https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 15 Aug 2026 07:41:50 -0400https://pop.com.sg/linear-algebra-identity-matrix-and-its-relation-to-eigenvalues-and-eig.htmlhttps://pop.com.sg/linear-algebra-identity-matrix-and-its-relation-to-eigenvalues-and-eig.htmlKindly help me understand this statement made by my prof. "The identity matrix I has the property that any non zero vector V is an eigenvector of eigenvalue 1." My assumption of this statement is......Sat, 15 Aug 2026 11:41:49 -0400https://pop.com.sg/what-precisely-is-the-difference-between-euclidean-geometry-and-non-eu.htmlhttps://pop.com.sg/what-precisely-is-the-difference-between-euclidean-geometry-and-non-eu.htmlI was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that non-Eucl......Sat, 15 Aug 2026 11:41:09 -0400https://pop.com.sg/can-someone-please-help-me-convert-this-triple-integral-to-spherical-c.htmlhttps://pop.com.sg/can-someone-please-help-me-convert-this-triple-integral-to-spherical-c.htmlI'm not sure how to approach the problem, though I know it needs to be broken up into two integrals before it can be evaluated based on the way the answer input is set up. I do not know how to start ...Sat, 15 Aug 2026 11:13:51 -0400https://pop.com.sg/why-are-exact-differential-equations-called-so.htmlhttps://pop.com.sg/why-are-exact-differential-equations-called-so.htmlAs the question says, why are exact differential equations called so? From Wikipedia, I got "The nomenclature of "exact differential equation" refers to the exact differential of a function". That ...Sat, 15 Aug 2026 11:13:51 -0400https://pop.com.sg/closure-of-the-set-of-all-polynomial-with-variable-x-in-0-1.htmlhttps://pop.com.sg/closure-of-the-set-of-all-polynomial-with-variable-x-in-0-1.htmlLet ${P}$ denote the set of all polynomial with variable $x\in [0,1]$, I need to know what is the closure of ${P}$ in $C[0,1]$? Well, Stone-Weierstrass theorem says: If $f\in C[0,1]$ then there exi......Sat, 15 Aug 2026 11:10:29 -0400https://pop.com.sg/what-are-the-mean-and-variance-of-the-log-of-a-random-variable.htmlhttps://pop.com.sg/what-are-the-mean-and-variance-of-the-log-of-a-random-variable.htmlHere's the problem. We have a random variable X that follows a Poisson law. If we take the log of this variable, what are the first two moments (mean and variance) of the law it follows? This looks ...Sat, 15 Aug 2026 11:02:54 -0400https://pop.com.sg/a-double-limit-question.htmlhttps://pop.com.sg/a-double-limit-question.htmlMaybe it's easy, but: Is it true that $$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$ If it is, could you help me prove it? Thanks...Sat, 15 Aug 2026 10:51:50 -0400https://pop.com.sg/recognizing-the-sequence-1-16-1-8-3-16-1-4-5-16.htmlhttps://pop.com.sg/recognizing-the-sequence-1-16-1-8-3-16-1-4-5-16.htmlWhat is the missing number? $$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \ \ \ [?]$$ $$A. \frac{5}{4}\quad B. \frac{3}{4}\quad C. \frac{5}{8}\quad D. \frac{3}{8}$$ Spo......Sat, 15 Aug 2026 10:45:05 -0400https://pop.com.sg/finding-a-spanning-set-for-a-null-space.htmlhttps://pop.com.sg/finding-a-spanning-set-for-a-null-space.htmlI have a matrix $A$ that is like this: \begin{equation} A = \pmatrix{ 1 & 2 & -3 & 1 & 5 \\ 1 & 3 & -1 & 4 & -2 \\ 1 & 1 & -5 & -2 & 12 \\ 1 & 4......Sat, 15 Aug 2026 10:44:41 -0400https://pop.com.sg/is-k-g-1-bg.htmlhttps://pop.com.sg/is-k-g-1-bg.htmlIs an Eilenberg-MacLane space $K(G,1)$ the same as the classifying space $BG$ for a group $G$ ?...Sat, 15 Aug 2026 10:44:01 -0400https://pop.com.sg/entire-function-with-vanishing-derivatives.htmlhttps://pop.com.sg/entire-function-with-vanishing-derivatives.htmlLet $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. And assume that at each point, one of it's derivatives vanishes. What can you say about $f$? A hint suggests that $f$ must be a poly......Sat, 15 Aug 2026 10:43:17 -0400https://pop.com.sg/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthu.htmlhttps://pop.com.sg/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthu.htmlI'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs......Sat, 15 Aug 2026 10:34:12 -0400https://pop.com.sg/question-concerning-linear-combinations-of-vectors-and-linear-independ.htmlhttps://pop.com.sg/question-concerning-linear-combinations-of-vectors-and-linear-independ.htmlMy question concerns the definition of linear combinations and a criteria for linear independence of a Set (either finite or infinite). Here is the following Definition and Criteria given: A vec......Sat, 15 Aug 2026 10:29:03 -0400https://pop.com.sg/is-there-a-term-do-refer-to-a-set-of-sets-in-math-to-distinguish-set-o.htmlhttps://pop.com.sg/is-there-a-term-do-refer-to-a-set-of-sets-in-math-to-distinguish-set-o.htmlIn mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. for example A = {2, 1, 3} B = {blue, white, red} C = {3, 