https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 02 Sep 2026 05:45:37 -0400https://pop.com.sg/why-is-pi-considered-irrational-if-it-can-be-expressed-as-ratio-of-cir.htmlhttps://pop.com.sg/why-is-pi-considered-irrational-if-it-can-be-expressed-as-ratio-of-cir.htmlPi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where b......Wed, 02 Sep 2026 09:41:52 -0400https://pop.com.sg/can-i-optimize-area-of-cylinder-with-no-givens.htmlhttps://pop.com.sg/can-i-optimize-area-of-cylinder-with-no-givens.htmlI have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads: A metal can is in the form of a cylinder. It has a bottom ......Wed, 02 Sep 2026 09:38:54 -0400https://pop.com.sg/example-of-an-inverse-semigroup.htmlhttps://pop.com.sg/example-of-an-inverse-semigroup.htmlAn inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...Wed, 02 Sep 2026 09:33:49 -0400https://pop.com.sg/what-039-s-the-intuition-behind-the-co-area-formula.htmlhttps://pop.com.sg/what-039-s-the-intuition-behind-the-co-area-formula.htmlI mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer. If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \ge......Wed, 02 Sep 2026 09:30:56 -0400https://pop.com.sg/how-to-rotate-a-vector-along-another-vector.htmlhttps://pop.com.sg/how-to-rotate-a-vector-along-another-vector.htmlI want to rotate a vector e.g $V = Ai + Bj + Ck$ along an another vector $U = xi + yj + zk$. The angle between $U$ and $V$ is $\theta$. How can I do that and how do I know the new value of $A$, $......Wed, 02 Sep 2026 09:20:21 -0400https://pop.com.sg/mean-and-variance-from-a-cumulative-distribution-function.htmlhttps://pop.com.sg/mean-and-variance-from-a-cumulative-distribution-function.htmlI'm trying to find the mean (expected value) and variance for the following distribution function: $F(x)=\begin{cases} 0 & \text{for } x \lt 0\\ x/4 & \text{for } 0 \le x \lt 1\\ ......Wed, 02 Sep 2026 09:14:39 -0400https://pop.com.sg/what-is-the-negation-of-the-implication-statement.htmlhttps://pop.com.sg/what-is-the-negation-of-the-implication-statement.htmlIn a course on logic and proofs the professor presented on the following lines to show an example of negation: $$ \neg (P \Rightarrow Q) \ \ \ \ \Longleftrightarrow \ \ \ \ P \wedge \neg Q $$ I ca......Wed, 02 Sep 2026 09:10:28 -0400https://pop.com.sg/area-between-two-curves-with-respect-to-y.htmlhttps://pop.com.sg/area-between-two-curves-with-respect-to-y.htmlLet $x_1$ be the red curve and $x_2$ be the blue curve. I've tried integrating $x_1-x_2$ as well as $x_2-x_1$ from $y=0$ to $y=9/2$ and both answers were marked wrong. What do I do here? I'm thinki......Wed, 02 Sep 2026 09:08:55 -0400https://pop.com.sg/hanson-wright-inequality-any-relation-between-the-original-quadratic-f.htmlhttps://pop.com.sg/hanson-wright-inequality-any-relation-between-the-original-quadratic-f.htmlHanson-Wright's inequality such as this provides a bound on the second-order chaos of the form $$P(x^\top A x -\mathbb{E}[x^\top A x] > t) $$ For example, $x\sim N(0, I_n)$ and $A$ is some $n\ti......Wed, 02 Sep 2026 09:03:52 -0400https://pop.com.sg/surface-integral-of-a-right-circular-cone.htmlhttps://pop.com.sg/surface-integral-of-a-right-circular-cone.htmlUse a surface integral to show that the surface area of a right circular cone of radius $R$ and height $h$ is $\pi R \sqrt{h^2+R^2}$. Hint -- Use the parametrization $x=r\cos\theta$, $y=r\sin\theta......Wed, 02 Sep 2026 08:58:46 -0400https://pop.com.sg/problems-that-are-easy-with-trigonometry-and-too-difficult-without-it.htmlhttps://pop.com.sg/problems-that-are-easy-with-trigonometry-and-too-difficult-without-it.htmlI was looking for examples that shows importance of trigonometry in an elementary level. Like if you wanted to surprise your high school students that how problems are made easy using trigonometry ......Wed, 02 Sep 2026 08:56:50 -0400https://pop.com.sg/does-euler-039-s-formula-give-e-ix-cos-x-i-sin-x.htmlhttps://pop.com.sg/does-euler-039-s-formula-give-e-ix-cos-x-i-sin-x.htmlDoes Eulers formula give $$e^{-ix}=\cos(x) -i\sin(x)$$ I know that $$e^{ix}=\cos(x)+i\sin(x)$$ But how does it work when we have a $-$ in front...Wed, 02 Sep 2026 08:54:40 -0400https://pop.com.sg/what-is-the-definition-of-a-circular-wedge.htmlhttps://pop.com.sg/what-is-the-definition-of-a-circular-wedge.htmlIn Ahlfors' Complex Analysis, chapter 3, section 4, the author claims that a region whose boundary consists of two