https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 20 Aug 2026 19:48:31 -0400https://pop.com.sg/converting-4-4-sqrt3-6-from-rectangular-to-cylindrical-coordinates.htmlhttps://pop.com.sg/converting-4-4-sqrt3-6-from-rectangular-to-cylindrical-coordinates.htmlI'm getting the wrong answer for $\theta$. What am I doing wrong? $$(4,-4{\sqrt3}, 6)$$ $$r =\sqrt{4^4 + (-4\sqrt{3})^2} = 8$$ $$z = 6$$ $$\tan\theta=\frac{-4\sqrt{3}}{4}$$ $$\tan\theta=-\sqrt{3......Thu, 20 Aug 2026 23:46:18 -0400https://pop.com.sg/should-coprime-numbers-both-be-prime.htmlhttps://pop.com.sg/should-coprime-numbers-both-be-prime.htmlTwo numbers $a$ and $b$ are coprime if and only if $(a, b) = 1$. $(4, 5) = 1$, are $4$ and $5$ coprime?...Thu, 20 Aug 2026 23:36:30 -0400https://pop.com.sg/ap-statistics-practice-question-about-linear-combinations.htmlhttps://pop.com.sg/ap-statistics-practice-question-about-linear-combinations.htmlThe question is as follows: A company ships gift baskets that contain apples and pears. The distributions of weight for the apples, the pears, and the baskets are each approximately normal. The me......Thu, 20 Aug 2026 23:36:28 -0400https://pop.com.sg/in-how-many-ways-can-12-distinct-pens-be-divided-equally-given-the-fol.htmlhttps://pop.com.sg/in-how-many-ways-can-12-distinct-pens-be-divided-equally-given-the-fol.htmlIn how many ways can $12$ distinct pens be divided equally 1) among $3$ children. options a) 12!/$(4!)^3$ b) 12!/$(4!)^3$ . 3! c) 12!/$(5!)^3$ d) 11!/$(4!)^3$ 2) into $3$ parcels. options a)......Thu, 20 Aug 2026 23:34:24 -0400https://pop.com.sg/calculating-mod-7.htmlhttps://pop.com.sg/calculating-mod-7.htmlProblem: Calculate $10^{10^{10}} \pmod 7$ According to Fermat's little theorem: $a^{p-1}\equiv1 \pmod p$, so $10^6\equiv1 \pmod 7$ and $10^n\equiv4 \pmod 6$, $n$ being any integer, why can we writ......Thu, 20 Aug 2026 23:33:50 -0400https://pop.com.sg/what-is-the-probability-that-none-of-the-events-happens.htmlhttps://pop.com.sg/what-is-the-probability-that-none-of-the-events-happens.htmlSuppose there are 3 events such that: P(A)=0.20 P(B)=0.10 P(C)=0.40 P(A ∩ B)=0.05 P(A ∩ C)=0.10 P(B ∩ C)=0.03 P(A ∩ B ∩ C)=0.01 What is the probability that none of the events happen......Thu, 20 Aug 2026 23:21:57 -0400https://pop.com.sg/how-can-we-prove-that-unitary-transformation-is-isometric.htmlhttps://pop.com.sg/how-can-we-prove-that-unitary-transformation-is-isometric.htmlI am studying Image Processing in which unitary transforms play an important role, one reason that I found for their use in image transformation is isometry(they preserve distance), I found a relev......Thu, 20 Aug 2026 23:15:52 -0400https://pop.com.sg/in-the-triangle-shown-what-is-the-degree-measure-of-angle-adb-duplicat.htmlhttps://pop.com.sg/in-the-triangle-shown-what-is-the-degree-measure-of-angle-adb-duplicat.htmlIn the triangle shown, what is the degree measure of $\angle ADB$ I have mainly tried angle-chasing, but it turns out that through plain angle-chasing you cannot find $\angle ADB$ or $\angle EBD$. I ...Thu, 20 Aug 2026 23:14:34 -0400https://pop.com.sg/distance-of-a-vector-from-a-subspace-linear-algebra.htmlhttps://pop.com.sg/distance-of-a-vector-from-a-subspace-linear-algebra.htmlI want to calculate the distance of the vector $x=(1,1,1,1)$ to the subspace $\{(1,0,2,0) , (0,1,0,2)\}$ I have solved this in 2 ways that I know of but the thing is, the results are different. ......Thu, 20 Aug 2026 23:13:06 -0400https://pop.com.sg/is-apple-ipad-tablet-good-for-mathematics-students-closed.htmlhttps://pop.com.sg/is-apple-ipad-tablet-good-for-mathematics-students-closed.htmlI am a math student. I'd like to find out if tablets (iPads, Galaxy Note 10.1) are worth the cost. How good are tablets for the purposes of reading textbooks as PDF and writing mathematics with a ......Thu, 20 Aug 2026 23:09:54 -0400https://pop.com.sg/handshaking-lemma-and-existence-of-the-graph.htmlhttps://pop.com.sg/handshaking-lemma-and-existence-of-the-graph.htmlHandshaking lemma in graph theory basically says that The degree sum is equal to twice the number of edges. The doubt I have is, does this condition enough to prove the existence of the graph. T......Thu, 20 Aug 2026 23:07:47 -0400https://pop.com.sg/cleverest-construction-of-a-dodecahedron-icosahedron.htmlhttps://pop.com.sg/cleverest-construction-of-a-dodecahedron-icosahedron.htmlOne can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedron, cube, and