https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 31 Aug 2026 19:25:55 -0400https://pop.com.sg/proving-trig-identities-complex-numbers.htmlhttps://pop.com.sg/proving-trig-identities-complex-numbers.htmlQuestion: Prove that if $z = \cos (\theta) + i\sin(\theta)$, then $$ z^n + {1\over z^n} = 2\cos(n\theta) $$ Hence prove that $\cos^6(\theta)$ $=$ $\frac 1{32}$$(\cos(6\theta)$ + ......Mon, 31 Aug 2026 23:23:57 -0400https://pop.com.sg/what-is-the-differentiation-operator.htmlhttps://pop.com.sg/what-is-the-differentiation-operator.htmlOn Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator. The link that underlies the words "differentiation operator" in fact gives......Mon, 31 Aug 2026 23:21:59 -0400https://pop.com.sg/proving-that-n-over-2-n-converges-to-0-duplicate.htmlhttps://pop.com.sg/proving-that-n-over-2-n-converges-to-0-duplicate.htmlI'm completely clueless on this one. I can easily calculate the limit using L'Hopital's rule, but proving that the series is converging to 0 is far more tricky. $$a_n= {n \over 2^n}$$ Any help?...Mon, 31 Aug 2026 23:15:25 -0400https://pop.com.sg/checking-whether-a-graph-is-planar.htmlhttps://pop.com.sg/checking-whether-a-graph-is-planar.htmlI have to check whether a graph is planar. The given type is $$ e ≤ 3v − 6 .$$ From Wikipedia: Note that these theorems provide necessary conditions for planarity that are not sufficient ...Mon, 31 Aug 2026 23:10:57 -0400https://pop.com.sg/positive-constant-scalar-definition.htmlhttps://pop.com.sg/positive-constant-scalar-definition.htmlIn French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one expl......Mon, 31 Aug 2026 23:07:33 -0400https://pop.com.sg/what-039-s-the-connection-between-the-indefinite-integral-and-the-defi.htmlhttps://pop.com.sg/what-039-s-the-connection-between-the-indefinite-integral-and-the-defi.htmlI want to understand the connection between the primitive function or antiderivative and the definite integral. My problem with this is the independent variable called t in the formula for the first ...Mon, 31 Aug 2026 23:01:08 -0400https://pop.com.sg/basic-open-sets-in-the-zariski-topology-are-also-compact.htmlhttps://pop.com.sg/basic-open-sets-in-the-zariski-topology-are-also-compact.htmlLet $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complement......Mon, 31 Aug 2026 22:58:43 -0400https://pop.com.sg/what-is-an-inductive-step.htmlhttps://pop.com.sg/what-is-an-inductive-step.htmlI have a set of problems I am trying to complete. They are all proofs by induction. I have made attempts at most of them and always get stuck doing the inductive step. Can someone explain in laym......Mon, 31 Aug 2026 22:56:12 -0400https://pop.com.sg/lists-versus-sets-in-linear-algebra.htmlhttps://pop.com.sg/lists-versus-sets-in-linear-algebra.htmlI’m currently learning linear algebra from “Linear Algebra Done Right” by Sheldon Axler. The author, in his proofs, makes use of lists of vectors, as opposed to the more conventional usage of sets of ...Mon, 31 Aug 2026 22:42:58 -0400https://pop.com.sg/what-exactly-are-eigen-things.htmlhttps://pop.com.sg/what-exactly-are-eigen-things.htmlWikipedia defines an eigenvector like this: An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at mos......Mon, 31 Aug 2026 22:28:50 -0400https://pop.com.sg/when-is-the-order-of-convergence-of-newton-raphson-greater-than-2.htmlhttps://pop.com.sg/when-is-the-order-of-convergence-of-newton-raphson-greater-than-2.htmlIf some function $f$ has a simple root at $x_*$ then I know that the order of convergence of the Newton-Raphson iteration is at least 2. But when is this order strictly greater than 2?...Mon, 31 Aug 2026 22:26:46 -0400https://pop.com.sg/theorem-1-8-viii-proof-of-mathematical-statistics-jun-shao.htmlhttps://pop.com.sg/theorem-1-8-viii-proof-of-mathematical-statistics-jun-shao.htmlI have some questions about the proof of Theorem 1.8(viii) of Jun Shao's Mathematical Statistics. The theorem is stating that If $X_n$ converges to $X$ in distribution, then for any $r > 0$, $li......Mon, 31 Aug 2026 22:21:26 -0400https://pop.com.sg/prove-s-is-a-supremum-iff-for-all-epsilon-0-there-exists-a-in-a-such-t.htmlhttps://pop.com.sg/prove-s-is-a-supremum-iff-for-all-epsilon-0-there-exists-a-in-a-such-t.htmlLet $A\subseteq\Bbb{R}$ is a nonempty set and $s\in \Bbb{R}$ is an upper bound. Prove $s$ is the supremum iff for all $\epsilon>0$ there exists $a\in A$ such that $a>s-\epsilon$. This is impo......Mon, 31 Aug 2026 22:19:29 -0400https://pop.com.sg/the-membership-table-for-a-u-b-c-a-c-u-b-c.htmlhttps://pop.com.sg/the-membership-table-for-a-u-b-c-a-c-u-b-c.htmlI have attached the image of my membership table. I have supposedly mistook three values I marked with green question marks. Can someone please tell me why they are wrong? the correct values are ones ...Mon, 31 Aug 2026 22:10:59 -0400https://pop.com.sg/understanding-quotient-groups.htmlhttps://pop.com.sg/understanding-quotient-groups.htmlAdmittedly this will be probably be a naive question, but here it goes: Is it possible to flesh out in simple terms, for someone with little background in group theory, what it means to take the ...Mon, 31 Aug 2026 22:08:53 -0400https://pop.com.sg/prove-8-cos-x-cos-2x-cos-3x-1-dfrac-sin-7x-sin-x.htmlhttps://pop.com.sg/prove-8-cos-x-cos-2x-cos-3x-1-dfrac-sin-7x-sin-x.htmlHow do you prove that $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?...Mon, 31 Aug 2026 21:43:23 -0400https://pop.com.sg/formula-for-the-diameter-of-an-emerging-beam.htmlhttps://pop.com.sg/formula-for-the-diameter-of-an-emerging-beam.htmlI have the following problem: A thin beam of white light with negligible radius is incident at $50.0^\circ$ on a $10.0 cm$ thick slab of clear plastic. The index of refraction for red light in t......Mon, 31 Aug 2026 21:38:59 -0400https://pop.com.sg/is-the-euclidean-plane-equal-to-mathbb-r-2.htmlhttps://pop.com.sg/is-the-euclidean-plane-equal-to-mathbb-r-2.htmlI understand what the Cartesian plane is and how one can identify the Euclidean plane and $\mathbb{R}^2$ in many different ways. My question comes down to this though: The Euclidean plane (as defi......Mon, 31 Aug 2026 21:38:11 -0400https://pop.com.sg/subordinate-matrix-norm-of-1-norm.htmlhttps://pop.com.sg/subordinate-matrix-norm-of-1-norm.htmlShow that for the vector norm $$\|x\|_1=\sum_{i=1}^n |x_i|$$ the subordinate matrix norm is $$ \|A\|_1=\max_{1\leq j\leq n} \sum_{i=1}^n |a_{ij}| $$ I've managed to narrow it down doing the followi......Mon, 31 Aug 2026 21:32:08 -0400https://pop.com.sg/what-does-secant-mean-in-mathematics.htmlhttps://pop.com.sg/what-does-secant-mean-in-mathematics.htmlI need your help. pleas I need some one to explain for me what does secant mean in mathematics? Clear example with images would be appreciated....Mon, 31 Aug 2026 21:25:10 -0400https://pop.com.sg/what-is-the-relationship-between-sets-sequences-and-functions.htmlhttps://pop.com.sg/what-is-the-relationship-between-sets-sequences-and-functions.htmlThere's a practice problem I've been working on that's been bothering me. I've been searching for answers here and elsewhere online but I can't the answer I'm looking for. The question is as follow......Mon, 31 Aug 2026 21:19:47 -0400https://pop.com.sg/what-does-the-sign-propto-mean.htmlhttps://pop.com.sg/what-does-the-sign-propto-mean.htmlI am a computer scientist, and one of my professors today used the symbol $\propto$. I tried to search that using google, but it returns no results, and I do not even know its name. So, I would li......Mon, 31 Aug 2026 21:19:19 -0400https://pop.com.sg/algebraic-and-geometric-multiplicities-of-eigenvalues-of-a-3-times-3-m.htmlhttps://pop.com.sg/algebraic-and-geometric-multiplicities-of-eigenvalues-of-a-3-times-3-m.html"Let A = \begin{pmatrix}-1&-1&-2\\ \:2&2&1\\ \:6&2&6\end{pmatrix} Find the characteristic polynomial of A. Find the distinct eigenvalues of A and their respective algebra......Mon, 31 Aug 2026 21:15:28 -0400https://pop.com.sg/find-a-function-of-depth-with-respect-to-time-when-filling-a-cone.htmlhttps://pop.com.sg/find-a-function-of-depth-with-respect-to-time-when-filling-a-cone.htmlThe task is about filling containers of different shape (cone, sphere, etc) – to find a function of depth $h(t)$ when filling a container with a constant rate of liquid supply. The problem is that......Mon, 31 Aug 2026 21:14:52 -0400https://pop.com.sg/burnside-039-s-lemma.htmlhttps://pop.com.sg/burnside-039-s-lemma.htmlI've been trying to understand what Burnside's Lemma is, and how to apply it, but the wiki page is confusing me. The problem I am trying to solve is: You have 4 red, 4 white, and 4 blue identical ...Mon, 31 Aug 2026 21:14:47 -0400https://pop.com.sg/what-is-the-meaning-of-orthogonal-projection-of-vectors.htmlhttps://pop.com.sg/what-is-the-meaning-of-orthogonal-projection-of-vectors.htmlThe vector $\dfrac{x\cdot y}{y\cdot y}y$ is called the projection of $x$ along $y$. I do not understand what it means, geometrically. Can anyone give me n specific example of what it means?...Mon, 31 Aug 2026 21:08:35 -0400https://pop.com.sg/what-is-an-algebraic-variety.htmlhttps://pop.com.sg/what-is-an-algebraic-variety.htmlI'm studying abelian varieties from Milne's book, but I'm having difficulty juggling different conventions and definitions of basic concepts, like those of algebraic and projective varieties. First......Mon, 31 Aug 2026 21:08:04 -0400https://pop.com.sg/derivatives-by-complex-number-and-conjugate.htmlhttps://pop.com.sg/derivatives-by-complex-number-and-conjugate.htmlSuppose we write a complex number $z=x+iy$. Then we have $\overline{z}=x-iy$, and therefore $x=\dfrac12(z+\overline{z})$, $y=-\dfrac12i(z-\overline{z})$. If the rules of calculus were applicabl......Mon, 31 Aug 2026 21:05:56 -0400https://pop.com.sg/does-a-bi-3-x-yi-3-implies-a-bi-x-yi.htmlhttps://pop.com.sg/does-a-bi-3-x-yi-3-implies-a-bi-x-yi.htmlDoes $(a + bi)^3 = (x+yi)^3 \implies a + bi = x + yi$, where $a,b,x,y \in \mathbb{R}?$ Please provide a proof/counterexample. I've tried brute force and ended up with: $$(a - x)(a^2 + ax + x^2) +......Mon, 31 Aug 2026 21:05:16 -0400https://pop.com.sg/when-can-we-say-that-a-mathrm-t-b-b-mathrm-t-a.htmlhttps://pop.com.sg/when-can-we-say-that-a-mathrm-t-b-b-mathrm-t-a.htmlI was looking at the derivation of the normal equation from here. Now, the author has used the fact that $A^{\mathrm T} B = B^{\mathrm T} A$ to reach the step shown in the below image. Can anyone ...Mon, 31 Aug 2026 20:58:58 -0400https://pop.com.sg/finding-cartesian-equation-from-parametric-trigonometric-equations.htmlhttps://pop.com.sg/finding-cartesian-equation-from-parametric-trigonometric-equations.htmlI'm trying to find the cartesian equation of the curve which is defined parametrically by: $$ x = 2\sin\theta, y = \cos^2\theta $$ Both approaches I take result in the same answer: $$ y = 1 - \s......Mon, 31 Aug 2026 20:47:17 -0400https://pop.com.sg/cantor-set-in-base-3.htmlhttps://pop.com.sg/cantor-set-in-base-3.htmlI'm trying to prove the cantor set $C$ is equivalent to the set of all numbers with ternary expansion of $2$'s and $0$'s. That is: Let $A_0=[0,1]$. $A_n$ is defined to equal $A_{n-1}$ with it's mi......Mon, 31 Aug 2026 20:45:58 -0400https://pop.com.sg/show-that-a-matrix-a-is-singular-if-and-only-if-0-is-an-eigenvalue.htmlhttps://pop.com.sg/show-that-a-matrix-a-is-singular-if-and-only-if-0-is-an-eigenvalue.htmlI can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition: A matrix $A$ is singular if and only if $0$ is an eigenvalue. Could anyone shed s......Mon, 31 Aug 2026 20:45:25 -0400https://pop.com.sg/rotating-rectangle-90-degrees-clockwise.htmlhttps://pop.com.sg/rotating-rectangle-90-degrees-clockwise.htmlI have a rectangle in the cartesian plane defined by the top left and Bottom right as $(3, 5),(5, 3)$ We rotate this around the origin clockwise by 90 degrees, what is the new top left and bottom ......Mon, 31 Aug 2026 20:40:05 -0400https://pop.com.sg/let-t-x-2y-xy-3-2-where-x-r-cos-theta-y-r-sin-theta-find-frac-partial-.htmlhttps://pop.com.sg/let-t-x-2y-xy-3-2-where-x-r-cos-theta-y-r-sin-theta-find-frac-partial-.htmlLet $T = x^2y-xy^3+2$ ; where $x = r\cos\theta, y=r\sin\theta$. Find $\dfrac{\partial T}{\partial r}$ and $\dfrac{\partial T}{\partial \theta}$. My answer: For $\dfrac{\partial T}{\partial r}$ I g......Mon, 31 Aug 2026 20:36:56 -0400https://pop.com.sg/how-to-integrate-int-x-n-e-x-dx.htmlhttps://pop.com.sg/how-to-integrate-int-x-n-e-x-dx.htmlHow can I solve this indefinite integral for an arbitrary integer $n>0$? $$ \int{x^n e^x dx}$$ I could partially integrate it for small $n$, but that's not really a solution. Edit: (TB) This ...Mon, 31 Aug 2026 20:35:32 -0400https://pop.com.sg/prob-15-chap-6-in-baby-rudin-if-f-is-a-real-continuously-differentiabl.htmlhttps://pop.com.sg/prob-15-chap-6-in-baby-rudin-if-f-is-a-real-continuously-differentiabl.htmlHere is Prob. 15, Chap. 6, in the book Principles of Mathematical Analysis, by Walter Rudin, 3rd edition: Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) =......Mon, 31 Aug 2026 20:33:14 -0400https://pop.com.sg/linear-transformation-from-r-3-to-r-2.htmlhttps://pop.com.sg/linear-transformation-from-r-3-to-r-2.htmlFind the matrix of the linear transformation $T\colon {\Bbb R}^3 \to {\Bbb R}^2$ such that $T(1,1,1) = (1,1)$, $T(1,2,3) = (1,2)$, $T(1,2,4) = (1,4)$. So far, I have only dealt with transformati......Mon, 31 Aug 2026 20:28:33 -0400https://pop.com.sg/infinity-plus-infinity.htmlhttps://pop.com.sg/infinity-plus-infinity.htmlLet $a \in \mathbb{C}$. Ahlfors says we let $a + \infty = \infty$ and $a \cdot \infty = \infty$. But we cannot define $\infty + \infty$ without violating the laws of arithimetic (i.e. field axioms)......Mon, 31 Aug 2026 20:20:12 -0400https://pop.com.sg/upper-bound-on-the-number-of-edges-in-the-farthest-point-voronoi-diagr.htmlhttps://pop.com.sg/upper-bound-on-the-number-of-edges-in-the-farthest-point-voronoi-diagr.htmlI'm trying to prove that the number of edges in the farthest-point voronoi diagram of $n$ points is upper-bounded by $2n-3$ (as claimed by Computational Geometry: Algorithms and Applications). I co......Mon, 31 Aug 2026 20:19:03 -0400https://pop.com.sg/diagonalizing-the-matrix-if-possible.htmlhttps://pop.com.sg/diagonalizing-the-matrix-if-possible.htmlDiagonalize the matrix $\begin{bmatrix}0&-4&-6\\-1&0&-3\\1&2&5\end{bmatrix}$ if possible So I know that I can check to see if this is diagonalizable by doing $A = PDP^{-1}$......Mon, 31 Aug 2026 20:17:20 -0400https://pop.com.sg/calculating-surface-area-of-a-sphere-with-cylindrical-coordinates.htmlhttps://pop.com.sg/calculating-surface-area-of-a-sphere-with-cylindrical-coordinates.htmlCalculating the surface area of a sphere of radius $a$ with spherical coordinates is quite easy ($4\pi a^2$). I'm trying to do the same with cylindrical coordinates, $\rho$, $\theta$, $z$, (just f......Mon, 31 Aug 2026 20:15:48 -0400https://pop.com.sg/does-zero-divide-zero.htmlhttps://pop.com.sg/does-zero-divide-zero.htmlI wanted to know is zero divisible by zero? I've read that division by zero is not allowed in mathematics, but for instance in Apostol's Introduction to analytic number theory, it states that only ......Mon, 31 Aug 2026 20:14:41 -0400https://pop.com.sg/how-is-ln-pronounced-by-english-speakers.htmlhttps://pop.com.sg/how-is-ln-pronounced-by-english-speakers.htmlI have always heard an expression like $\ln (x^2)$ pronounced aloud as "ell-enn ex squared". That is, the name of the function $\ln$ is read aloud as a two-letter abbreviation. However, I recentl......Mon, 31 Aug 2026 20:04:42 -0400https://pop.com.sg/length-of-a-parabolic-curve.htmlhttps://pop.com.sg/length-of-a-parabolic-curve.htmlI just wanted to know how I can find the length of a curve given by $f(x) = x^2$ from $x=0$ to $x=1$. For appproximation, the length is a bit larger than the hypotenuse of isosceles right triangle......Mon, 31 Aug 2026 20:00:50 -0400https://pop.com.sg/epsilon-delta-derivative-proof-of-x-n.htmlhttps://pop.com.sg/epsilon-delta-derivative-proof-of-x-n.htmlI'm currently trying to prove the power rule using the epsilon-delta definition of a derivative. I've already done it for the basic limit definition, but I thought it might be a helpful exercise to......Mon, 31 Aug 2026 19:50:15 -0400https://pop.com.sg/average-path-length-of-unconnected-graphs.htmlhttps://pop.com.sg/average-path-length-of-unconnected-graphs.htmlI'm reading graph theory and I found a concept named average path length. Basically, it is the average of all the shortest lengths between any two nodes of a graph. In case two nodes are not connec......Mon, 31 Aug 2026 19:48:24 -0400https://pop.com.sg/cos-2-alpha-cos-2-beta-cos-2-gamma-1-duplicate.htmlhttps://pop.com.sg/cos-2-alpha-cos-2-beta-cos-2-gamma-1-duplicate.htmlLet be $\alpha, \beta, \gamma$ the angles between a generic direction in 3D and the axes $x,y,z$, respectively. Prove that $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =1$. PS: the 2D case is tr......Mon, 31 Aug 2026 19:42:28 -0400https://pop.com.sg/definition-of-an-affine-subspace.htmlhttps://pop.com.sg/definition-of-an-affine-subspace.htmlI am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me. I cite: A subset $B$ of a $\mathbb{R}$-affine space $A$ ...Mon, 31 Aug 2026 19:40:02 -0400https://pop.com.sg/integral-int-0-1-frac-arctan-x-x-2-x-1-dx.htmlhttps://pop.com.sg/integral-int-0-1-frac-arctan-x-x-2-x-1-dx.htmlAfter seeing this integral I've