https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 05 Sep 2026 17:29:49 -0400https://pop.com.sg/what-does-s-t-mean.htmlhttps://pop.com.sg/what-does-s-t-mean.htmlEnglish is my second language and I have a question. What does "s.t." mean? $ \text{min} \quad f(x) = (x_1−2)^2+(x_2−1)^2 $ $ \text{s.t.}\qquad g_{1}(x) = x_{1} - 2x_{2} + 1 = 0 $ $ \qquad\qquad......Sat, 05 Sep 2026 21:28:22 -0400https://pop.com.sg/equation-for-a-smooth-step-function.htmlhttps://pop.com.sg/equation-for-a-smooth-step-function.htmlhttp://i.stack.imgur.com/t1PlV.jpg The equation must be smooth and satisfy the following two conditions: When $x \leq -10, y=-50$ When $x \geq 10, y=50$ Is there an equation (or multiple equatio......Sat, 05 Sep 2026 21:22:10 -0400https://pop.com.sg/how-are-these-triangles-similar.htmlhttps://pop.com.sg/how-are-these-triangles-similar.htmlI am trying to understand a proof of Pythagorean's Theorem. I am trying to see how all three triangles are similar. So far I have: (1) $\triangle ABC$ and $\triangle DBA$ have side $AB$ in commo......Sat, 05 Sep 2026 21:22:06 -0400https://pop.com.sg/why-dividing-by-zero-still-works.htmlhttps://pop.com.sg/why-dividing-by-zero-still-works.htmlToday, I was at a class. There was a question: If $x = 2 +i$, find the value of $x^3 - 3x^2 + 2x - 1$. What my teacher did was this: $x = 2 + i \;\Rightarrow \; x - 2 = i \; \Rightarrow \; (x ......Sat, 05 Sep 2026 21:19:21 -0400https://pop.com.sg/could-someone-please-explain-double-angle-identities.htmlhttps://pop.com.sg/could-someone-please-explain-double-angle-identities.htmlI don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc. I tried to explain the $\sin(2\theta)$ double-angle ...Sat, 05 Sep 2026 21:12:23 -0400https://pop.com.sg/how-many-ways-can-the-word-arranger-be-arranged.htmlhttps://pop.com.sg/how-many-ways-can-the-word-arranger-be-arranged.htmlI have tried this problem by doing 8 factorial over two times two factorial, which I got 10080, which doesn't seem to be right since the answer on the answer key is pretty far from my answer, can s......Sat, 05 Sep 2026 21:11:58 -0400https://pop.com.sg/stokes-039-s-first-problem-rayleigh-plate-u-0-t-t-n.htmlhttps://pop.com.sg/stokes-039-s-first-problem-rayleigh-plate-u-0-t-t-n.htmlSo basically we have Stoke's first Problem/Rayleigh plate, meaning an impulsively starting Plate with the boundary conditions $u(0,t)=t^n$ and $u(\infty,t)=0$. In theory it is easy to solve the PDE......Sat, 05 Sep 2026 21:08:40 -0400https://pop.com.sg/why-is-the-number-of-ways-of-choosing-0-items-from-n-items-1.htmlhttps://pop.com.sg/why-is-the-number-of-ways-of-choosing-0-items-from-n-items-1.htmlIt seems easy to grasp that number of ways of choosing $n$ items from $n$ items is 1. But I am unable to understand why is it 1 for choosing 0 items....Sat, 05 Sep 2026 21:08:37 -0400https://pop.com.sg/express-sin-4-theta-by-formulae-involving-sin-and-cos-and-its-powers.htmlhttps://pop.com.sg/express-sin-4-theta-by-formulae-involving-sin-and-cos-and-its-powers.htmlI have an assignment question that says "Express $\sin 4\theta$ by formulae involving $\sin$ and $\cos$ and its powers." I'm told that $\sin 2\theta = 2 \sin\theta \cos\theta$ but I don't know how......Sat, 05 Sep 2026 21:06:05 -0400https://pop.com.sg/number-of-possible-combinations-of-5-from-the-set-of-numbers-1-10.htmlhttps://pop.com.sg/number-of-possible-combinations-of-5-from-the-set-of-numbers-1-10.htmlI am trying to figure out how to set up this problem or any similar one. If you have 10 balls numbered from 1 to 10, and you pick 5 balls, what is the probability that you will have picked ball#1? In ...Sat, 05 Sep 2026 21:01:30 -0400https://pop.com.sg/is-1-divided-by-3-equal-to-0-333-duplicate.htmlhttps://pop.com.sg/is-1-divided-by-3-equal-to-0-333-duplicate.htmlI have been taught that $\frac{1}{3}$ is 0.333.... However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had infinite threes, as 0.333... is suppo......Sat, 05 Sep 2026 20:57:33 -0400https://pop.com.sg/vectors-vs-cartesian-coordinates.htmlhttps://pop.com.sg/vectors-vs-cartesian-coordinates.htmlWhat is the difference, if any, between a Cartesian coordinate and a vector? Is it that a vector always has direction and magnitude, whilst Cartesian coordinates do not?...Sat, 05 Sep 2026 20:55:45 -0400https://pop.com.sg/evaluating-the-given-integral-by-changing-to-polar-coordinates-why-is-.htmlhttps://pop.com.sg/evaluating-the-given-integral-by-changing-to-polar-coordinates-why-is-.htmlEvaluate