octahedron all admit very intuitive ...Thu, 20 Aug 2026 23:07:25 -0400https://pop.com.sg/the-product-or-composition-of-permutation-groups-in-two-line-notation.htmlhttps://pop.com.sg/the-product-or-composition-of-permutation-groups-in-two-line-notation.htmlThis question has been asked before, I know: Product of Permutations However, his did not resolve my problem. Here's an example I've been looking at, which is to find the product of two permutat......Thu, 20 Aug 2026 22:56:14 -0400https://pop.com.sg/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-fals.htmlhttps://pop.com.sg/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-fals.htmlI am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: $$\begin{array}{|c|c|c|} \...Thu, 20 Aug 2026 22:44:46 -0400https://pop.com.sg/when-is-it-possible-to-lift-a-function-to-its-covering-space.htmlhttps://pop.com.sg/when-is-it-possible-to-lift-a-function-to-its-covering-space.htmlGiven a continuous function $f: X \to Y$ a covering $p: E \to Y$, what are there conditions for me to say that there exists a lift $g: X \to E$ s.t. $p \circ g = f$? Under what conditions is this l......Thu, 20 Aug 2026 22:39:43 -0400https://pop.com.sg/curl-of-a-vector-in-spherical-coordinates.htmlhttps://pop.com.sg/curl-of-a-vector-in-spherical-coordinates.htmlThe curl of a Vector function in curvilinear coordinate system is given by $$ \nabla \times A = \frac 1 {h_1 h_2 h_3} \begin{vmatrix} h_1 \hat e_1 & h_2 \hat e_2 & h_3 \hat e_3\\ \partial \...Thu, 20 Aug 2026 22:35:55 -0400https://pop.com.sg/how-to-show-that-a-function-is-unbounded.htmlhttps://pop.com.sg/how-to-show-that-a-function-is-unbounded.htmlHow to prove that the function $f:(0,2)\to\mathbb{R}, f(x)=\frac{1}{x}$ is unbounded. I know for a function is unbounded if: $\forall M>0 \exists x\text{ such that }|f(x)|>M$...Thu, 20 Aug 2026 22:34:06 -0400https://pop.com.sg/what-039-s-the-limit-of-a-square-root-function.htmlhttps://pop.com.sg/what-039-s-the-limit-of-a-square-root-function.html$$\lim_{x \to \sqrt{3}^{-}} \sqrt{x^2-3}$$ What's the answer of this limit? There are two hyppothesis: $0$ and undefined. Undefined Because the square root of a negative number is undefined, $0$ ...Thu, 20 Aug 2026 22:29:35 -0400https://pop.com.sg/a-matrix-defines-a-self-adjoint-operator-if-and-only-if-it-is-symmetri.htmlhttps://pop.com.sg/a-matrix-defines-a-self-adjoint-operator-if-and-only-if-it-is-symmetri.htmlI know this is probably a silly question, but I'm stuck with it. $\forall x,y \in \mathbb{R}^{n}$, and $A$ real matrix ($n\times n$), then $\langle Ax,y\rangle=\langle x,Ay\rangle \iff A^{T}=A$ ......Thu, 20 Aug 2026 22:19:01 -0400https://pop.com.sg/express-arcsin-x-in-terms-of-arccos-x-solve-the-equation-2-arctan-x-ar.htmlhttps://pop.com.sg/express-arcsin-x-in-terms-of-arccos-x-solve-the-equation-2-arctan-x-ar.htmlExpress $\arcsin(x)$ in terms of $\arccos(x)$. Using the same, solve the equation $$ 2\,\tan^{-1}x = \sin^{-1} x + \cos^{-1} x $$ I'm not sure if I am on the right track, but here is what i di......Thu, 20 Aug 2026 22:18:01 -0400https://pop.com.sg/why-the-emphasis-on-projective-space-in-algebraic-geometry.htmlhttps://pop.com.sg/why-the-emphasis-on-projective-space-in-algebraic-geometry.htmlI have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what lit......Thu, 20 Aug 2026 22:14:44 -0400https://pop.com.sg/equation-of-a-3d-curve-shaped-like-a-logarithmic-spiral.htmlhttps://pop.com.sg/equation-of-a-3d-curve-shaped-like-a-logarithmic-spiral.htmlI'm not exactly certain of the mathematical description of this surface (if I were I wouldn't have a question), but I basically want to make a "3D spiral" which is basically a sine wave "wrapped" a......Thu, 20 Aug 2026 22:13:52 -0400https://pop.com.sg/how-to-find-the-focus-of-a-cassini-oval.htmlhttps://pop.com.sg/how-to-find-the-focus-of-a-cassini-oval.htmlYou are given some Cassini oval with foci somewhere on the x-axis. How to construct the foci using a compass and straightedge if only the coordinate axes and the oval are given? The foci are the same ...Thu, 20 Aug 2026 22:11:51 -0400https://pop.com.sg/what-is-the-difference-between-the-frobenius-norm-and-the-2-norm-of-a-.htmlhttps://pop.com.sg/what-is-the-difference-between-the-frobenius-norm-and-the-2-norm-of-a-.htmlGiven a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are ident......Thu, 20 Aug 2026 22:06:20 -0400https://pop.com.sg/repeated-squaring-techniques.htmlhttps://pop.com.sg/repeated-squaring-techniques.htmlHere is the problem and the solution my text gives me. When I first approached this problem, I looked up "repeated squaring" online and tried the following algorithm from khan academy. Find the b......Thu, 20 Aug 2026 22:06:07 -0400https://pop.com.sg/epsilon-delta-limit-proofs-at-and-going-to-infinity.htmlhttps://pop.com.sg/epsilon-delta-limit-proofs-at-and-going-to-infinity.htmlSo I understand the concept of epsilon delta limit proofs with linear functions, easy enough, and I am still shaky about doing it with non linear but I am slowly understanding that. I don't quite ...Thu, 20 Aug 2026 21:49:14 -0400https://pop.com.sg/finding-a-parameterization-of-the-paraboloid-900z-25x-2-36y-2.htmlhttps://pop.com.sg/finding-a-parameterization-of-the-paraboloid-900z-25x-2-36y-2.htmlQuestion: Find a parameterization of the paraboloid $900z = 25x^2 + 36y^2$. My Work $25x^2 + 36y^2 = 900z$ $\implies (5x)^2 + (6y)^2 = (30\sqrt{z})^2$ Can we can represent this equation using ...Thu, 20 Aug 2026 21:48:09 -0400https://pop.com.sg/infinitite-products-in-complex-analysis.htmlhttps://pop.com.sg/infinitite-products-in-complex-analysis.htmlIn the text, "Functions of a Complex Variable" The notion of infinite products were discussed which are defined in $(1.)$. What makes an infinite product representation of holomorphic function so ...Thu, 20 Aug 2026 21:36:47 -0400https://pop.com.sg/ln-x-ln-y-kt-make-x-the-subject.htmlhttps://pop.com.sg/ln-x-ln-y-kt-make-x-the-subject.htmlSo I thought you take the inverse function of the whole expression getting: $x - y = e^{kt}$ and so your final answer would be $x = e^{kt} + y$ but according to the answers in the book $x = ye^{kt}$. ...Thu, 20 Aug 2026 21:27:38 -0400https://pop.com.sg/factoring-quadratic-trinomials.htmlhttps://pop.com.sg/factoring-quadratic-trinomials.htmlI'm currently doing some homework, but I'm COMPLETELY stuck on one problem. I need to factor the following trinomial: $$5x^2+7xy+2y^2$$ How can I solve this problem? I have no idea what to do bec......Thu, 20 Aug 2026 21:19:05 -0400https://pop.com.sg/what-will-be-the-remainder-when-2-31-is-divided-by-5.htmlhttps://pop.com.sg/what-will-be-the-remainder-when-2-31-is-divided-by-5.htmlThe question is given in the title: Find the remainder when $2^{31}$ is divided by $5$. My friend explained me this way: $2^2$ gives $-1$ remainder. So, any power of $2^2$ will give $-1$ ...Thu, 20 Aug 2026 21:18:44 -0400https://pop.com.sg/proof-of-the-usual-topology-on-r.htmlhttps://pop.com.sg/proof-of-the-usual-topology-on-r.htmlI'm trying to prove that the usual topology on R is in fact a topology for R. I've read through a few books on topology and most of them seem to skip over the proof and list it as "trivial". My math ...Thu, 20 Aug 2026 21:12:35 -0400https://pop.com.sg/how-many-bits-do-i-need-to-store-a-given-fraction.htmlhttps://pop.com.sg/how-many-bits-do-i-need-to-store-a-given-fraction.htmlUnlike integers, decimal fractions cannot be directly represented in binary. Therefore, what is the procedure to find how many bits to use to express a given binary fraction are sufficient? Basica......Thu, 20 Aug 2026 21:08:22 -0400https://pop.com.sg/motivation-for-definition-of-quotient-map-and-passing-to-the-quotient.htmlhttps://pop.com.sg/motivation-for-definition-of-quotient-map-and-passing-to-the-quotient.htmlI'm a bit confused by the definition of of a quotient map between topological spaces. What is the reason for constructing a topology in this way? I am familiar with quotients from algebra and even so ...Thu, 20 Aug 2026 21:07:34 -0400https://pop.com.sg/projection-matrices.htmlhttps://pop.com.sg/projection-matrices.htmlI have found these two apparently contradicting remarks about projection matrices: A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection matrix......Thu, 20 Aug 2026 21:02:17 -0400https://pop.com.sg/how-many-minutes-in-1-day.htmlhttps://pop.com.sg/how-many-minutes-in-1-day.htmlThere are 24*60 minutes in a day (ignoring the imperfections of the natural world, the Earth and Sun). So there are 24*60 valid 24 hour times (excluding seconds) on a digital clock. Each of these......Thu, 20 Aug 2026 20:54:42 -0400https://pop.com.sg/things-to-check-for-with-a-orthogonal-matrix-or-vector.htmlhttps://pop.com.sg/things-to-check-for-with-a-orthogonal-matrix-or-vector.htmlWhat are there different rules