https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 26 Aug 2026 02:13:17 -0400https://pop.com.sg/what-is-the-difference-between-a-forest-and-a-spanning-forest.htmlhttps://pop.com.sg/what-is-the-difference-between-a-forest-and-a-spanning-forest.htmlIf a graph is labelled as a forest it does not contain any cycles, meaning it consists of all trees, which I realize can even be a single node (since that is technically a tree). If a graph is lab......Wed, 26 Aug 2026 06:07:33 -0400https://pop.com.sg/reduction-formula-for-int-cosh-n-x-dx.htmlhttps://pop.com.sg/reduction-formula-for-int-cosh-n-x-dx.htmlGiven $n\in\Bbb N$, $n > 2$, find the reduction formula for: $$ \int \cosh^n x dx $$ I've derived an answer, but the signs do not match, so my question would be: where is the mistake in my ...Wed, 26 Aug 2026 06:05:22 -0400https://pop.com.sg/integral-int-infty-infty-x-2-e-x-2-mathrm-dx.htmlhttps://pop.com.sg/integral-int-infty-infty-x-2-e-x-2-mathrm-dx.htmlI don't know how to evaluate it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates. $$\int_{-\infty}^\infty x^2 e^......Wed, 26 Aug 2026 06:01:41 -0400https://pop.com.sg/generalization-x-text-sup-q-in-mathbb-q-q-x.htmlhttps://pop.com.sg/generalization-x-text-sup-q-in-mathbb-q-q-x.htmlHow do I prove that $x=\text{sup}\{q\in \mathbb{Q}:q<x\}$? Provided that $x\in\mathbb{R}$......Wed, 26 Aug 2026 06:00:32 -0400https://pop.com.sg/random-math-questions-modular-arithmetic-notation.htmlhttps://pop.com.sg/random-math-questions-modular-arithmetic-notation.htmlI found this amazing wall clock picture on the internet but I really don't know a few things. I don't know what's $B'_L$, 3, why $2^{-1}\equiv 4[7]$ and the one with the black and white ci......Wed, 26 Aug 2026 06:00:08 -0400https://pop.com.sg/how-does-a-branch-cut-define-a-branch.htmlhttps://pop.com.sg/how-does-a-branch-cut-define-a-branch.htmlI am studying complex analysis and I have problem understanding the concept of branch cut. The lecturer draw this as some curve that starts from a point and goes on and on in some direction (for e......Wed, 26 Aug 2026 05:44:53 -0400https://pop.com.sg/what-is-a-convex-polyhedron.htmlhttps://pop.com.sg/what-is-a-convex-polyhedron.htmlI was doing a spot of light reading (crystallography), when the term "convex" polyhedron came up in a a section (very prominently) in conjunction with something else called the "Euler ...Wed, 26 Aug 2026 05:38:15 -0400https://pop.com.sg/is-a-hypersurface-really-defined-by-an-arbitrary-polynomial.htmlhttps://pop.com.sg/is-a-hypersurface-really-defined-by-an-arbitrary-polynomial.htmlIn An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...Wed, 26 Aug 2026 05:32:27 -0400https://pop.com.sg/why-is-euler-039-s-number-2-71828-and-not-anything-else-closed.htmlhttps://pop.com.sg/why-is-euler-039-s-number-2-71828-and-not-anything-else-closed.htmlWhy is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$? I know that $e$ is the base of natural logarithms. I know about areas on hyperbola xy=1 and I know its formula: $$e =\......Wed, 26 Aug 2026 05:23:36 -0400https://pop.com.sg/why-the-slope-of-a-line-is-change-in-y-divided-by-change-in-x-duplicat.htmlhttps://pop.com.sg/why-the-slope-of-a-line-is-change-in-y-divided-by-change-in-x-duplicat.htmlWhy the slope of a line isn't the change in x divided by the change in y?...Wed, 26 Aug 2026 05:17:42 -0400https://pop.com.sg/product-of-two-independent-poisson-variables.htmlhttps://pop.com.sg/product-of-two-independent-poisson-variables.htmlSuppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ \operatorname{Poisson}(\lambda_1)$ and $X_2 ∼ \operatorname{Poisson}(\lambda_2)$. What will be the probability ...Wed, 26 Aug 2026 05:14:53 -0400https://pop.com.sg/if-my-average-speed-is-50-mph-is-it-better-to-go-10-faster-or-5-mph-fa.htmlhttps://pop.com.sg/if-my-average-speed-is-50-mph-is-it-better-to-go-10-faster-or-5-mph-fa.htmlBACKGROUND Suppose we have a journey between $A$ and $B$ where different sections of roads have different speed limits and our average speed is $50$ mph. If when I repeat the journey I travel $10......Wed, 26 Aug 2026 05:10:26 -0400https://pop.com.sg/show-marsaglia-polar-method-produce-the-standard-normal-distribution.htmlhttps://pop.com.sg/show-marsaglia-polar-method-produce-the-standard-normal-distribution.htmlSuppose $X, Y$ are two independent