the given integral by changing to polar coordinates. $\int\int_D x^2y \, dA$, where $D$ is the top half of the disk with center the origin and radius $5$. when i plugged everything in, I ......Sat, 05 Sep 2026 20:52:32 -0400https://pop.com.sg/basic-examples-of-monoids.htmlhttps://pop.com.sg/basic-examples-of-monoids.htmlWhat are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply ...Sat, 05 Sep 2026 20:46:56 -0400https://pop.com.sg/show-that-blaschke-factors-satisfy-inequality-duplicate.htmlhttps://pop.com.sg/show-that-blaschke-factors-satisfy-inequality-duplicate.htmlTheorem: Let $w, z \in \mathbb{C}$ such that $\bar{z}w \neq 1$. Then $$ |z| < 1 \land |w| < 1 \Rightarrow \left| \frac{w-z}{1-\bar{w}z}\right| < 1$$ I tried to make $w = u +iv, z = x + iy......Sat, 05 Sep 2026 20:43:00 -0400https://pop.com.sg/trigonometric-proof-involving-sec-and-cosec.htmlhttps://pop.com.sg/trigonometric-proof-involving-sec-and-cosec.htmlHow should I proceed to prove the following: $$\sec^2 A +\csc^2 A = \sec^2 A \cdot \csc^2 A$$ This is what I've attempted: $$LHS = 1 + \tan^2 A + 1 + \cot^2= 2+ 1\cot^2 + \cot^2$$ However, this ......Sat, 05 Sep 2026 20:38:05 -0400https://pop.com.sg/why-are-equations-written-by-equating-something-to-zero.htmlhttps://pop.com.sg/why-are-equations-written-by-equating-something-to-zero.htmlA linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all these equations written as $\dots = 0 $? Why do ...Sat, 05 Sep 2026 20:37:38 -0400https://pop.com.sg/prove-set-inclusivity.htmlhttps://pop.com.sg/prove-set-inclusivity.htmlI need help proving the following Suppose $A,B \subseteq X$. Prove that if $X \setminus B \subseteq X \setminus A$, then $A \subseteq B$. I really need to see an example of how to do these types of ...Sat, 05 Sep 2026 20:33:38 -0400https://pop.com.sg/an-example-of-kirchhoff-equation.htmlhttps://pop.com.sg/an-example-of-kirchhoff-equation.htmlI'm studying about a simply kind of Kirchhoff equation in one-dimensional that means $$\begin{cases} u_{tt}(x,t)-\left(1+\int_{0}^{1}{u^2_x(x,t)dx}\right)u_{xx}(x,t)=|u(x,t)|^{p-2}u(x,t),0<x<......Sat, 05 Sep 2026 20:31:46 -0400https://pop.com.sg/finding-the-associated-unit-eigenvector.htmlhttps://pop.com.sg/finding-the-associated-unit-eigenvector.htmlBackground Find the eigenvalues $λ_1<λ_2$ and two associated unit eigenvectors of the symmetric matrix $$A = \begin{bmatrix}-7&12\\12&11\end{bmatrix}$$ My work so far $$A = \begin{bmatri......Sat, 05 Sep 2026 20:29:32 -0400https://pop.com.sg/user-carsten-s-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-carsten-s-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 05 Sep 2026 20:18:57 -0400https://pop.com.sg/drawing-an-ellipse.htmlhttps://pop.com.sg/drawing-an-ellipse.htmlYou are given a pen, pencil, pair of compasses, scissors, a ruler and a piece of paper. I require a natural way to draw an ellipse. This can be simple, from using a circle piece of paper, to a co......Sat, 05 Sep 2026 20:13:29 -0400https://pop.com.sg/calculating-inverse-of-trig-functions-without-calculator-closed.htmlhttps://pop.com.sg/calculating-inverse-of-trig-functions-without-calculator-closed.htmlWe have $\sin 25°\approx 0.42.$ How to calculate $\arcsin0. 42\approx 25°$ without the help of calculator? And how does calculator evaluate the inverse functions value....Sat, 05 Sep 2026 20:12:15 -0400https://pop.com.sg/qr-factorization-of-complex-matrix.htmlhttps://pop.com.sg/qr-factorization-of-complex-matrix.htmlIf you have two complex numbers $a,b$ how can you find the QR factorization of $ M = \begin{bmatrix} aI_n\\ bI_n \end{bmatrix} $, I can't seem to be able to do it. I tried an ear......Sat, 05 Sep 2026 20:10:22 -0400https://pop.com.sg/gay-speed-dating-problem.htmlhttps://pop.com.sg/gay-speed-dating-problem.htmlHere's an interesting problem that I came up with the other night. With straight speed dating, (assuming the number of men and women are equal) the number of iterations that need to be made before......Sat, 05 Sep 2026 20:07:38 -0400https://pop.com.sg/when-does-the-tangent-line-to-the-sine-curve-pass-through-the-origin.htmlhttps://pop.com.sg/when-does-the-tangent-line-to-the-sine-curve-pass-through-the-origin.htmlI am trying to find values $a$ and $w$ for which the line $y=ax$ is tangent to the curve $y=\sin(x)$ at $x=w$. One immediate solution is $a=1$ and $w=0$, but I would like $a<0$ and $w\in (\pi,3......Sat, 05 Sep 2026 19:55:56 -0400https://pop.com.sg/are-there-different-kinds-of-invalid-arguments.htmlhttps://pop.com.sg/are-there-different-kinds-of-invalid-arguments.htmlin my semantics class we had two examples of invalid arguments: $$\begin{array}{rl} & p \lor q \\ & \neg p \\ \hline \therefore & \neg q \end{array}$$ and $$\begin{arr......Sat, 05 Sep 2026 19:27:05 -0400https://pop.com.sg/proving-that-the-number-sqrt-3-7-sqrt-50-sqrt-3-7-5-sqrt-2-is-rational.htmlhttps://pop.com.sg/proving-that-the-number-sqrt-3-7-sqrt-50-sqrt-3-7-5-sqrt-2-is-rational.htmlI've been struggling to show that the number $\sqrt[3]{7 + \sqrt{50}} + \sqrt[3]{7 - 5\sqrt{2}}$ is rational. I would like to restructure it to prove it, but I can't find anything besides $\sqrt{50......Sat, 05 Sep 2026 19:21:22 -0400https://pop.com.sg/calc-of-potential-function-using-line-integral.htmlhttps://pop.com.sg/calc-of-potential-function-using-line-integral.htmlI tried to calculate a potential function using the two major methods (by partial differentiation/integration, and by line integral). The two worked with all of the examples that I tried, but then I ...Sat, 05 Sep 2026 19:16:20 -0400https://pop.com.sg/what-is-the-reason-we-usually-don-039-t-use-formal-proofs-in-mathemati.htmlhttps://pop.com.sg/what-is-the-reason-we-usually-don-039-t-use-formal-proofs-in-mathemati.htmlIs the reason for not using formal proofs very often in mathematics because it is usually too lengthy for a person to make such a proofs or the reason is that it is simply not possible for everythi......Sat, 05 Sep 2026 19:09:11 -0400https://pop.com.sg/usefulness-of-the-characteristic-function-set-theory.htmlhttps://pop.com.sg/usefulness-of-the-characteristic-function-set-theory.htmlFirst a definition for those who need it: If $S$ is a set and $S_0$ is a subset of $S$ then the characteristic function of $S_0$ as a subset of $S$ is the function $f$, with domain $S$ defined by $......Sat, 05 Sep 2026 18:37:12 -0400https://pop.com.sg/solving-the-following-volterra-integral-equation.htmlhttps://pop.com.sg/solving-the-following-volterra-integral-equation.htmlHow do I solve the following Volterra, non-homogeneous, $1st$ kind Integral Equation : $$ \dfrac{x^2}{2}=\int_0^x (1-x^2+t^2)u(t) dt$$ I know I cannot apply Laplace Transform because the kernel......Sat, 05 Sep 2026 18:21:35 -0400https://pop.com.sg/the-jth-factorial-moment-of-x-geo-p-duplicate.htmlhttps://pop.com.sg/the-jth-factorial-moment-of-x-geo-p-duplicate.htmlDefine $E[{X \choose j}]=E[X(X-1)(X-2) \dots (X-j+1)]/j\,!$ which $G_X(s)$ is a probability generating function. $E[X(X-1)(X-2) \dots (X-j+1)]=G_X^{(j)}(s=1)$ $E[{X \choose j}]$ is called the jth ...Sat, 05 Sep 2026 18:15:23 -0400https://pop.com.sg/three-simultaneously-diagonalizable-matrices.htmlhttps://pop.com.sg/three-simultaneously-diagonalizable-matrices.htmlI have three symmetric square matrices $M$, $G$, and $S$ with the following properties: $S$: symmetric and positive semi-definite. $M$: Fully diagonal with positive entries. $G$: is a subset of ......Sat, 05 Sep 2026 18:14:16 -0400https://pop.com.sg/prove-by-mathematical-induction-that-n-3-n-is-divisible-by-3-for-all-n.htmlhttps://pop.com.sg/prove-by-mathematical-induction-that-n-3-n-is-divisible-by-3-for-all-n.htmlI'm working on a task where I'm a bit unsure if the answer I've got is correct. Here is the task: Show by induction that the following assertion is true for all natural numbers $n$ $n^3 ......Sat, 05 Sep 2026 18:11:53 -0400https://pop.com.sg/vector-magnitude-notation.htmlhttps://pop.com.sg/vector-magnitude-notation.htmlIs the follow equality true? $$\left| \overrightarrow {u}\right| =u$$ I wonder, because on AP Physics formula sheets, sometimes the magnitude of a vector is clearly denoted, while other times the ...Sat, 05 Sep 2026 18:11:28 -0400https://pop.com.sg/calculating-the-norm-of-a-matrix-in-matlab.htmlhttps://pop.com.sg/calculating-the-norm-of-a-matrix-in-matlab.htmlSo I've been given the matrix A A = [1 2; 3 1] and the question is: If v is an error vector with ||v|| = 0.01, give an upper bound on ||A^30v||. ||A^30v|| = ||A^30|| * ||v||, correct? I used th......Sat, 05 Sep 2026 18:10:05 -0400https://pop.com.sg/is-the-nash-embedding-theorem-a-special-case-of-the-whitney-embedding-.htmlhttps://pop.com.sg/is-the-nash-embedding-theorem-a-special-case-of-the-whitney-embedding-.htmlThe Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean sp......Sat, 05 Sep 2026 18:03:28 -0400https://pop.com.sg/square-roots-variables-with-exponents.htmlhttps://pop.com.sg/square-roots-variables-with-exponents.htmlAlright, so let me get this straight: $\sqrt{x^2} = |x|$ $\sqrt{x^3} = x\sqrt{x}$ $\sqrt{x^4} = x^2$ $\sqrt{x^6} = |x^3|$ Are these correct?...Sat, 05 Sep 2026 17:51:17 -0400https://pop.com.sg/over-which-interval-does-f-x-have-a-negative-average-rate-of-change.htmlhttps://pop.com.sg/over-which-interval-does-f-x-have-a-negative-average-rate-of-change.htmlI'm an adult learner of maths (self-studying). I've been struggling with this question and would like some help. I've been given 4 different x,y coordinates (indicated as stars on the image below) ......Sat, 05 Sep 2026 17:50:55 -0400https://pop.com.sg/characteristic-equation-of-differential-equation.htmlhttps://pop.com.sg/characteristic-equation-of-differential-equation.htmlGiven $x''+3x'+2x=4.$ ($''=2nd $ derivative, $'=1st$ derivative) Determine the characteristic equation of this differential equation. I'm having a hard time doing this because of that $4$. Any h......Sat, 05 Sep 2026 17:42:59 -0400https://pop.com.sg/why-is-x-4-16x-2-4-irreducible-in-mathbb-q-x.htmlhttps://pop.com.sg/why-is-x-4-16x-2-4-irreducible-in-mathbb-q-x.htmlDetermine whether $X^4-16X^2+4$ is irreducible in $\mathbb{Q}[X]$. To solve this problem, I reasoned that since $X^4-16X^2+4$ has no rational roots hence irreducible. But there is a hint to this ...Sat, 05 Sep 2026 17:40:37 -0400https://pop.com.sg/understanding-a-riddle-how-many-flowers-in-a-bouquet-if-all-but-two-ar.htmlhttps://pop.com.sg/understanding-a-riddle-how-many-flowers-in-a-bouquet-if-all-but-two-ar.htmlThis is quite a popular riddle in interviews and in general: In a bouquet of flowers, all but two are roses, all but two are tulips, and all but two are daisies. How many flowers are in the bouque......Sat, 05 Sep 2026 17:37:09 -0400https://pop.com.sg/what-is-the-best-way-to-arrange-trapezoidable-bricks-to-form-a-giant-c.htmlhttps://pop.com.sg/what-is-the-best-way-to-arrange-trapezoidable-bricks-to-form-a-giant-c.htmlI have some bricks that form a trapezoid shape. They have the following dimensions: a = 4.25 in b = 9.05 in c = 2.375 in d = 9.05 in I want to arrange these such that they form a giant lawn circle......Sat, 05 Sep 2026 17:36:02 -0400https://pop.com.sg/surface-area-of-a-cylinders-and-prisms.htmlhttps://pop.com.sg/surface-area-of-a-cylinders-and-prisms.htmlA cylinder has a diameter of 9cm and a height of 25cm. What is the surface area of the cylinder if it has a top and a base?...Sat, 05 Sep 2026 17:31:05 -0400https://pop.com.sg/geometry-find-angle-x-in-triangle.htmlhttps://pop.com.sg/geometry-find-angle-x-in-triangle.htmlI have not been able to find a euclidean geometry solution to this, but any other solutions are also appreciated. Let ABC be a triangle with AB=CD and angles as marked in the diagram. Find the mea......Sat, 05 Sep 2026 17:27:06 -0400https://pop.com.sg/what-books-must-every-math-undergraduate-read.htmlhttps://pop.com.sg/what-books-must-every-math-undergraduate-read.htmlI'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working......Sat, 05 Sep 2026 17:19:16 -0400https://pop.com.sg/when-to-connect-points-on-a-graph.htmlhttps://pop.com.sg/when-to-connect-points-on-a-graph.htmlSay I have a graph that has a set of points on it. The points represent some data. When should I connect the points with a straight line, when should I connect the points with a curved approximat......Sat, 05 Sep 2026 16:57:55 -0400https://pop.com.sg/filling-in-the-blanks-on-a-magic-triangle.htmlhttps://pop.com.sg/filling-in-the-blanks-on-a-magic-triangle.htmlI am struggling with this problem. I have tried out a few numbers with no luck, and I have a feeling there is a slick method to this that I am missing, rather than just guesswork. Any hints, would be ...Sat, 05 Sep 2026 16:54:51 -0400https://pop.com.sg/what-does-it-mean-when-a-function-is-finite.htmlhttps://pop.com.sg/what-does-it-mean-when-a-function-is-finite.htmlWhen someone says a real valued function $f(x)$ on $\mathbb{R}$ is finite, does it mean that $|f(x)| \leq M$ for all $x \in \mathbb{R}$ with some $M$ independent of $x$?...Sat, 05 Sep 2026 16:54:50 -0400https://pop.com.sg/brezis-functional-analysis-exercise-2-4.htmlhttps://pop.com.sg/brezis-functional-analysis-exercise-2-4.htmlLet $\alpha: E \times F \to \mathbb{R}$ ($E,F$ banach spaces) be a linear functional with the following property: For fixed $x \in E$, $\alpha$ is a continuous map. For fixed $y \in F$, $\alpha$ is