https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 17 Aug 2026 10:49:23 -0400https://pop.com.sg/sketching-the-contour-map-of-a-function-with-several-variables.htmlhttps://pop.com.sg/sketching-the-contour-map-of-a-function-with-several-variables.htmlConsider the surface given by the function $f(x,y) = \frac{4}{\sqrt{x^2+y^2-9}}$. Draw the level curves for the values $c = 1$ and $c = 2$. Be sure to label your axes to indicate the scale. Note: W......Mon, 17 Aug 2026 14:47:02 -0400https://pop.com.sg/prove-divergence-of-sum-ln-1-1-n.htmlhttps://pop.com.sg/prove-divergence-of-sum-ln-1-1-n.htmlWe have seen the harmonic series is a divergent series whose terms approach $0$. Show that $$\sum_{n = 1}^\infty \text{ln}\left(1 + \frac{1}{n}\right)$$ is another series with this property. Deno......Mon, 17 Aug 2026 14:46:09 -0400https://pop.com.sg/find-the-general-form-of-the-solution-to-the-system-of-equations-below.htmlhttps://pop.com.sg/find-the-general-form-of-the-solution-to-the-system-of-equations-below.htmlFind the general form of the solution to the system of equations below. \begin{align} 2x_1-x_2+8x_3-x_4&=0\\ -4x_1+3x_2-18x_3+x_4&=0\\ 2x_1+x_2+4x_3-3x_4&=0. \end{align} My attempt: ......Mon, 17 Aug 2026 14:36:28 -0400https://pop.com.sg/anyone-have-a-good-proof-for-the-second-part-of-ftc.htmlhttps://pop.com.sg/anyone-have-a-good-proof-for-the-second-part-of-ftc.htmlDoes anyone have a good proof for the second part of the fundamental theorem of calculus? I haven't been able to find any good videos on it so far so I'd like someone to write it down and I can throw ...Mon, 17 Aug 2026 14:08:59 -0400https://pop.com.sg/lax-wendroff-method-for-linear-advection-stability-analysis.htmlhttps://pop.com.sg/lax-wendroff-method-for-linear-advection-stability-analysis.htmlQuestion 1: Consider the wave equation $$ u_t + c(x) u_x = 0 , $$ where $x\in \Omega \subset \Bbb R$ and $c(x)$ is a function of $x$. (a) Show that the Lax-Wendroff scheme for this PDE is......Mon, 17 Aug 2026 14:02:52 -0400https://pop.com.sg/how-do-i-integrate-e-sqrt-x-using-the-substitution-rule.htmlhttps://pop.com.sg/how-do-i-integrate-e-sqrt-x-using-the-substitution-rule.htmlI've been trying to integrate that function, but it looks like I'm missing something, so can any one please show me how to integrate $e^\sqrt{x}$, in order to correct my procedure. Thanks...Mon, 17 Aug 2026 13:46:38 -0400https://pop.com.sg/given-a-quadrilateral-prove-it-is-a-square.htmlhttps://pop.com.sg/given-a-quadrilateral-prove-it-is-a-square.htmlConsider this game. You have a perfect ruler and a perfect protractor. You are also able to observe perfectly if two lines are parallel. Is it possible to prove that a given quadrilateral......Mon, 17 Aug 2026 13:42:44 -0400https://pop.com.sg/proofs-in-category-theory-monicity-of-composition-of-monic-arrows.htmlhttps://pop.com.sg/proofs-in-category-theory-monicity-of-composition-of-monic-arrows.htmlI've just started working on proofs in category theory and I'm interested in thinking about the structure of a proof in terms of this domain. This means thinking about the proof in terms of arrows ......Mon, 17 Aug 2026 13:33:16 -0400https://pop.com.sg/what-quality-of-a-distribution-describes-the-spikiness-of-its-density-.htmlhttps://pop.com.sg/what-quality-of-a-distribution-describes-the-spikiness-of-its-density-.htmlI'm a programmer, not a math guy, so please answer in English. ;) Suppose I have a multi-modal univariate distribution like: .. . .. ........... .. . ........Mon, 17 Aug 2026 13:31:57 -0400https://pop.com.sg/the-maths-required-for-an-economics-degree.htmlhttps://pop.com.sg/the-maths-required-for-an-economics-degree.htmlI have a degree in computer science and I wanted to do another degree in economics. However, my maths have been weak since high school always scoring slightly above passing rate. During my course of ...Mon, 17 Aug 2026 13:31:44 -0400https://pop.com.sg/function-of-three-variables-hessian-zero.htmlhttps://pop.com.sg/function-of-three-variables-hessian-zero.htmlGiven the function $f : \mathbb{R}^3 \to \mathbb{R}$ defined by $f(x,\,y,\,z) = (x - 1)^4 - (x - y)^4 - (y - z)^4$ it presents a unique critical point of coordinates $(1,\,1,\,1)$. Unfortunately, ......Mon, 17 Aug 2026 13:19:17 -0400https://pop.com.sg/equating-coefficients-meaning-and-example.htmlhttps://pop.com.sg/equating-coefficients-meaning-and-example.htmlI am trying to understand what "Equating the Coefficients " means I am given the following: $$ t_1 \left(t_2+\frac{1}{x+1}\right)+\frac{2 t_2}{x}=t_1^2 b_2'+t_1 \left(2 b_2 t_1'+b_1'\right)+b_1 t......Mon, 17 Aug 2026 13:18:32 -0400https://pop.com.sg/questions-tagged-lambert-w.htmlhttps://pop.com.sg/questions-tagged-lambert-w.