variables that uniformly distributed on the interval $[-1,1]$. If $s = x^2 + y^2 < 1$, show the result of $\frac{x}{\sqrt{s}} \sqrt{-2\ln(s)}$ comes from a sta......Wed, 26 Aug 2026 05:03:07 -0400https://pop.com.sg/how-is-the-graph-coloring-problem-np-complete.htmlhttps://pop.com.sg/how-is-the-graph-coloring-problem-np-complete.htmlAn NP-Complete problem can be checked efficiently, but has no known way of being solved in polynomial time. Then, how is the graph coloring problem (http://en.wikipedia.org/wiki/Graph_coloring_pro......Wed, 26 Aug 2026 05:01:31 -0400https://pop.com.sg/show-that-the-perimeter-of-an-octagon-is-8-sqrt-2-1.htmlhttps://pop.com.sg/show-that-the-perimeter-of-an-octagon-is-8-sqrt-2-1.htmlMy question is as follows: The top of a table is made in the shape of a regular octagon by cutting the congruent isosceles triangles from the corners of a $1$ m square piece of wood. Show that the ...Wed, 26 Aug 2026 04:47:44 -0400https://pop.com.sg/finding-the-limit-when-denominator-0.htmlhttps://pop.com.sg/finding-the-limit-when-denominator-0.htmlI'm having trouble solving this limit: $$\lim_{x \to -2^-} \frac{1}{(x + 2)^2}$$ I can't find a way to rationalize the denominator. Also, is there a way to do it without plugging in -2.001 and st......Wed, 26 Aug 2026 04:47:01 -0400https://pop.com.sg/prove-that-if-a-and-b-are-mutually-exclusive-then-p-a-leq-p-b-c.htmlhttps://pop.com.sg/prove-that-if-a-and-b-are-mutually-exclusive-then-p-a-leq-p-b-c.htmlSuppose $\textbf{P}(A\cap B) = 0$. Prove $\textbf{P}(A) \leq \textbf{P}(B^{c})$ using the axioms of probability. I know the axioms are: $\textbf{P}(A)\geq 0$. $\textbf{P}(\Omega) = 1$. If A and......Wed, 26 Aug 2026 04:41:00 -0400https://pop.com.sg/find-the-composition-of-a-piecewise-function.htmlhttps://pop.com.sg/find-the-composition-of-a-piecewise-function.htmlLet $$f(x)= \begin{cases} 5 &\text{, x $\geq$ 0}\\ x^2 &\text{, x < 0} \end{cases}$$ and $$g(x)= \begin{cases} \sqrt{x} &\text{, x > 3}\\ 4 &\text{, x $<$ 3} \end{cases}$$ ......Wed, 26 Aug 2026 04:37:40 -0400https://pop.com.sg/show-that-the-dimention-of-the-intersection-of-projective-linear-sub-s.htmlhttps://pop.com.sg/show-that-the-dimention-of-the-intersection-of-projective-linear-sub-s.htmlProposition: Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap M)\geq d_1+d_2-n$ (we consider the dimenti......Wed, 26 Aug 2026 04:36:48 -0400https://pop.com.sg/difference-between-imaginary-part-and-imaginary-number.htmlhttps://pop.com.sg/difference-between-imaginary-part-and-imaginary-number.htmlIs there a difference between imaginary part and imaginary number? Because our teacher says there is. I mean in $5+3i$ , is it right to say the imaginary part is $3i$ and imaginary number is $3$?......Wed, 26 Aug 2026 04:34:58 -0400https://pop.com.sg/find-a-circle-perpendicular-to-two-other-circles.htmlhttps://pop.com.sg/find-a-circle-perpendicular-to-two-other-circles.htmlHere are two circles $C_1 = [ x^2 + y^2 = 1]$ and $C_2 = [ (x-2)^2 + y^2 = 3 ]$. The radii are $1$ and $3$ and the two circles are orthogonal to each other $C_1 \cdot C_2 = 0$. What is the equati......Wed, 26 Aug 2026 04:34:09 -0400https://pop.com.sg/finding-the-local-trivilization-of-canonical-line-bundle.htmlhttps://pop.com.sg/finding-the-local-trivilization-of-canonical-line-bundle.htmlI am going through Nakahara's "Geometry, topology and physics" by myself and I got stuck on a calculation. I have the total space $$L=\{(p,v)\in\mathbb{C}P^n\times\mathbb{C}^{n+1}|v=ap,a\in\math......Wed, 26 Aug 2026 04:32:13 -0400https://pop.com.sg/express-cos-tan-1-x-in-terms-of-x.htmlhttps://pop.com.sg/express-cos-tan-1-x-in-terms-of-x.htmlExpress $\cos(\tan^{-1}(x))$ in terms of $x$. My Attempt: Let $\tan^{-1}(x)=A$. $$x=\tan (A)$$ Then, $$\cos(\tan^{-1}(x))=\cos (A)$$...Wed, 26 Aug 2026 04:30:10 -0400https://pop.com.sg/composition-of-function-with-itself.htmlhttps://pop.com.sg/composition-of-function-with-itself.htmlFor the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider the following statements: $A$ : For ...Wed, 26 Aug 2026 04:18:25 -0400https://pop.com.sg/boolean-algebra-sum-of-products-form.htmlhttps://pop.com.sg/boolean-algebra-sum-of-products-form.htmlI have a logic circuit where the output can be represented with the following boolean expression (1)$\overline {xy}$ + x $\bar y z$ + $\overline {\bar x + z} $ + y Using truth tables I found the ...Wed, 26 Aug 2026 04:05:49 -0400https://pop.com.sg/simultaneous-diagonalization-of-commuting-linear-transformations.htmlhttps://pop.com.sg/simultaneous-diagonalization-of-commuting-linear-transformations.htmlLet $V$ be a vector space of finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that if $TS=ST$ every eigenspace $V_\lambda$ of $S$ is $T$-inv......Wed, 26 Aug 2026 04:04:31 -0400https://pop.com.sg/derived-categories-of-filtered-modules.htmlhttps://pop.com.sg/derived-categories-of-filtered-modules.htmlFor a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which decl......Wed, 26 Aug 2026 04:02:36 -0400https://pop.com.sg/sum-of-sets-of-measure-zero.htmlhttps://pop.com.sg/sum-of-sets-of-measure-zero.htmlLet $A$ and $B$ be two subsets of $\Bbb R$ of measure zero. Is it true that the Minkowski sum $A+B = \{ a + b \mid a \in A, b \in B \}$ has measure zero as well? I think so but I can't prove it. The ...Wed, 26 Aug 2026 03:59:58 -0400https://pop.com.sg/how-to-evaluate-the-integral-int-e-2x-sin-2x-dx-closed.htmlhttps://pop.com.sg/how-to-evaluate-the-integral-int-e-2x-sin-2x-dx-closed.htmlHow can I evaluate the following integral? $$\int e^{2x} \sin^2x\ dx$$...Wed, 26 Aug 2026 03:58:37 -0400https://pop.com.sg/universal-hashing-family.htmlhttps://pop.com.sg/universal-hashing-family.htmlI'm not really sure this question is proper for MathExchange but I have found it more suitable than StackOverFlow. So, I'm taking the course of Data Structures and Algorithms and we got some examp......Wed, 26 Aug 2026 03:53:12 -0400https://pop.com.sg/proof-of-intersection-and-union-of-set-a-with-empty-set.htmlhttps://pop.com.sg/proof-of-intersection-and-union-of-set-a-with-empty-set.htmlI need to prove the following: Prove that $A\cup \!\, \varnothing \!\,=A$ and $A\cap \!\, \varnothing \!\,=\varnothing \!\,$ It's my understanding that to prove equality, I must prove that both are ...Wed, 26 Aug 2026 03:45:43 -0400https://pop.com.sg/calculating-the-square-root-of-2.htmlhttps://pop.com.sg/calculating-the-square-root-of-2.htmlSince $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so th......Wed, 26 Aug 2026 03:32:14 -0400https://pop.com.sg/convergent-or-divergent-integral.htmlhttps://pop.com.sg/convergent-or-divergent-integral.htmlConvergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do......Wed, 26 Aug 2026 03:29:23 -0400https://pop.com.sg/differentiation-using-d-or-delta.htmlhttps://pop.com.sg/differentiation-using-d-or-delta.htmlAre the symbols $d$ and $\delta$ equivalent in expressions like $dy/dx$? Or do they mean something different? Thanks...Wed, 26 Aug 2026 03:27:24 -0400https://pop.com.sg/limacon-cardioid-orientation-confusion.htmlhttps://pop.com.sg/limacon-cardioid-orientation-confusion.htmlI'm learning about polar equations and am trying to graph the curve r = 3 + 3 sin(theta), with theta between 0 and 2pi. Based on what I've learned and the image below, since a = b = 3, it would see......Wed, 26 Aug 2026 03:26:52 -0400https://pop.com.sg/can-you-predict-the-remainder-when-111-1-100-ones-is-divided-by-111111.htmlhttps://pop.com.sg/can-you-predict-the-remainder-when-111-1-100-ones-is-divided-by-111111.htmlWe have 111...11 a hundred times. We begin with 1 in our quotient. We have 93 ones remaining. We add six zeros according to the division algorithm to get 1000000 in our quotient. We add another on......Wed, 26 Aug 2026 03:20:48 -0400https://pop.com.sg/limits-at-infinity-involving-e.htmlhttps://pop.com.sg/limits-at-infinity-involving-e.htmlI am working with vector functions and one of the problems has $e$ in it. I always thought that $$\lim_{x\to\infty}e^{ax} = \infty$$ but the book says otherwise and when I went all the way back to ...Wed, 26 Aug 2026 03:12:20 -0400https://pop.com.sg/for-u-substitution-of-a-definite-integral-is-it-acceptable-to-have-the.htmlhttps://pop.com.sg/for-u-substitution-of-a-definite-integral-is-it-acceptable-to-have-the.htmlFor example, I have the problem: $$ \int_{-1}^1x^4\left(1-x^2\right)^2dx $$ I set $\,u=1-x^2\,$, which gives $\,x^2=1-u\,$, $\,x=\pm\sqrt{1-u}\,$ and $\,-\frac{1}{2}du=xdx$. For the upper and lower ...Wed, 26 Aug 2026 03:05:14 -0400https://pop.com.sg/inverse-of-random-variable.htmlhttps://pop.com.sg/inverse-of-random-variable.htmlLet's say we have a random variable $X$ of which the