decided to give a try to calculate: $$I=\int_0^1 \frac{\arctan x}{x^2-x-1}dx$$ That is because it's common for many integrals to have a combination of a polynomial i......Mon, 31 Aug 2026 19:29:03 -0400https://pop.com.sg/contour-integral-for-x-3-e-x-1.htmlhttps://pop.com.sg/contour-integral-for-x-3-e-x-1.htmlWhat contour and integrand do we use to evaluate $$ \int_0^\infty \frac{x^3}{e^x-1} dx $$ Or is this going to need some other technique?...Mon, 31 Aug 2026 19:13:01 -0400https://pop.com.sg/find-diagonal-of-inverse-matrix.htmlhttps://pop.com.sg/find-diagonal-of-inverse-matrix.htmlI have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the chole......Mon, 31 Aug 2026 19:12:07 -0400https://pop.com.sg/how-to-simplify-log-base-2-and-log-base-4.htmlhttps://pop.com.sg/how-to-simplify-log-base-2-and-log-base-4.htmlHow do I simplify the following expression: $$\log_2(2x+1) - 5\log_4(x^2) + 4\log_2(x)$$ That's it, please help me ok?...Mon, 31 Aug 2026 19:04:55 -0400https://pop.com.sg/simplify-sin-1-tan-x.htmlhttps://pop.com.sg/simplify-sin-1-tan-x.htmlI'm aware of how we can simplify functions which have $Arc$ as an argument . For example $\sin(\cos^{-1}(x)) = \sqrt{1-x^2}$ but what about cases which $Arc$ is out of the parentheses ? For instance ...Mon, 31 Aug 2026 18:50:01 -0400https://pop.com.sg/construct-a-bijective-map-and-prove-it-is-bijective.htmlhttps://pop.com.sg/construct-a-bijective-map-and-prove-it-is-bijective.htmlS is a finite set with more than 1 element. Construct a bijective map $f:S\times \mathbb{Z}\rightarrow\mathbb{Z}$ and prove it is bijective. ($\mathbb{Z}$ is the set of all integers) I'm at a los......Mon, 31 Aug 2026 18:45:45 -0400https://pop.com.sg/integral-int-limits-0-pi-sin-x-cos-x-dx.htmlhttps://pop.com.sg/integral-int-limits-0-pi-sin-x-cos-x-dx.html$$\int \limits _0 ^\pi |\sin x + \cos x|\; dx$$ If I divide integral in two parts $\int\limits_0^{\frac{\pi}{2}}{(\sin x + \cos x)\,dx}$ and $\int\limits_{\frac{\pi}{2}}^\pi{(\sin x - \cos x)\,dx......Mon, 31 Aug 2026 18:32:37 -0400https://pop.com.sg/finding-general-formula-for-a-recursion-function.htmlhttps://pop.com.sg/finding-general-formula-for-a-recursion-function.htmlIf we have a recursion relation defined as $a_n = 3a_{n-1}+1$ with $a_1=1$ then find the general formula for $a_n$ in terms of $n$ with a(1) = 1. So far I have: $a_n = 3a_{n-2}+1+1 = 3a_{n-3}+1+1......Mon, 31 Aug 2026 18:27:07 -0400https://pop.com.sg/proving-the-fundamental-period-of-tangent.htmlhttps://pop.com.sg/proving-the-fundamental-period-of-tangent.htmlI'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that $\pi$ is a fundamental period of the tangent function. I ......Mon, 31 Aug 2026 18:20:27 -0400https://pop.com.sg/probability-of-drawing-5-cards-from-a-deck-of-52-that-will-have-the-sa.htmlhttps://pop.com.sg/probability-of-drawing-5-cards-from-a-deck-of-52-that-will-have-the-sa.htmlA standard deck of cards has 52 members consisting of 4 suits each with 13 members. Five cards are dealt from the randomly mixed deck. What is the probability that all cards are the same suit? E......Mon, 31 Aug 2026 18:13:17 -0400https://pop.com.sg/prove-convergent-sequences-are-bounded.htmlhttps://pop.com.sg/prove-convergent-sequences-are-bounded.htmlI don't understand this one part in the proof for convergent sequences are bounded. Proof: Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have: $n > N \...Mon, 31 Aug 2026 18:00:45 -0400https://pop.com.sg/isomorphism-classes-of-abelian-groups-of-certain-order.htmlhttps://pop.com.sg/isomorphism-classes-of-abelian-groups-of-certain-order.htmlworking on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space......Mon, 31 Aug 2026 18:00:02 -0400https://pop.com.sg/independence-of-a-random-variable-x-from-itself.htmlhttps://pop.com.sg/independence-of-a-random-variable-x-from-itself.htmlIn our lecture on probability, my professor made the comment that "a random variable X is not independent from itself." (Here he was specifically talking about discrete random variables.) I asked h......Mon, 31 Aug 2026 17:44:58 -0400https://pop.com.sg/what-is-the-difference-between-p-a-b-and-p-a-b.htmlhttps://pop.com.sg/what-is-the-difference-between-p-a-b-and-p-a-b.htmlI feel that $p(a,b)$ = the probability that event $a$ and $b$ happen at the same time. $p(a|b)$ = the probability that event $a$ happens due to the event $b$ happens. For me, I think the meanin......Mon, 31 Aug 2026 17:42:39 -0400https://pop.com.sg/asymptotic-approximation-of-the-arctangent.htmlhttps://pop.com.sg/asymptotic-approximation-of-the-arctangent.htmlThat is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x))......Mon, 31 Aug 2026 17:39:52 -0400https://pop.com.sg/use-cylindrical-coordinates-to-find-the-volume-of-a-solid.htmlhttps://pop.com.sg/use-cylindrical-coordinates-to-find-the-volume-of-a-solid.htmlTo Find: use cylindrical coordinates to find the volume of a solid that is enclosed between the surface $r^2+z^2=20$ and the surface $z=r^2$: $$\iiint_R f(x,y,z)\,dv=\iiint_R f(r\cos\theta,r\sin\t......Mon, 31 Aug 2026 17:38:13 -0400https://pop.com.sg/definition-of-maximal-and-minimal-duplicate.htmlhttps://pop.com.sg/definition-of-maximal-and-minimal-duplicate.htmlPossible Duplicate: difference between maximal element and greatest element When I first encountered the terms maximal and minimal, I confused them with maximum and minimum. Many of my classma......Mon, 31 Aug 2026 17:36:44 -0400https://pop.com.sg/difference-between-basis-and-subbasis-in-a-topology.htmlhttps://pop.com.sg/difference-between-basis-and-subbasis-in-a-topology.htmlI was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?...Mon, 31 Aug 2026 17:29:02 -0400https://pop.com.sg/definition-of-an-induced-matrix-norm.htmlhttps://pop.com.sg/definition-of-an-induced-matrix-norm.htmlCould someone explain the second equality in the definition of a induced matrix norm to me? Let $\| \cdot \|$ be a norm for $\cdot \in \mathbb{R}^n$, then the induced matrixnorm for $A\in \ma......Mon, 31 Aug 2026 17:27:02 -0400https://pop.com.sg/polynomial-whose-roots-adjoin-to-a-field-mathbb-q-sqrt6-to-give-an-ext.htmlhttps://pop.com.sg/polynomial-whose-roots-adjoin-to-a-field-mathbb-q-sqrt6-to-give-an-ext.htmlI want to find a polynomial whose roots adjoin to a field $\mathbb Q(\sqrt6)$ to give an extension field $\mathbb Q(\sqrt2,\sqrt3)$such that $[\mathbb Q(\sqrt2,\sqrt3):\mathbb Q(\sqrt6)]=2$. Is th......Mon, 31 Aug 2026 17:26:20 -0400https://pop.com.sg/finding-eccentricity-from-the-rotating-ellipse-formula.htmlhttps://pop.com.sg/finding-eccentricity-from-the-rotating-ellipse-formula.htmlI see that from a normal ellipse formula, we can acquire the eccentricity via this formula here. However, for this formula (1): $A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$ When parameter $......Mon, 31 Aug 2026 17:19:43 -0400https://pop.com.sg/proof-of-arctan-2-pi-2-arctan-1-2.htmlhttps://pop.com.sg/proof-of-arctan-2-pi-2-arctan-1-2.htmlI am studying undergraduate complex analysis, and in my Textbook the author claimed that $$\arctan(2)=\frac{\pi}{2}-\arctan\left(\frac{1}{2}\right)$$ when he was doing an example regarding to ...Mon, 31 Aug 2026 16:58:51 -0400https://pop.com.sg/hyperbolic-curve-and-hyperbola.htmlhttps://pop.com.sg/hyperbolic-curve-and-hyperbola.htmlDef: A hyperbolic curve is an algebraic curve obtained by removing $r$ points from a smooth, proper curve of genus $g,$ where $g$ and $r$ are nonnegative integers such that $2g−2+r > 0.$ How ......Mon, 31 Aug 2026 16:57:47 -0400https://pop.com.sg/solving-right-triangle-given-two-sides.htmlhttps://pop.com.sg/solving-right-triangle-given-two-sides.htmlI have a right triangle whose base has length 40 cm and whose hypotenuse has length 43 cm. How can I determine the height and the measures of the remaining two angles?...Mon, 31 Aug 2026 16:54:44 -0400https://pop.com.sg/what-is-the-exact-definition-of-a-reflexive-relation.htmlhttps://pop.com.sg/what-is-the-exact-definition-of-a-reflexive-relation.htmlWell... The three definitions I got for reflexive, symmetric, and transitive relations are: R is said to be reflexive on A if $\forall x\in A : xRx$. R is symmetric if $\forall x,y\in A : xRy \...Mon, 31 Aug 2026 16:48:33 -0400https://pop.com.sg/determine-the-signs-of-the-partial-derivatives-for-the-function-f-whos.htmlhttps://pop.com.sg/determine-the-signs-of-the-partial-derivatives-for-the-function-f-whos.htmlMy question is how am I supposed to determine what function that is? I think I could handle the partial derivatives after that.. I couldn't find an example showing how to identify this graph anywhe......Mon, 31 Aug 2026 16:38:19 -0400https://pop.com.sg/does-det-a-b-det-a-det-b-hold.htmlhttps://pop.com.sg/does-det-a-b-det-a-det-b-hold.htmlWell