https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 25 Aug 2026 17:27:58 -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 -0400https://pop.com.sg/analytic-method-for-determining-if-a-function-is-one-to-one.htmlhttps://pop.com.sg/analytic-method-for-determining-if-a-function-is-one-to-one.htmlIn algebra, we learn that if a function $ f(x) $ has a one-to-one mapping, then we can find the inverse function $ f^{-1}(x) $. The method that I have seen taught is the "horizontal line test": if ......Tue, 25 Aug 2026 20:54:59 -0400https://pop.com.sg/why-is-two-to-the-power-of-zero-equal-to-binary-one.htmlhttps://pop.com.sg/why-is-two-to-the-power-of-zero-equal-to-binary-one.htmlProbably a simple question and possibly not asked very well. What I want to know is.. In binary, a decimal value of 1 is also 1. It can be expressed as $x = 1 \times 2^0$ Question: Why is two to......Tue, 25 Aug 2026 20:52:57 -0400https://pop.com.sg/what-are-the-applications-of-euler-039-s-formula.htmlhttps://pop.com.sg/what-are-the-applications-of-euler-039-s-formula.htmlI have just seen the proof for Euler's theorem, and i have to say, i am pretty amazed! But I still don't understand why it is useful, or intuitive, at all. Idk where to even begin. I have seen the ......Tue, 25 Aug 2026 20:46:16 -0400https://pop.com.sg/converting-a-number-from-base-10-to-base-8.htmlhttps://pop.com.sg/converting-a-number-from-base-10-to-base-8.htmlI'm really confused and I can't get the concept. How is 14 in base 8 = 16? And how does 8 in base 8 = 10? In case of 14 isn't 1 * 8 + 4 * 1 = 12? Shouldn't any number in base 8 be lower than a ......Tue, 25 Aug 2026 20:30:17 -0400https://pop.com.sg/expectation-of-bar-x-2.htmlhttps://pop.com.sg/expectation-of-bar-x-2.htmlIf all enumerated $X$s are observations from a population with a population $\mu$ and variance $\sigma^2$: Why is this true? $$\mathbb E\!\left(\bar X ^2\right) = \frac{\sigma^2}{n}+\mu^2$$ So......Tue, 25 Aug 2026 20:24:44 -0400https://pop.com.sg/does-an-overdetermined-system-always-have-no-solutions-closed.htmlhttps://pop.com.sg/does-an-overdetermined-system-always-have-no-solutions-closed.htmlWhat is the problem with over-determined systems in linear algebra? Do they always have no solution? Is there a proof of that?...Tue, 25 Aug 2026 20:23:47 -0400https://pop.com.sg/what-does-this-double-sided-arrow-longleftrightarrow-mean.htmlhttps://pop.com.sg/what-does-this-double-sided-arrow-longleftrightarrow-mean.htmlWhat is $\longleftrightarrow$ used for in mathematics? I know about $\iff$ being used for "If and only if". Are they the same thing? I was watching a YouTube video that said: $$\sum^{\infty}_{n=1}......Tue, 25 Aug 2026 20:16:12 -0400https://pop.com.sg/change-of-variables-for-definite-integrals.htmlhttps://pop.com.sg/change-of-variables-for-definite-integrals.htmlFirst of all I would like to start off by asking why do they have different change of variable formulas for definite integrals than indefinite...why cant we just integrate using U substitution as we ...Tue, 25 Aug 2026 20:11:34 -0400https://pop.com.sg/what-does-spherical-convex-fuction-mean.htmlhttps://pop.com.sg/what-does-spherical-convex-fuction-mean.htmlLet $M$ be a riemannian manifold. A function $f: M \mapsto \mathbb{R} $ is called spherical convex, if \begin{equation} \sin(\lvert xz \rvert) f(y) \leq \sin(\lvert xy\rvert) f(z) + \sin( \lvert yz\...Tue, 25 Aug 2026 20:09:23 -0400https://pop.com.sg/sample-standard-deviation-vs-population-standard-deviation.htmlhttps://pop.com.sg/sample-standard-deviation-vs-population-standard-deviation.htmlI have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values: Sample Population I ...Tue, 25 Aug 2026 20:04:50 -0400https://pop.com.sg/natural-numbers-in-set-theory-is-0-1-2.htmlhttps://pop.com.sg/natural-numbers-in-set-theory-is-0-1-2.htmlThe set of natural numbers $\mathbb{N}$ in set theory is defined with the axiom of infinity as the smallest inductive set and then it is usually proven that $\mathbb{N}$ satisfies the Peano axioms ......Tue, 25 Aug 2026 19:56:24 -0400https://pop.com.sg/finding-points-in-which-the-first-derivative-does-not-exist-of-a-set.htmlhttps://pop.com.sg/finding-points-in-which-the-first-derivative-does-not-exist-of-a-set.htmlHow