distribution is unknown. Now there are these general rules like $E[X + Y] = E[X] + E[Y]$ etc. But what if we would define $ \quad Y = \dfrac{1}......Wed, 26 Aug 2026 03:04:21 -0400https://pop.com.sg/stardew-valley-farm-recreational-math-problem.htmlhttps://pop.com.sg/stardew-valley-farm-recreational-math-problem.htmlIn the game Stardew Valley, farming is essential. In order to plant seeds, one must till the soil using a hoe. A golden hoe is an upgraded hoe that allows the player to till 1, 3, or 5 squares of d......Wed, 26 Aug 2026 02:57:03 -0400https://pop.com.sg/what-is-the-smallest-20-digit-number-with-at-least-13-distinct-prime-f.htmlhttps://pop.com.sg/what-is-the-smallest-20-digit-number-with-at-least-13-distinct-prime-f.htmlThe number $10^{19}+1035830$ is a $20$-digit number with $12$ distinct prime factors. I am not sure whether it is the smallest example because to save time I only considered numbers with at least $3$ ...Wed, 26 Aug 2026 02:52:05 -0400https://pop.com.sg/why-shift-the-result-of-subtracting-vectors.htmlhttps://pop.com.sg/why-shift-the-result-of-subtracting-vectors.htmlWhy do we shift the resulting vector after subtracting any two vectors a and b? I learned that when you add any two vectors a and b, the sum vector looks like the one that can be seen in the picture ...Wed, 26 Aug 2026 02:42:12 -0400https://pop.com.sg/newton-method-to-solve-nonlinear-system.htmlhttps://pop.com.sg/newton-method-to-solve-nonlinear-system.htmlHow can I solve this problem: ${e^x}^y +{x}^2+y-1.2=0 \\ {x}^2 +{y}^2+x-0.55=0$ for various loaded inputs with start points $(x_0 , y_0) = 0.6, 0.5$ Could anybody help me. Thank you in advance...Wed, 26 Aug 2026 02:40:34 -0400https://pop.com.sg/what-is-orthogonal-transformation.htmlhttps://pop.com.sg/what-is-orthogonal-transformation.htmlWhen $A^{T}A = AA^{T} = I$, I am told it is an orthogonal transformation $A$. But don't really get what it means. Hope to hear some explanations. $\begin{bmatrix}cos\theta & sin\theta \\ -sin\......Wed, 26 Aug 2026 02:28:19 -0400https://pop.com.sg/n-th-derivative-of-e-x-sin-x.htmlhttps://pop.com.sg/n-th-derivative-of-e-x-sin-x.htmlCan someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - e^{-......Wed, 26 Aug 2026 02:27:19 -0400https://pop.com.sg/inverse-of-percentage.htmlhttps://pop.com.sg/inverse-of-percentage.htmlWhat is the best way to cacluate the inverse of a percentage? For example: 100 * .85 = 85 85 * x = 100 x = 100 / 85 x = 1.17647 Also, 279 * .85 = 237.15 237.15 * x = 279 x = 279 / 237.15 x = 1.......Wed, 26 Aug 2026 02:24:50 -0400https://pop.com.sg/what-is-the-difference-between-a-definition-and-a-theorem.htmlhttps://pop.com.sg/what-is-the-difference-between-a-definition-and-a-theorem.htmlThis may get into a discussion, but I have a homework problem and it tells me there is a difference between a definition and a theorem. I don't know how to differentiate the two in this question: ...Wed, 26 Aug 2026 02:24:19 -0400https://pop.com.sg/a-family-has-two-children-one-child-is-a-girl-what-is-the-probability-.htmlhttps://pop.com.sg/a-family-has-two-children-one-child-is-a-girl-what-is-the-probability-.htmlMy initial thought process: Sample space: GG, GB, BG, BB. I then crossed out BG because it's the same as GB because order doesn't matter here. And because we know one is a girl, there leaves two ...Wed, 26 Aug 2026 02:18:32 -0400https://pop.com.sg/uniqueness-of-solution-of-differential-equation.htmlhttps://pop.com.sg/uniqueness-of-solution-of-differential-equation.htmlConsider the differential equation $ y'(x)=f(x,y(x)) $ with initial condition $y(x_0)=y_0$. According to some theorem, if $f$ is continuous, then there exists at least one solution that satisfies the ...Wed, 26 Aug 2026 02:14:02 -0400https://pop.com.sg/find-the-area-of-the-region-that-is-enclosed-by-the-cardioid-r-2-2-sin.htmlhttps://pop.com.sg/find-the-area-of-the-region-that-is-enclosed-by-the-cardioid-r-2-2-sin.htmlWe just learned polar integration, so I know that's how we're supposed to do it. I have a problem though: I'm getting a negative answer. What I did: Using the graph, which is: I figured out that......Wed, 26 Aug 2026 02:10:17 -0400https://pop.com.sg/set-of-natural-numbers-subset-of-the-set-of-real-numbers-closed.htmlhttps://pop.com.sg/set-of-natural-numbers-subset-of-the-set-of-real-numbers-closed.htmlThe natural