circular arcs with common end points is either a "circular wedge" or its complemen......Wed, 02 Sep 2026 08:30:35 -0400https://pop.com.sg/how-do-i-properly-read-a-clinometer.htmlhttps://pop.com.sg/how-do-i-properly-read-a-clinometer.htmlIf the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?...Wed, 02 Sep 2026 08:27:16 -0400https://pop.com.sg/understanding-quantifier-elimination.htmlhttps://pop.com.sg/understanding-quantifier-elimination.htmlI am struggling to understand Quantifier Elimination as it is treated in Hodges' "A Shorter Model Theory". The relevant definitions are : Definition: Take $K$ to be a class of $L$-struct......Wed, 02 Sep 2026 08:26:00 -0400https://pop.com.sg/why-is-the-method-of-prime-factorization-help-in-finding-the-lcm-dupli.htmlhttps://pop.com.sg/why-is-the-method-of-prime-factorization-help-in-finding-the-lcm-dupli.htmlwhy does finding the prime factors help in finding the lcm of two or more numbers. I know how to do it but why is this the case? Edit I found this answer from just thinking on my own.I don't kno......Wed, 02 Sep 2026 08:24:04 -0400https://pop.com.sg/definition-laplace-operator-on-h-1-0-omega.htmlhttps://pop.com.sg/definition-laplace-operator-on-h-1-0-omega.htmlLet $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Is the Laplace operador defined in the space $H^2(\Omega)$ as a sum of second weak derivatives? How is the Laplace operador defined in the......Wed, 02 Sep 2026 08:22:09 -0400https://pop.com.sg/proof-of-congruence-with-mod-3.htmlhttps://pop.com.sg/proof-of-congruence-with-mod-3.htmlOk so I've tried making this one work for a while and keep getting stuck. I have the following: Prove: ($x^2$ is congruent to $x\mod3$) $\iff$ ($x$ is not congruent to $2\mod3$). So I know it is bi-...Wed, 02 Sep 2026 08:19:49 -0400https://pop.com.sg/pigeonhole-principle-formula-using-propositonal-logic.htmlhttps://pop.com.sg/pigeonhole-principle-formula-using-propositonal-logic.htmlAccording to the Pigeonhole Principle, if we try to place $n+1$ pigeons in $n$ pigeonholes, then at least one pigeonhole must have two or more pigeons. For $i \in \{1, 2, \dots, n+1\}$ and $j \in \......Wed, 02 Sep 2026 08:12:48 -0400https://pop.com.sg/what-is-the-mathematical-term-and-symbol-for-division-without-remainde.htmlhttps://pop.com.sg/what-is-the-mathematical-term-and-symbol-for-division-without-remainde.htmlWhat is the mathematical term and symbol for division without remainders? Take for example 322 / 100. I know 322 / 100 is division and the result is 3.22. I know 322 % 100 is a modulo and the resul......Wed, 02 Sep 2026 08:02:00 -0400https://pop.com.sg/express-99-2-3-as-a-fraction-no-calculator.htmlhttps://pop.com.sg/express-99-2-3-as-a-fraction-no-calculator.htmlMy 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is ...Wed, 02 Sep 2026 07:58:24 -0400https://pop.com.sg/please-verify-my-proof-sqrt-18-is-irrational.htmlhttps://pop.com.sg/please-verify-my-proof-sqrt-18-is-irrational.htmlThe following is one of my exam questions and I would like to know whether my proof is correct or not. Prove $\sqrt{18}$ is irrational using proof by contradiction. Following is my proof......Wed, 02 Sep 2026 07:52:34 -0400https://pop.com.sg/vertical-pipe-through-a-45-roof-how-to-determine-the-roof-hole-039-s-s.htmlhttps://pop.com.sg/vertical-pipe-through-a-45-roof-how-to-determine-the-roof-hole-039-s-s.htmlI have a tent roof that is at a 45° angle. In the tent, I have a woodstove that has a vertical stove pipe. I want to cut a hole in the roof of the tent that will be the exact shape of the protru......Wed, 02 Sep 2026 07:45:36 -0400https://pop.com.sg/mixtures-and-alligations-acid-concentration.htmlhttps://pop.com.sg/mixtures-and-alligations-acid-concentration.htmlFrom a vessel containing $1$ litre of pure acid, $100$ ml pure acid was drawn out in each of the beakers A and B. The acid in both the beakers was diluted by adding water in different proportions. ......Wed, 02 Sep 2026 07:43:55 -0400https://pop.com.sg/keeping-the-arc-length-constant-between-points-in-a-spiral.htmlhttps://pop.com.sg/keeping-the-arc-length-constant-between-points-in-a-spiral.htmlI'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is s = r * th......Wed, 02 Sep 2026 07:31:09 -0400https://pop.com.sg/find-max-min-volume-of-rectangular-box-using-lagrange-multipliers.htmlhttps://pop.com.sg/find-max-min-volume-of-rectangular-box-using-lagrange-multipliers.htmlCan anyone