do I find the point in which the first derivative does not exist (in order to find the extremas) without the use of limits? Is it correct to find the first derivative and find it's domain and ......Tue, 25 Aug 2026 19:52:12 -0400https://pop.com.sg/calculate-the-convolution-of-the-product-of-a-unit-step-function-and-t.htmlhttps://pop.com.sg/calculate-the-convolution-of-the-product-of-a-unit-step-function-and-t.htmlRequest Please check my work. I am not certain how to calculate the convolution of the unit step function. Given: Find the convolution of $f(t)=t$ and $g(t)=u(t-1)$. $$h(t)=(f*g)(t)=\int_0^t f(......Tue, 25 Aug 2026 19:47:56 -0400https://pop.com.sg/area-of-a-circle-inscribed-in-a-polygon.htmlhttps://pop.com.sg/area-of-a-circle-inscribed-in-a-polygon.htmlIf a circle is inscribed in a polygon, show that, $$\dfrac{\text{(Area of inscribed circle)}}{\text{(Perimeter of inscribed circle)}} = \dfrac{\text{(Area of Polygon)}}{\text{(Perimeter of Poly......Tue, 25 Aug 2026 19:23:33 -0400https://pop.com.sg/supremum-and-infimum-example-closed.htmlhttps://pop.com.sg/supremum-and-infimum-example-closed.htmlcan anyone give me a good example of how to find supremum and infimum not lim sup and lim inf. I try to find some good example online but it keep given me lim sup and lim inf. I just need one good ...Tue, 25 Aug 2026 19:12:45 -0400https://pop.com.sg/calculate-matrix-by-using-cayley-hamilton-theorem.htmlhttps://pop.com.sg/calculate-matrix-by-using-cayley-hamilton-theorem.htmlCalculate matrix $B = A^{10}-3A^9-A^2+4A$ using Cayley-Hamilton theorem on $A$. $$A = \begin{pmatrix} 2 & 2 & 2 & 5 \\ -1 & -1 & -1 & -5 \\ -2 & -2 & -1 & ......Tue, 25 Aug 2026 19:12:12 -0400https://pop.com.sg/change-origin-of-shapes-in-graph.htmlhttps://pop.com.sg/change-origin-of-shapes-in-graph.htmlThis might seem pretty simple, but keep in mind I'm in middle school. I graphed a spiral using $r = \theta$. The problem is, the origin is at $(0, 0)$. How can I change the origin to something lik......Tue, 25 Aug 2026 19:09:14 -0400https://pop.com.sg/implicit-differentiation-how-chain-rule-is-applied-vs-explicit-differe.htmlhttps://pop.com.sg/implicit-differentiation-how-chain-rule-is-applied-vs-explicit-differe.htmlPlease assume this is my initial exposure to calculus and derivatives. I am having difficulty making the connection between the application of the chain rule to explicit differentiation and that of ...Tue, 25 Aug 2026 19:02:36 -0400https://pop.com.sg/what-is-the-condition-that-determines-a-proper-integral.htmlhttps://pop.com.sg/what-is-the-condition-that-determines-a-proper-integral.htmlI am reading Problems in Calculus of One Variable by I.A.Maron and the author seems to make the following statements: $$\int_{0}^{\pi/2}\frac{\sin t}{\sqrt t}\text{dt}\text{ is proper integral sin......Tue, 25 Aug 2026 18:56:08 -0400https://pop.com.sg/find-the-value-of-x-and-y-given-this-equation.htmlhttps://pop.com.sg/find-the-value-of-x-and-y-given-this-equation.htmlSo I have a College Admission test tomorrow and I am hoping that you could help me understand how to arrive at the solution to this: 1.) Given the following equations: $$3x-y=30\\ 5x-3y=10$$ ......Tue, 25 Aug 2026 18:53:38 -0400https://pop.com.sg/how-to-calculate-exp-x-using-taylor-series.htmlhttps://pop.com.sg/how-to-calculate-exp-x-using-taylor-series.htmlWe know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...Tue, 25 Aug 2026 18:47:29 -0400https://pop.com.sg/how-to-evaluate-powers-of-powers-i-e-2-3-4-in-absence-of-parentheses.htmlhttps://pop.com.sg/how-to-evaluate-powers-of-powers-i-e-2-3-4-in-absence-of-parentheses.htmlIf you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you ...Tue, 25 Aug 2026 18:38:52 -0400https://pop.com.sg/what-does-a-dot-in-a-circle-mean.htmlhttps://pop.com.sg/what-does-a-dot-in-a-circle-mean.htmlI'm looking at some formulas involving matrices (in the context of machine learning, but I'm not sure it's relevant) and I came across $\odot$. What could this mean? The context is $M \odot N$, whe......Tue, 25 Aug 2026 18:30:54 -0400https://pop.com.sg/how-many-minutes-does-it-take-n-moles-to-dig-m-holes.htmlhttps://pop.com.sg/how-many-minutes-does-it-take-n-moles-to-dig-m-holes.htmlRecently, I