numbers are said to be a subset of the real numbers but how is this possible since in the set of natural numbers division is not allowed....Wed, 26 Aug 2026 01:52:49 -0400https://pop.com.sg/limit-of-product-of-sequences-is-the-product-of-the-limits-of-the-sequ.htmlhttps://pop.com.sg/limit-of-product-of-sequences-is-the-product-of-the-limits-of-the-sequ.htmlI want to prove that if: $$\lim_{n \to \infty}s_n = L_1, \lim_{n \to \infty}t_n = L_2$$ then $$\lim_{n \to \infty}(s_n t_n) = L_1L_2$$ where $s_n, t_n$ are complex sequences (I'm working through......Wed, 26 Aug 2026 01:49:31 -0400https://pop.com.sg/find-a-solution-for-23x-5-mod-60.htmlhttps://pop.com.sg/find-a-solution-for-23x-5-mod-60.htmlThe topic is euclidean algorithm and GCD. This is a simple question, but I'm just stuck finding an inverse for 23 in $Z_{60}$. I performed the algorithm and proved that 23 and 60 are coprime, so th......Wed, 26 Aug 2026 01:46:16 -0400https://pop.com.sg/does-s-spans-v-mean-every-v-in-v-is-a-finite-linear-combination-of-ele.htmlhttps://pop.com.sg/does-s-spans-v-mean-every-v-in-v-is-a-finite-linear-combination-of-ele.htmlDefine $\mathbb{R}^{\infty}:=\lbrace (a_1,a_2,a_3,\ldots) \mid a_i \in \mathbb{R} \rbrace$ and let $\phi_i((a_1,a_2,a_3,\ldots))=a_i$. Show that $S=\lbrace \phi_i \mid i \geq 1 \rbrace$ does not sp......Wed, 26 Aug 2026 01:24:55 -0400https://pop.com.sg/what-do-we-mean-by-039-defining-a-function-039.htmlhttps://pop.com.sg/what-do-we-mean-by-039-defining-a-function-039.htmlFirst I will start by quote from Wikipedia about function's defining methods : There are many other ways of defining functions. Examples include piecewise definitions, induction or recursion, ...Wed, 26 Aug 2026 01:16:44 -0400https://pop.com.sg/linearizing-equations-of-motion.htmlhttps://pop.com.sg/linearizing-equations-of-motion.htmlI have been looking at the operations of a quadrotor drone. I am reading the maths behind it and in one section it mentions: "These equations of motion are linearized with respect to an equili......Wed, 26 Aug 2026 01:04:31 -0400https://pop.com.sg/discriminant-of-a-higher-degree-polynomials.htmlhttps://pop.com.sg/discriminant-of-a-higher-degree-polynomials.htmlI searched a lot in Google and even tried myself to determine the nature of roots of 4th and 3rd degree standard equations.Is their any way to know nature of roots of an equation such as $(b^2-4ac)......Wed, 26 Aug 2026 01:01:33 -0400https://pop.com.sg/how-can-i-calculate-the-centroid-of-polygon.htmlhttps://pop.com.sg/how-can-i-calculate-the-centroid-of-polygon.htmlWhat is the way to calculate the centroid of polygon? I have a concave polygon of 16 points, and I want know the centroid of that. thanks...Wed, 26 Aug 2026 01:01:01 -0400https://pop.com.sg/what-are-the-conditions-of-co-planer-vectors.htmlhttps://pop.com.sg/what-are-the-conditions-of-co-planer-vectors.htmlUnder what condition three vectors A, B, C will be co-planer? I want to learn this rules and theorem....Wed, 26 Aug 2026 00:58:26 -0400https://pop.com.sg/can-someone-please-explain-to-me-why-cotangent-graphs-look-the-way-the.htmlhttps://pop.com.sg/can-someone-please-explain-to-me-why-cotangent-graphs-look-the-way-the.htmlCan someone please explain to me why cotangent graphs look the way they do? I want to know why they basically look like mirror reflected tangent graphs. I get that if $\tan\theta=y/x$, then $\cot\......Wed, 26 Aug 2026 00:57:39 -0400https://pop.com.sg/the-limit-of-a-fraction.htmlhttps://pop.com.sg/the-limit-of-a-fraction.htmlWhat is the value of $$\lim_{x\to\infty}\frac{\sqrt{x^4-1}}{2x^2+3x-1}$$ I tried to factorise the denominator or the numerator but it takes me nowhere. My teacher said it equals to $\frac{1}{2}$. B......Wed, 26 Aug 2026 00:51:20 -0400https://pop.com.sg/what-do-mathematicians-mean-when-they-say-form.htmlhttps://pop.com.sg/what-do-mathematicians-mean-when-they-say-form.htmlAs in differential form, modular form, quadratic form? I'm sorry if this is a really silly question....Wed, 26 Aug 2026 00:46:14 -0400https://pop.com.sg/how-to-express-this-truth-set.htmlhttps://pop.com.sg/how-to-express-this-truth-set.htmlHow would you express this truth set in mathematical terms? $4 < x^2 \le 9, X \in \mathbb{R}$...Wed, 26 Aug 2026 00:41:55 -0400https://pop.com.sg/is-sqrt-2-sqrt-3-rational-or-irrational-duplicate.htmlhttps://pop.com.sg/is-sqrt-2-sqrt-3-rational-or-irrational-duplicate.htmlI