help me solve the problem below? This is question number 14.8.42 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the maximum and minimum volumes of a ...Wed, 02 Sep 2026 07:25:58 -0400https://pop.com.sg/determining-all-the-elements-of-s4.htmlhttps://pop.com.sg/determining-all-the-elements-of-s4.htmlWhat is an easy way to determine the elements of $S_{4}$? While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this. I trie......Wed, 02 Sep 2026 07:25:34 -0400https://pop.com.sg/union-of-intersections-and-intersection-of-unions.htmlhttps://pop.com.sg/union-of-intersections-and-intersection-of-unions.htmlWould someone be able to help me understand the (concept of) "union of intersections" and the "intersection of unions". More specifically, how to approach (prove) the following equality of two sets......Wed, 02 Sep 2026 07:20:38 -0400https://pop.com.sg/a-letter-is-picked-at-random-from-the-alphabet.htmlhttps://pop.com.sg/a-letter-is-picked-at-random-from-the-alphabet.htmlJust a book problem I need help on: A letter is picked at random from the alphabet. Find the probability that the letter is contained in the word "house" or in the word "phone". I know this probl......Wed, 02 Sep 2026 07:17:43 -0400https://pop.com.sg/cos-cos-x-sin-sin-x.htmlhttps://pop.com.sg/cos-cos-x-sin-sin-x.htmlShow that $\forall x \in \mathbb{R}\ \cos (\cos x) > \sin (\sin x)$ I've tried the obvious thing to do: $\varphi: x \mapsto \cos(\cos x) - \sin (\sin x)$ $\varphi ' (x) = \sin (\cos x) \sin ......Wed, 02 Sep 2026 07:14:18 -0400https://pop.com.sg/incremental-averaging.htmlhttps://pop.com.sg/incremental-averaging.htmlIs there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance? For example you have a = [4 6 3 9 4 12 4 18] and you wa......Wed, 02 Sep 2026 07:06:19 -0400https://pop.com.sg/find-initial-value-and-growth-in-exponential-growth-model.htmlhttps://pop.com.sg/find-initial-value-and-growth-in-exponential-growth-model.htmlI was given two models each with specific time($t$) and end amount($P$). $100$ after $3$ hours and $400$ after $5$ hours. Using the exponential growth model I get these $100=Pe^{3r}$ and $400=Pe^{5......Wed, 02 Sep 2026 06:58:56 -0400https://pop.com.sg/how-to-find-an-angle-in-range-0-360-between-2-vectors.htmlhttps://pop.com.sg/how-to-find-an-angle-in-range-0-360-between-2-vectors.htmlI know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the r......Wed, 02 Sep 2026 06:57:49 -0400https://pop.com.sg/on-the-number-of-ways-to-draw-kissing-circles.htmlhttps://pop.com.sg/on-the-number-of-ways-to-draw-kissing-circles.htmlSo I was watching Numberphile with Neil Sloane of OEIS fame on the number of ways to make circles intersect. During the intro, he explicitly forbid kissing (touching, tangent) circles. This made me......Wed, 02 Sep 2026 06:44:53 -0400https://pop.com.sg/finding-the-domain-of-arcsin-function.htmlhttps://pop.com.sg/finding-the-domain-of-arcsin-function.htmlI have to find the domain of: $ f(x) = 10 \frac{2}{3\arcsin x^2}$ Assumption 1: $-1 \leq 3\arcsin x^2 \leq 1$ $-1 \leq x^2 \leq 1$ $x \in [-1;1]$ Assumption 2: $3\arcsin x^2 \neq 0$ $x^2 \ne......Wed, 02 Sep 2026 06:36:22 -0400https://pop.com.sg/the-intuition-behind-slant-asymptotes.htmlhttps://pop.com.sg/the-intuition-behind-slant-asymptotes.htmlI'm asking this question to get a better understanding of oblique asymptotes. As regards vertical asymptotes, I know that they represent the numbers that make a function undefined. An example of t......Wed, 02 Sep 2026 06:35:24 -0400https://pop.com.sg/find-a-recurrence-relation-for-the-number-of-sequences-of-1s-3s-and-5s.htmlhttps://pop.com.sg/find-a-recurrence-relation-for-the-number-of-sequences-of-1s-3s-and-5s.html(a) Find a recurrence relation for the number of sequences of $ \ 1s, \ 3s, \ and \ \ 5s \ $ whose terms sum to $ \ n $ (b) Repeat part $ \ (a) $ with the added condition that no $ \ 5 \ $ ......Wed, 02 Sep 2026 06:34:46 -0400https://pop.com.sg/the-integral-form-of-inner-product-closed.htmlhttps://pop.com.sg/the-integral-form-of-inner-product-closed.htmlI want to know how scientists know that the inner product of f and g equal to integration from $$\int_a^b f(x)g(x)\ dx.