came across this problem: 5 moles dig 4 holes in 3 minutes. How many minutes will it take 9 moles to dig 6 holes? I decided that I should just use proportions and calculate it. $\i......Tue, 25 Aug 2026 18:13:37 -0400https://pop.com.sg/do-we-count-only-distinct-roots-in-descartes-039-rule-of-signs.htmlhttps://pop.com.sg/do-we-count-only-distinct-roots-in-descartes-039-rule-of-signs.htmlDescartes' rule of signs states that numbet of positive roots of a polynomial $P(x) = a_nx^n + ... + a_1x + a_0$ is not greater than the number of sign changes of coefficients. By Gauss we know that ...Tue, 25 Aug 2026 18:12:39 -0400https://pop.com.sg/variance-reduction-by-conditioning-how-to-know-over-what-variable-to-c.htmlhttps://pop.com.sg/variance-reduction-by-conditioning-how-to-know-over-what-variable-to-c.htmlI have a question related to variance reduction by conditioning. If I have to RV´s $X$ and $Y$ with means $1$ and $2$ respectevely and I want to approach the value of $P(X+Y \gt 4)$ by simulating. ...Tue, 25 Aug 2026 18:09:15 -0400https://pop.com.sg/what-is-a-perfect-circle.htmlhttps://pop.com.sg/what-is-a-perfect-circle.htmlPeople often play around with the word "perfect circle" but I have always been curious as to what exactly is a "perfect circle". It is easy to imagine, but to me, even when given ...Tue, 25 Aug 2026 18:01:08 -0400https://pop.com.sg/how-should-i-think-about-what-it-means-for-a-manifold-to-be-orientable.htmlhttps://pop.com.sg/how-should-i-think-about-what-it-means-for-a-manifold-to-be-orientable.htmlLet M be a smooth manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that for all $\alpha, \beta$, $\textrm{det}(J(\phi_{\alpha} \......Tue, 25 Aug 2026 17:51:51 -0400https://pop.com.sg/direct-approach-to-the-closed-graph-theorem.htmlhttps://pop.com.sg/direct-approach-to-the-closed-graph-theorem.htmlIn the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, an......Tue, 25 Aug 2026 17:49:18 -0400https://pop.com.sg/is-there-any-symbol-for-intersects.htmlhttps://pop.com.sg/is-there-any-symbol-for-intersects.htmlGiven two polygons $A$ and $B$ the intersection of them is represented by the notation $A \cap B$ that returns a geometry (more precisely the set of intersecting points) resulting from the intersec......Tue, 25 Aug 2026 17:41:32 -0400https://pop.com.sg/can-n-2-4n-be-a-perfect-square.htmlhttps://pop.com.sg/can-n-2-4n-be-a-perfect-square.htmlDo there exist positive integers $n,k$ such that $n^2+4n=k^2$? I'm not sure how to attack this question. I was able to get that $n=\frac{-4\pm\sqrt{16+4k^2}}{2}$, but I don't think it is of any use....Tue, 25 Aug 2026 17:37:09 -0400https://pop.com.sg/definition-of-a-markov-process-in-continuous-state-space.htmlhttps://pop.com.sg/definition-of-a-markov-process-in-continuous-state-space.htmlThe Markov property is usually defined as something like $$ P(X_n\in A|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2}, \dots, X_0=x_0)=P(X_n\in A|X_{n-1}=x_{n-1}). $$ My question is, what do the sides of this ...Tue, 25 Aug 2026 17:30:31 -0400https://pop.com.sg/is-this-a-viable-new-proof-for-pythagoras-theorem.htmlhttps://pop.com.sg/is-this-a-viable-new-proof-for-pythagoras-theorem.htmlA triangle with side lengths $a$, $b$, $c$ and a height ($h$) that intercepts the hypotenuse ($c$) such that $c$ is split into two side lengths, $c = m + n$, we can find Pythagorean theorem using s......Tue, 25 Aug 2026 17:29:01 -0400https://pop.com.sg/complex-logarithm.htmlhttps://pop.com.sg/complex-logarithm.htmlI have $e^{jk}$ and I want to take a logarithm from it, $\log(e^{jk})$ must be $jk$, right? here some example I have tried to do with matlab. $$\log(e^{j2})=j2$$ $$\log(e^{j3})=j3$$ but for $e^{j4}......Tue, 25 Aug 2026 17:22:37 -0400https://pop.com.sg/why-are-percentage-increases-and-decreases-not-equal.htmlhttps://pop.com.sg/why-are-percentage-increases-and-decreases-not-equal.htmlE.g. 2/3 and 3/2 is 66% and 150%. Why is there a 16% difference between the 2 (34% and and 50%) and why does the direction of the percentage change (positive or negative change things). I can do the ...Tue, 25 Aug 2026 17:20:40 -0400https://pop.com.sg/how-many-solutions-are-there-to-the-equation-x-1-x-2-x-3-x-4-x-5-21.htmlhttps://pop.com.sg/how-many-solutions-are-there-to-the-equation-x-1-x-2-x-3-x-4-x-5-21.htmlQuestion How many solutions are there to the equation $x_1 + x_2 + x_3 + x_4 + x_5 = 21$, such that all of $x_{i}$ where $i=1,2,3,4,5$ are non negative and $0\leq x_1 \leq 3$ $1\leq x_2 \lt 4$ and ......Tue, 25 Aug 2026 17:20:01 -0400https://pop.com.sg/help-with-binary-division-problem-dividing-1001011010000-by-1001.htmlhttps://pop.com.sg/help-with-binary-division-problem-dividing-1001011010000-by-1001.htmlI have this binary division practice problem, I am not sure how to deal with a leading 1 when pulling down the next bit. Here is my process, I'll try to make it clear to where my confusion comes ab......Tue, 25 Aug 2026 17:15:36 -0400https://pop.com.sg/differential-equation-x-039-039-kx.htmlhttps://pop.com.sg/differential-equation-x-039-039-kx.htmlInitial condition $x(0)=1$, $k>0$ \begin{eqnarray} \lambda^2=-k \end{eqnarray} \begin{eqnarray} \lambda=\pm\sqrt{-k} \end{eqnarray} I have no idea how to erase this $k$ what should be the nex......Tue, 25 Aug 2026 16:58:34 -0400https://pop.com.sg/find-the-probability-distribution-of-x.htmlhttps://pop.com.sg/find-the-probability-distribution-of-x.htmlLet $X$ be a random variable with cumulative distribution function: $$F(x)=\begin{cases} 0 & : x < -2 \\ 1/8 & : -2 \leq x < -1 \\ 3/8 & : -1 \leq x < 0 \\ 5/8 & : 0......Tue, 25 Aug 2026 16:35:31 -0400https://pop.com.sg/what-is-a-diameter-of-smaller-circle.htmlhttps://pop.com.sg/what-is-a-diameter-of-smaller-circle.htmlI have two coins $A$ and $B$. The diameter of coins A is $d_A=18$ mm. The smaller coin $B$ rolls around $A$ without sliding. Both coins have marks on the edge. At the beginning of the movement, the......Tue, 25 Aug 2026 16:31:45 -0400https://pop.com.sg/volume-of-regular-octahedron.htmlhttps://pop.com.sg/volume-of-regular-octahedron.htmlAccording to several sources on the internet, the volume of a regular octahedron with unit edge lengths is approximately 0.47. I haven't seen an explanation for why this is so yet. When I tried to ......Tue, 25 Aug 2026 16:31:38 -0400https://pop.com.sg/calculating-logs-in-your-head.htmlhttps://pop.com.sg/calculating-logs-in-your-head.htmlI quite often need to estimate $\log_2(x)$ for positive integer values of $x$. I find it frustrating that I have to rely on a calculator/computer to do this. By way of contrast, I can happily com......Tue, 25 Aug 2026 16:30:37 -0400https://pop.com.sg/mutually-exclusive-events-can-be-independent-flipping-a-coin-example-d.htmlhttps://pop.com.sg/mutually-exclusive-events-can-be-independent-flipping-a-coin-example-d.htmlFlipping a coin we get either head or tail can't get both so events are mutually exclusive. i.e P(A and B) =0. but flipping the same coin twice may result in either head or tail and result of flipp......Tue, 25 Aug 2026 16:28:58 -0400https://pop.com.sg/proving-associativity-of-a-certain-binary-aperation-in-any-complemente.htmlhttps://pop.com.sg/proving-associativity-of-a-certain-binary-aperation-in-any-complemente.htmlIf, in a Boolean lattice $(X,\vee,\wedge,0,1,')$ (i.e. a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+z)$ ra......Tue, 25 Aug 2026 16:24:55 -0400https://pop.com.sg/hamiltonian-graph-and-2-connected-graph.htmlhttps://pop.com.sg/hamiltonian-graph-and-2-connected-graph.htmlA non-complete graph is called 2-connected if it stays connected after removing a vertex (and all edges which are incident to that vertex). Show that a Hamiltonian graph is 2-connected. I'm having ...Tue, 25 Aug 2026 16:13:25 -0400https://pop.com.sg/definition-of-the-mathematical-proof.htmlhttps://pop.com.sg/definition-of-the-mathematical-proof.htmlHow do we define a mathematical proof? Is it a series of arguments? Is it a series of conclusions? Is it manipulation of formulas? Is it a mixture of laws of logic and axioms,theorems or defini......Tue, 25 Aug 2026 16:12:38 -0400https://pop.com.sg/count-the-number-of-solutions-of-the-inequality-x-y-z-leq-n.htmlhttps://pop.com.sg/count-the-number-of-solutions-of-the-inequality-x-y-z-leq-n.htmlProblem Given $A, B, C $ and $N$. How many integer solutions are there of the following inequality: $$x + y + z \leq N$$ where $0 \leq x \leq A, 0 \leq