have proven that the sum of any two irrational numbers is not always irrational and that a rational plus an irrational is irrational, however I am not sure how to prove this specific case. I was ...Wed, 26 Aug 2026 00:33:03 -0400https://pop.com.sg/understanding-the-definition-of-nowhere-dense-sets-in-abbott-039-s-und.htmlhttps://pop.com.sg/understanding-the-definition-of-nowhere-dense-sets-in-abbott-039-s-und.htmlFirst of all, I am sorry for asking a question about understanding a definition in a book named Understanding Analysis. But it is my first time to encounter basic topology, so I hope you can excuse......Wed, 26 Aug 2026 00:31:25 -0400https://pop.com.sg/eigenvectors-and-eigenvalues-of-hessian-matrix.htmlhttps://pop.com.sg/eigenvectors-and-eigenvalues-of-hessian-matrix.htmlBecause the Hessian matrix is real and symmetric, we can decompose it into a set of real eigenvalues and an orthogonal basis of eigenvectors. The second derivative in a specific direction ...Wed, 26 Aug 2026 00:29:01 -0400https://pop.com.sg/does-the-1-360-of-a-circle-have-dimension.htmlhttps://pop.com.sg/does-the-1-360-of-a-circle-have-dimension.htmlA circle has area$= \pi r^2$ , so the $1/360$ has area of $\pi r^2/360$ therefore 1degree is equals to $\pi r^2/360$ ?...Wed, 26 Aug 2026 00:19:03 -0400https://pop.com.sg/ode-039-s-repeated-complex-roots.htmlhttps://pop.com.sg/ode-039-s-repeated-complex-roots.htmlI got a question about his ODE: $(D-3)^3(D^2+25)^2(D-2)y = 7x^2e^{3x} + x\sin (5x)+4x^3$. So, in the solutions of the homogeneous eq. appears a complex root ($r = \pm5i$) with multiplicity 2. What ......Wed, 26 Aug 2026 00:12:36 -0400https://pop.com.sg/why-does-2-2-equal-1-2-2.htmlhttps://pop.com.sg/why-does-2-2-equal-1-2-2.htmlMy intuitive way of thinking about it is that it is $2/2/2$ or $2/2^2$, So why then is it $1/2^2$? what is the flaw in my thinking?...Wed, 26 Aug 2026 00:09:20 -0400https://pop.com.sg/boolean-algebra-simplification-pos.htmlhttps://pop.com.sg/boolean-algebra-simplification-pos.htmlI'm given this expression $$ (x+y'+z')(x'+z') $$ the $'$ meaning not. I have to simplify this to 3 literals and show my answer as a product of sums. Every calculator I check says the answer is $(x......Wed, 26 Aug 2026 00:01:44 -0400https://pop.com.sg/finding-the-catenary-curve-with-given-arclength-through-two-given-poin.htmlhttps://pop.com.sg/finding-the-catenary-curve-with-given-arclength-through-two-given-poin.htmlI need to find a general way of expressing a catenary, and I couldn't find anything online. Is it possible to explicitly find an explicit equation of a catenary which goes through $P_1 = (x_1,y_1)......Tue, 25 Aug 2026 23:51:05 -0400https://pop.com.sg/what-is-the-probability-of-p-a-039-or-b-039.htmlhttps://pop.com.sg/what-is-the-probability-of-p-a-039-or-b-039.htmlP(A) = 0.7, P(B) = 0.2 assuming independent events what is the probability of not A or not B? I thought I could substitute P(A') = 1-P(A) and P(B') = 1-P(B) so: P(A' or B') = 1-P(A) + 1-P(B) = 0.3 ......Tue, 25 Aug 2026 23:50:52 -0400https://pop.com.sg/does-the-converse-of-angle-bisector-theorem-true.htmlhttps://pop.com.sg/does-the-converse-of-angle-bisector-theorem-true.htmlMy question is clearly stated in the title. By that theorem, I am not referring to the one saying “points on the angle bisector of an angle are equidistant from the sides”. It is the one that divi......Tue, 25 Aug 2026 23:49:14 -0400https://pop.com.sg/making-sense-of-the-definition-of-polyhedron.htmlhttps://pop.com.sg/making-sense-of-the-definition-of-polyhedron.htmlA polyhedron is defined as the solution set of a finite number of linear equalities and inequalities: $$P = \{x | a^T_j x ≤ b_j , j = 1, . . . , m, c^T_j x = d_j , j = 1, . . . , p\}$$ A polyhedro......Tue, 25 Aug 2026 23:39:09 -0400https://pop.com.sg/find-the-number-of-permutations-with-the-given-property.htmlhttps://pop.com.sg/find-the-number-of-permutations-with-the-given-property.htmlI want to solve the problem If the problem were to find the number of elements conjugate to $\tau$, I had to look for the permutations with the same structure. But, here the question is different.......Tue, 25 Aug 2026 23:35:13 -0400https://pop.com.sg/can-critical-points-occur-at-endpoints-e-g-f-x-frac-1-x-at-the-interva.htmlhttps://pop.com.sg/can-critical-points-occur-at-endpoints-e-g-f-x-frac-1-x-at-the-interva.htmlGiven