$$...Wed, 02 Sep 2026 06:31:56 -0400https://pop.com.sg/plotting-frac-1-ln-x.htmlhttps://pop.com.sg/plotting-frac-1-ln-x.htmlI need assistance in plotting the graph of $\frac{1}{\ln x}$. wolframalpha gives this. How to plot this function (both real and imaginary part) using calculus?...Wed, 02 Sep 2026 06:30:37 -0400https://pop.com.sg/what-is-differential-probability-closed.htmlhttps://pop.com.sg/what-is-differential-probability-closed.htmlI am trying to understand the following statement (taken from a context related to volumetric image rendering): Density denotes the differential probability that a ray interacts with the volumetric “...Wed, 02 Sep 2026 06:14:31 -0400https://pop.com.sg/to-show-that-function-is-constant-duplicate.htmlhttps://pop.com.sg/to-show-that-function-is-constant-duplicate.htmlLet $f$ be defined on $\mathbb{R}$ and suppose that |$f(x)$ - $f(y)$| $\leq$ $(x-y)^2$ $x,y \in\mathbb{R}$. Here I have to show that $f$ is a constant function. I think I have to show that $f'(x)$......Wed, 02 Sep 2026 06:02:42 -0400https://pop.com.sg/the-concept-physical-meaning-interpretations-behind-the-bessel-039-s-i.htmlhttps://pop.com.sg/the-concept-physical-meaning-interpretations-behind-the-bessel-039-s-i.htmlIf $\{\boldsymbol{\varphi}_1, \boldsymbol{\varphi}_2, ... \}$ is an orthonormal system in $\mathcal{H}$, then the Bessel's inequality is: $$\sum_{j=1}^{\infty} |\langle \mathbf{x},\boldsymbol{\...Wed, 02 Sep 2026 05:51:12 -0400https://pop.com.sg/proof-of-the-existence-of-all-phone-numbers-in-pi-duplicate.htmlhttps://pop.com.sg/proof-of-the-existence-of-all-phone-numbers-in-pi-duplicate.htmlBy way of motivating example, someone came up with a completely evil user interface where the user was required to select their ten digit phone number by moving a sliding window over the digits of ......Wed, 02 Sep 2026 05:48:45 -0400https://pop.com.sg/proving-that-the-number-of-vertices-of-odd-degree-in-any-graph-g-is-ev.htmlhttps://pop.com.sg/proving-that-the-number-of-vertices-of-odd-degree-in-any-graph-g-is-ev.htmlI'm having a bit of a trouble with the below question Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$. ...Wed, 02 Sep 2026 05:45:46 -0400https://pop.com.sg/are-the-notations-lg-2-n-and-lg-n-2-the-same-thing.htmlhttps://pop.com.sg/are-the-notations-lg-2-n-and-lg-n-2-the-same-thing.htmlI have a problem that is has the notation $lg^2 n$ and I just want to verify that it actually means / is the same as $(lg n)^2$ If it is not the same please tell me how to evaluate $lg^2 n$ (always ...Wed, 02 Sep 2026 05:44:48 -0400https://pop.com.sg/method-to-find-colossally-abundant-numbers.htmlhttps://pop.com.sg/method-to-find-colossally-abundant-numbers.htmlColossally abundant numbers are positive integers $n$ for which there exists a positive exponent $\epsilon$ such that $$\frac{\sigma(n)}{n^{1+\epsilon}}>\frac{\sigma(m)}{m^{1+\epsilon}}$$ for ......Wed, 02 Sep 2026 05:31:53 -0400https://pop.com.sg/cardinality-of-a-set-containing-sets.htmlhttps://pop.com.sg/cardinality-of-a-set-containing-sets.htmlI've just started learning basic set theory and am puzzled by this question I came up with: What is the cardinality of {1, {2,3}}? Do I treat sets within sets as just one element and so the answe......Wed, 02 Sep 2026 05:27:25 -0400https://pop.com.sg/derivative-of-x-2-y-2-25.htmlhttps://pop.com.sg/derivative-of-x-2-y-2-25.htmlI can find the derivative of the above using implicit differentiation but was interested in finding it using the explicit function way. Nudged by “dx”: $y=(25-x^2)^{1/2}\\$ $dy= (25-(x+dx)^2)^{1......Wed, 02 Sep 2026 05:25:30 -0400https://pop.com.sg/understanding-purpose-of-dual-vector-space.htmlhttps://pop.com.sg/understanding-purpose-of-dual-vector-space.htmlI am a beginner in differential geometry and have questions/confusion about dual vector space. I took a look at this and this questions. But both did not resolve my question. We have standard ...Wed, 02 Sep 2026 05:21:58 -0400https://pop.com.sg/penrose-tile-generator.htmlhttps://pop.com.sg/penrose-tile-generator.htmlDoes anyone know if there's a client or web app that generates Penrose patterns which can then be converted to a tileable rectangular background image for web site? I found this http://stephencoll......Wed, 02 Sep 2026 05:15:08 -0400https://pop.com.sg/how-do-you-determine-if-a-matrix-is-invertible-by-investigating-the-eq.htmlhttps://pop.com.sg/how-do-you-determine-if-a-matrix-is-invertible-by-investigating-the-eq.htmlHow do you determine if a matrix is invertible by investigating the equation Ax = I? For a 3x3 matrix (A) with the following Row 1: 1, 0, 1 Row 2: 1, 1, 0 Row 3: 0, 1, 1 I know the identit......Wed, 02 Sep 2026 