a ...Sat, 05 Sep 2026 16:45:32 -0400https://pop.com.sg/are-all-polynomial-expressions-an-algebraic-expression.htmlhttps://pop.com.sg/are-all-polynomial-expressions-an-algebraic-expression.htmlI mean to ask that, real numbers can be written in the form of constant polynomials like, 4x^0, 2x^0 etc. If all polynomial expressions are algebraic expressions then 4x^0 = 4x^0 should be a ...Sat, 05 Sep 2026 16:40:31 -0400https://pop.com.sg/how-to-find-all-primitive-triples-a-b-c-pythagorean-triples.htmlhttps://pop.com.sg/how-to-find-all-primitive-triples-a-b-c-pythagorean-triples.htmlhow to find all primitive triples when one value of $(a,b,c)$ is given? For example in this case $a = 45$. What is the procedure to find the primitive triples ? Conditions for primitive triples ......Sat, 05 Sep 2026 16:34:48 -0400https://pop.com.sg/solving-system-of-equations-with-complex-numbers.htmlhttps://pop.com.sg/solving-system-of-equations-with-complex-numbers.htmlEquation 1$$ \frac{V_{1}}{5} + \frac{V_{1}-V_{2}}{10+j6} - 10\angle45^\circ = 0 $$ Equation 2 $$ -4V_{1} + \frac{V_{2}-V_{1}}{10+j6} + \frac{V_{3}}{-j2} + \frac{V_{3}}{8+j7} = 0 $$ Equation 3 ......Sat, 05 Sep 2026 16:16:32 -0400https://pop.com.sg/critiques-on-proof-showing-sqrt-12-is-irrational.htmlhttps://pop.com.sg/critiques-on-proof-showing-sqrt-12-is-irrational.htmlMy only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ $$\Big(\f......Sat, 05 Sep 2026 16:14:14 -0400https://pop.com.sg/find-the-length-of-de.htmlhttps://pop.com.sg/find-the-length-of-de.htmlLet $ABC$ be a triangle with $AB<AC$. Let $D$ be on the side $AC$ such that $AD=AB$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $ID$ ......Sat, 05 Sep 2026 16:12:26 -0400https://pop.com.sg/is-0-1-km-equals-to-100-0-m-in-context-to-significant-figures.htmlhttps://pop.com.sg/is-0-1-km-equals-to-100-0-m-in-context-to-significant-figures.htmlThe number of significant digits in 0.1km is 1 and we know that 1 km = 1000m. This then implies that the 0.1 km is equal to 100.0 m, but the rule says the change in units doesn't affect the number of ...Sat, 05 Sep 2026 16:08:09 -0400https://pop.com.sg/computing-limit-of-1-1-n-n-2.htmlhttps://pop.com.sg/computing-limit-of-1-1-n-n-2.htmlHow can I compute the limit $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n^2}$? Of course $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n} = e$, and then $\left(1+\frac{1}{n}\right......Sat, 05 Sep 2026 15:59:31 -0400https://pop.com.sg/how-to-prove-that-frac-ln-5-ln-2-is-irrational.htmlhttps://pop.com.sg/how-to-prove-that-frac-ln-5-ln-2-is-irrational.htmlI have to show that $\frac{\ln 5}{\ln 2}$ is irrational I have tried the following: Assume it's rational so $\frac{\ln 5}{\ln 2} = \frac{p}{q}$ which becomes $\log _2\left(5\right)=\frac{p}{q}$ The......Sat, 05 Sep 2026 15:58:06 -0400https://pop.com.sg/help-required-in-finding-determinant-using-characteristic-equation.htmlhttps://pop.com.sg/help-required-in-finding-determinant-using-characteristic-equation.htmlA problem in my textbook proceeds as follows: Find $\det(A)$ given that $A$ has $p(\lambda)$ as its characteristic polynomial a) $p(\lambda)=\lambda^3-2\lambda^2+\lambda+5$ b) $p(\lambda)=\lamb......Sat, 05 Sep 2026 15:56:44 -0400https://pop.com.sg/sketch-the-graph-x-e-t-sin-t-t-ge-0.htmlhttps://pop.com.sg/sketch-the-graph-x-e-t-sin-t-t-ge-0.htmlMy graph is always negative though, and that doesn't make sense cause $t$ is supposed to be positive. I substituted $x$ as $y$ and $t$ as $x$....Sat, 05 Sep 2026 15:52:25 -0400https://pop.com.sg/sketch-the-phase-line-of-ode-frac-dx-dt-f-t-x.htmlhttps://pop.com.sg/sketch-the-phase-line-of-ode-frac-dx-dt-f-t-x.htmlI was asked to sketch the phase line of scalar ODE, and using the phase lines, determine whether all equilibrium point(s) are sinks, sources, or nodes. Here is the scalar ODE: $$\frac{dx}{dt}=f(t,x......Sat, 05 Sep 2026 15:41:54 -0400https://pop.com.sg/finding-the-length-of-a-helix.htmlhttps://pop.com.sg/finding-the-length-of-a-helix.htmlDoes anyone have any clue on how to start this question off? The highly scary helter skelter at the fair is a cylindrical tower of height 30 metres and circumference 8 metres. The slide is wound ...Sat, 05 Sep 2026 15:38:57 -0400https://pop.com.sg/three-altitudes-of-a-triangle-are-concurrent.htmlhttps://pop.com.sg/three-altitudes-of-a-triangle-are-concurrent.htmlI have been told that this well known fact can be shown using only Euclid's propositions from books one to