that an extreme value can only occur at a critical point and in the following case $$f(x) = \frac{1}{x} \quad [1,4]$$ we definitely have two absolute extreme values (maximum and minimum), are......Tue, 25 Aug 2026 23:32:08 -0400https://pop.com.sg/how-do-i-find-all-angles-of-4-sided-polygon-given-side-lengths.htmlhttps://pop.com.sg/how-do-i-find-all-angles-of-4-sided-polygon-given-side-lengths.htmlI have a program that lets users draw custom 4-sided shapes using java 2d. I want to calculate the angles inside the shapes so I can rotate text to the proper angle and label each side. I am tryin......Tue, 25 Aug 2026 23:29:38 -0400https://pop.com.sg/use-the-limit-definition-to-find-the-slope-of-the-tangent-line-at-a-po.htmlhttps://pop.com.sg/use-the-limit-definition-to-find-the-slope-of-the-tangent-line-at-a-po.htmlUse the limit definition to find the slope of the tangent line to the graph of $f(x)=\sqrt{8x+1}$ at the point $(6,7)$. I sort of get how to do this but the square root is making it difficult. I k......Tue, 25 Aug 2026 23:28:02 -0400https://pop.com.sg/prove-that-a-fraction-that-has-terminating-decimals-has-its-denominato.htmlhttps://pop.com.sg/prove-that-a-fraction-that-has-terminating-decimals-has-its-denominato.htmlthis is a number systems question and I am currently trying to prove this statement : "A fraction $\frac{a}{b}$ (with $a, b \in \mathbb{Z}$, where $b\neq0$) in lowest terms has a terminating decim......Tue, 25 Aug 2026 23:25:03 -0400https://pop.com.sg/why-is-integration-the-inverse-of-differentiation.htmlhttps://pop.com.sg/why-is-integration-the-inverse-of-differentiation.htmlWhy is integration the inverse of differentiation, I mean why do I get the same function when I integrate and then differentiate the result?...Tue, 25 Aug 2026 23:07:15 -0400https://pop.com.sg/find-a-polynomial-solution-for-the-legendre-equation.htmlhttps://pop.com.sg/find-a-polynomial-solution-for-the-legendre-equation.htmlThe Legendre equation $$(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0$$ has a polynomial solution $P_n$ when $\alpha$ is a non-negative number $n$. Find $P_1$ satisfying $P_1(1)=1$ and then find the gene......Tue, 25 Aug 2026 22:55:39 -0400https://pop.com.sg/how-to-prove-that-dfrac-d-dx-e-kx-ke-kx.htmlhttps://pop.com.sg/how-to-prove-that-dfrac-d-dx-e-kx-ke-kx.htmlSo how to prove that $\dfrac{d}{dx}e^{kx}=ke^{kx}$? What I did is to think about $e^{kx}$ not as a single function but as a composition of functions, that is $h(x)=e^{kx}=f[g(x)]=f[e^x]=e^{kx}$ w......Tue, 25 Aug 2026 22:45:51 -0400https://pop.com.sg/outer-product-of-two-matrices.htmlhttps://pop.com.sg/outer-product-of-two-matrices.htmlHow would I go about calculating the outer product of two matrices of 2 dimensions each? From what I can find, outer product seems to be the product of two vectors, $u$ and the transpose of another ...Tue, 25 Aug 2026 22:42:50 -0400https://pop.com.sg/definition-of-divergence.htmlhttps://pop.com.sg/definition-of-divergence.htmlWe see in certain places that $$\operatorname{div} \mathbf{F}|_p = \lim_{V \to \{p\}} \iint_{\partial V} \frac{\mathbf{F} \cdot \mathbf{\hat n}}{|V|} \: dS.$$ But what is $V$? Neighborhoods of $p$? ...Tue, 25 Aug 2026 22:27:28 -0400https://pop.com.sg/double-and-triple-integrals-does-the-order-of-integration-matter-what-.htmlhttps://pop.com.sg/double-and-triple-integrals-does-the-order-of-integration-matter-what-.htmlI have not found anything about it, that's why I am asking it here. As far as I know, the volume is: $$V=\iiint_E dxdydz$$ But does the order of dx, dy, dz matter? For example, if i can define the......Tue, 25 Aug 2026 22:14:17 -0400https://pop.com.sg/determining-the-truth-value-of-a-statement.htmlhttps://pop.com.sg/determining-the-truth-value-of-a-statement.htmlI am stuck with the following question, Determine the truth value of each of the following statments(a statement is a sentence that evaluates to either true or false but you can not be indec......Tue, 25 Aug 2026 22:13:48 -0400https://pop.com.sg/notation-of-the-summation-of-a-set-of-numbers.htmlhttps://pop.com.sg/notation-of-the-summation-of-a-set-of-numbers.htmlGiven a set of numbers $S=\{x_1,\dotsc,x_{|S|}\}$, where $|S|$ is the size of the set, what would be the appropriate notation for