05:05:20 -0400https://pop.com.sg/finite-sub-cover-for-0-1.htmlhttps://pop.com.sg/finite-sub-cover-for-0-1.htmlWhile learning topology one learns about compact set. The standard definition is: A set $X$ is said to be compact if open cover has a finite subcover. Since $[0,1]$ is compact, if we take a open ......Wed, 02 Sep 2026 05:02:49 -0400https://pop.com.sg/sin-properties-why-is-this-undefined.htmlhttps://pop.com.sg/sin-properties-why-is-this-undefined.htmlWhy is sin(-π/2)^(1/2) undefined? sin(-π/2) is -1, so I thought the the answer would be the same, -1^(1/2)=-1? But my calculator and the answers sheet both say undefined......Wed, 02 Sep 2026 05:00:18 -0400https://pop.com.sg/show-that-the-polynomial-p-0-p-1-ldots-p-m-are-linearly-dependent.htmlhttps://pop.com.sg/show-that-the-polynomial-p-0-p-1-ldots-p-m-are-linearly-dependent.htmlLet $p_0, p_1, \ldots, p_m $ polynomials in $\mathbb{P}_m$ with the property that $p_j(1)=0\ \forall j$. Show that the polynomial $p_0, p_1, \ldots,p_m $ are linearly dependent. My approach for t......Wed, 02 Sep 2026 04:27:56 -0400https://pop.com.sg/how-do-you-simplify-fractions-that-have-exponents.htmlhttps://pop.com.sg/how-do-you-simplify-fractions-that-have-exponents.htmlI'm not sure how to simplify a fraction with a large exponent for example: $$\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$$...Wed, 02 Sep 2026 04:19:52 -0400https://pop.com.sg/tips-and-tricks-for-proofs-involving-algebraic-equation-manipulation.htmlhttps://pop.com.sg/tips-and-tricks-for-proofs-involving-algebraic-equation-manipulation.htmlI recently completed a solid chunk of my first truly formal math book on Set Theory and decided to move on to Michael Spivak's Calculus (as I heard it was a good place to begin one's journey into ...Wed, 02 Sep 2026 04:19:19 -0400https://pop.com.sg/how-to-find-the-area-bounded-by-the-curve-and-x-axis-closed.htmlhttps://pop.com.sg/how-to-find-the-area-bounded-by-the-curve-and-x-axis-closed.htmlThe text says: Determine the area bounded by the curve and $x$-axis. Could somebody solve or at least explain how I should solve these problems. $y=x^3+1.5x^2$ $y=-x^3-3x^2$ $y=x^2+x+2$ and $y=-x^......Wed, 02 Sep 2026 04:16:31 -0400https://pop.com.sg/finding-the-constant-k-from-density-function.htmlhttps://pop.com.sg/finding-the-constant-k-from-density-function.htmlWondering how to find the value of the constant k for this. I know you have to integrate, so I integrated both functions, however I'm not quite sure on the steps afterwards. Any help would be greatly ...Wed, 02 Sep 2026 04:06:11 -0400https://pop.com.sg/solving-inequalities-on-both-sides-with-complex-numbers.htmlhttps://pop.com.sg/solving-inequalities-on-both-sides-with-complex-numbers.htmlI need to sketch this region $\left \{ z\in\mathbb{C}| |z-i|\leq |z-1| \right \}$. I'd like some assistance with solving this inequality because I think that's where I'm going wrong. To solve the ...Wed, 02 Sep 2026 04:05:53 -0400https://pop.com.sg/when-does-the-limit-of-a-sum-become-an-integral.htmlhttps://pop.com.sg/when-does-the-limit-of-a-sum-become-an-integral.htmlIn many maths and physics texts and courses, I've been told in many cases that the limit of a sum becomes an integral, i.e. (very roughly): $$\lim_{n\to\infty} \sum_{x=0}^n f(x) = \int_0^\infty f(......Wed, 02 Sep 2026 04:04:38 -0400https://pop.com.sg/difference-between-pure-and-applied-mathematics.htmlhttps://pop.com.sg/difference-between-pure-and-applied-mathematics.htmlSo, I have been looking into college courses to see what I want my major to be, and I noticed that M.I.T offers two specified types of Mathematics. Pure and Applied. What is the major differences i......Wed, 02 Sep 2026 03:40:45 -0400https://pop.com.sg/why-do-we-add-a-zero-to-dividend-during-long-division.htmlhttps://pop.com.sg/why-do-we-add-a-zero-to-dividend-during-long-division.htmlSuppose we want to divide 3 by 2. By Long Division we first write 1 as the first digit as the quotient, then we subtract 2 from 3, then we add a zero to the remainder 1, and then add a decimal point ...Wed, 02 Sep 2026 03:33:49 -0400https://pop.com.sg/find-the-gradient-and-equation-of-the-normal-to-the-curve-y-x-3-x-5-at.htmlhttps://pop.com.sg/find-the-gradient-and-equation-of-the-normal-to-the-curve-y-x-3-x-5-at.htmlI was wondering if anyone can help with this question.I'm just not sure how to find the gradients and equations of these questions. If you could help