three, and cyclic quadrilaterals. I can't figure out how to start, which quadrilateral sh......Sat, 05 Sep 2026 15:33:43 -0400https://pop.com.sg/are-there-infinite-many-primes-p-such-that-2p-1-is-also-prime.htmlhttps://pop.com.sg/are-there-infinite-many-primes-p-such-that-2p-1-is-also-prime.htmlI did a search online and found a similar notion called Sophie Germain prime, which by definition is a prime $p$ such that $2p+1$ is also prime. Sophie Germain primes are conjectured to be of infin......Sat, 05 Sep 2026 15:29:51 -0400https://pop.com.sg/how-many-degrees-of-freedom-does-a-line-have-on-a-plane.htmlhttps://pop.com.sg/how-many-degrees-of-freedom-does-a-line-have-on-a-plane.htmlUsing $y=ax+b$ I can get a line for every point $(a,b)$ but there are still some lines left $(x=c)$. So it seems that lines are more than points so they have more degrees of freedom than $2$. How m......Sat, 05 Sep 2026 15:27:54 -0400https://pop.com.sg/integral-of-product-of-two-functions-in-terms-of-the-integral-of-the-o.htmlhttps://pop.com.sg/integral-of-product-of-two-functions-in-terms-of-the-integral-of-the-o.htmlProblem: Let $f$ and $g$ be two continuous functions on $[ a,b ]$ and assume $g$ is positive. Prove that $$\int_{a}^{b}f(x)g(x)dx=f(\xi )\int_{a}^{b}g(x)dx$$ for some $\xi$ in $[ a,b ]$. Here i......Sat, 05 Sep 2026 15:14:39 -0400https://pop.com.sg/proof-that-the-sum-of-two-gaussian-variables-is-another-gaussian.htmlhttps://pop.com.sg/proof-that-the-sum-of-two-gaussian-variables-is-another-gaussian.htmlThe sum of two Gaussian variables is another Gaussian. It seems natural, but I could not find a proof using Google. What's a short way to prove this? Thanks! Edit: Provided the two variables are ...Sat, 05 Sep 2026 14:52:51 -0400https://pop.com.sg/how-many-4-digit-even-numbers-have-all-4-digits-distinct.htmlhttps://pop.com.sg/how-many-4-digit-even-numbers-have-all-4-digits-distinct.htmlHow many $4$ digit even numbers have all $4$ digits distinct ? This question has been asked if we replace even with positive. I was solving this Problem like: For first position $1-9$ (No $0$) ......Sat, 05 Sep 2026 14:45:33 -0400https://pop.com.sg/how-to-increase-mean-of-set-of-numbers.htmlhttps://pop.com.sg/how-to-increase-mean-of-set-of-numbers.htmlLets say: $X = \{x_1, x_2, x_3, ... \} $ be a set of Real numbers in range $(R_1, R_2)$ and $m =$ mean of $x$ If I have to increase mean of set $X$ by $3$, each number in the set has to be increa......Sat, 05 Sep 2026 14:41:51 -0400https://pop.com.sg/why-do-we-study-polish-spaces.htmlhttps://pop.com.sg/why-do-we-study-polish-spaces.htmlIn descriptive set theory, a lot of space is devoted to properties of Polish spaces. (A Polish space is a topological space, which is separable and completely metrizable.) I would like to know why......Sat, 05 Sep 2026 14:40:57 -0400https://pop.com.sg/tensor-calculus.htmlhttps://pop.com.sg/tensor-calculus.htmlI am currently a 3rd year undergraduate electronic engineering student. I have completed a course in dynamics, calculus I, calculus II and calculus III. I've started self studying tensor calculus,......Sat, 05 Sep 2026 14:40:54 -0400https://pop.com.sg/when-an-expression-has-double-brackets-in-it-x-what-type-of-expression.htmlhttps://pop.com.sg/when-an-expression-has-double-brackets-in-it-x-what-type-of-expression.htmlI am curious to know what type but I know that one is greatest integer parent function. There is another one that explains it and I want to know what another function represents that?...Sat, 05 Sep 2026 14:40:09 -0400https://pop.com.sg/why-aren-039-t-repeating-decimals-irrational-but-something-like-pi-is.htmlhttps://pop.com.sg/why-aren-039-t-repeating-decimals-irrational-but-something-like-pi-is.htmlWe use closest representations for both of them, but they are not completely true. $\frac{22}7$ and $3.14$ are not exactly $\pi$ but we use them as the best option available. $\frac13$ is $0.