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 17 Aug 2026 13:17:48 -0400https://pop.com.sg/understanding-convolution-summation-in-discrete-time-signals.htmlhttps://pop.com.sg/understanding-convolution-summation-in-discrete-time-signals.htmlGeneral definition of convolution states: $$ u(n)*s(n) = \sum_k u(k)s(n-k) $$ However, unable to grasp the fundamental over here, I am wondering what summation would the following represent: $u(n-......Mon, 17 Aug 2026 13:09:03 -0400https://pop.com.sg/calculating-generalized-eigenvectors.htmlhttps://pop.com.sg/calculating-generalized-eigenvectors.htmli'm a bit confused when trying to calculating generalized eigenvectors if there is more than one eigenvectors for a specific eigenvalue. Example: $A=\begin{pmatrix}0&1&-1&-1\\0&0&a......Mon, 17 Aug 2026 13:03:25 -0400https://pop.com.sg/proof-of-the-power-series-1-x-2-x-3-ldots-x-n-frac-1-1-x-duplicate.htmlhttps://pop.com.sg/proof-of-the-power-series-1-x-2-x-3-ldots-x-n-frac-1-1-x-duplicate.htmlCan anyone show me the proof of this equation: $$ \lim_{n \to \infty} 1 + x + x^2 + x^3 + \ldots + x^n = \frac{1}{1-x}, $$ where $|x|<1$. Edit: I have then additionally written $x$ on the lef......Mon, 17 Aug 2026 13:03:14 -0400https://pop.com.sg/asymptotic-behavior-of-polynomials.htmlhttps://pop.com.sg/asymptotic-behavior-of-polynomials.htmlThe following question is taken from Introduction to Algorithms by CLRS, Chapter $3.$ Let $$p(n) = \sum_{i=0}^d a_i n^i,$$ where $a_d>0,$ be a degree-$d$ polynomial in $n$, and let $k$ b......Mon, 17 Aug 2026 13:01:18 -0400https://pop.com.sg/what-is-the-normal-derivative-of-a-pde.htmlhttps://pop.com.sg/what-is-the-normal-derivative-of-a-pde.htmlKreyszig's introduction of the normal derivative $u_n = \partial u / \partial n$ leaves me a bit puzzled as to what variable $n$ is if $u$ is a function of $x$ and $y$. The trailing example uses a ...Mon, 17 Aug 2026 12:58:53 -0400https://pop.com.sg/can-039-t-figure-out-this-trig-proof-cos-x-y-cos-x-y-cos-2-x-sin-2-y.htmlhttps://pop.com.sg/can-039-t-figure-out-this-trig-proof-cos-x-y-cos-x-y-cos-2-x-sin-2-y.html$$\cos(x+y)\cos(x-y)=\cos^2(x)-\sin^2(y)$$ I use the sum and difference formula to reduce the right hand side of the equation to $$[\cos(x)\cos(y)-\sin(x)\sin(y)][\cos(x)\cos(y)+\sin(x)\sin(y)]$$ ......Mon, 17 Aug 2026 12:53:48 -0400https://pop.com.sg/proof-of-the-laplace-expansion.htmlhttps://pop.com.sg/proof-of-the-laplace-expansion.htmlI just learned about Laplace Expansion for determinant calculation in high-school, they taught me how to calculate minors cofactors and everything but they did not include the proof in the book. I ......Mon, 17 Aug 2026 12:53:13 -0400https://pop.com.sg/what-is-the-number-of-non-increasing-4-digit-numbers.htmlhttps://pop.com.sg/what-is-the-number-of-non-increasing-4-digit-numbers.htmlThis problem, though quite simple, has stumped my teenage mind. How many 4-digit numbers are there whose digits are non-increasing? This seemed quite simple at first, meaning I have the digits [9 8......Mon, 17 Aug 2026 12:49:09 -0400https://pop.com.sg/how-to-evaluate-this-surface-integral-iint-t-e-y-x-y-x-da-where-t-is-t.htmlhttps://pop.com.sg/how-to-evaluate-this-surface-integral-iint-t-e-y-x-y-x-da-where-t-is-t.htmlI am given the following problem: Use a change of variables to evaluate $\int \int_T e^{(y-x)/(y+x)} dA$ where $T$ is the triangular region with vertices $(0,0)$, $(1,0)$ and $(0,1)$. Here is w......Mon, 17 Aug 2026 12:46:25 -0400https://pop.com.sg/satisfying-hypotheses-for-l-039-hospital-039-s-rule.htmlhttps://pop.com.sg/satisfying-hypotheses-for-l-039-hospital-039-s-rule.htmlI have that the following hypothesis need to be satisfied in order to use L'Hospital's rule for, \begin{equation} \lim_{x\to a}\frac{f(x)}{g(x)} \end{equation} $f(x)$ and $g(x)$ have to be func......Mon, 17 Aug 2026 12:46:15 -0400https://pop.com.sg/single-variable-calculus-reference-recommendations.htmlhttps://pop.com.sg/single-variable-calculus-reference-recommendations.htmlThis question is a generalization of the common question asking for calculus references. It is here to abstract away the repetition, and give a canonical resource for calculus references. I'm l......Mon, 17 Aug 2026 12:41:29 -0400https://pop.com.sg/finding-the-center-and-radius-of-a-circle-given-a-general-degree-2-equ.htmlhttps://pop.com.sg/finding-the-center-and-radius-of-a-circle-given-a-general-degree-2-equ.htmlI