it would be greatly appreciated. $$y=x^3+x+5$$...Wed, 02 Sep 2026 03:30:31 -0400https://pop.com.sg/limit-points-and-interior-points.htmlhttps://pop.com.sg/limit-points-and-interior-points.htmlI am reading Rudin's book on real analysis and am stuck on a few definitions. First, here is the definition of a limit/interior point (not word to word from Rudin) but these definitions are worde......Wed, 02 Sep 2026 03:29:15 -0400https://pop.com.sg/calculus-maximum-volume-equation.htmlhttps://pop.com.sg/calculus-maximum-volume-equation.htmlI tried my best to come up with an answer but I failed. I would really appreciate if someone can help me with this: Question: To carry a suitcase on an airplace, the length + width + height must......Wed, 02 Sep 2026 03:27:17 -0400https://pop.com.sg/is-there-a-name-for-the-point-of-a-exponential-curve-where-the-y-axis-.htmlhttps://pop.com.sg/is-there-a-name-for-the-point-of-a-exponential-curve-where-the-y-axis-.htmlIt's been hard to come up with a question title that makes sense so please bear with me. On an exponential curve there's a point on the x axis where the y axis starts increasing significantly. The......Wed, 02 Sep 2026 03:20:36 -0400https://pop.com.sg/derivative-of-determinant-of-a-matrix.htmlhttps://pop.com.sg/derivative-of-determinant-of-a-matrix.htmlI would like to know how to calculate: $$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big).$$...Wed, 02 Sep 2026 02:55:48 -0400https://pop.com.sg/evaluating-lim-x-to0-frac-2-cot-2x-cot-x-sin-2x.htmlhttps://pop.com.sg/evaluating-lim-x-to0-frac-2-cot-2x-cot-x-sin-2x.htmlWhat is the value of $\lim_{x\to0}\frac{2\cot 2x-\cot x}{\sin 2x}$ ? $1)\text{zero}\qquad\qquad2)\frac12\qquad\qquad3)-\frac12\qquad\qquad4)-1$ Here is my work: $$\lim_{x\to0}\frac{2\cot 2x-\cot x......Wed, 02 Sep 2026 02:35:36 -0400https://pop.com.sg/is-zero-positive-or-negative.htmlhttps://pop.com.sg/is-zero-positive-or-negative.htmlFollow up to this question. Is $0$ a positive number?...Wed, 02 Sep 2026 02:32:40 -0400https://pop.com.sg/why-is-mathbb-f-3-x-x-2-1-cong-mathbb-f-9.htmlhttps://pop.com.sg/why-is-mathbb-f-3-x-x-2-1-cong-mathbb-f-9.htmlI want to show that $\mathbb Z[i]/(3) \cong \mathbb F_9$, and I know that $\mathbb Z[i]/(3) \cong \mathbb Z[x]/(3, x^2 + 1) \cong \mathbb F_3[x]/(x^2 + 1)$. How do we know that $\mathbb F_3[x]/(x^......Wed, 02 Sep 2026 02:28:10 -0400https://pop.com.sg/given-system-of-equations-a-b-2-ab-4-solve-a-2-b-2-and-a-3.htmlhttps://pop.com.sg/given-system-of-equations-a-b-2-ab-4-solve-a-2-b-2-and-a-3.htmlI am trying to solve $a^2+b^2$ and $a^3$ given $a+b = 2, ab=4$. I have the key with the answers $a^2+b^2=-4$ and $a^3=-8$ but am wondering which steps to take to get to that answer. My understand......Wed, 02 Sep 2026 02:09:47 -0400https://pop.com.sg/set-of-positive-integers-with-unique-sums.htmlhttps://pop.com.sg/set-of-positive-integers-with-unique-sums.htmlWhat I'm looking for is the name of a type of number set. Given a number T (for total) and a set of positive integers S, I want to uniquely identify the subset of S that sums to T. All sets contain......Wed, 02 Sep 2026 02:05:19 -0400https://pop.com.sg/list-of-ways-to-tell-if-degree-sequence-is-impossible-for-a-simple-gra.htmlhttps://pop.com.sg/list-of-ways-to-tell-if-degree-sequence-is-impossible-for-a-simple-gra.htmlI'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$ is not possible because there are only 3 vertices in total so one can't have degree 3. The ......Wed, 02 Sep 2026 02:02:50 -0400https://pop.com.sg/in-a-venn-diagram-where-are-other-number-sets-located.htmlhttps://pop.com.sg/in-a-venn-diagram-where-are-other-number-sets-located.htmlI remember of this image I've learned at school: I've heard about other number (which I'm not really sure if they belong to a new set) such as quaternions, p-adic numbers. Then I got three questio......Wed, 02 Sep 2026 01:44:29 -0400https://pop.com.sg/uniqueness-of-adjoint-functors-up-to-isomorphism.htmlhttps://pop.com.sg/uniqueness-of-adjoint-functors-up-to-isomorphism.htmlSuppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...Wed, 02 Sep 2026 01:31:24 -0400https://pop.com.sg/completeness-of-probability-space-and-p-null-set.htmlhttps://pop.com.sg/completeness-of-probability-space-and-p-null-set.htmlCompleteness of probability space is often equivalent to the sigma algebra of the space contains all $P$-null sets. Why does this definition make intuitive sense? Why