https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 27 Aug 2026 15:16:43 -0400https://pop.com.sg/correct-interpretation-of-confidence-interval.htmlhttps://pop.com.sg/correct-interpretation-of-confidence-interval.htmlI think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained. My understan......Thu, 27 Aug 2026 19:15:19 -0400https://pop.com.sg/product-rule-for-gradient.htmlhttps://pop.com.sg/product-rule-for-gradient.htmlI have two functions scalar functions of vector $\textbf{x}$, $f(\textbf{x})$ and $g(\textbf{x})$. I know the gradient of $f(\textbf{x})$ (i.e. $\triangledown f(\textbf{x})$) and I want to find the ...Thu, 27 Aug 2026 19:08:57 -0400https://pop.com.sg/how-can-i-simplify-and-verify-the-logical-equivalence-using-these-laws.htmlhttps://pop.com.sg/how-can-i-simplify-and-verify-the-logical-equivalence-using-these-laws.html∼(p ∨∼q) ∨ (∼p ^ ~ q) ≡ ~p Please help I don't know where to start. These are the laws I need to list in each step when simplifying. Commutative laws: p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p Associative l......Thu, 27 Aug 2026 18:55:18 -0400https://pop.com.sg/linearizing-logarithmic-function.htmlhttps://pop.com.sg/linearizing-logarithmic-function.htmlI have a given set of data points (y,x) with uncertainties. When I plot those points on a graph, the trendline appears to follow the equation y = c + a*ln(x). I want to be able to find the uncer......Thu, 27 Aug 2026 18:46:28 -0400https://pop.com.sg/what-is-this-angle-in-a-right-triangle-with-sides-of-length-5-12-and-1.htmlhttps://pop.com.sg/what-is-this-angle-in-a-right-triangle-with-sides-of-length-5-12-and-1.htmlHow do I find the missing adjacent angle to leg b in a right triangle with the following side lengths: leg a = 5, leg b = 12, and hypotenuse = 13. Thanks...Thu, 27 Aug 2026 18:45:51 -0400https://pop.com.sg/finding-remainder-using-fermat-039-s-little-theorm.htmlhttps://pop.com.sg/finding-remainder-using-fermat-039-s-little-theorm.htmlUse Fermat's Little Theorem to find the remainder of $5^{15}$ divided by $1337$. I know Fermat's Little Theorem, but failed to understand how to use it in this. output.944...Thu, 27 Aug 2026 18:45:35 -0400https://pop.com.sg/proof-of-stars-and-bars-formula.htmlhttps://pop.com.sg/proof-of-stars-and-bars-formula.htmlI am trying to prove a formula (for ways of distributing $n$ identical balls among $r$ persons when each person may get any number of balls) ${n+r-1}\choose{r-1}$. But I am not able to prove it. I ......Thu, 27 Aug 2026 18:45:01 -0400https://pop.com.sg/finding-the-parametric-form-of-a-standard-equation.htmlhttps://pop.com.sg/finding-the-parametric-form-of-a-standard-equation.htmlI need to find the parametric form of $3x - 2y + 10 = 0$. I found that the parametric form for this equation could be : \begin{align}x &= t\\ y &= 5 + \frac{3}{2} t \end{align} I did thi......Thu, 27 Aug 2026 18:39:06 -0400https://pop.com.sg/finding-domain-of-sqrt-x-12-x-9-x-4-x-1.htmlhttps://pop.com.sg/finding-domain-of-sqrt-x-12-x-9-x-4-x-1.htmlHow to find the domain of this function $\sqrt{x^{12} - x^9 + x^4 - x + 1}$? Its really hard to find all 12 roots and plot them or use tricks like "wavy curve" to check where the expression in the ...Thu, 27 Aug 2026 18:38:30 -0400https://pop.com.sg/equilibrium-probability-in-a-markov-chain-closed.htmlhttps://pop.com.sg/equilibrium-probability-in-a-markov-chain-closed.htmlcan someone please tell me what an equilibrium probability in a markov chain is and what the relation between the equlibrium probability and equilibrium distribution is? Thanks...Thu, 27 Aug 2026 18:31:17 -0400https://pop.com.sg/correct-graph-of-arccot-x.htmlhttps://pop.com.sg/correct-graph-of-arccot-x.htmlThere are 2 possible ways of choosing an interval of cot x graph for obtaining the graph of arccot x. Why do we choose the interval (0,$\pi$) instead of the first choice? This may be because the gr......Thu, 27 Aug 2026 18:29:29 -0400https://pop.com.sg/how-to-compute-a-linear-fractional-transformation-that-maps-a-circle-t.htmlhttps://pop.com.sg/how-to-compute-a-linear-fractional-transformation-that-maps-a-circle-t.htmlHow to compute a linear fractional transformation that maps a circle to a given circle? For example, let $C_1$ be the circle $|z+(2+i)|=1$ and $C_2$ be the circle $|z-5|=7$. How to find a linear ...Thu, 27 Aug 2026 18:27:55 -0400https://pop.com.sg/either-john-isn-039-t-stupid-and-he-is-lazy-or-he-is-stupid-he-039-s-s.htmlhttps://pop.com.sg/either-john-isn-039-t-stupid-and-he-is-lazy-or-he-is-stupid-he-039-s-s.htmlI'm reading How To Prove it by Daniel J. Velleman, and I came across the following example which sounds odd to me. Either John isn't stupid and he is lazy, or he is stupid. He's Stupid. Ther......Thu, 27 Aug 2026 18:20:04 -0400https://pop.com.sg/how-to-factorize-x-2-10yz-2xz-2xy-3y-2-3z-2.htmlhttps://pop.com.sg/how-to-factorize-x-2-10yz-2xz-2xy-3y-2-3z-2.htmlHow to factorize $$x^2+10yz-2xz-2xy-3y^2-3z^2$$ It is expanded and we should make them into parts and factorize each part individually. the last answer is $$(x+y-3z)(x-3y+z)$$ but how to get it ?...Thu, 27 Aug 2026 18:17:00 -0400https://pop.com.sg/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence.htmlhttps://pop.com.sg/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence.htmlI tried Googling "formula for sum of quadratic sequence", which did not give me anything useful. I just want an explicit formula for figuring out a sum for a quadratic sequence. For example, how wo......Thu, 27 Aug 2026 18:12:13 -0400https://pop.com.sg/variance-of-conditional-variance.htmlhttps://pop.com.sg/variance-of-conditional-variance.htmlIf we have two random variables X and Y which share a joint pdf and pmf (there is a discrete and a continuous scenario), how do we calculate: Var[Var[X|Y]] - I looked at the Theory of Total Varian......Thu, 27 Aug 2026 18:09:18 -0400https://pop.com.sg/proof-of-caratheodory-039-s-theorem-for-convex-sets-using-radon-039-s-.htmlhttps://pop.com.sg/proof-of-caratheodory-039-s-theorem-for-convex-sets-using-radon-039-s-.htmlI am self-studying some discrete geometry / convex analysis. Many descriptions of Caratheodory's Theorem for convex sets mention that Radon's Lemma can be used to simplify the proof, but I haven't......Thu, 27 Aug 2026 17:51:52 -0400https://pop.com.sg/finding-the-solution-set-for-an-absolute-value-equation-algebraically.htmlhttps://pop.com.sg/finding-the-solution-set-for-an-absolute-value-equation-algebraically.htmle.g. the equation $|x-2|=3$, geometrically it's easy to solve drawing the number line. I get the solution set {-1,5} which I understand. but I was looking at some notes I found online of how to so......Thu, 27 Aug 2026 17:49:02 -0400https://pop.com.sg/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-.htmlhttps://pop.com.sg/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-.htmlHere's the problem: Show that if $H$ and $K$ are subgroups of one abelian group $G$, then $G_1=\{hk\mid h\in H\text{ and } k\in K\}$ is a subgroup of $G$ I know that "abelian" means tha......Thu, 27 Aug 2026 17:41:25 -0400https://pop.com.sg/fitch-system-or-alternatives.htmlhttps://pop.com.sg/fitch-system-or-alternatives.htmlIs there a good resource for how to understand the rules of a Fitch proof system, or an alternative? I'm currently trying to get a sense or how things are proved in natural deduction, which differs......Thu, 27 Aug 2026 17:34:09 -0400https://pop.com.sg/icosahedral-symmetry-as-permutation-group.htmlhttps://pop.com.sg/icosahedral-symmetry-as-permutation-group.htmlHopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the icosa......Thu, 27 Aug 2026 17:29:04 -0400https://pop.com.sg/when-to-use-the-contrapositive-to-prove-a-statment.htmlhttps://pop.com.sg/when-to-use-the-contrapositive-to-prove-a-statment.htmlMy question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the fo......Thu, 27 Aug 2026 17:27:11 -0400https://pop.com.sg/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct.htmlhttps://pop.com.sg/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct.htmlSo I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. Are my examples correct? symmetric & antisymmetric R ={(1,1),(2,2),(3,3)} not symmetr......Thu, 27 Aug 2026 17:18:12 -0400https://pop.com.sg/using-method-of-moments-on-binomial-distribution-closed.htmlhttps://pop.com.sg/using-method-of-moments-on-binomial-distribution-closed.htmlHi, I am stuck on this particular question. Any help? Thanks....Thu, 27 Aug 2026 17:17:09 -0400https://pop.com.sg/recurrence-relation-merge-sort.htmlhttps://pop.com.sg/recurrence-relation-merge-sort.htmlWe know the recurrence relation for normal merge sort. It is T(n) = 2T(n/2) + n. After solving it we can get T(n) = cnlogn. I would like to know the recurrence relation for K way merge sort i.e. in......Thu, 27 Aug 2026 17:11:30 -0400https://pop.com.sg/integral-from-infinity-to-infinity.htmlhttps://pop.com.sg/integral-from-infinity-to-infinity.htmlMy physics professor today wrote on the blackboard: $$ \int_{\infty}^{\infty} f(x) dx = 0 $$ for every function $f$. And the proof he gave was: $$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a......Thu, 27 Aug 2026 17:11:11 -0400https://pop.com.sg/a-parallelogram-and-a-circle.htmlhttps://pop.com.sg/a-parallelogram-and-a-circle.htmlThe perimeter of a parallelogram $ABCD$ is $30$ $cm$. If $D$ and $M$ which is the midpoint of $\overline{DC}$ lie on the circle with diameter $\overline{AB}$, what is the perimeter of $ABMD$? We k......Thu, 27 Aug 2026 16:58:59 -0400https://pop.com.sg/why-is-two-the-only-even-number-that-is-prime.htmlhttps://pop.com.sg/why-is-two-the-only-even-number-that-is-prime.htmlThe other prime numbers are all odd numbers such as $5, 11, 127,$ and $37$. So, why is $2$ the only prime even number there is? Is it because it only has 1 and itself that way, even though it's ......Thu, 27 Aug 2026 16:58:54 -0400https://pop.com.sg/relaxed-menage-problem-sitting-arrangement-for-couples.htmlhttps://pop.com.sg/relaxed-menage-problem-sitting-arrangement-for-couples.htmlThe original problem is this: How many ways are there to seat n couples around a circular table such that no couple sits next to each other? For an answer, I looked at the solution here: https://...Thu, 27 Aug 2026 16:55:17 -0400https://pop.com.sg/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-t.htmlhttps://pop.com.sg/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-t.htmlProvided we have this truth table where "$p\implies q$" means "if $p$ then $q$": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&......Thu, 27 Aug 2026 16:50:59 -0400https://pop.com.sg/how-do-you-rotate-a-vector-by-a-unit-quaternion.htmlhttps://pop.com.sg/how-do-you-rotate-a-vector-by-a-unit-quaternion.htmlGiven a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ......Thu, 27 Aug 2026 16:48:09 -0400https://pop.com.sg/finding-limit-of-multivariable-function-using-the-squeeze-theorem.htmlhttps://pop.com.sg/finding-limit-of-multivariable-function-using-the-squeeze-theorem.htmlIn my textbook (Stewart's Calculus), the video tutor solutions for some problems use the squeeze theorem to determine the limit of a function. For example: Find $$\lim_{(x, y) \to (0, 0)} \frac{x......Thu, 27 Aug 2026 16:38:44 -0400https://pop.com.sg/proof-of-the-continuous-image-of-a-connected-set-is-connected.htmlhttps://pop.com.sg/proof-of-the-continuous-image-of-a-connected-set-is-connected.htmlNone of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question! In Rudin Theorem 4.22, we know that ......Thu, 27 Aug 2026 16:38:21 -0400https://pop.com.sg/proving-the-inverse-if-any-of-a-lower-triangular-matrix-is-lower-trian.htmlhttps://pop.com.sg/proving-the-inverse-if-any-of-a-lower-triangular-matrix-is-lower-trian.htmlThe inverse of a non-singular lower triangular matrix is lower triangular. Construct a proof of this fact as follows. Suppose that $L$ is a non-singular lower triangular matrix. If $b \in \mathbb{R......Thu, 27 Aug 2026 16:15:36 -0400https://pop.com.sg/factorising-fourth-power-polynomial-with-5-terms-duplicate.htmlhttps://pop.com.sg/factorising-fourth-power-polynomial-with-5-terms-duplicate.htmlI've searched all over the internet and cannot seem to factorise this polynomial. $x^4 - 2x^3 + 8x^2 - 14x + 7$ The result should be $(x − 1)(x^3 − x^2 + 7x − 7)$ What are the steps to get to that ...Thu, 27 Aug 2026 16:05:11 -0400https://pop.com.sg/what-does-the-ln-or-1n-mean-in-this-equation.htmlhttps://pop.com.sg/what-does-the-ln-or-1n-mean-in-this-equation.htmlWhat does the ln mean here? It's from this page. https://en.wikipedia.org/wiki/Universally_unique_identifier#Collisions Finally, what could I have done to find out for myself? (What did I miss?) I......Thu, 27 Aug 2026 15:54:57 -0400https://pop.com.sg/calculate-sides-of-right-triangle-with-hypotenuse-and-area-or-perimete.htmlhttps://pop.com.sg/calculate-sides-of-right-triangle-with-hypotenuse-and-area-or-perimete.htmlI'm trying to find if it is possible to find the lengths of the base and height of a right triangle with only the hypotenuse and the area (or the perimeter) of the triangle. I would have just figured ...Thu, 27 Aug 2026 15:53:12 -0400https://pop.com.sg/what-does-ll-mean.htmlhttps://pop.com.sg/what-does-ll-mean.htmlI saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically. http://en.wikipedia.org/wiki/German_tank_problem EDIT: It looks like this can use TeX commands......Thu, 27 Aug 2026 15:50:57 -0400https://pop.com.sg/notation-for-repeated-division.htmlhttps://pop.com.sg/notation-for-repeated-division.htmlFor addition, there is $\sum$ For multiplication, there is $\prod$ What about for division? Something like ${/}_{i=1}^n a_i = a_1 \div a_2\ \div\ ...\ \div\ a_n$?...Thu, 27 Aug 2026 15:47:35 -0400https://pop.com.sg/finding-formula-for-nth-partial-sum.htmlhttps://pop.com.sg/finding-formula-for-nth-partial-sum.htmlI need to find a formula for the nth partial sum of the series: $ 2 + \frac 23 + \frac 29 + \frac {2}{27} + .... + \frac {2}{3^{n-1}} + ... $ then I need to use the formula to find the series' su......Thu, 27 Aug 2026 15:41:06 -0400https://pop.com.sg/related-rates-question-a-man-starts-walking-north-at-4-ft-s-from-a-poi.htmlhttps://pop.com.sg/related-rates-question-a-man-starts-walking-north-at-4-ft-s-from-a-poi.htmlA man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes......Thu, 27 Aug 2026 15:29:16 -0400https://pop.com.sg/checking-for-validity-using-truth-table-shortcut.htmlhttps://pop.com.sg/checking-for-validity-using-truth-table-shortcut.htmlI have the following: p ⊃ (q ≡ r) (q ≡ r) ⊃ p ∴ [(~p v q) · ( q ⊃ p) ≡ r I'm trying to check for validity so I created a truth table to check: My understanding is that the argument is ...Thu, 27 Aug 2026 15:28:09 -0400https://pop.com.sg/what-is-category-theory-useful-for.htmlhttps://pop.com.sg/what-is-category-theory-useful-for.htmlOkay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says ......Thu, 27 Aug 2026 15:11:46 -0400https://pop.com.sg/integration-of-fundamental-solution-of-laplace-039-s-equation.htmlhttps://pop.com.sg/integration-of-fundamental-solution-of-laplace-039-s-equation.htmlI am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of ...Thu, 27 Aug 2026 15:03:25 -0400https://pop.com.sg/how-to-find-the-derivative-of-the-function-f-x-2x-3-4-x-2-x-1-5.htmlhttps://pop.com.sg/how-to-find-the-derivative-of-the-function-f-x-2x-3-4-x-2-x-1-5.htmlCan someone please tell me how to find the derivative of this function? I've been working on this one problem since yesterday and I still can't find the answer... $$f(x) = (2x-3)^4 (x^2 + x + 1)^5......Thu, 27 Aug 2026 15:01:23 -0400https://pop.com.sg/what-is-a-smooth-surface.htmlhttps://pop.com.sg/what-is-a-smooth-surface.htmlWhat is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand t......Thu, 27 Aug 2026 14:49:46 -0400https://pop.com.sg/what-exactly-does-it-mean-for-a-function-to-be-well-behaved.htmlhttps://pop.com.sg/what-exactly-does-it-mean-for-a-function-to-be-well-behaved.htmlOften in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of......Thu, 27 Aug 2026 14:49:28 -0400https://pop.com.sg/what-039-s-the-correct-definition-of-conjugate-and-do-we-identify-them.htmlhttps://pop.com.sg/what-039-s-the-correct-definition-of-conjugate-and-do-we-identify-them.htmlWhat does a "conjugate" in math mean? I know the definition says if we have $x+y$, then $x-y$ is its conjugate. So, $5+ \sqrt{5}$ and $5-\sqrt{5}$ are conjugate of one another. Are $8$ and $2$ ...Thu, 27 Aug 2026 14:41:26 -0400https://pop.com.sg/trace-norm-properties.htmlhttps://pop.com.sg/trace-norm-properties.htmlLet $\|A\|_1=\operatorname{trace}(\sqrt{A^* A})$. I already proved that for arbitrary unitary matrices $U$ and $V$, $\|UAV^*\|_1=\|A\|_1$ and $\|A\|_1=\sigma_1+\dots+\sigma_k$. Now I would like to ......Thu, 27 Aug 2026 14:40:27 -0400https://pop.com.sg/sd-of-a-bernoulli-trial.htmlhttps://pop.com.sg/sd-of-a-bernoulli-trial.htmlWhy nothing is mentioned about the standard deviation of a Bernoulli trial ? Does it even make sense if I try