am trying to find the center and radius of the circle with equation $x^2 + y^2 -6x + 10y + 9 = 0$...Mon, 17 Aug 2026 12:38:42 -0400https://pop.com.sg/is-10-closer-to-infinity-than-1.htmlhttps://pop.com.sg/is-10-closer-to-infinity-than-1.htmlThis may be considered a philosophical question but is the number "10" closer to infinity than the number "1"?...Mon, 17 Aug 2026 12:29:20 -0400https://pop.com.sg/why-is-a-genus-1-curve-smooth-and-is-it-still-true-for-a-non-zero-genu.htmlhttps://pop.com.sg/why-is-a-genus-1-curve-smooth-and-is-it-still-true-for-a-non-zero-genu.htmlIn the very commonly used J. Silverman's AEC an elliptic curve is defined as a genus 1 projective curve with a fixed point 0. In all the other books I looked at it is defined to be (also) smooth. B......Mon, 17 Aug 2026 12:28:18 -0400https://pop.com.sg/finding-ratio-of-cevian-lines.htmlhttps://pop.com.sg/finding-ratio-of-cevian-lines.htmlI am preparing for an exam and doing some pratice problems. So I'm having a difficult time with this problem. At first I thought the ratio was 2:1 and then I also thought I would be able to use the......Mon, 17 Aug 2026 12:25:39 -0400https://pop.com.sg/number-of-5-card-charlie-hands-in-blackjack.htmlhttps://pop.com.sg/number-of-5-card-charlie-hands-in-blackjack.htmlA five-card Charlie in blackjack is when you have a total of 5 cards and you do not exceed a point total of 21. How many such hands are there? Of course, the natural next question concerns six-card ...Mon, 17 Aug 2026 12:21:04 -0400https://pop.com.sg/finding-hitting-probability-from-markov-chain.htmlhttps://pop.com.sg/finding-hitting-probability-from-markov-chain.htmlI have a Markov chain with states {1,2,3,4,5} which has the following transition matrix: $$P= \begin{bmatrix} 0.3 & 0 & 0.7 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0.5 & ......Mon, 17 Aug 2026 12:13:18 -0400https://pop.com.sg/user-oxodo-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-oxodo-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 17 Aug 2026 12:11:31 -0400https://pop.com.sg/determining-if-a-statement-is-tautology-contingent-or-contradiction.htmlhttps://pop.com.sg/determining-if-a-statement-is-tautology-contingent-or-contradiction.htmlSay we are given the following statement A ≡ B where A is some logic statement and B is some logic statement. Determine if it is tautology, contingent or contradiction. I know the definition of ......Mon, 17 Aug 2026 12:06:36 -0400https://pop.com.sg/how-do-you-find-a-derivative-of-y-1-cos-x-sin-x.htmlhttps://pop.com.sg/how-do-you-find-a-derivative-of-y-1-cos-x-sin-x.htmlThe question asks to find the derivative of the function $1-\cos(x)\sin(x)$, and I thought maybe using some derivative rules I could, but I don't know where to start....Mon, 17 Aug 2026 11:58:06 -0400https://pop.com.sg/why-do-equations-work-and-how-do-they-relate-to-each-other.htmlhttps://pop.com.sg/why-do-equations-work-and-how-do-they-relate-to-each-other.htmlOk, so I understand that an equation is something like 15 = 15 , and that the only criteria as far as I can tell for it being an equation is that both sides are equal to each other. I have a few ...Mon, 17 Aug 2026 11:54:07 -0400https://pop.com.sg/what-is-the-radial-direction.htmlhttps://pop.com.sg/what-is-the-radial-direction.htmlI'm currently in an electrostatics course; and wanting to rip my hair out. The question says: A sphere of radius 𝑎 is polarized such that the polarization at 𝒓 within the sphere is given by 𝑷 = ......Mon, 17 Aug 2026 11:45:04 -0400https://pop.com.sg/determine-which-positive-integers-have-exactly-36-positive-divisors.htmlhttps://pop.com.sg/determine-which-positive-integers-have-exactly-36-positive-divisors.htmlDetermine which positive integers have exactly 36 positive divisors. Let $n=p_1^a{_2}p_1^a{_2}...p_s^a{_s}$. Then $\tau(n)= \prod_{j=1}^{s} a_j+1$. We need $\tau(n)=36$. So we list the factor pair......Mon, 17 Aug 2026 11:36:15 -0400https://pop.com.sg/in-the-given-figure-pqrs-is-a-parallelogram.htmlhttps://pop.com.sg/in-the-given-figure-pqrs-is-a-parallelogram.htmlIn the given figure, $PQRS$ is a parallelogram. $PQ$ is produced to $L$ so that $QL=PQ$. The line $SL$ cuts $QR$ at $O$. Prove that: $\triangle PQS=2\triangle ROL$. Attempt $$PL=LQ$$ $$ Q \ \tex......Mon, 17 Aug 2026 11:31:57 -0400https://pop.com.sg/show-that-any-non-cyclic-group-of-order-9-is-isomorphic-to-mathbb-z-3-.htmlhttps://pop.com.sg/show-that-any-non-cyclic-group-of-order-9-is-isomorphic-to-mathbb-z-3-.htmlShow that any