considering two $n \times n$ matrices does the following hold true: $$\det(A+B) = \det(A) + \det(B)$$ Can there be said anything about $\det(A+B)$? If $A/B$ are symmetric (or maybe even of th......Mon, 31 Aug 2026 16:37:19 -0400https://pop.com.sg/find-angle-in-triangle-with-incenter.htmlhttps://pop.com.sg/find-angle-in-triangle-with-incenter.htmlIn a triangle $ABC$ if $AD$, $BE$, $CF$ are the angle bisectors of $\angle A$, $\angle B$ $\angle C$ respectively with incenter $I$ and $\angle A=120^\circ$ Find $\angle DFI$ It's $30^\circ$. If $......Mon, 31 Aug 2026 16:33:22 -0400https://pop.com.sg/how-to-prove-sin-x-sin-y-2-sin-left-frac-x-y-2-right-cos-left-frac-x-y.htmlhttps://pop.com.sg/how-to-prove-sin-x-sin-y-2-sin-left-frac-x-y-2-right-cos-left-frac-x-y.htmlI am stuck at one task, it's to proof the following equation using addition theorems. From a draft I understood that the equation is correct or true. But that's it. What's a good way to explain it ...Mon, 31 Aug 2026 16:30:46 -0400https://pop.com.sg/multivariable-calculus-hard-problems-with-solutions.htmlhttps://pop.com.sg/multivariable-calculus-hard-problems-with-solutions.htmlI'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not. For example, a typical problem goes like t......Mon, 31 Aug 2026 16:23:44 -0400https://pop.com.sg/does-the-existence-of-orthocenter-proof-also-work-for-obtuse-triangles.htmlhttps://pop.com.sg/does-the-existence-of-orthocenter-proof-also-work-for-obtuse-triangles.htmlI need to prove the existence of an orthocenter for an obtuse triangle. I tried proving the existence of an orthocenter, meaning a point where the heights of $\triangle ABC$, where $[AB]=c, [AC]=b......Mon, 31 Aug 2026 16:05:19 -0400https://pop.com.sg/proving-the-klein-4-group-is-abelian.htmlhttps://pop.com.sg/proving-the-klein-4-group-is-abelian.htmlIf ker $f = \{ (1), (12)(34), (14)(23), (13)(24) \}$, which is the Klein 4 group, how do I prove that it is an Abelian group? Do I just show that each element in the ker $f$ is commutative to prove......Mon, 31 Aug 2026 15:58:18 -0400https://pop.com.sg/simple-diffy-q-problem.htmlhttps://pop.com.sg/simple-diffy-q-problem.htmlSo as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem: $$xy' = 2y$$ where I know the solution is $x^2$ but......Mon, 31 Aug 2026 15:58:17 -0400https://pop.com.sg/general-solution-for-a-nonhomogeneous-differential-equation.htmlhttps://pop.com.sg/general-solution-for-a-nonhomogeneous-differential-equation.htmlI have to find the general solution of this eq : $$y''-4y'+5y=e^{2s} $$ I have found the general solution of the homogeneous part of this eq. $$Y_h= e^{2s} ( C_1 \cos s - C_2 \sin s ) $$ I ...Mon, 31 Aug 2026 15:57:41 -0400https://pop.com.sg/can-someone-explain-the-concept-of-fundamental-domain-to-me.htmlhttps://pop.com.sg/can-someone-explain-the-concept-of-fundamental-domain-to-me.htmlHi for a Geometries course we're dealing with fundamental and I don't quite understand the definition of a fundamental domain. The definition in my book is that a fundamental domain: is a region......Mon, 31 Aug 2026 15:51:25 -0400https://pop.com.sg/why-are-there-3-solutions-in-this-trigonometric-equation-sin-theta-0.htmlhttps://pop.com.sg/why-are-there-3-solutions-in-this-trigonometric-equation-sin-theta-0.htmlmy question is Why are there 3 solutions in this trigonometric equation $sin\theta = 0$? I thought since the range for the angle is between 0 and 360 it has 2 solutions? I tried and I have got 2 ...Mon, 31 Aug 2026 15:47:55 -0400https://pop.com.sg/the-significance-of-linear-approximation.htmlhttps://pop.com.sg/the-significance-of-linear-approximation.htmlI want to know what makes linear approximation so important (or useful). What I am aware of in my current state of limited understanding is that linear approximation is one of the applications of a ...Mon, 31 Aug 2026 15:47:30 -0400https://pop.com.sg/constant-of-integration-of-the-integrating-factor.htmlhttps://pop.com.sg/constant-of-integration-of-the-integrating-factor.htmlI was reading my notes and at some point I write that in solving a first order linear DE using the integrating factor the final solution should have only one arvitrary constant arising from the ...Mon, 31 Aug 2026 15:44:02 -0400https://pop.com.sg/how-to-compute-dottie-number-accurately.htmlhttps://pop.com.sg/how-to-compute-dottie-number-accurately.htmlDottie