\bar......Sat, 05 Sep 2026 14:38:34 -0400https://pop.com.sg/int-e-mean-what-does-int-mean.htmlhttps://pop.com.sg/int-e-mean-what-does-int-mean.htmlIn real analysis what is Int symbol mean? like Int(E) , int (A U B) ?? I want to know what is Int mean in real analysis a few example will be also good thank you...Sat, 05 Sep 2026 14:36:31 -0400https://pop.com.sg/arbitrary-matrices-solving.htmlhttps://pop.com.sg/arbitrary-matrices-solving.htmlI'm trying to solve some equations with arbitrary matrices. My Problem is that I don't know what way to solve the equation should be taken. Examples: Solve for $X$. $X,A$ are arbitrary matrices. ......Sat, 05 Sep 2026 14:26:19 -0400https://pop.com.sg/find-the-derivative-at-the-point-1-ln-pi-4-of-x-arctan-x-e-y.htmlhttps://pop.com.sg/find-the-derivative-at-the-point-1-ln-pi-4-of-x-arctan-x-e-y.htmlFind the derivative at the point $(1, \ln(\pi/4)$ of $x \arctan x = e^y$. So I think this may be an implicit differentiation problem. I'm not exactly sure where to start. I tried differentiating......Sat, 05 Sep 2026 14:25:26 -0400https://pop.com.sg/uniqueness-proof-discrete-math-help.htmlhttps://pop.com.sg/uniqueness-proof-discrete-math-help.htmlWhile reading I came across Uniqueness Proofs. Where a theorem asserts the existence of a unique element with a particular property. In order to prove this two steps are needed, Prove existence and......Sat, 05 Sep 2026 14:24:05 -0400https://pop.com.sg/proving-the-volume-of-a-cone.htmlhttps://pop.com.sg/proving-the-volume-of-a-cone.htmlAbstract definition of a cone: locus of points of line segments join a single point $v$, called vertex with set of coplanar points called base. Height of cone is distance from vertex to plane. If I ...Sat, 05 Sep 2026 14:20:19 -0400https://pop.com.sg/calculating-the-number-of-subintervals-required-for-the-difference-bet.htmlhttps://pop.com.sg/calculating-the-number-of-subintervals-required-for-the-difference-bet.htmlI'm having trouble with calculating the minimum number of subintervals required for the difference between the upper and lower Riemann sums to be a particular value. So say I have the following ...Sat, 05 Sep 2026 14:12:26 -0400https://pop.com.sg/derivation-of-cubic-formula-closed.htmlhttps://pop.com.sg/derivation-of-cubic-formula-closed.htmlHow do we derive the so called cubic formula without using Cardano's method or substitution? I would like to see a step by step proof of where wolfram alpha derives this answer. And also explain wh......Sat, 05 Sep 2026 14:09:20 -0400https://pop.com.sg/topological-invariants.htmlhttps://pop.com.sg/topological-invariants.htmlDo continuous maps necessarily preserve topological invariants? Or is it necessary for the maps to be homeomorphisms? Are there simple examples where continuous maps do not preserve these invariants?...Sat, 05 Sep 2026 14:05:56 -0400https://pop.com.sg/what-is-the-integral-of-function-f-x-sin-x-x.htmlhttps://pop.com.sg/what-is-the-integral-of-function-f-x-sin-x-x.htmlI want to know if the function sinx/x is integrable and if it is, then what's its integral? My high school book says its a non-integrable function while WolframAlpha says its integral is Si(x) + ...Sat, 05 Sep 2026 14:03:47 -0400https://pop.com.sg/question-from-putnam-039-89-primes-of-the-form-101-ldots01.htmlhttps://pop.com.sg/question-from-putnam-039-89-primes-of-the-form-101-ldots01.htmlI'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general in......Sat, 05 Sep 2026 13:59:16 -0400https://pop.com.sg/math-conversion-question-1km-0-6214mi-and-1gal-3-78l.htmlhttps://pop.com.sg/math-conversion-question-1km-0-6214mi-and-1gal-3-78l.htmlIn Europe, gasoline efficiency is measured in km/L. If your car's gas mileage is 35.0mi/gal , how many liters of gasoline would you need to buy to complete a 142-km trip in Europe? Use the following ...Sat, 05 Sep 2026 13:47:59 -0400https://pop.com.sg/finding-the-parallel-sides-of-a-trapezoid-given-all-side-lengths-and-h.htmlhttps://pop.com.sg/finding-the-parallel-sides-of-a-trapezoid-given-all-side-lengths-and-h.htmlSuppose that we are given side lengths $a, b, c, d$ of a trapezoid. We know that two of them are parallel, and all values are different. Moreover, we are given the height $h$ from the base (dista......Sat, 05 Sep 2026 13:38:22 -0400https://pop.com.sg/how-can-num2str-function-cut-off-some-digits-in-matlab.htmlhttps://pop.com.sg/how-can-num2str-function-cut-off-some-digits-in-matlab.htmlI am reading the book " MATLAB: A practical introductoin programming and problem solving", written by Stormy Attaway and getting curious with the function "num2str". On page 256, the author used the ...Sat, 05 Sep 2026 13:29:53 -0400https://pop.com.sg/obtain-equation-of-right-circular-cylinder.htmlhttps://pop.com.sg/obtain-equation-of-right-circular-cylinder.htmlObtain equation of right circular cylinder on the circle through points $(a,0,0),(0,b,0)$ and $(0,0,c)$ as the guiding curve. I proceeded by considering this circle as intersection of sphere from the ...Sat, 05 Sep 2026 