non-cyclic group of order $9$ is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$ The basic idea to show this is to take any non-cyclic group $G$ of order $9$ and apply the theorem b......Mon, 17 Aug 2026 11:30:25 -0400https://pop.com.sg/symbol-for-finite.htmlhttps://pop.com.sg/symbol-for-finite.htmlI understand there is a symbol for infinite. Is there one for finite? I searched and found there is none. How is finite represented symbolically?...Mon, 17 Aug 2026 11:21:30 -0400https://pop.com.sg/completeness-can-be-omitted-from-banach-fixed-point-theorem.htmlhttps://pop.com.sg/completeness-can-be-omitted-from-banach-fixed-point-theorem.htmlIn Kreyszig's Functional Analysis, page no. 303, exercise no. 3 says that completeness cannot be omitted from Banach's Fixed Point Theorem. But if we take $f(x)=x^2$ from an incomplete metric space......Mon, 17 Aug 2026 11:19:53 -0400https://pop.com.sg/is-there-a-way-to-write-the-convolution-form-of-a-derivative.htmlhttps://pop.com.sg/is-there-a-way-to-write-the-convolution-form-of-a-derivative.htmlIn image filtering and computer simulations, derivatives can be applied to a image by performing a convolution of the image/data with an appropriate kernel matrix. For example, in 1D this could be ......Mon, 17 Aug 2026 11:16:05 -0400https://pop.com.sg/derivative-of-ln-x-e-duplicate.htmlhttps://pop.com.sg/derivative-of-ln-x-e-duplicate.htmlIn Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ......Mon, 17 Aug 2026 11:03:37 -0400https://pop.com.sg/markov-process-vs-markov-chain-vs-random-process-vs-stochastic-process.htmlhttps://pop.com.sg/markov-process-vs-markov-chain-vs-random-process-vs-stochastic-process.htmlI'm trying to understand each of the above terms, and I'm having a lot of trouble deciphering the difference between them. According to Wikipeda: A Markov chain is a memoryless, random process. A ...Mon, 17 Aug 2026 11:03:10 -0400https://pop.com.sg/infinite-algebraic-extension-of-mathbb-q.htmlhttps://pop.com.sg/infinite-algebraic-extension-of-mathbb-q.htmlI have this problem in a exercise list: "Prove that $K=\mathbb{Q}(2^{\frac{1}{2}},2^{\frac{1}{3}}, 2^{\frac{1}{4}}, \ldots)$ is an algebraic extension, but not a finite extension of $\mathbb{Q}$."......Mon, 17 Aug 2026 11:02:28 -0400https://pop.com.sg/what-is-the-formula-for-the-birthday-problem.htmlhttps://pop.com.sg/what-is-the-formula-for-the-birthday-problem.htmlWhat is the formula for the birthday problem? I can't seem to find JUST the formula. I don't care how it works at this point, I just want to know what to do. Can someone please just tell me? Most ...Mon, 17 Aug 2026 10:57:23 -0400https://pop.com.sg/plotting-the-complex-exponential-function-and-it-039-s-period-by-hand.htmlhttps://pop.com.sg/plotting-the-complex-exponential-function-and-it-039-s-period-by-hand.htmlI'm reading a bit about the exponential function $e_k: \mathbb{R} \longrightarrow \mathbb{C}$ given by $$e_{k}: x \mapsto e^{i k x}=\cos k x+i \sin k x$$ The period is given by $2\pi/|k|$. I do no......Mon, 17 Aug 2026 10:56:13 -0400https://pop.com.sg/how-to-solve-trigonometric-equation-sin-x-x-cdot-cos-x-0.htmlhttps://pop.com.sg/how-to-solve-trigonometric-equation-sin-x-x-cdot-cos-x-0.htmlI'm facing the problem of solving $$\sin(x)+x \cdot \cos(x)=0$$ using $$\tan(x)=\sin(x)/\cos(x)$$ I end at $$x+\tan(x)=0$$ on the other hand, I also tried $\cos(x)= \pm \sqrt{1-\sin......Mon, 17 Aug 2026 10:52:47 -0400https://pop.com.sg/lotus-law-of-the-unconscious-statistician-for-non-one-to-one-functions.htmlhttps://pop.com.sg/lotus-law-of-the-unconscious-statistician-for-non-one-to-one-functions.htmlMy textbook, Introduction to Probability by Blitzstein and Hwang, defines LOTUS (law of the unconscious statistician) as follows: Theorem 4.5.1 (LOTUS). If $X$ is a discrete r.v. and $g$ is a func......Mon, 17 Aug 2026 10:49:23 -0400https://pop.com.sg/find-a-quintic-polynomial-function-given-roots-and-y-intercept.htmlhttps://pop.com.sg/find-a-quintic-polynomial-function-given-roots-and-y-intercept.htmlI need to find an expression that satisfies the qualifying conditions for a quintic polynomial. $f(0)=3$ and $f(-2)=f(\frac{1}{2})=f(1)=0$. With this information, I found that the zeros are $2, ......Mon, 17 Aug 2026 10:43:38 -0400https://pop.com.sg/finding-the-slope-of-a-tangent-line-to-polar-curve.htmlhttps://pop.com.sg/finding-the-slope-of-a-tangent-line-to-polar-curve.htmlQuestion: Find the slope of the tangent line to the graph of $r = e^\theta - 4$ at $\theta = \frac{\pi}{4}$. $$x = r\cos \theta = (e^\theta - 4)\cos\theta$$ $$y = r\sin \theta = (e^\theta - 4)\sin\...Mon, 17 Aug 2026 10:43:06 -0400https://pop.com.sg/area-of-a-triangle-adjacent-to-two-similar-triangles.htmlhttps://pop.com.sg/area-of-a-triangle-adjacent-to-two-similar-triangles.htmlThis GMAT problem states: In the figure above $AD = 4$, $AB = 3$ and $CD = 9$. What is the area of triangle $AEC$ The solution states: to find the base we need to see that triangles $AEB$ and ......Mon, 17 Aug 2026 10:39:49 -0400https://pop.com.sg/if-squaring-a-number-means-multiplying-that-number-with-itself-then-sh.htmlhttps://pop.com.sg/if-squaring-a-number-means-multiplying-that-number-with-itself-then-sh.htmlIf squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? For example the square of $2$ is $2^2=2 \cdot 2=4 ......Mon, 17 Aug 2026 10:24:13 -0400https://pop.com.sg/finding-all-solutions-to-tan-5x-9-tan-x-0.htmlhttps://pop.com.sg/finding-all-solutions-to-tan-5x-9-tan-x-0.htmlI am stuck when it comes to finding the end value of a trig function. I have the following question: $$ \tan^5x - 9\tan{x} = 0 $$ I worked the problem and got: $$ \tan x = 0\\ \tan^4x-9 = 0\\ x ......Mon, 17 Aug 2026 10:22:35 -0400https://pop.com.sg/when-to-use-total-probability-rule-and-bayes-039-theorem.htmlhttps://pop.com.sg/when-to-use-total-probability-rule-and-bayes-039-theorem.htmlConsider the following information about travelers on vacation: $40$% check work email, $30$% use a cell phone, $25$% bring a laptop with them, $23$% both check work email and use a cell phone, and......Mon, 17 Aug 2026 10:11:41 -0400https://pop.com.sg/prove-casting-out-nines-of-an-integer-is-equivalent-to-that-integer-mo.htmlhttps://pop.com.sg/prove-casting-out-nines-of-an-integer-is-equivalent-to-that-integer-mo.htmlLet $s(x)$ be an abstraction for casting out nines of integer $x$. For all integers $x$, prove $s(x) \equiv x$ mod $9$ I'm not asking for an answer more of a way to attack this problem. Can't thin......Mon, 17 Aug 2026 09:54:10 -0400https://pop.com.sg/is-my-general-formula-for-polynomial-multiplication-right.htmlhttps://pop.com.sg/is-my-general-formula-for-polynomial-multiplication-right.htmlIs the following equation for polynomial multiplication correct? $ p(x)=p_{0} + \dots + p_{m}x^m \\ q(x)=q_{0} + \dots + q_{n}x^n \\ $ If $m \geq n$ then $i := m$. If $n \gt m$ then $i := n$. $ ......Mon, 17 Aug 2026 09:51:53 -0400https://pop.com.sg/use-the-alternative-form-of-the-derivative-to-find-the-derivative-at-x.htmlhttps://pop.com.sg/use-the-alternative-form-of-the-derivative-to-find-the-derivative-at-x.htmlUse the alternative form of the derivative to find the derivative at $x=c$ if $f(x)=x^3+4x$ and $c=2$. I keep getting stuck with the answer being $0$, no matter how I try to solve it. If someone ......Mon, 17 Aug 2026 09:47:59 -0400https://pop.com.sg/a-matrix-to-the-power-of-zero-gives-identity-matrix-even-if-it-doesn-0.htmlhttps://pop.com.sg/a-matrix-to-the-power-of-zero-gives-identity-matrix-even-if-it-doesn-0.htmlIf one matrix whose determinant is equal to 0 which means it doesn't have an inverse. Then how is possible to find the value of the matrix to the power of 0 equal to identity matrix when multiplyin......Mon, 17 Aug 2026 09:41:11 -0400https://pop.com.sg/magnitude-and-direction-of-a-vector-5-6.htmlhttps://pop.com.sg/magnitude-and-direction-of-a-vector-5-6.htmlFind the magnitude and direction of the vector $<-5,6>$ I found the magnitude: $||v||=\sqrt{(-5)^2+6^2}=\sqrt{25+36}=\sqrt{61}$ In the direction this is what I did: $\theta=\tan^{-1}(\frac{6}......Mon, 17 Aug 2026 09:38:03 -0400https://pop.com.sg/when-do-you-use-the-quadratic-formula.htmlhttps://pop.com.sg/when-do-you-use-the-quadratic-formula.htmlI am revising for a mathematics exam and am looking over simultaneous equations. I was curious as to when I use the quadratic formula and when I don't? I realize there are multiple ways to solve a ...Mon, 17 Aug 2026 09:36:13 -0400https://pop.com.sg/how-to-find-lim-limits-x-to-0-frac-sin4x-sin2x.htmlhttps://pop.com.sg/how-to-find-lim-limits-x-to-0-frac-sin4x-sin2x.htmlHow to find the limit $$\lim\limits_{x \to 0}\dfrac{\sin4x}{\sin2x}\,?$$ Should I do $$\lim\limits_{x \to 0}\dfrac{\dfrac{\sin4x}{4x}}{\dfrac{\sin2x}{2x}}\,?