to visualize it ?...Thu, 27 Aug 2026 14:29:43 -0400https://pop.com.sg/euclidean-norm-and-matrix-question.htmlhttps://pop.com.sg/euclidean-norm-and-matrix-question.htmlIf $A$ is a real $n \times n$ matrix and $x \in \mathbb{R}^n$ then is it true that $|Ax| \le |A||x|$? Here $|x|$ is the usual euclidean norm for vectors and $|A| = \sqrt{\Sigma_{i,j} a_{ij}^2}$...Thu, 27 Aug 2026 14:28:32 -0400https://pop.com.sg/notation-is-a-monotonic-function-of.htmlhttps://pop.com.sg/notation-is-a-monotonic-function-of.htmlThe symbols $\sim$ and $\propto$ are used to denote direct proportionality of two quantities. Sometimes, a weaker statement is enough or desired, for example when one can only assume that an incre......Thu, 27 Aug 2026 14:22:10 -0400https://pop.com.sg/trigonometric-expression-for-1-cos-alpha-cos-2-alpha-cdots-cos-n-alpha.htmlhttps://pop.com.sg/trigonometric-expression-for-1-cos-alpha-cos-2-alpha-cdots-cos-n-alpha.htmlThis question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question t......Thu, 27 Aug 2026 14:12:14 -0400https://pop.com.sg/how-frac-1-x-is-integrable-if-it-is-not-a-continuous-function.htmlhttps://pop.com.sg/how-frac-1-x-is-integrable-if-it-is-not-a-continuous-function.htmlIf function is not continuous then it means it's not differentiable and that would mean that it should not be integrable as differentiation is the reverse process of integration. I am talking about ...Thu, 27 Aug 2026 14:04:28 -0400https://pop.com.sg/ackermann-039-s-function-is-mu-recursive.htmlhttps://pop.com.sg/ackermann-039-s-function-is-mu-recursive.htmlIn my book there is the following proof that Ackermann's function is $\mu$-recursive: We propose to show that Ackermann's funcition is $\mu$-recursive. The first part of the job is to devise a sc......Thu, 27 Aug 2026 13:46:41 -0400https://pop.com.sg/prove-that-cot-2-pi-7-cot-2-2-pi-7-cot-2-3-pi-7-5.htmlhttps://pop.com.sg/prove-that-cot-2-pi-7-cot-2-2-pi-7-cot-2-3-pi-7-5.htmlProve that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...Thu, 27 Aug 2026 13:43:09 -0400https://pop.com.sg/definition-of-degree-matrix-of-directed-graph.htmlhttps://pop.com.sg/definition-of-degree-matrix-of-directed-graph.htmlim sorry if this is a trivial question. As the title stated, how is the degree matrix of a directed graph defined? I looked up wikipedia of degree matrix but there was no defitnition of it. Instead......Thu, 27 Aug 2026 13:35:35 -0400https://pop.com.sg/how-many-axis-of-symmetry-of-the-cube-are-there.htmlhttps://pop.com.sg/how-many-axis-of-symmetry-of-the-cube-are-there.htmlIn my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there? The teacher gives me the definition: Definition: If we rotate a 3-dimension object around......Thu, 27 Aug 2026 13:31:01 -0400https://pop.com.sg/sum-of-divergent-and-convergent-series.htmlhttps://pop.com.sg/sum-of-divergent-and-convergent-series.html$\sum_{n=1}^{\infty}x_n$ is a convergent series and $\sum_{n=1}^{\infty}y_n$ is a divergent series. Prove their sum diverges. My attempt: Suppose $\sum_{n=1}^{\infty}x_n + y_n$ converges. Since ......Thu, 27 Aug 2026 13:30:28 -0400https://pop.com.sg/what-is-the-meaning-of-the-term-distance-preserving.htmlhttps://pop.com.sg/what-is-the-meaning-of-the-term-distance-preserving.htmlI've been looking for a definition of this term in everyday words but I can't come across one, the use of the word is in the context of Isometrics....Thu, 27 Aug 2026 13:24:22 -0400https://pop.com.sg/what-is-the-use-of-calculus.htmlhttps://pop.com.sg/what-is-the-use-of-calculus.htmlI know this may seem like a really broad question, but I will narrow it down. I really want to know the purpose of some of the things my teacher is emphasizing in my calc class. For example why it so ...Thu, 27 Aug 2026 13:15:38 -0400https://pop.com.sg/real-world-examples-of-continuous-uniform-distribution-on-0-1.htmlhttps://pop.com.sg/real-world-examples-of-continuous-uniform-distribution-on-0-1.htmlCan someone give me real world examples of uniform distribution on [0,1] of a continuous random variable, because I could not make out one....Thu, 27 Aug 2026 13:14:09 -0400https://pop.com.sg/on-the-proof-of-fleury-039-s-algorithm-question-2.htmlhttps://pop.com.sg/on-the-proof-of-fleury-039-s-algorithm-question-2.htmlOn