y \leq B, 0 \leq z \leq C$? When $A + ......Tue, 25 Aug 2026 16:01:49 -0400https://pop.com.sg/inversion-of-rotation-matrix.htmlhttps://pop.com.sg/inversion-of-rotation-matrix.htmlFor example, I have a two-dimensional rotation matrix $$ \begin{bmatrix} 0.5091 & -0.8607 \\ 0.8607 & \phantom{-}0.5091 \end{bmatrix} $$ and I have a vector I'd like to ...Tue, 25 Aug 2026 16:00:50 -0400https://pop.com.sg/how-to-factorize-x-3-7x-6.htmlhttps://pop.com.sg/how-to-factorize-x-3-7x-6.htmlHow do you factorize this polynomial: $${x^3 - 7x + 6}$$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a pre-......Tue, 25 Aug 2026 15:56:02 -0400https://pop.com.sg/how-to-simplify-a-n-b-n.htmlhttps://pop.com.sg/how-to-simplify-a-n-b-n.htmlHow to simplify $a^n - b^n$? If it would be $(a+b)^n$, then I could use the binomial theorem, but it's a bit different, and I have no idea how to solve it. Thanks in advance....Tue, 25 Aug 2026 15:50:35 -0400https://pop.com.sg/finding-domain-and-range-of-inverse-function-without-finding-inverse.htmlhttps://pop.com.sg/finding-domain-and-range-of-inverse-function-without-finding-inverse.htmlSo i found a question which states: Without finding the inverse, state the domain and range of $f^{-1}$ $f(x) = (x-1)/(x-4)$ where $ x≠4$ how can i find the domain and range of the function's ...Tue, 25 Aug 2026 15:32:38 -0400https://pop.com.sg/how-do-i-calculate-derivative-of-sgn-x.htmlhttps://pop.com.sg/how-do-i-calculate-derivative-of-sgn-x.htmlWe know $|x| = \sqrt(x^2)$, determine the second derivative $\frac{d^2}{dx^2}|x|$, So the first derivative is sgn(x), but how do I get the second?...Tue, 25 Aug 2026 15:29:38 -0400https://pop.com.sg/proving-aaa-sas.htmlhttps://pop.com.sg/proving-aaa-sas.htmlDoes anyone have a clue how I can prove that (Angle - Angle - Angle) <-> (Side - Angle - Side) for the similarity of triangles? I don´t know where to start? /Alex...Tue, 25 Aug 2026 15:25:44 -0400https://pop.com.sg/find-the-global-maximum-and-global-minimum.htmlhttps://pop.com.sg/find-the-global-maximum-and-global-minimum.html$ f(x) = x^3 - 6x^2 - 15x + 8 $ for the interval $ [-2,6] $ My answer was : $$ f'(x) = 3(x^2-4x-5) $$ $$ 3(x-5)(x+1) $$ so the critical points are $ x =5\; and \;x=-1 $ plugging them back with ......Tue, 25 Aug 2026 15:24:13 -0400https://pop.com.sg/why-is-the-divergence-of-curl-expected-to-be-zero.htmlhttps://pop.com.sg/why-is-the-divergence-of-curl-expected-to-be-zero.htmlI was reading the proof of Helmholtz decomposition theorem where I found the relation between the rotational and the irrotational fields are not symmetric. And by that I mean if the divergence of the ...Tue, 25 Aug 2026 15:23:28 -0400https://pop.com.sg/compute-the-limit-as-x-approaches-infinity.htmlhttps://pop.com.sg/compute-the-limit-as-x-approaches-infinity.html$$\lim_{x\to \infty} \frac{6x^4+4}{(x^2-2)(6x^2-1)}$$ Based on the way the function behaves I can say it will reach 1, but I can't seem to be able to calculate the limit. Thanks for the help!...Tue, 25 Aug 2026 15:17:50 -0400https://pop.com.sg/find-the-area-enclosed-by-circles-r-3-cos-theta-and-r-3-sin-theta.htmlhttps://pop.com.sg/find-the-area-enclosed-by-circles-r-3-cos-theta-and-r-3-sin-theta.htmlFind the area enclosed by circles $r=3\cos\theta$ and $r=3\sin\theta$ I know that the graph of $r=3\sin\theta$ intersects at $\theta=0$ and at $\theta=\pi$ why aren't these two bounds used and why......Tue, 25 Aug 2026 15:15:52 -0400https://pop.com.sg/prove-pascal-039-s-rule-algebraically.htmlhttps://pop.com.sg/prove-pascal-039-s-rule-algebraically.htmlI am trying to prove Pascal's Rule algebraically but I'm stuck on simplifying the numerator. This is the last step that I have, but I'm not sure where to go from here $$=\frac{\left[(k-1)(n-k)!