$$ This doesn't seem to look right, c......Mon, 17 Aug 2026 09:30:59 -0400https://pop.com.sg/deriving-second-order-transfer-function-from-spring-mass-damper-system.htmlhttps://pop.com.sg/deriving-second-order-transfer-function-from-spring-mass-damper-system.htmlI am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...Mon, 17 Aug 2026 09:26:27 -0400https://pop.com.sg/degeneracy-in-simplex-algorithm.htmlhttps://pop.com.sg/degeneracy-in-simplex-algorithm.htmlAccording to my understanding, Degeneracy in a linear optimization problem, occurs when the same extreme point of a bounded feasible region $X$ can be represented by more than one basis, that is not ...Mon, 17 Aug 2026 09:16:04 -0400https://pop.com.sg/find-corresponding-number-in-a-range-by-index.htmlhttps://pop.com.sg/find-corresponding-number-in-a-range-by-index.htmlThere's a array: array(1,2,3,...12, 1,2,3,...9,9,10,...12, 1,2,3,...12, 1,2,3,...12, 1,2,3,...6,6,7,8,...12, 1,2,3,3,4,...12, ...); All numbers are in the range 1-12, one number maybe occur twi......Mon, 17 Aug 2026 09:15:43 -0400https://pop.com.sg/least-common-multiple-in-euclidean-algorithm.htmlhttps://pop.com.sg/least-common-multiple-in-euclidean-algorithm.htmlI want to prove that in last step of Euclidean algorithm we have lcm representation (by last step I mean the step with zero representation as $0 = x * E_0 + y * E_1$, where we apply euclidean algo......Mon, 17 Aug 2026 09:09:41 -0400https://pop.com.sg/how-do-you-integrate-e-x-2.htmlhttps://pop.com.sg/how-do-you-integrate-e-x-2.htmlI know that $\int{\frac{1}{x}}dx$ is simply $\ln{(x)}+c$ (-which is clearly unrelated to the problem but I just thought I would share anyway) but I am not sure how to approach $e^{x{^2}}$. Perhaps a ...Mon, 17 Aug 2026 09:09:03 -0400https://pop.com.sg/prove-using-rolle-039-s-theorem-that-an-equation-has-exactly-one-real-.htmlhttps://pop.com.sg/prove-using-rolle-039-s-theorem-that-an-equation-has-exactly-one-real-.htmlProve that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof. Since $f(x) = x^7+x^5+x^3+1$ is a polynomial then it is continu......Mon, 17 Aug 2026 09:08:18 -0400https://pop.com.sg/cauchy-binet-formula-proof-intuition-determinants.htmlhttps://pop.com.sg/cauchy-binet-formula-proof-intuition-determinants.htmlI am having trouble proving the Cauchy-Binet Theorem. I jotted down how far I got in the proof, but I just find myself stuck. Any guidance would be greatly appreciated! I understand that $\begin......Mon, 17 Aug 2026 09:06:24 -0400https://pop.com.sg/why-is-the-berry-paradox-a-paradox-at-all.htmlhttps://pop.com.sg/why-is-the-berry-paradox-a-paradox-at-all.htmlThe Berry Paradox arises from first assigning every combination of 12 english words to an integer and then asking for "the smallest positive integer not definable in fewer than twelve words". This ...Mon, 17 Aug 2026 09:05:05 -0400https://pop.com.sg/express-in-interval-notation-where-f-is-differentiable-graph.htmlhttps://pop.com.sg/express-in-interval-notation-where-f-is-differentiable-graph.htmlThe graph of the function is cut off in the vertical direction because the function escapes to negative infinity. I know [2,-1)U(-1,1)U(1,3)U(3,4], but the system is telling me that my second set is ...Mon, 17 Aug 2026 09:04:13 -0400https://pop.com.sg/number-of-positive-real-roots-in-equation-x-6-x-1-0-closed.htmlhttps://pop.com.sg/number-of-positive-real-roots-in-equation-x-6-x-1-0-closed.htmlHow can I find the number of positive real roots in the equation $$x^6 - x - 1 =0$$ This question is given in my analysis book and I have no idea how to start...Mon, 17 Aug 2026 09:01:08 -0400https://pop.com.sg/finding-the-gcd-of-three-numbers.htmlhttps://pop.com.sg/finding-the-gcd-of-three-numbers.htmlI have found an algorithm called the Binary GCD/Stein Algorithm which takes two non-negative integers and finds the greatest common factor of the two. What I am trying to do is take three numbers ......Mon, 17 Aug 2026 08:58:12 -0400https://pop.com.sg/finding-eigenvalues-of-block-2x2-matrix.htmlhttps://pop.com.sg/finding-eigenvalues-of-block-2x2-matrix.htmlIn my studies of probability theory, I have recently come across the following eigenvalue problem as part of my research: Let $ \Sigma $ be a given positive definite $ d \times d $ matrix, and l......Mon, 17 Aug 2026 08:57:01 -0400https://pop.com.sg/word-problem-involving-time-and-square-meters.htmlhttps://pop.com.sg/word-problem-involving-time-and-square-meters.htmlBetty paints twice as fast as Dan. Working together Dan and Betty can paint, $2,400$ square feet in $4$ hours. Another employee sue, joined their painting team. Working together, Dan, Betty and......Mon, 17 Aug 2026 08:55:39 -0400https://pop.com.sg/finding-an-expression-for-the-general-term-of-a-taylor-series.htmlhttps://pop.com.sg/finding-an-expression-for-the-general-term-of-a-taylor-series.htmlI am working on a homework problem that asks the