pages 42-43 in [1], it says: We conclude our introduction to Eulerian graphs with an algorithm for constructing an Eulerian trail in a give Eulerian graph. The method is know as Fleury's ...Thu, 27 Aug 2026 13:09:57 -0400https://pop.com.sg/what-does-the-term-arbitrary-number-mean-in-math.htmlhttps://pop.com.sg/what-does-the-term-arbitrary-number-mean-in-math.htmlWhat does "arbitrary number" means in math? I've often seen this vocabulary but I can't find the meaning of it. For example, I'll see phrases like "arbitrary positive integer"....Thu, 27 Aug 2026 13:00:54 -0400https://pop.com.sg/computing-t-beta-notation.htmlhttps://pop.com.sg/computing-t-beta-notation.htmlI am having what I think is a notational issue with the following problem: For each of the following linear operators $T$ on a vector space $V$ and ordered bases $\beta$, compute $[T]_\beta$, and ...Thu, 27 Aug 2026 12:59:00 -0400https://pop.com.sg/trisecting-angles-which-can-be-trisected.htmlhttps://pop.com.sg/trisecting-angles-which-can-be-trisected.htmlI'm aware that, in general, arbitrary angles cannot be trisected by a compass and straightedge, but also that it is possible for some specific angles. I've been googling for a while to find: Which ...Thu, 27 Aug 2026 12:52:05 -0400https://pop.com.sg/what-is-the-value-of-xyz.htmlhttps://pop.com.sg/what-is-the-value-of-xyz.htmlIf $a,b$ and $c$ are not equal to $0$ and $1$ and if $a^x=b,b^y=c,c^z=a$,then $xyz=?$ We have tried to solve by equation,but it can't produce the desired result....Thu, 27 Aug 2026 12:47:16 -0400https://pop.com.sg/dice-game-probability-theory.htmlhttps://pop.com.sg/dice-game-probability-theory.htmlThere are many dice problems on here so there could be duplicate. If there is I apologize. Here is the question. The game is given as follows: you are given 5 dice, and the goal of the game is to ......Thu, 27 Aug 2026 12:42:53 -0400https://pop.com.sg/what-is-a-function.htmlhttps://pop.com.sg/what-is-a-function.htmlI have been quite confused by the definition of functions and their uses.. First of all can one define functions in a clear understandable way, with a clear explanation of their uses, how they work......Thu, 27 Aug 2026 12:40:31 -0400https://pop.com.sg/sum-of-frac-1-n-2-for-n-1-2-3-duplicate.htmlhttps://pop.com.sg/sum-of-frac-1-n-2-for-n-1-2-3-duplicate.htmlPossible Duplicate: Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$. I just got the "New and Revised" edition of "Mathematics: The New Golden Age", by Keith Devlin. On p. 64 it ......Thu, 27 Aug 2026 12:38:55 -0400https://pop.com.sg/prove-two-angles-add-up-to-90-degrees.htmlhttps://pop.com.sg/prove-two-angles-add-up-to-90-degrees.html$\triangle ABC$ is inscribed within circle $O$. $D$ is the midpoint of $AC$. $E$ is on $AB$ such that $ED/EB=CD/CB$. $CE$ intersects circle $O$ at $F$. Prove that $\angle EDF + \angle CDB = 90^{\ci......Thu, 27 Aug 2026 12:38:54 -0400https://pop.com.sg/the-antiderivative-of-sin-1-x.htmlhttps://pop.com.sg/the-antiderivative-of-sin-1-x.htmlHow to prove that the function $f(x)=\sin\frac{1}{x}$ for $x\neq 0,f(0)=0$ has an antiderivative? This means $F(x)=\int^{x}_{0}\sin(1/t)dt$ has derivative $0$ at $x=0$, but I have no idea how to pr......Thu, 27 Aug 2026 12:36:47 -0400https://pop.com.sg/why-can-we-use-geometric-proofs-in-algebra.htmlhttps://pop.com.sg/why-can-we-use-geometric-proofs-in-algebra.htmlFor example, whenever I search for a proof of the Pythagorean theorem, I get a drawing of a geometric proof, yet we use the Pythagorean theorem to algebraically compute distance between points in an ...Thu, 27 Aug 2026 12:35:27 -0400https://pop.com.sg/understanding-the-definition-of-a-splitting-field.htmlhttps://pop.com.sg/understanding-the-definition-of-a-splitting-field.htmlThe splitting field of a polynomial $p$ with coefficients over a field $K$ is defined as the smallest field that contains $K$, in which the polynomial can split into linear factors. I just want t......Thu, 27 Aug 2026 12:30:17 -0400https://pop.com.sg/new-pattern-question-2-7-26-101-400-closed.htmlhttps://pop.com.sg/new-pattern-question-2-7-26-101-400-closed.htmlSame Mom trying to help daughter here. Once I know how to explain this pattern, can you tell me what I'd call this area of math so I can go