https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 25 Aug 2026 21:10:04 -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 -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 -0400