following: "Find an expression for the general term of each of the series below. Use $n$ as your index, and pick your general term so that the sum ......Mon, 17 Aug 2026 08:43:07 -0400https://pop.com.sg/show-the-function-is-decreasing-in-an-interval.htmlhttps://pop.com.sg/show-the-function-is-decreasing-in-an-interval.htmlLet $s>1, s\in\mathbb{R}$, and let $f$ be a function defined by $$f(x)=\frac{ln(x)}{x^s}, x>0$$ Determine the monotone intervals of $f$. I note that $f(x)$'s domain is $(0,\infty)$. I then......Mon, 17 Aug 2026 08:42:43 -0400https://pop.com.sg/calculate-limit-of-sequence-a-n-1-3-1-a-n.htmlhttps://pop.com.sg/calculate-limit-of-sequence-a-n-1-3-1-a-n.htmlI have been given the following problem: $$ a_1 = 1, a_{n+1} = 3-\frac{1}{a_n}\\ \text{ Find: }\lim\limits_{n \to \infty} a_n $$ I have already proved that the function is always increasing and bou......Mon, 17 Aug 2026 08:35:21 -0400https://pop.com.sg/limit-of-the-geometric-sequence.htmlhttps://pop.com.sg/limit-of-the-geometric-sequence.htmlI have the following homework question: For what values of $r$ does $\lim_{n\rightarrow \infty} r^n$ exist? What does it converge to? Correct me if I am wrong, but if $r$ is greater than or equa......Mon, 17 Aug 2026 08:27:36 -0400https://pop.com.sg/calculate-regular-or-equilateral-triangle-altitude-with-radius-only-po.htmlhttps://pop.com.sg/calculate-regular-or-equilateral-triangle-altitude-with-radius-only-po.htmlI need to calculate the altitude of a regular triangle (equilateral) but i only have the radius (polygon radius) available (http://www.mathopenref.com/polygonradius.html). I have been searching for ...Mon, 17 Aug 2026 08:23:48 -0400https://pop.com.sg/can-a-sum-of-a-finite-number-of-exponentially-growing-numbers-be-calcu.htmlhttps://pop.com.sg/can-a-sum-of-a-finite-number-of-exponentially-growing-numbers-be-calcu.htmlThis is a question based on the exponential growth pattern of a game mechanic in World of Warcraft: The Heart of Azeroth item in the game has a level that can be increased by gathering Azerite, wit......Mon, 17 Aug 2026 08:20:04 -0400https://pop.com.sg/idempotent-matrix.htmlhttps://pop.com.sg/idempotent-matrix.htmlI have a question: Let A be an idempotent matrix. Show that I-A is idempotent Show that I+A is nonsingular and (I+A)^(-1) = I- ${1\over 2}$A I solved 1: (I - A)^2 = (I - A)(I - A) = I^2 - ......Mon, 17 Aug 2026 08:19:56 -0400https://pop.com.sg/how-to-find-the-area-of-a-quarter-circle-given-only-the-perimeter-of-a.htmlhttps://pop.com.sg/how-to-find-the-area-of-a-quarter-circle-given-only-the-perimeter-of-a.htmlI have been looking for a formula for these types of problems but I still haven't gotten an answer. The question you guys can use can be found here. All I need is the formula but if you want you can ...Mon, 17 Aug 2026 08:01:06 -0400https://pop.com.sg/how-to-find-the-value-of-this-summation-equation.htmlhttps://pop.com.sg/how-to-find-the-value-of-this-summation-equation.htmlThe question is: $$\sum_{i=1}^n (i^2+3i+4)$$ I get that $$\sum_{i=1}^n i^2 = \frac{n(n+1)(n+2)}{6}$$ and $$3\sum_{i=1}^n i = \frac{3n(n+1)}{2}$$ so one would get I'll call this form1: $$\frac......Mon, 17 Aug 2026 07:50:59 -0400https://pop.com.sg/shadow-prices-in-linear-programming.htmlhttps://pop.com.sg/shadow-prices-in-linear-programming.htmlI am quite confused about the meaning of shadow price from explanations on the internet. It can be understood as the value of a change in revenue if the constraint is relaxed, or how much you woul......Mon, 17 Aug 2026 07:49:05 -0400https://pop.com.sg/finding-the-span-of-3-vectors.htmlhttps://pop.com.sg/finding-the-span-of-3-vectors.htmlLet $V = \mathbb R^3$, a vector space over the reals. Find the span $W$ of $\{(1, 2, 1), (3, −1, −4), (0, 7, 7)\}$ in the form $\{(x, y, z) ∈ V \mid ax + by + cz = 0\}$ for some $a, b, c$. Find a b......Mon, 17 Aug 2026 07:46:55 -0400https://pop.com.sg/inverse-of-the-curl.htmlhttps://pop.com.sg/inverse-of-the-curl.htmlGiven $\nabla \cdot u = 0$ and $w = \nabla \times u$ equation 9 of http://projecteuclid.org/euclid.cmp/1103941230 has the identity $u = - \nabla \times (\nabla^{-1} w)$. What is $\nabla^{-1}$ exac......Mon, 17 Aug 2026 07:42:37 -0400https://pop.com.sg/equation-of-hyperbola-with-eccentricity-frac32.htmlhttps://pop.com.sg/equation-of-hyperbola-with-eccentricity-frac32.htmlA hyperbola whose transverse axis is along the major axis of the conic $\frac{2x^2}{3}+\frac{y^2}{4}=4$ and has vertices on the foci of this conic. If the eccentricity of the hyperbola is $\frac32$......Mon, 17 