13:23:15 -0400https://pop.com.sg/find-the-area-of-a-triangle-given-the-coordinates-of-its-vertices.htmlhttps://pop.com.sg/find-the-area-of-a-triangle-given-the-coordinates-of-its-vertices.htmlThe following is a GRE practice test question which I don't understand. According to this general argument, it appears that any arbitrary point $(x, 7)$ would have "height" equal to 10 but then if......Sat, 05 Sep 2026 13:11:03 -0400https://pop.com.sg/conic-sections-parabola-what-is-p.htmlhttps://pop.com.sg/conic-sections-parabola-what-is-p.htmlHelp my teacher says $p$ can't be negative because it's distance. I watched TOCT's tutorial (The Organic Chemistry Tutor) in YouTube about parabola and he said "$(x-h)^2 = 4p(y-k)$ if $p$ is posi......Sat, 05 Sep 2026 12:50:59 -0400https://pop.com.sg/matrix-addition-multiplication-with-different-sizes.htmlhttps://pop.com.sg/matrix-addition-multiplication-with-different-sizes.htmlI have the following two matrices: $$A=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ and }B=\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$ So I have two matrixes with ...Sat, 05 Sep 2026 12:47:26 -0400https://pop.com.sg/what-is-the-exact-definition-of-an-inverse-function.htmlhttps://pop.com.sg/what-is-the-exact-definition-of-an-inverse-function.htmlI recently started reading a book which has challenged my understanding of functions and inverse functions quite a bit (mostly the format of the notations that are used); I know that we write a fun......Sat, 05 Sep 2026 12:47:09 -0400https://pop.com.sg/trigonometric-quadratic-formula-and-other-trig-solutions-for-roots-of-.htmlhttps://pop.com.sg/trigonometric-quadratic-formula-and-other-trig-solutions-for-roots-of-.htmlI was experimenting with trig functions and their connections to roots of polynomials when I derived a sort of quadratic formula: $$ax^2+bx+c=0\implies x=n\cos\left(\frac12\arccos\left(\frac{b^2......Sat, 05 Sep 2026 12:41:47 -0400https://pop.com.sg/is-there-a-power-series-with-0-radius-of-convergence.htmlhttps://pop.com.sg/is-there-a-power-series-with-0-radius-of-convergence.htmlQuestion: Is there a power series with $0$ radius of convergence?(doesn't converge anywhere) I asked my teacher this question and he replied in negative, however he didn't mention any proof for......Sat, 05 Sep 2026 12:39:33 -0400https://pop.com.sg/order-of-integration-in-triple-integral.htmlhttps://pop.com.sg/order-of-integration-in-triple-integral.htmlIs there any hard and fast rule for what order you integrate for triple integrals. I know of Fubini's theorem but surely this doesn't cover all cases of triple integrals. Say for example I have, $$\...Sat, 05 Sep 2026 12:39:13 -0400https://pop.com.sg/explanation-of-line-element-formula-dl-2-dx-2-dy-2.htmlhttps://pop.com.sg/explanation-of-line-element-formula-dl-2-dx-2-dy-2.htmlI found this in a physics textbook without justification: $$dl^2 = dx^2 +dy^2,$$ where I presume that $l = \sqrt{x^2+y^2}$. Why is this so? By my calculations I obtain $$ dl = \dfrac{\partial l}{\...Sat, 05 Sep 2026 12:28:39 -0400https://pop.com.sg/grouping-numbers-that-are-close-to-each-other.htmlhttps://pop.com.sg/grouping-numbers-that-are-close-to-each-other.htmlLet's say you have a set of numbers S and you want to obtain subsets of $S$, $s_1,s_2,\ldots, s_i$, where $i < N$. Is there a particular operation that will group the numbers that are "close to......Sat, 05 Sep 2026 12:28:17 -0400https://pop.com.sg/using-only-the-field-axioms-of-real-numbers-prove-that-1-1-1.htmlhttps://pop.com.sg/using-only-the-field-axioms-of-real-numbers-prove-that-1-1-1.htmlUsing only the field axioms of real numbers prove that $(-1)(-1) = 1$ (1) I start with an obvious fact:$$0 = 0$$ (2) Add $(-1)$ to both sides of the equation: $$0 + (-1) = 0+ (-1)$$ (3) Zero is the ...Sat, 05 Sep 2026 12:25:08 -0400https://pop.com.sg/how-do-we-know-the-ratio-between-circumference-and-diameter-is-the-sam.htmlhttps://pop.com.sg/how-do-we-know-the-ratio-between-circumference-and-diameter-is-the-sam.htmlThe number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same......Sat, 05 Sep 2026 12:18:50 -0400https://pop.com.sg/notation-for-a-constant-vs-a-variable.htmlhttps://pop.com.sg/notation-for-a-constant-vs-a-variable.htmlI have a straight line $y=f(x)$, i.e. $$ y=kx+m $$ $k, m$ are constants. $x$ is a variable. But is it correct to write $$ k,m\in\mathbb R $$ and $$ x\in \mathbb R \qquad? $$ Is this notation corre......Sat, 05 Sep 2026 12:17:51 -0400