+k(......Tue, 25 Aug 2026 15:15:24 -0400https://pop.com.sg/why-are-proofs-not-written-as-collections-of-logic-symbols-but-are-ins.htmlhttps://pop.com.sg/why-are-proofs-not-written-as-collections-of-logic-symbols-but-are-ins.htmlMathematical proofs are written as sentences and not as collections of logic symbols. Through logical operations, it is much easier for me to visualize what the symbols are trying to tell us rather......Tue, 25 Aug 2026 15:10:49 -0400https://pop.com.sg/proof-of-the-inclusion-exclusion-formula-in-probability.htmlhttps://pop.com.sg/proof-of-the-inclusion-exclusion-formula-in-probability.htmlLet $(\Omega,\mathcal F, P)$ be a probability space and let $A_1,A_2,...,A_n$ be events in $\mathcal F$. Prove the following inclusion-exclusion formula $P(\bigcup_{i=1}^nA_i)=\sum_{k=1}^n$ $\sum_{\...Tue, 25 Aug 2026 15:03:25 -0400https://pop.com.sg/show-that-the-eigenvalues-of-a-unitary-matrix-have-modulus-1.htmlhttps://pop.com.sg/show-that-the-eigenvalues-of-a-unitary-matrix-have-modulus-1.htmlShow that the eigenvalues of a unitary matrix have modulus $1$. I know that a unitary matrix can be defined as a square complex matrix $A$, such that $$AA^*=A^*A=I$$ where $A^*$ is the conjugate ...Tue, 25 Aug 2026 14:58:37 -0400https://pop.com.sg/surface-area-of-a-cube-as-a-function-of-its-volume.htmlhttps://pop.com.sg/surface-area-of-a-cube-as-a-function-of-its-volume.htmlExpressing the surface area of a cube as a function of its volume is like plugging in the surface area into volume? Analyzing $\text{S.A.}=6s^2$, and $V=s^3$ I would assume that the Surface area is ...Tue, 25 Aug 2026 14:51:23 -0400https://pop.com.sg/how-to-prove-that-frac-pi-2-sin-pi-z-2-sum-n-infty-n-neq-0-n-infty-1-z.htmlhttps://pop.com.sg/how-to-prove-that-frac-pi-2-sin-pi-z-2-sum-n-infty-n-neq-0-n-infty-1-z.htmlThe following question was part of my complex analysis assignment and I am not able to prove it. How to prove that $$\frac {\pi^2 } { \sin(\pi z)^2 } = \sum_{n=-\infty , n \neq0 }^{n=\infty} \fra......Tue, 25 Aug 2026 14:40:28 -0400https://pop.com.sg/how-to-find-the-maximum-value-of-f-x-y-39-x-2-y-2-on-the-line-x-6y-37.htmlhttps://pop.com.sg/how-to-find-the-maximum-value-of-f-x-y-39-x-2-y-2-on-the-line-x-6y-37.htmlso I'm totally lost how to find this question find the maximum value of $f(x,y) = 39 - x^2 - y^2$ on the line $x+6y=37$ I know how to find local max and min of multivariable equations, but now I ......Tue, 25 Aug 2026 14:35:39 -0400https://pop.com.sg/density-of-the-sum-of-two-exponential-random-variable.htmlhttps://pop.com.sg/density-of-the-sum-of-two-exponential-random-variable.htmlFor independent random variables X ∼ Exp(1) and Y ∼ Exp(2), find the density of the random variable Z = X + Y . My work: For any exponential distribution with parameter $\lambda$ the function is $......Tue, 25 Aug 2026 14:21:25 -0400https://pop.com.sg/dividing-an-infinite-power-series-by-another-infinite-power-series.htmlhttps://pop.com.sg/dividing-an-infinite-power-series-by-another-infinite-power-series.htmlLet's say I have two power series $\,\mathrm{F}\left(x\right) = \sum_{n = 0}^{\infty}\,a_{n}\,x^{n}$ and $\,\mathrm{G}\left(x\right) = \sum_{n = 0}^{\infty}\,b_{n}\,x^{n}$. If I define the functio......Tue, 25 Aug 2026 14:18:17 -0400https://pop.com.sg/graham-039-s-number-why-so-big.htmlhttps://pop.com.sg/graham-039-s-number-why-so-big.htmlCan someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !...Tue, 25 Aug 2026 14:16:06 -0400https://pop.com.sg/what-is-the-simplest-proof-of-the-pythagorean-theorem-you-know-duplica.htmlhttps://pop.com.sg/what-is-the-simplest-proof-of-the-pythagorean-theorem-you-know-duplica.htmlMaybe enough so to explain it to children....Tue, 25 Aug 2026 14:14:23 -0400https://pop.com.sg/why-doesn-039-t-my-ti-84-simplify-fractions.htmlhttps://pop.com.sg/why-doesn-039-t-my-ti-84-simplify-fractions.htmlSay we have: $$\frac 12 + \frac 13 + \frac 14$$ When I calculate this on my TI-84 Plus CE-T, it gives $$\frac {13}{12}$$ Instead of $$1\frac {1}{12}$$ Is there any reason for it doing so an......Tue, 25 Aug 2026 14:14:17 -0400https://pop.com.sg/proof-the-levi-civita-symbol-is-a-tensor.htmlhttps://pop.com.sg/proof-the-levi-civita-symbol-is-a-tensor.htmlA tensor of rank $n$ has components $T_{ij\cdots k}$ (with $n$ indices) with respect to each basis $\{\mathbf{e}_i\}$ or coordinate system $\{x_i\}$, and satisfies the following rule of change of b......Tue, 25 Aug 2026 14:10:23 -0400https://pop.com.sg/what-is-sqrt-36.htmlhttps://pop.com.sg/what-is-sqrt-36.htmlI am working on a math problem $(x^{2} - 6x + 45)$ where I must utilize the "completing the square" method, and was able to break down the problem to $(x-3)^{2} = -36.