would it be "incomplete" if the ...Wed, 02 Sep 2026 01:29:21 -0400https://pop.com.sg/how-can-a-gradient-be-thought-of-as-a-function.htmlhttps://pop.com.sg/how-can-a-gradient-be-thought-of-as-a-function.htmlIf a function $f(x,y)$ would output a value in a third dimension, $z=f(x,y)$ for example. How can we treat the gradient of $f$ as a function in $x$ and $y$, when the output of the gradient is a vec......Wed, 02 Sep 2026 01:20:27 -0400https://pop.com.sg/determine-if-the-sequence-an-ln-n-n-1-n-is-convergent.htmlhttps://pop.com.sg/determine-if-the-sequence-an-ln-n-n-1-n-is-convergent.htmlI have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!...Wed, 02 Sep 2026 01:17:24 -0400https://pop.com.sg/notation-for-every-odd-integer-number.htmlhttps://pop.com.sg/notation-for-every-odd-integer-number.htmlI have this equation: $$f(x)=\tan(x)$$ I found the vertical asymptotes to be: $$x=\frac{\pi}{2}k$$ What is the proper notation for that k is equal to every odd number integer(negative,posi......Wed, 02 Sep 2026 01:14:42 -0400https://pop.com.sg/tangent-normal-and-binormal-vectors.htmlhttps://pop.com.sg/tangent-normal-and-binormal-vectors.htmlLet $C$ be the graph of vector function $r(s)$ parametrized by arc length with $r'(0)= ({2\over3},{1\over3},{2\over3})$ and $r''(0)=(-3,12,-3)$. I need to find tangent vector $T(0)$, normal vec......Wed, 02 Sep 2026 01:13:39 -0400https://pop.com.sg/geometric-meanings-of-hyperbolic-cosine-and-sine.htmlhttps://pop.com.sg/geometric-meanings-of-hyperbolic-cosine-and-sine.htmlIn euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry. Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosine and sine ?...Wed, 02 Sep 2026 01:08:00 -0400https://pop.com.sg/how-many-5-digit-zip-codes-can-there-be-with-exactly-two-digits-the-sa.htmlhttps://pop.com.sg/how-many-5-digit-zip-codes-can-there-be-with-exactly-two-digits-the-sa.htmlHow many 5-digit United States zip codes can there be with exactly one pair of digits? This includes both zip codes that correspond to an actual city, town, incorporated community, ghost town, etc.......Wed, 02 Sep 2026 01:06:40 -0400https://pop.com.sg/proof-that-the-three-interior-angles-of-a-triangle-is-congruent-to-a-s.htmlhttps://pop.com.sg/proof-that-the-three-interior-angles-of-a-triangle-is-congruent-to-a-s.htmlI'm trying to essentially prove that the interior angles of a triangle add up to 180 degrees. However, I'm trying to prove it without mentioning measurements of an angle. I think I understand the p......Wed, 02 Sep 2026 01:06:22 -0400https://pop.com.sg/a-constant-function-on-matlab-closed.htmlhttps://pop.com.sg/a-constant-function-on-matlab-closed.htmlclear clc eta = 100; a= 0; xi=0.016; axis([0 0.01 0 1.1]); x = 0:0.0001:0.01; N = -a+1.6*(1/(xi*eta)); y=N; plot(x,y,'LineWidth',2); set(gca,'FontSize',10); Hi. I'd like to plot the function $N(x......Wed, 02 Sep 2026 01:04:09 -0400https://pop.com.sg/applications-of-the-law-of-the-iterated-logarithm.htmlhttps://pop.com.sg/applications-of-the-law-of-the-iterated-logarithm.htmlThe law of the iterated logarithm says that if $X_n$ is a sequence of iid random variables with zero expectation and unit variance, then the partial sums sequence $S_n = \sum_{i = 1}^n X_i$ satisfies ...Wed, 02 Sep 2026 01:03:01 -0400https://pop.com.sg/adding-exponents-with-unknowns-with-the-same-base.htmlhttps://pop.com.sg/adding-exponents-with-unknowns-with-the-same-base.htmlHow do you add numbers with the same base but with unknown exponents? For example, $2^{x+2} + 2^{x+2}$ I understand that you do $2^x \cdot 2^2$ but I get stuck here. I don't know what to do from he......Wed, 02 Sep 2026 01:01:13 -0400https://pop.com.sg/function-map-notation.htmlhttps://pop.com.sg/function-map-notation.htmlPlease forgive my ignorance, if I've phrased my question improperly. I'm not sure what the appropriate terminology is; that's the basis of my question. So, I'm not sure if I'm even remotely close i......Wed, 02 Sep 2026 00:47:35 -0400https://pop.com.sg/find-the-equation-of-the-normal-to-the-curve.htmlhttps://pop.com.sg/find-the-equation-of-the-normal-to-the-curve.htmlGood day everyone. The normal to the curve $y=x^2-2x$ at a point P is parallel to the straight line $y=-2x+5$ .Find the equation of the normal to the curve at point P. What does it question mean? ...Wed, 02 Sep 2026 00:38:28 -0400https://pop.com.sg/how-to-simplify-frac-sin-alpha-sin-beta-cos-beta-cos-alpha-cdot-cos-al.htmlhttps://pop.com.sg/how-to-simplify-frac-sin-alpha-sin-beta-cos-beta-cos-alpha-cdot-cos-al.htmlThere is this problem: $$\left(\frac{\sin\alpha - \sin\beta}{\cos\beta -\cos\alpha}\right)\cdot \cos\alpha \cdot \cos\beta = \frac{1}{\tan\alpha -\tan\beta}$$ I started as $$\left(\frac{\sin\alpha ......Wed, 02 Sep 2026 00:36:00 -0400https://pop.com.sg/uniqueness-of-payout-of-nash-equilibrium-in-a-2-player-zero-sum-game.htmlhttps://pop.com.sg/uniqueness-of-payout-of-nash-equilibrium-in-a-2-player-zero-sum-game.htmlI'm convinced that in a 2 player zero sum game, any two NE strategies must have the same expected payout for both players. I'm even more convinced that this is not true for 3 or more players. Where......Wed, 02 Sep 2026 00:16:46 -0400https://pop.com.sg/how-to-show-that-this-matrix-is-positive-semidefinite.htmlhttps://pop.com.sg/how-to-show-that-this-matrix-is-positive-semidefinite.htmlUsing the definition, show that the following matrix is positive semidefinite. $$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$ In other words, if the ...Wed, 02 Sep 2026 00:11:20 -0400https://pop.com.sg/creating-a-tridiagonal-matrix-in-matlab.htmlhttps://pop.com.sg/creating-a-tridiagonal-matrix-in-matlab.htmlHow can I create a tridiagonal matrix that I can use for Crout factorization? And, I don't have any codes on how to create one since I am new to matlab. Ok, please help me understand what does the ...Wed, 02 Sep 2026 00:05:31 -0400https://pop.com.sg/approximation-vs-interpolation.htmlhttps://pop.com.sg/approximation-vs-interpolation.htmlSorry if this is a silly question, i'm just getting back into math after a long time away. My question is regarding approximation and interpolation. In which cases is it appropriate for one technique ...Wed, 02 Sep 2026 00:05:10 -0400https://pop.com.sg/must-a-comeager-set-be-dense.htmlhttps://pop.com.sg/must-a-comeager-set-be-dense.htmlSay, on a complete separable metric space. Separable probably doesn't matter. It's easy to see the opposite; if $D$ is a dense $G_\delta$ set, it's a countable intersection of open sets which con......Wed, 02 Sep 2026 00:03:11 -0400https://pop.com.sg/is-there-a-general-matrix-to-reflect-about-the-line-y-mx-c.htmlhttps://pop.com.sg/is-there-a-general-matrix-to-reflect-about-the-line-y-mx-c.htmlI have been looking into matrix transformations and found the following matrix to reflect about the line $y=(\tan\theta)x$. $$R = \begin{bmatrix} \cos(2\theta)& \sin(2\theta)\\ \s......Tue, 01 Sep 2026 23:58:16 -0400https://pop.com.sg/whats-the-rule-for-putting-up-a-plus-minus-sign-when-taking-under-root.htmlhttps://pop.com.sg/whats-the-rule-for-putting-up-a-plus-minus-sign-when-taking-under-root.htmlGiven a simple equation.... $\ (x+1)^2 =21 $ if we take the under root of both sides , we get $\ x+1 = \pm \sqrt{21} $ why dont we get a $ \pm $ on the left hand side ?...Tue, 01 Sep 2026 23:52:49 -0400https://pop.com.sg/how-do-you-find-the-altitude-in-a-pyramid-sat-math-question.htmlhttps://pop.com.sg/how-do-you-find-the-altitude-in-a-pyramid-sat-math-question.htmlThe pyramid shown above has altitude h and a square base of side m. The four edges that meet at V, the vertex of the pyramid, each have length e. If e = m, what is the value of h in terms of m? A) $\...Tue, 01 Sep 2026 23:49:13 -0400https://pop.com.sg/mean-expected-value-or-expectation-of-a-constant.htmlhttps://pop.com.sg/mean-expected-value-or-expectation-of-a-constant.htmlAccording to Wikipedia, "the expected value of a constant is equal to the constant itself; i.e., if $c$ is a constant, then $\mathbb E[c] = c$." I am currently having a hard time picturing what this ...Tue, 01 Sep 2026 23:42:42 -0400https://pop.com.sg/how-to-obtain-the-gradient-in-polar-coordinates.htmlhttps://pop.com.sg/how-to-obtain-the-gradient-in-polar-coordinates.htmlI'm not sure on how to find the gradient in polar coordinates. The thing that troubles me the most is how to find the unit vectors $\hat{r}$ and $\hat{\theta}$. My approach for the rest is expressi......Tue, 01 Sep 2026 23:40:23 -0400https://pop.com.sg/calculate-the-tangent-plane-of-x-2y-xy-2-at-a-b-1-1.htmlhttps://pop.com.sg/calculate-the-tangent-plane-of-x-2y-xy-2-at-a-b-1-1.htmlWrite the equation of the tangent plane at the point $(a, b, f(a, b))$ on the graph of $f$. The equation to the tangent plane for partial derivatives is given as: $$z = f(a.b)+(x-a)\frac{\partial f}{\...Tue, 01 Sep 2026 23:16:45 -0400