re-teach myself? I feel like I could have done this in ......Thu, 27 Aug 2026 12:21:11 -0400https://pop.com.sg/calculus-question-with-circle-and-string-tracing-an-area.htmlhttps://pop.com.sg/calculus-question-with-circle-and-string-tracing-an-area.htmlThe figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the ...Thu, 27 Aug 2026 12:19:56 -0400https://pop.com.sg/find-an-equation-of-the-line-tangent-to-the-graph-of-at-the-point-wher.htmlhttps://pop.com.sg/find-an-equation-of-the-line-tangent-to-the-graph-of-at-the-point-wher.htmlSo I was given the following prompt when studying for a test: "A curve $C$ is defined by the parametric equations $x(t)=3+t^2$ and $y(t)=t^3+5t$. Find an equation of the line tangent to the gr......Thu, 27 Aug 2026 12:14:24 -0400https://pop.com.sg/how-to-translate-a-vector-and-then-rotate-by-a-point.htmlhttps://pop.com.sg/how-to-translate-a-vector-and-then-rotate-by-a-point.htmlI am trying to do this problem: Identify the combination formed by first translating by the vector $(2,0)$ and then rotating by $90$ degrees about $(0,0)$. but I'm a bit confused so, I searc......Thu, 27 Aug 2026 12:13:14 -0400https://pop.com.sg/minimum-velocity.htmlhttps://pop.com.sg/minimum-velocity.htmlThere is a particle moving along the $x$-axis at any time $t \geq 0$. The velocity of this particle is given by $$v(t) = \cos(πt) - t(6-2\pi).$$ I am supposed to find the particle's minimum velo......Thu, 27 Aug 2026 12:12:13 -0400https://pop.com.sg/complex-analysis-liouville-039-s-theorem-application-and-polynomial-de.htmlhttps://pop.com.sg/complex-analysis-liouville-039-s-theorem-application-and-polynomial-de.htmlExercise : Let $f:\mathbb C \to \mathbb C$ be an entire function. (a) If $|f(z)| \geq 1$ $\forall z \in \mathbb C$, show that the function $f$ is constant in $\mathbb C.$ (b) If : $$\lim_{|z|\......Thu, 27 Aug 2026 12:03:30 -0400https://pop.com.sg/number-of-possible-combinations-of-the-enigma-machine-plugboard.htmlhttps://pop.com.sg/number-of-possible-combinations-of-the-enigma-machine-plugboard.htmlThis is a question about basic combinatorics. I recently watched again a youtube video about the Enigma cipher machine (in the Numberphile channel, https://www.youtube.com/watch?v=G2_Q9FoD-oQ), wh......Thu, 27 Aug 2026 11:39:23 -0400https://pop.com.sg/projection-of-triangle-in-3d-to-2d-and-reverse.htmlhttps://pop.com.sg/projection-of-triangle-in-3d-to-2d-and-reverse.htmlIn a software I am developing, I use a triangle defined in 3D by three points $A$, $B$, and $C$. In order to display this triangle on the screen, I use a simple projection as follow (this is basica......Thu, 27 Aug 2026 11:35:53 -0400https://pop.com.sg/finding-the-dihedral-angle-of-a-octahedron.htmlhttps://pop.com.sg/finding-the-dihedral-angle-of-a-octahedron.htmlI am trying to find the dihedral angle of a regular octahedron. All the triangles on a octahedron are equilateral triangles. So, when I drop the perpendicular from the top of the octahedron to the ......Thu, 27 Aug 2026 11:27:38 -0400https://pop.com.sg/questions-about-aleph-aleph-null.htmlhttps://pop.com.sg/questions-about-aleph-aleph-null.htmlNote: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly. I rece......Thu, 27 Aug 2026 11:19:17 -0400https://pop.com.sg/integer-expression.htmlhttps://pop.com.sg/integer-expression.htmlExpression $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}$$ takes an integer value $2$ when $a=t, b=t, c=3t$. Are there any other integer values of variables for integer value of expression?...Thu, 27 Aug 2026 11:15:32 -0400https://pop.com.sg/when-is-mathbb-b-mathbb-x-compact.htmlhttps://pop.com.sg/when-is-mathbb-b-mathbb-x-compact.htmlLet $\mathbb{X}$ be a (real or complex) Banach Space and let $\mathbb{B}(\mathbb{X})$ be the space of all bounded linear operators of $\mathbb{X}$. Let $\tau_u$ be the Uniform Topology on $\mathb......Thu, 27 Aug 2026 11:14:20 -0400https://pop.com.sg/what-does-s-1-2-3-mean.htmlhttps://pop.com.sg/what-does-s-1-2-3-mean.htmlApparently, if one adds an asterisk to the right side of a set definition, it means the set to the left can be built out of elements in the set to the right. How is this so? What does the asterisk ...Thu, 27 Aug 2026 11:09:34 -0400https://pop.com.sg/sin-6-6-or-sin-6-6.htmlhttps://pop.com.sg/sin-6-6-or-sin-6-6.htmlFor