$ I know I have to ......Tue, 25 Aug 2026 13:55:05 -0400https://pop.com.sg/definition-of-exp-x.htmlhttps://pop.com.sg/definition-of-exp-x.htmlI've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia ......Tue, 25 Aug 2026 13:54:25 -0400https://pop.com.sg/expressing-logarithms-as-ratios-of-natural-logarithms.htmlhttps://pop.com.sg/expressing-logarithms-as-ratios-of-natural-logarithms.html$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an abstract ...Tue, 25 Aug 2026 13:42:14 -0400https://pop.com.sg/explanation-of-the-rational-root-theorem.htmlhttps://pop.com.sg/explanation-of-the-rational-root-theorem.htmlI just discovered the rational root theorem and I feel like I can understand it if I can get past the notational jargon presented in Wikipedia. Here's what I think I understood. If $r$ is a rational ...Tue, 25 Aug 2026 13:39:12 -0400https://pop.com.sg/limited-function.htmlhttps://pop.com.sg/limited-function.htmlLet $f:[0,+\infty) \rightarrow \mathbb{R}$ be a function bounded on each bounded interval. Prove that if $\lim_{x \rightarrow +\infty } [f(x+1)-f(x)] = L$, then $\lim_{x \rightarrow +\infty} \frac......Tue, 25 Aug 2026 13:34:51 -0400https://pop.com.sg/formula-for-probability-of-specific-die-outcomes.htmlhttps://pop.com.sg/formula-for-probability-of-specific-die-outcomes.htmlRoll some number, N, of the same kind of dice, say d6. Picking a target outcome, say 1, what is the FORMULA to determine the chance of getting 1s as a result, which is to say, how many chances of ......Tue, 25 Aug 2026 13:32:39 -0400https://pop.com.sg/why-balanced-ternary-exists-what-is-the-problem-with-ternary.htmlhttps://pop.com.sg/why-balanced-ternary-exists-what-is-the-problem-with-ternary.htmlRecently, I countered a problem of representing data in ternary. I came to know there exists ternary and balanced ternary representation. It is my best understanding that balanced ternary helps in ...Tue, 25 Aug 2026 13:30:54 -0400https://pop.com.sg/prove-that-a-subseteq-b-and-b-is-an-ideal-of-r.htmlhttps://pop.com.sg/prove-that-a-subseteq-b-and-b-is-an-ideal-of-r.htmlLet $R$ be a commutative ring with unity, let $A$ be an ideal of $R$ and let $a \in R$. Define $B = \{b \in R ~|~ ab \in A \}$. Prove that $A \subseteq B$ (where $A$ does not have to be the proper ......Tue, 25 Aug 2026 13:03:25 -0400https://pop.com.sg/probability-of-duplicate-guid.htmlhttps://pop.com.sg/probability-of-duplicate-guid.htmlA GUID (globally unique identifier) is a 32 character hexadecimal string: http://en.wikipedia.org/wiki/Globally_Unique_Identifier If you randomly generate 2, the chance of them being the same is ...Tue, 25 Aug 2026 13:02:18 -0400https://pop.com.sg/how-do-you-solve-an-equation-involving-the-weierstrass-p-function.htmlhttps://pop.com.sg/how-do-you-solve-an-equation-involving-the-weierstrass-p-function.htmlWhat $z\in \mathbb{C}$ satisfy \begin{eqnarray*} \wp(z) = \wp(nz) \end{eqnarray*} if $n\in \mathbb{N}$? The Weierstrass p-function is defined as \begin{eqnarray*} \wp(z)=\frac{1}{z^2}+\sum_{\omega......Tue, 25 Aug 2026 12:56:30 -0400https://pop.com.sg/sufficient-conditions-on-the-isomorphism-of-two-groups.htmlhttps://pop.com.sg/sufficient-conditions-on-the-isomorphism-of-two-groups.htmlLet $G_1, G_2$ be two groups with at least one nontrivial proper subgroup each. Let $S_1, S_2$ be the sets of proper subgroups of, respectively $G_1, G_2$. Suppose there exists a bijective functi......Tue, 25 Aug 2026 12:43:18 -0400https://pop.com.sg/can-the-entropy-of-a-random-variable-with-countably-many-outcomes-be-i.htmlhttps://pop.com.sg/can-the-entropy-of-a-random-variable-with-countably-many-outcomes-be-i.htmlConsider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by $$H(X) = \sum_i -p_i \log p_i.