Aug 2026 07:40:12 -0400https://pop.com.sg/how-to-estimate-a-definite-integral-using-taylor-series.htmlhttps://pop.com.sg/how-to-estimate-a-definite-integral-using-taylor-series.htmlI have a problem that asks Evaluate to five decimal places using the taylor series for the definite integral. $$\int_{1}^{2} \frac{e^x}{x} dx$$ I don't get how to do this, given that the minute ......Mon, 17 Aug 2026 07:27:43 -0400https://pop.com.sg/distribution-and-absolute-value.htmlhttps://pop.com.sg/distribution-and-absolute-value.htmlI have a question about distribution and absolute values. I was solving a problem and was wondering if it would be okay to distribute a number into an absolute value with two terms. For example $3|......Mon, 17 Aug 2026 07:17:00 -0400https://pop.com.sg/what-is-the-definition-of-a-relation.htmlhttps://pop.com.sg/what-is-the-definition-of-a-relation.htmlLet $S$ be a set. An equivalence relation on $S$ is a subset $R\subseteq S\times S$ satisfying that $(s,s)\in R$ for all $s\in S$, $(s,t)\in R \implies (t,s)\in R$, $(s,t)\in R\; \land\; (t,u)\in R\...Mon, 17 Aug 2026 07:10:47 -0400https://pop.com.sg/should-we-use-parentheses-or-brackets-around-functions.htmlhttps://pop.com.sg/should-we-use-parentheses-or-brackets-around-functions.htmlShould we use parentheses or brackets around functions? Is there a widely accepted rule/style? For example, which of the following is the preferred way of writing formulas? $(f(x) + g(x)) * h(x)$......Mon, 17 Aug 2026 07:09:44 -0400https://pop.com.sg/weak-convergence-of-product-of-weakly-and-strongly-convergent-l-2-sequ.htmlhttps://pop.com.sg/weak-convergence-of-product-of-weakly-and-strongly-convergent-l-2-sequ.htmlthere is one question bothering me for quite a while now. Let $a_{n},b_{n}\in L^{2}:a_{n}\stackrel{L^{2}}{\rightharpoonup} a\in L^{2} $ weakly $ b_{n}\stackrel{L^{2}}{\rightarrow} b \in L^{2}$ str......Mon, 17 Aug 2026 06:55:50 -0400https://pop.com.sg/help-with-converting-the-derivative-of-cos-3-x.htmlhttps://pop.com.sg/help-with-converting-the-derivative-of-cos-3-x.htmlI needed to find the derivative of $\cos^3(x)$ and find it in a list of given derivatives, but I could not find $-3\sin(x)\cos^2(x)$, so I waited until I did more problems and the only one left was......Mon, 17 Aug 2026 06:44:55 -0400https://pop.com.sg/linear-algebra-notation-a-dagger.htmlhttps://pop.com.sg/linear-algebra-notation-a-dagger.htmlI'm a little confused as to some linear algebra notation and would appreciate some clarification. There exists an $n\times n$ diagonal matrix $Q$ where the entries along the diagonal are the eigen......Mon, 17 Aug 2026 06:38:18 -0400https://pop.com.sg/projecting-a-surface-segment-of-a-cone-onto-a-2d-plane.htmlhttps://pop.com.sg/projecting-a-surface-segment-of-a-cone-onto-a-2d-plane.htmlFirstly, I'd like to apologise - I do not know the correct terms for what I am asking. Assume that the top/bottom of the highlighted portion there is actually aligned with the base. To help expla......Mon, 17 Aug 2026 06:32:29 -0400https://pop.com.sg/extensions-and-kazhdan-039-s-property-t.htmlhttps://pop.com.sg/extensions-and-kazhdan-039-s-property-t.htmlIs Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?...Mon, 17 Aug 2026 06:27:57 -0400https://pop.com.sg/the-langlands-program-for-beginners.htmlhttps://pop.com.sg/the-langlands-program-for-beginners.htmlAssuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program a......Mon, 17 Aug 2026 06:10:34 -0400https://pop.com.sg/proof-for-combination-formula-n-choose-k.htmlhttps://pop.com.sg/proof-for-combination-formula-n-choose-k.htmlI have been looking at this problem for a long time. Can anyone prove the combination formula using factorials N choose K? In case anyone does not know how to list all combinations in a set, you ......Mon, 17 Aug 2026 06:03:03 -0400https://pop.com.sg/understanding-the-ideal-generated-by-a-polynomial.htmlhttps://pop.com.sg/understanding-the-ideal-generated-by-a-polynomial.htmlSo my class on ring theory recently began, and I'm having a bit of trouble understanding ideals that are generated by polynomials. For an arbitrary ring, I know the definition of such an ideal, tha......Mon, 17 Aug 2026 06:00:50 -0400https://pop.com.sg/finding-the-inverse-of-a-number-under-a-certain-modulus.htmlhttps://pop.com.sg/finding-the-inverse-of-a-number-under-a-certain-modulus.htmlHow does one get the inverse of 7 modulo 11? I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure. Likewise, I have the same problem finding the inverse o......Mon, 17 Aug 2026 05:49:54 -0400