small values of $\theta$, $\sin\theta = \theta$ this must be done in what units radians or degrees Sin (6°)=6 or sin(6)=6...Thu, 27 Aug 2026 11:07:06 -0400https://pop.com.sg/cryptography-solve-x-2-equiv-331-pmod-385-using-modular-arithmetic.htmlhttps://pop.com.sg/cryptography-solve-x-2-equiv-331-pmod-385-using-modular-arithmetic.htmlHow can I find (3) congruence equations to solve $$x^2\equiv331\pmod{385}$$ using Legendre and Jacobi Symbols and use the Chinese Remainder Theorem to combine the solutions to those equations to ...Thu, 27 Aug 2026 11:05:42 -0400https://pop.com.sg/angle-between-two-surfaces.htmlhttps://pop.com.sg/angle-between-two-surfaces.htmlI have the next question: Find the angle between two surfaces: $x^2+y^2+z^2=9$ and $z=x^2+y^2-3$ at the point $(2,-1,2)$. I have the next formula: $$\cos\theta =\frac{\nabla \Phi _{1}\cdot\nabla......Thu, 27 Aug 2026 10:55:11 -0400https://pop.com.sg/when-simplifying-sin-arctan-x-why-is-negative-x-not-considered.htmlhttps://pop.com.sg/when-simplifying-sin-arctan-x-why-is-negative-x-not-considered.htmlLet $u = \arctan(x)$, hence $x = \tan(u)$ for $u$ belongs in $(-\frac\pi2, \frac\pi2)$. Since $u$ belongs in $(-\frac\pi2, \frac\pi2)$, we consider $\sin(u)$ where $u$ belongs in $(-\frac\pi2, \fra......Thu, 27 Aug 2026 10:54:34 -0400https://pop.com.sg/can-i-solve-this-differential-equation-at-a-point.htmlhttps://pop.com.sg/can-i-solve-this-differential-equation-at-a-point.htmlI want to solve this differential equation evaluted at $x=0$. I will start with an example with $m=2$ $$ f(0)+f'(0)+f''(0)=2\lambda_{1}\lambda_{2}+4 $$ where $m$,$\lambda_{1}$ and $\lambda_{1}$ are ...Thu, 27 Aug 2026 10:49:18 -0400https://pop.com.sg/does-bounded-in-probability-imply-convergence-in-probability.htmlhttps://pop.com.sg/does-bounded-in-probability-imply-convergence-in-probability.htmlA random variable, $X_n$, is defined to be bounded in probability if there exists an $M$ and $N$ for which \begin{align*} \mathbb{P}\big(|X_n| < M\big) > 1 - \epsilon,\ \forall n > N,\ \fo......Thu, 27 Aug 2026 10:48:24 -0400https://pop.com.sg/proof-of-clarkson-039-s-inequality.htmlhttps://pop.com.sg/proof-of-clarkson-039-s-inequality.htmlTrying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + \left|\left|f-g\right|......Thu, 27 Aug 2026 10:43:45 -0400https://pop.com.sg/what-are-some-of-the-best-textbooks-on-multi-objective-optimization.htmlhttps://pop.com.sg/what-are-some-of-the-best-textbooks-on-multi-objective-optimization.htmlI am looking for textbooks on non-linear multi-objective optimization that are rigorous. For instance, I am most comfortable with Arkadi Nemirovski style. I am looking for a book on nonlinear multi-...Thu, 27 Aug 2026 10:35:23 -0400https://pop.com.sg/why-is-homology-unable-to-distinguish-between-planar-and-non-planar-gr.htmlhttps://pop.com.sg/why-is-homology-unable-to-distinguish-between-planar-and-non-planar-gr.htmlIntuitively, one can think of non-planar graphs as having a "two-dimensional hole" (Edit: this is both unclear and not entirely, if at all, correct, see comments below question), thus necessitating......Thu, 27 Aug 2026 10:26:06 -0400https://pop.com.sg/does-zeta-3-have-a-connection-with-pi.htmlhttps://pop.com.sg/does-zeta-3-have-a-connection-with-pi.htmlThe problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Severa......Thu, 27 Aug 2026 10:25:26 -0400https://pop.com.sg/is-there-a-formula-for-the-roots-of-a-quintic-equation.htmlhttps://pop.com.sg/is-there-a-formula-for-the-roots-of-a-quintic-equation.htmlI can get my head around this so someone explain it please. $(1)$ From Galois theory it is known there is no formula to solve a general quintic equation. But it is known a general quintic can be ......Thu, 27 Aug 2026 10:23:36 -0400https://pop.com.sg/reference-for-harmonic-analysis.htmlhttps://pop.com.sg/reference-for-harmonic-analysis.htmlI'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering Uni......Thu, 27 Aug 2026 10:19:58 -0400https://pop.com.sg/are-all-eigenvectors-of-any-matrix-always-orthogonal.htmlhttps://pop.com.sg/are-all-eigenvectors-of-any-matrix-always-orthogonal.htmlI have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to s......Thu, 27 Aug 2026 10:19:38 -0400