the sum of this set of numbers? Is it $$\sum_{x_i \in S} x_i \qquad\...Tue, 25 Aug 2026 21:58:45 -0400https://pop.com.sg/relative-rate-of-change.htmlhttps://pop.com.sg/relative-rate-of-change.htmlProblem: Volume of a cubic box is $V = L^3$. How are the relative rates of change of $V$ and $L$ related? This problem seems really simple, but I can't understand the concept of a relative rate of ......Tue, 25 Aug 2026 21:56:07 -0400https://pop.com.sg/real-numbers-raised-to-imaginary-powers-closed.htmlhttps://pop.com.sg/real-numbers-raised-to-imaginary-powers-closed.htmlWhat is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples......Tue, 25 Aug 2026 21:50:01 -0400https://pop.com.sg/meaning-and-types-of-geometry.htmlhttps://pop.com.sg/meaning-and-types-of-geometry.htmlI heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do yo......Tue, 25 Aug 2026 21:49:49 -0400https://pop.com.sg/fourier-dimension-of-a-measure-restricted-to-an-open-set.htmlhttps://pop.com.sg/fourier-dimension-of-a-measure-restricted-to-an-open-set.htmlSuppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\ri......Tue, 25 Aug 2026 21:44:13 -0400https://pop.com.sg/how-do-you-pronounce-tilde-r.htmlhttps://pop.com.sg/how-do-you-pronounce-tilde-r.htmlI found $\tilde{R}$ in a mathematical text, and I would like to know how this is pronounced. I tried to search on the internet but was not able to find anything related....Tue, 25 Aug 2026 21:41:17 -0400https://pop.com.sg/find-the-values-of-x-for-which-this-series-converges.htmlhttps://pop.com.sg/find-the-values-of-x-for-which-this-series-converges.htmlFind the values of x for which $$\sum_{n = 1}^{\infty} \frac{2^n}{x^n}$$ converges. This is what I'm thinking: I tried graphing it with a really big n number to get an idea on how it may look, a......Tue, 25 Aug 2026 21:36:48 -0400https://pop.com.sg/is-202-303-greater-or-303-202.htmlhttps://pop.com.sg/is-202-303-greater-or-303-202.htmlFind without use of calculator which of the two numbers is greater $202^{303}$ or $303^{202}$. I think we have to do this with calculus because I got this question from my calculus book. I tried ...Tue, 25 Aug 2026 21:31:37 -0400https://pop.com.sg/the-matrix-of-a-projection-can-never-be-invertible.htmlhttps://pop.com.sg/the-matrix-of-a-projection-can-never-be-invertible.htmlI am currently studying linear transformations in order to refresh my knowledge of linear algebra. One statement in my textbook (by David Poole) is: When considering linear transformations from $\...Tue, 25 Aug 2026 21:29:22 -0400https://pop.com.sg/implicitly-differentiate-x-3-y-3-3xy-closed.htmlhttps://pop.com.sg/implicitly-differentiate-x-3-y-3-3xy-closed.htmlI have this practice problem before a test. Use implicit differentiation to find $dy/dx$ for the equation $$x^3+y^3=3xy.$$ I have no idea how to do this, I didn't understand my lecturer. Can you guys ...Tue, 25 Aug 2026 21:28:57 -0400https://pop.com.sg/how-do-boolean-valued-functions-work.htmlhttps://pop.com.sg/how-do-boolean-valued-functions-work.htmlConsider this function: $$P: X\to \{true, false\}.$$ There's nothing in that expression that says when $X$ is true and when it is not true. How do these work?...Tue, 25 Aug 2026 21:19:56 -0400https://pop.com.sg/show-that-the-ideal-i-2-1-sqrt-13-is-a-maximal-ideal-in-mathbb-z-sqrt-.htmlhttps://pop.com.sg/show-that-the-ideal-i-2-1-sqrt-13-is-a-maximal-ideal-in-mathbb-z-sqrt-.htmlShow that the ideal $I = (2,1+\sqrt{-13})$ is a maximal ideal in $\mathbb{Z}[\sqrt{-13}]$? Is it principal? To show that the ideal is maximal, I know that $I \text{ maximal} \iff \mathbb{Z}[\sqrt{......Tue, 25 Aug 2026 21:18:41 -0400https://pop.com.sg/solving-heat-equation-under-neumann-boundary-conditions.htmlhttps://pop.com.sg/solving-heat-equation-under-neumann-boundary-conditions.htmlI am having trouble solving following equation \begin{align*} u_t - u_{xx} = 0 & \; ;0<x<L, t>0 \\ u(0, x)=x &\; ; 0<x<L\\ u_x(t, 0) = u_x(t, L) = 0&\; ; t >0 \end......Tue, 25 Aug 2026 21:06:32 -0400https://pop.com.sg/equation-of-a-circle-in-a-complex-plane.htmlhttps://pop.com.sg/equation-of-a-circle-in-a-complex-plane.htmlIn the book I'm reading, one of the exercises starts with mentioning that $\left|\frac{z-z_1}{z-z_2}\right|=c$, where $c\neq 1$ is a constant, is an equation of the circle. It seems clear to me t......Tue, 25 Aug 2026 20:57:39 -0400