number is root of this equation : $cos \alpha = \alpha$, $\alpha \approx 0.73908513321516064165531208767\dots$. I wonder how can I compute it ? I have tried to do it with an approximating f......Mon, 31 Aug 2026 15:43:19 -0400https://pop.com.sg/order-of-elements-in-bbb-z-24.htmlhttps://pop.com.sg/order-of-elements-in-bbb-z-24.htmlWhat elements of $\Bbb Z_{24}$ are order $2$? Order $3$? Order $4$? Order $6$? Order $2: 12$ Order $3: 8,16$ Order $4: 6,18$ Order $6: 4$ I had listed element $20$ as order $6$, but erased it ......Mon, 31 Aug 2026 15:31:33 -0400https://pop.com.sg/finding-domain-of-a-function-with-a-natural-logarithm-at-the-denominat.htmlhttps://pop.com.sg/finding-domain-of-a-function-with-a-natural-logarithm-at-the-denominat.htmlI have the function: $y = f(x) = \frac{x}{\ln x}$ The function is undefined for the conditions: a denominator of a fraction being zero. a logarithm being negative or equal to zero. Hence, is ......Mon, 31 Aug 2026 15:29:37 -0400https://pop.com.sg/cardinality-of-the-set-of-all-prime-factors-of-120-closed.htmlhttps://pop.com.sg/cardinality-of-the-set-of-all-prime-factors-of-120-closed.htmlI need to find the cardinality of the set of all prime factors of $120$. Will it be $16$? Since the set of all prime factors of $120$ will always have $16$ elements i.e $\{ 1, 2, 3, 4, 5, 6, 8, 10......Mon, 31 Aug 2026 15:28:20 -0400https://pop.com.sg/the-wald-test-with-poisson-distribution.htmlhttps://pop.com.sg/the-wald-test-with-poisson-distribution.htmlLet $X_1,\ldots, X_n\sim \operatorname{Poisson}(\lambda)$. Let $\lambda_w>0$ be given, I am trying to find the size $\alpha$ Wald test for $H_0$: $\lambda=\lambda_w$ vs $H_1$: $\lambda\neq \lamb......Mon, 31 Aug 2026 15:11:26 -0400https://pop.com.sg/find-the-residues-of-frac-1-sin-pi-z-closed.htmlhttps://pop.com.sg/find-the-residues-of-frac-1-sin-pi-z-closed.htmlUse Euler's formula $\sin \pi z =\frac{ e^{i\pi z} - e^{-i\pi z}}{2i}$ to find the residue of $\frac{1}{\sin \pi z}$. Show that the complex zeros of $\sin \pi z$ are exactly at the integers, and ......Mon, 31 Aug 2026 15:06:52 -0400https://pop.com.sg/probability-of-winning-a-game-of-craps.htmlhttps://pop.com.sg/probability-of-winning-a-game-of-craps.htmlThe dice game craps is played as follows: The player throws 2 dice, and if the sum is 7 or 11, he/she wins. If the sum is 2, 3, or 12, he/she loses. If the sum is anything else, then he/she continues ...Mon, 31 Aug 2026 15:05:09 -0400https://pop.com.sg/converting-an-equation-from-cartesian-to-polar-form.htmlhttps://pop.com.sg/converting-an-equation-from-cartesian-to-polar-form.htmlI'm trying to convert $x^2 +y^2 =(2-x)^2$ into a polar equation in the form $r=f(\theta)$. The answer is apparently $r=\frac{2}{1+cos(\theta)}$, but I can't seem to get this....Mon, 31 Aug 2026 14:51:09 -0400https://pop.com.sg/how-to-evaluate-the-complex-logarithms-log-i-and-log-3-4i.htmlhttps://pop.com.sg/how-to-evaluate-the-complex-logarithms-log-i-and-log-3-4i.htmlHow to evaluate $$\log(i)?$$ Also, how to evaluate $$\log(3+4i)?$$ I am reading complex analysis and I know that logarithm is a multibranched function and is periodic. What I have for the definitio......Mon, 31 Aug 2026 14:47:23 -0400https://pop.com.sg/non-isosceles-triangles-with-vertices-in-a-20-sided-regular-polygon.htmlhttps://pop.com.sg/non-isosceles-triangles-with-vertices-in-a-20-sided-regular-polygon.htmlSuppose $A_1A_2 . . . A_{20}$ is a $20-$sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but whose sides are no......Mon, 31 Aug 2026 14:46:15 -0400https://pop.com.sg/random-sample-vs-simple-random-sample.htmlhttps://pop.com.sg/random-sample-vs-simple-random-sample.htmlI am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...Mon, 31 Aug 2026 14:42:59 -0400https://pop.com.sg/how-do-find-if-a-relation-is-a-function-algebraically.htmlhttps://pop.com.sg/how-do-find-if-a-relation-is-a-function-algebraically.htmlIs there a way to see if a relation is a function without having to do a "vertical line test" (where you draw a vertical line on the graph and if there line touches two points then it's not a funct......Mon, 31 Aug 2026 14:42:48 -0400https://pop.com.sg/what-does-smooth-curve-mean.htmlhttps://pop.com.sg/what-does-smooth-curve-mean.htmlIn this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?...Mon, 31 Aug 2026 14:42:02 -0400