concrete quantifier elimination procedure with a concrete theory, especially one that's a standard example of what a constructive argument for quantifier ...Sat, 29 Aug 2026 13:09:33 -0400https://pop.com.sg/spherical-triangle-given-two-angles-and-one-length-find-another-length.htmlhttps://pop.com.sg/spherical-triangle-given-two-angles-and-one-length-find-another-length.htmlLet $[u,v,w]$ be a positively oriented spherical triangle in $S^2$ If $a=\cos^{-1}(\frac13)$, $\beta = \frac{2\pi}{3}$, $\gamma = \frac{\pi}{3}$, find $b$. We have that: $a=\cos^{-1}(\frac13)\imp......Sat, 29 Aug 2026 13:09:01 -0400https://pop.com.sg/ratios-make-me-feel-like-an-idiot-help-me-mix-up-some-coca-cola.htmlhttps://pop.com.sg/ratios-make-me-feel-like-an-idiot-help-me-mix-up-some-coca-cola.htmlI may be over-complicating things, but something doesn't seem right (and I swear this isn't homework, I'm friggin 30 years old :P). I want to see what it will cost me to make a TWO liter of Coca-C......Sat, 29 Aug 2026 13:04:07 -0400https://pop.com.sg/what-is-wrong-with-this-solution-of-2x-ln-2-3y-ln-3.htmlhttps://pop.com.sg/what-is-wrong-with-this-solution-of-2x-ln-2-3y-ln-3.htmlWhat is wrong with this solution of Solve for $ (2x)^{\ln2 }= (3y)^{\ln3 }$ $3^{\ln x} = 2^{\ln y} $ Here is my solution- $ (2x)^{\ln2 }= (3y)^{\ln3 }$ $ 2^{\ln2x }= 3^{\ln3y }$ $ 2^{(\ln ......Sat, 29 Aug 2026 12:56:41 -0400https://pop.com.sg/proof-of-the-associative-law-and-the-commutative-law.htmlhttps://pop.com.sg/proof-of-the-associative-law-and-the-commutative-law.htmlThe associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily. We can prove$^2$ the ...Sat, 29 Aug 2026 12:40:53 -0400https://pop.com.sg/if-ax-2-bx-7-15-x-2-cx-14-for-all-values-of-x-and-a-b-8-what-are-the-t.htmlhttps://pop.com.sg/if-ax-2-bx-7-15-x-2-cx-14-for-all-values-of-x-and-a-b-8-what-are-the-t.htmlPlease solve the following problem and show your work: If $(ax + 2)( bx + 7) = 15 x^2 + cx + 14$ for all values of $x$, and $a + b =8$, what are the two possible values for $c$ ?...Sat, 29 Aug 2026 12:25:09 -0400https://pop.com.sg/maclaurin-series-from-sin-x-to-cos-x-using-derivative.htmlhttps://pop.com.sg/maclaurin-series-from-sin-x-to-cos-x-using-derivative.htmlI understand how to find the MacLaurin series for $\sin(x)$. $$\sum_{n=1}^\infty \frac{x^{2n-1}\cdot\!(-1)^{n-1}}{(2n-1)!}$$ Now I am trying to find the MacLaurin series for $\cos(x)$ by taking the ...Sat, 29 Aug 2026 12:22:48 -0400https://pop.com.sg/is-1e-005-1e-5.htmlhttps://pop.com.sg/is-1e-005-1e-5.htmlTotally new to Matlab, I encountered the notation $1e-005$ without proper context to guess its meaning. After googling, I found that it may represent the number $0.00001$. However, I know that $1e-5$ ...Sat, 29 Aug 2026 12:13:39 -0400https://pop.com.sg/how-to-solve-x-log-x-10-6.htmlhttps://pop.com.sg/how-to-solve-x-log-x-10-6.htmlI am trying to solve $$x\log(x) = 10^6$$ but can't find an elegant solution. Any ideas ?...Sat, 29 Aug 2026 12:04:14 -0400https://pop.com.sg/is-there-a-difference-between-theorem-proof-and-theorem-demonstration.htmlhttps://pop.com.sg/is-there-a-difference-between-theorem-proof-and-theorem-demonstration.htmlWhen referring to proving that a theorem is true, I have seen both words used interchangeably. Do they mean the same thing ?...Sat, 29 Aug 2026 12:01:57 -0400https://pop.com.sg/why-is-cos135-circ-negative-when-length-is-always-positive.htmlhttps://pop.com.sg/why-is-cos135-circ-negative-when-length-is-always-positive.htmlConsider the following diagram. I am told that $\cos 45^{\circ}$ = $\frac{1}{\sqrt 2}$. I understand this. I am next told taught that $\cos 135^{\circ}$ = $\cos 45^{\circ}$ in 2nd quadrant. And f......Sat, 29 Aug 2026 11:59:11 -0400