$$ Is it possibl......Tue, 25 Aug 2026 12:15:31 -0400https://pop.com.sg/where-can-i-find-a-list-of-large-prime-numbers-closed.htmlhttps://pop.com.sg/where-can-i-find-a-list-of-large-prime-numbers-closed.htmlA repository of say 13 digit prime, 15 digit primes etc....Tue, 25 Aug 2026 11:58:54 -0400https://pop.com.sg/is-basis-change-ever-useful-in-practical-linear-algebra.htmlhttps://pop.com.sg/is-basis-change-ever-useful-in-practical-linear-algebra.htmlIn layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?...Tue, 25 Aug 2026 11:55:21 -0400https://pop.com.sg/how-to-factor-x-2-2xy-y-2.htmlhttps://pop.com.sg/how-to-factor-x-2-2xy-y-2.htmlI know the formula $x^2 + 2xy + y^2 = (x + y)^2$ by heart and in order to understand it better I am trying to solve the formula myself; however, I don't understand how step 4 is derived from step 3......Tue, 25 Aug 2026 11:47:22 -0400https://pop.com.sg/integral-of-1-x-2-without-power-rule.htmlhttps://pop.com.sg/integral-of-1-x-2-without-power-rule.htmlI was wondering if it was possible to evaluate the following integral without using the power rule for negative exponents \begin{equation*} \int \frac{1}{x^2} \; dx \end{equation*} When using ...Tue, 25 Aug 2026 11:39:09 -0400https://pop.com.sg/math-logic-symbol-for-instance-of.htmlhttps://pop.com.sg/math-logic-symbol-for-instance-of.htmlI saw a discussion of a possible symbol for "example," but I need "example/instance of." There is of course $a \in S$ which is a is a member of/in S, but is there a specific way of stating $a "...Tue, 25 Aug 2026 11:32:29 -0400https://pop.com.sg/i-want-an-intutive-theoretical-understanding-of-the-sin-omega-t-f-clos.htmlhttps://pop.com.sg/i-want-an-intutive-theoretical-understanding-of-the-sin-omega-t-f-clos.htmlWhat is the meaning of $x(t)=x_0\cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ phase constant? I know how to solve the mathematical problems whic......Tue, 25 Aug 2026 11:22:58 -0400https://pop.com.sg/translation-of-weber-039-s-lehrbuch-der-algebra-vol-1-2-3.htmlhttps://pop.com.sg/translation-of-weber-039-s-lehrbuch-der-algebra-vol-1-2-3.htmlI have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would ......Tue, 25 Aug 2026 11:17:26 -0400https://pop.com.sg/what-039-s-the-difference-between-a-b-and-a-b.htmlhttps://pop.com.sg/what-039-s-the-difference-between-a-b-and-a-b.htmlIs there any difference between $]a,b[$ and $(a,b)$? If there is no difference, what would be the motivation of using $]a,b[$ over $(a,b)$?...Tue, 25 Aug 2026 11:16:57 -0400https://pop.com.sg/axiomatic-definition-of-groups.htmlhttps://pop.com.sg/axiomatic-definition-of-groups.htmlTo define natural numbers one can either: use the Peano axioms in second-order logic; encode them in set theory as von Neumann ordinals. The relation between those two definitions of natural num......Tue, 25 Aug 2026 11:15:22 -0400https://pop.com.sg/use-laplace-transform-to-solve-the-following-initial-value-problems.htmlhttps://pop.com.sg/use-laplace-transform-to-solve-the-following-initial-value-problems.htmlUse Laplace transform to solve the following initial–value problem. $y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$ Answer $s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(......Tue, 25 Aug 2026 11:06:40 -0400https://pop.com.sg/how-much-does-the-owner-lose.htmlhttps://pop.com.sg/how-much-does-the-owner-lose.htmlA thief stole a $\$100$ from the owner without him knowing. The same thief comes back inside the store $5$ minutes later and spends $\$70$ of the money. The owner then gives him back $\$30$ in chan......Tue, 25 Aug 2026 11:04:44 -0400https://pop.com.sg/simple-double-summation.htmlhttps://pop.com.sg/simple-double-summation.htmlI've seen how nesting works with a simple $(i+j)$ but this problem below is tripping me up. It's either because of the multipliers or because they each start at zero but I get 60, and the answer I ...Tue, 25 Aug 2026 10:59:18 -0400https://pop.com.sg/probability-density-of-the-maximum-of-samples-from-a-uniform-distribut.htmlhttps://pop.com.sg/probability-density-of-the-maximum-of-samples-from-a-uniform-distribut.htmlSuppose $$X_1, X_2, \dots, X_n\sim Unif(0, \theta), iid$$ and suppose $$\hat\theta = \max\{X_1, X_2, \dots, X_n\}$$ How would I find the probability density of $\hat\theta$? Thank you!...Tue, 25 Aug 2026 10:55:53 -0400https://pop.com.sg/how-to-quickly-tell-if-a-discontinuity-is-removable-or-non-removable.htmlhttps://pop.com.sg/how-to-quickly-tell-if-a-discontinuity-is-removable-or-non-removable.htmlGiven any function where you're asked to find all points of discontinuity and to classify those points, is there a quick way to look at it and tell if that discontinuity is removable or not?...Tue, 25 Aug 2026 10:52:43 -0400