you need to check for with an orthogonal matrix or vector? Why does an orthogonal matrix need to nxn but a orthogonal vector does not? $$\begin{bmatrix} 1\\ 0\\ 1......Thu, 20 Aug 2026 20:52:00 -0400https://pop.com.sg/confused-in-the-term-closed-in-closed-subgroup.htmlhttps://pop.com.sg/confused-in-the-term-closed-in-closed-subgroup.htmlwell, In Brian C Halls Book, I am not getting the definition of Matrix Lie group, as he says : A matrix Lie Group is any subgroup $G$ of $GL_n(\mathbb{C})$ with the following property: If $A_m$ is ......Thu, 20 Aug 2026 20:49:00 -0400https://pop.com.sg/smooth-boundary-is-a-smooth-manifold.htmlhttps://pop.com.sg/smooth-boundary-is-a-smooth-manifold.htmlLet $M\subset \mathbb{R}^n$ be a $k$-dimensional manifold and $X\subset M$ a subset. The boundary of $X$ in $M$, denoted by $\partial_M X$, is the set of all $x\in X$ such that each neighborhoud of......Thu, 20 Aug 2026 20:37:13 -0400https://pop.com.sg/definition-of-compact-set-subset.htmlhttps://pop.com.sg/definition-of-compact-set-subset.htmlI found one exercise in my book Let $X$ be a compact subset of a metric space $M$. Prove that $X$ is closed. In the definitions, the book only mentions compact space and never compact set. An ...Thu, 20 Aug 2026 20:31:19 -0400https://pop.com.sg/how-to-figure-out-the-log-of-a-number-without-a-calculator.htmlhttps://pop.com.sg/how-to-figure-out-the-log-of-a-number-without-a-calculator.htmlI have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is there a formu......Thu, 20 Aug 2026 20:28:56 -0400https://pop.com.sg/find-a-value-to-make-a-function-continuous.htmlhttps://pop.com.sg/find-a-value-to-make-a-function-continuous.htmlI've got the following question: State the value of $a$ that makes the function continuous, where $$f(x) = \begin{cases} x^2 + 3x + 5, && \text{when $x > 1$}\\ a, && \text{whe......Thu, 20 Aug 2026 20:24:47 -0400https://pop.com.sg/householder-transformation-calculation.htmlhttps://pop.com.sg/householder-transformation-calculation.htmlLet $H=I-2ww^t$ be the Householder transformation such that $H\begin{bmatrix}-24\\0\\7\end{bmatrix}=\begin{bmatrix}25\\0\\0\end{bmatrix}$ what is w? Now I know H is involutory and orthog......Thu, 20 Aug 2026 20:21:32 -0400https://pop.com.sg/solutions-to-the-under-damped-harmonic-oscillator-equation.htmlhttps://pop.com.sg/solutions-to-the-under-damped-harmonic-oscillator-equation.htmlFor the damped harmonic oscillator equation $$\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\frac{k}{m}x=0$$ we get that the general solution is $$x(t)=Ae^{-\gamma t}e^{i\omega_d t}+Be^{-\gamma t}e^{-i\...Thu, 20 Aug 2026 20:14:48 -0400https://pop.com.sg/fourier-transform-of-cos-x-on-finite-interval.htmlhttps://pop.com.sg/fourier-transform-of-cos-x-on-finite-interval.htmlLet $f(x)=\cos x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$, and $0$ everywhere else. Then $\tilde f(\omega)$ becomes $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos (x) e^{-i \omega x} dx=\frac{2\c......Thu, 20 Aug 2026 20:10:35 -0400https://pop.com.sg/is-it-possible-to-solve-for-two-unknowns-from-one-equation.htmlhttps://pop.com.sg/is-it-possible-to-solve-for-two-unknowns-from-one-equation.htmlIs it possible to solve for two unknowns using only one equation? For example: $x+3y=32$ Where $x$ and $y$ are integers. Thanks :)...Thu, 20 Aug 2026 19:58:21 -0400https://pop.com.sg/algebraic-proof-of-de-morgan-039-s-law-with-three-sets.htmlhttps://pop.com.sg/algebraic-proof-of-de-morgan-039-s-law-with-three-sets.htmlGiven: $A'$ $\cup$ $B'$ $\cup$ $C'$ $=$ $(A$ $\cap$ $B$ $\cap$ $C$ $)'$ Problem: Show how the identity above can be proved using two steps of De Morgan's Law along with some other basic set rule......Thu, 20 Aug 2026 19:46:19 -0400https://pop.com.sg/is-every-positive-definite-matrix-also-positive-semidefinite.htmlhttps://pop.com.sg/is-every-positive-definite-matrix-also-positive-semidefinite.htmlI am in trouble with the definitions of positive definite and positive semidefinite matrices. By definition, does the following implication hold? $$\mbox{positive definite} \implies \mbox{positive ...Thu, 20 Aug 2026 19:38:24 -0400https://pop.com.sg/what-is-the-area-of-a-triangle-with-sides-sqrt-5-sqrt-10-sqrt-13.htmlhttps://pop.com.sg/what-is-the-area-of-a-triangle-with-sides-sqrt-5-sqrt-10-sqrt-13.htmlI found a "fun algebra problem" that asks you to find the area of a triangle whose sides are $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$. After some algebra hell trying to work with Heron's formula, I pl......Thu, 20 Aug 2026 19:23:51 -0400https://pop.com.sg/difference-between-homomorphisms-and-homomorphic-images.htmlhttps://pop.com.sg/difference-between-homomorphisms-and-homomorphic-images.htmlI was reading Group Homomorphisms from Contemporary Abstract Algebra - Gallian. In pg-216, 8th ed (pg-200, 4th ed), the author writes - "We know that the number of homomorphic images of a cyclic ......Thu, 20 Aug 2026 19:21:42 -0400https://pop.com.sg/meaning-of-exponential-map.htmlhttps://pop.com.sg/meaning-of-exponential-map.htmlI've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do wit......Thu, 20 Aug 2026 19:19:14 -0400https://pop.com.sg/what-is-a-symmetric-group.htmlhttps://pop.com.sg/what-is-a-symmetric-group.htmlThis one has been bugging me for a while - I have a number of definitions for a symmetric group, none of which I understand. Clearly, $S_n$ is a group, which I follow. Now, the wikipedia definition......Thu, 20 Aug 2026 19:16:20 -0400https://pop.com.sg/matrix-mapping-from-r2-to-r3-determine-matrix-and-size-question.htmlhttps://pop.com.sg/matrix-mapping-from-r2-to-r3-determine-matrix-and-size-question.htmlLet $T$ be a linear mapping from $\Bbb R^2$ to $\Bbb R^3$ . $T$ is represented by a matrix $A$ (‘standard matrix’). What is the size of this matrix? Determine $A$ if we know that $$T\left(\begin{b......Thu, 20 Aug 2026 19:16:17 -0400https://pop.com.sg/decrypting-rsa-when-given-n-and-e-but-not-d.htmlhttps://pop.com.sg/decrypting-rsa-when-given-n-and-e-but-not-d.htmlThe cipher text is O1v3nFbVCbuZLUeJDZO9L9 using the base-64 alphabet N=15241604814814604814814604814814609737853 e=47 I know to work out d you factorize n which equals p and q then you get (p-1)(q-......Thu, 20 Aug 2026 19:10:14 -0400https://pop.com.sg/unsolvable-differential-equation.htmlhttps://pop.com.sg/unsolvable-differential-equation.htmlIs there an analytic solution to the following equation? $$ \frac{dx}{dy} = \frac{x^2 - y^2}{x^2 + y^2} $$ I believe the answer is 'no'. It isn't separable or exact I've had trouble finding any ...Thu, 20 Aug 2026 19:04:25 -0400https://pop.com.sg/which-non-negative-matrices-have-negative-eigenvalues.htmlhttps://pop.com.sg/which-non-negative-matrices-have-negative-eigenvalues.htmlIt's easy to proof by counterexample that non-negative matrices can have negative eigenvalues. For example, the following matrix has -1 as an eigenvalue: $$ A = \begin{bmatrix} 0 & 0 &......Thu, 20 Aug 2026 18:54:05 -0400https://pop.com.sg/prove-that-a-2-b-2-is-rational-if-a-b-and-ab-are-rational-duplicate.htmlhttps://pop.com.sg/prove-that-a-2-b-2-is-rational-if-a-b-and-ab-are-rational-duplicate.htmlTrying to prove or disprove this statement: For all real numbers A, B, if $AB$ is rational and $A+B$ is rational, then $A^2$+$B^2$ is rational Ive hit a wall as Im assuming A and B must either be ...Thu, 20 Aug 2026 18:37:25 -0400https://pop.com.sg/frac-1-6-frac-5-6-12-frac-5-8-6-12-18-frac-5-8-11-6-12-18-24-cdots.htmlhttps://pop.com.sg/frac-1-6-frac-5-6-12-frac-5-8-6-12-18-frac-5-8-11-6-12-18-24-cdots.htmlHow to find the sum $$\frac {1}{6} + \frac {5}{6.12} + \frac {5. 8}{6.12.18} + \frac {5.8.11}{6.12.18.24} + \cdots.$$ I have tried to make this series in the form $1 + n x + \frac {n (n + 1......Thu, 20 Aug 2026 18:34:34 -0400https://pop.com.sg/how-to-derive-inverse-hyperbolic-trigonometric-functions.htmlhttps://pop.com.sg/how-to-derive-inverse-hyperbolic-trigonometric-functions.html$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of......Thu, 20 Aug 2026 18:22:00 -0400https://pop.com.sg/why-rationalize-the-denominator.htmlhttps://pop.com.sg/why-rationalize-the-denominator.htmlIn grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "ther......Thu, 20 Aug 2026 18:21:10 -0400https://pop.com.sg/magnitude-of-the-sum-of-vectors-in-3d-space.htmlhttps://pop.com.sg/magnitude-of-the-sum-of-vectors-in-3d-space.htmlI have three vectors $u_1,u_2,u_3$, $$ ||u_1||=3\\ ||u_2||=1\\ ||u_3||=2\\ \angle u_1,u_2=\frac{\pi}{3}\\ \angle u_3,u_1=\frac{\pi}{4}\\ \angle u_3,u_2=\frac{\pi}{4}. $$ I am asked to calculate the ...Thu, 20 Aug 2026 18:20:35 -0400https://pop.com.sg/borel-sigma-algebra-definition.htmlhttps://pop.com.sg/borel-sigma-algebra-definition.htmlDefinition: The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals $\ (a, b]$, where $\ a<b$ in $\mathbb R$ (We ......Thu, 20 Aug 2026 18:17:00 -0400https://pop.com.sg/what-is-the-definition-of-differentiability.htmlhttps://pop.com.sg/what-is-the-definition-of-differentiability.htmlSome places define it as: If the Left hand derivative and the Right hand derivative at a point are equal then the function is said to be differentiable at that point. Others define it based on ......Thu, 20 Aug 2026 18:11:48 -0400https://pop.com.sg/books-for-advanced-algebra-and-advanced-geometry-amc-12-duplicate.htmlhttps://pop.com.sg/books-for-advanced-algebra-and-advanced-geometry-amc-12-duplicate.htmlI will be taking the American Math Contest. Various topics are covered in this test including advanced geometry and algebra. It would be great if any of you could provide books/references which con......Thu, 20 Aug 2026 18:09:22 -0400https://pop.com.sg/what-is-gcd-12345-54321.htmlhttps://pop.com.sg/what-is-gcd-12345-54321.htmlWhat is $\gcd(12345,54321)$? I noticed that after trying $\gcd(12,21),\gcd(123,321),$ and $\gcd(1234,4321)$ that they are all less then or equal to $3$. That leads me to question if there is an ea......Thu, 20 Aug 2026 18:07:53 -0400https://pop.com.sg/why-are-abelian-groups-of-interest-what-is-their-usefulness.htmlhttps://pop.com.sg/why-are-abelian-groups-of-interest-what-is-their-usefulness.htmlI am reading about Abelian groups So apparently it is a set, with an associative binary operation, and identity element, an inverse operation and the binary operation must also be symmetric. But it......Thu, 20 Aug 2026 18:07:13 -0400https://pop.com.sg/jordan-region-problem.htmlhttps://pop.com.sg/jordan-region-problem.htmlWhat is a Jordan Region? I can't find the definition anywhere. The question asked, if $E\subset \mathbb{R}^n$, bounded and with finitely many accumulation points, then $E$ is Jordan region....Thu, 20 Aug 2026 18:07:02 -0400https://pop.com.sg/complex-numbers-closed-under-division.htmlhttps://pop.com.sg/complex-numbers-closed-under-division.htmlMy question is very simple that is: Is Complex number closed under division? Can we consider this 0+0i as complex numbers? or it is not a complex number....Thu, 20 Aug 2026 18:04:42 -0400https://pop.com.sg/2-dimensional-laplace-039-s-equation-in-polar-coordinates.htmlhttps://pop.com.sg/2-dimensional-laplace-039-s-equation-in-polar-coordinates.htmlThe problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdo......Thu, 20 Aug 2026 17:55:29 -0400https://pop.com.sg/triangles-flagpoles-and-heights-oh-my.htmlhttps://pop.com.sg/triangles-flagpoles-and-heights-oh-my.htmlHere is a math question i got from school: On a horizontal plane, there are two flagpoles. One is 20m, and the other is 10m. There is a wire connected from the top of each flagpole, to the bottom ......Thu, 20 Aug 2026 17:52:08 -0400https://pop.com.sg/what-is-a-central-projection.htmlhttps://pop.com.sg/what-is-a-central-projection.htmlWhat is a central projection in the context of a C*-algebra or a W*-algebra? Unfortunately while I see results on this concept, I haven't been able to find a definition. Thanks!...Thu, 20 Aug 2026 17:38:20 -0400https://pop.com.sg/radius-of-convergence-for-binomial-series.htmlhttps://pop.com.sg/radius-of-convergence-for-binomial-series.htmlThe question is simple: determine the radius of convergence for the given series. I have done this for other problems using the following rule which our book described: If the given series is $$......Thu, 20 Aug 2026 17:24:01 -0400https://pop.com.sg/number-of-squares-on-a-rectangular-board-that-are-neither-in-the-4th-r.htmlhttps://pop.com.sg/number-of-squares-on-a-rectangular-board-that-are-neither-in-the-4th-r.htmlA rectangular game board is composed of identical squares arranged in a rectangular array of $r$ rows and $r+1$ columns. The $r$ rows are numbered from $1$ through $r$, and the $r+1$ columns are ...Thu, 20 Aug 2026 17:23:47 -0400https://pop.com.sg/proof-by-mathematical-induction-related-with-geometry.htmlhttps://pop.com.sg/proof-by-mathematical-induction-related-with-geometry.htmlQ:Consider $n+2$ distinct point from the circumference of a circle.If consecutive points along the circle are joined by line segments creating a polygon with $n+2$ sides then the sum of interior an......Thu, 20 Aug 2026 17:17:17 -0400https://pop.com.sg/difference-between-fourier-transform-and-wavelets.htmlhttps://pop.com.sg/difference-between-fourier-transform-and-wavelets.htmlWhile understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency wherea......Thu, 20 Aug 2026 17:10:24 -0400https://pop.com.sg/help-understanding-how-ln-and-e-cancel.htmlhttps://pop.com.sg/help-understanding-how-ln-and-e-cancel.htmlI realise cancel may be the wrong term and inverse may be more appropriate but these is one situation I really don't get…or rather haven't found a suitable explanation. Most sources I have come acr......Thu, 20 Aug 2026 17:08:11 -0400https://pop.com.sg/pentagons-and-triangles.htmlhttps://pop.com.sg/pentagons-and-triangles.htmlIf you have a regular pentagon with all equal sides and you split that pentagon into 5 triangles (inside of that pentagon) would the triangles be equalateral or not?...Thu, 20 Aug 2026 17:05:26 -0400https://pop.com.sg/growth-of-exponential-functions-vs-polynomial.htmlhttps://pop.com.sg/growth-of-exponential-functions-vs-polynomial.htmlWill $2^x$ take over $x^{1000}$ ? I thought that exponential functions had the fastest growth rate, however, graphing it on wolfram alpha made it seem as if the initial behaviors of the two functi......Thu, 20 Aug 2026 17:05:17 -0400https://pop.com.sg/integral-int-0-1-ln-x-dx.htmlhttps://pop.com.sg/integral-int-0-1-ln-x-dx.htmlI have a question about the integral of $\ln x$. When I try to calculate the integral of $\ln x$ from 0 to 1, I always get the following result. $\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1) ......Thu, 20 Aug 2026 16:59:11 -0400https://pop.com.sg/using-integral-definition-to-solve-this-integral.htmlhttps://pop.com.sg/using-integral-definition-to-solve-this-integral.htmlI'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...Thu, 20 Aug 2026 16:58:55 -0400https://pop.com.sg/best-way-to-find-reduced-row-echelon-form-rref-of-a-matrix.htmlhttps://pop.com.sg/best-way-to-find-reduced-row-echelon-form-rref-of-a-matrix.htmlI'm sitting here doing rref problems and many of them seem so tedious. Any tricks out there to achieve rref with less effort or am I stuck with rewriting the matrix for every 2/3 operations? I kno......Thu, 20 Aug 2026 16:58:18 -0400https://pop.com.sg/how-to-find-an-inverse-of-the-following-function.htmlhttps://pop.com.sg/how-to-find-an-inverse-of-the-following-function.html$$f(x)=x^3+1$$ To find inverse, from what I've learned we change the y to x $$x=y^3+1$$ solve for y $$x-1=y^3$$ Should I cube root the x-1 for this? if i did that I still would not get the answer ...Thu, 20 Aug 2026 16:44:42 -0400https://pop.com.sg/is-there-a-simple-method-to-do-lu-decomposition-by-hand.htmlhttps://pop.com.sg/is-there-a-simple-method-to-do-lu-decomposition-by-hand.htmlToday my professor in numerical analysis pointed out that in the exam we will probably have to do LU decomposition by hand. I understand how the decomposition works theoretically, but when it comes ...Thu, 20 Aug 2026 16:43:24 -0400https://pop.com.sg/is-a-matrix-multiplied-with-its-transpose-something-special.htmlhttps://pop.com.sg/is-a-matrix-multiplied-with-its-transpose-something-special.htmlIn my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$?...Thu, 20 Aug 2026 16:30:51 -0400https://pop.com.sg/why-arccos-2-have-value-in-complex-numbers-even-though-its-d-f-1-1.htmlhttps://pop.com.sg/why-arccos-2-have-value-in-complex-numbers-even-though-its-d-f-1-1.htmlI've seen that arccos has $D_f [-1,1]$ however when putting arccos(2) into wolframalpha, I get the complex number result if I understand it correctly. Why is it so? If true, should not $D_f$ be ...Thu, 20 Aug 2026 16:28:51 -0400https://pop.com.sg/how-to-find-out-x-in-a-trinomial-closed.htmlhttps://pop.com.sg/how-to-find-out-x-in-a-trinomial-closed.htmlHow can I find out what X equals in this? $$x^2 - 2x - 3 = 117$$ How would I get started? I'm truly stuck....Thu, 20 Aug 2026 16:28:47 -0400https://pop.com.sg/subset-vs-subseteq-when-not-referring-to-strict-inclusion.htmlhttps://pop.com.sg/subset-vs-subseteq-when-not-referring-to-strict-inclusion.htmlInspired by the confusion in the comments on this question: I always thought that the standard was to read $\subset$ as "is a strict subset of", and $\subseteq$ could mean proper or improper inclu......Thu, 20 Aug 2026 16:26:50 -0400https://pop.com.sg/exponential-distribution-moment-generating-function-to-find-the-mean.htmlhttps://pop.com.sg/exponential-distribution-moment-generating-function-to-find-the-mean.htmlWith mean = 2 with exponential distribution Calculate $ E(200 + 5Y^2 + 4Y^3) = 432 $ $E(200) = 200 $ $E(5Y^2) = 5E(Y^2) = 5(8) = 40 $ $E(4Y^3) = 4E(Y^3) = 4(48) = 192 $ $E(Y^2) = V(Y) + [E(Y......Thu, 20 Aug 2026 16:21:05 -0400https://pop.com.sg/what-is-the-formal-definition-of-ordered-tree.htmlhttps://pop.com.sg/what-is-the-formal-definition-of-ordered-tree.htmlHi I understand that for rooted tree, the definition is as follow: Is a directed graph $D=<V,E>$ $\exists v\in V\rightarrow \deg^-(v)=0$ which indicate the in-degree of $v$ is $0$ $\forall v......Thu, 20 Aug 2026 16:16:46 -0400https://pop.com.sg/find-an-area-of-a-rectangle-by-its-diagonal-and-quotient-of-its-sides.htmlhttps://pop.com.sg/find-an-area-of-a-rectangle-by-its-diagonal-and-quotient-of-its-sides.htmlOkay, I've found no formulas for this one online but I'm pretty sure that the area of a rectangle is calculable from its diagonal and ratio (quotient of its sides). Think of it: with a constant ...Thu, 20 Aug 2026 16:09:29 -0400https://pop.com.sg/why-is-dot-product-multiplication-commutative.htmlhttps://pop.com.sg/why-is-dot-product-multiplication-commutative.htmlSay I have a unit vector $\hat{u}$ and another vector $\vec{t}$ of magnitude 4 units, then I connect the vectors tail to tail with an acute angle between them. I don't understand how the dot produc......Thu, 20 Aug 2026 15:59:00 -0400https://pop.com.sg/what-is-an-indeterminate-in-a-polynomial-ring.htmlhttps://pop.com.sg/what-is-an-indeterminate-in-a-polynomial-ring.htmlI am currently studying polynomial ring. I have a basic doubt. What is $x$ in a polynomial? A polynomial is an expression $\sum_{k = 0}^n a_k x^k$, where $n,k \in \mathbb{N}$ and $a_k \in R$ where......Thu, 20 Aug 2026 15:55:13 -0400https://pop.com.sg/is-the-center-of-g-the-same-as-the-centralizer-of-g-in-g.htmlhttps://pop.com.sg/is-the-center-of-g-the-same-as-the-centralizer-of-g-in-g.htmlIs the center, $Z(G)$, of a group $G$ the same as the centralizer, $C(g)$, of an element $g\in G$? I have proven that $C(g)\leq G\forall g\in G$ but my homework, in a later problem, asks me to prov......Thu, 20 Aug 2026 15:52:44 -0400https://pop.com.sg/triple-integrals-and-cylindrical-coordinates.htmlhttps://pop.com.sg/triple-integrals-and-cylindrical-coordinates.htmlmy question is can cylindrical coordinates be somehow changed in order? I mean the standard cylindrical coordinates suppose a line of length $r$ that makes an angle $\theta$ with the $x$ axis and l......Thu, 20 Aug 2026 15:47:38 -0400https://pop.com.sg/find-the-spectral-decomposition-of-a.htmlhttps://pop.com.sg/find-the-spectral-decomposition-of-a.html$$ A= \begin{pmatrix} -3 & 4\\ 4 & 3 \end{pmatrix} $$ So i am assuming that i must find the evalues and evectors of this matrix first, and that is exactly what i did. The evalues are $5......Thu, 20 Aug 2026 15:43:20 -0400https://pop.com.sg/converting-piecewise-cdf-to-pdf.htmlhttps://pop.com.sg/converting-piecewise-cdf-to-pdf.htmlFor instance, I have the following CDF that I am trying to find the mean for, by firstly finding the PDF and then integrating over the interval. $$F(x) = \begin{cases} 0 & x\leq -2 ......Thu, 20 Aug 2026 15:23:57 -0400https://pop.com.sg/calculate-the-value-of-a-derivative-at-the-origin.htmlhttps://pop.com.sg/calculate-the-value-of-a-derivative-at-the-origin.htmlI have the following question in a course: An example of the logistic function is defined by $$\varphi(v)=\frac{1}{1+e^{-av}}$$ whose limiting values are $0$ and $1$. Show that the derivative of $\...Thu, 20 Aug 2026 15:21:04 -0400https://pop.com.sg/memorizing-the-unit-circle.htmlhttps://pop.com.sg/memorizing-the-unit-circle.htmlI know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. What is the best way to memorize the ...Thu, 20 Aug 2026 15:11:27 -0400https://pop.com.sg/combinations-of-up-to-n-out-of-m-elements.htmlhttps://pop.com.sg/combinations-of-up-to-n-out-of-m-elements.htmlGiven a set of $m$ unique elements to choose from, and using at most $n$ elements in a combination, how many combinations can I have? A combination can repeat an element more than once, as long as ......Thu, 20 Aug 2026 14:58:12 -0400https://pop.com.sg/how-to-diagonalize-this-matrix.htmlhttps://pop.com.sg/how-to-diagonalize-this-matrix.htmlCan someone show me step-by-step how to diagonalize this matrix? I'm trying to teach myself differential equations + linear algebra, but I'm stumped on how to do this. I'd really appreciate if some......Thu, 20 Aug 2026 14:51:30 -0400