5, 6} D = {A, C} all the ...Sat, 15 Aug 2026 10:28:46 -0400https://pop.com.sg/what-is-the-difference-between-a-point-and-a-vector.htmlhttps://pop.com.sg/what-is-the-difference-between-a-point-and-a-vector.htmlI understand that a vector has direction and magnitude whereas a point doesn't. However, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do c......Sat, 15 Aug 2026 10:24:29 -0400https://pop.com.sg/is-a-simplified-function-the-same-as-the-original.htmlhttps://pop.com.sg/is-a-simplified-function-the-same-as-the-original.htmlIs a simplified function the same as the original? Example: Let $f(x) = \frac{ax}{x}$, and $g(x) = a$ where $a$ and $x$ are real numbers. Does $g$ = $f$?...Sat, 15 Aug 2026 10:20:59 -0400https://pop.com.sg/good-book-for-self-study-of-continued-fractions.htmlhttps://pop.com.sg/good-book-for-self-study-of-continued-fractions.htmlDoes anyone have a recommendation for a rigorous while readable book to use for the self study of continued fractions? PS - As examples of "rigorous while readable book" for self-learning, A. Pr......Sat, 15 Aug 2026 10:20:18 -0400https://pop.com.sg/average-length-of-the-longest-segment.htmlhttps://pop.com.sg/average-length-of-the-longest-segment.htmlThis post is related to a previous SE post If a 1 meter rope …. concerning average length of a smallest segment. A rope of 1m is divided into three pieces by two random points. Find the average ...Sat, 15 Aug 2026 10:13:46 -0400https://pop.com.sg/splitting-a-circle-into-3-equal-parts-using-2-lines.htmlhttps://pop.com.sg/splitting-a-circle-into-3-equal-parts-using-2-lines.htmlThe picture probably explains my question best. I need to find a way to divide a circle into 3 parts of equal area with only 2 lines that intersect each other on the outline of the circle. Also I n......Sat, 15 Aug 2026 10:10:32 -0400https://pop.com.sg/how-to-solve-equations-of-this-form-x-x-n.htmlhttps://pop.com.sg/how-to-solve-equations-of-this-form-x-x-n.htmlHow would I go about solving equations of this form: $$ x^x = n $$ for values of n that do not have obvious solutions through factoring, such as $27$ ($3^3$) or $256$ ($4^4$). For instance, how ......Sat, 15 Aug 2026 10:06:56 -0400https://pop.com.sg/finding-average-speed-using-the-mean-value-theorem-for-integrals.htmlhttps://pop.com.sg/finding-average-speed-using-the-mean-value-theorem-for-integrals.htmlI am learning how to use the equation $\frac {1}{b-a} \int^b_af(x)dx$. More secifically I dont understand how to apply it to the following problem: A car starting from rest accelerates at a rate ......Sat, 15 Aug 2026 10:01:06 -0400https://pop.com.sg/characterizations-of-local-topology-of-mathbb-r-n.htmlhttps://pop.com.sg/characterizations-of-local-topology-of-mathbb-r-n.htmlWhat topological properties uniquely characterize the local topology of $\mathbb{R}^n$? I know $\mathbb{R}$ is the unique complete ordered field, but that's partly algebraic. Better if the ...Sat, 15 Aug 2026 09:58:40 -0400https://pop.com.sg/verify-using-the-definition-of-convergence-of-a-sequence-that-the-foll.htmlhttps://pop.com.sg/verify-using-the-definition-of-convergence-of-a-sequence-that-the-foll.html(a) lim $\displaystyle\frac{2n^2}{n^3+3} = 0$ (b) lim $\displaystyle\frac{\text{sin}(n^2)}{n^{\frac{1}{3}}} = 0$ For (a) I have: $$\bigg|\displaystyle\frac{2n^2}{n^3+3} - 0\bigg| < \epsilon$$ ......Sat, 15 Aug 2026 09:57:52 -0400https://pop.com.sg/what-is-the-difference-between-a-trace-and-a-contour-in-calculus.htmlhttps://pop.com.sg/what-is-the-difference-between-a-trace-and-a-contour-in-calculus.htmlAs far as I can tell they're exactly the same thing, but the notes here discuss them as if they are separate: The final topic in this section is that of traces.  In some ways these are similar to ...Sat, 15 Aug 2026 09:55:11 -0400https://pop.com.sg/solving-double-derivative-frac-d-2y-dx-2-f-y-by-integration.htmlhttps://pop.com.sg/solving-double-derivative-frac-d-2y-dx-2-f-y-by-integration.htmlI want to ask if a differential equation of second order can be solved by integration? Like equations of the type $\dfrac{d^2y}{dx^2} = f(y)$. I know this can be solved by making equations of the f......Sat, 15 Aug 2026 09:53:01 -0400https://pop.com.sg/plus-or-minus-minus-or-plus-it-is-the-same.htmlhttps://pop.com.sg/plus-or-minus-minus-or-plus-it-is-the-same.htmlAre "Plus or Minus" or "Minus or Plus" the same ? $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$x = \frac{-b\mp \sqrt{b^2-4ac}}{2a}$$ $$x^2=9 \implies x = \pm 3 \quad \text{or}\quad x = \mp3$$ These ...Sat, 15 Aug 2026 09:47:41 -0400https://pop.com.sg/counting-all-6-digit-x-numbers-s-t-x-123456.htmlhttps://pop.com.sg/counting-all-6-digit-x-numbers-s-t-x-123456.htmlWe want to count all the $6$ digits numbers (including ones with $0$'s as digits) that are strictly greater than $123456$ (or even better, some general method of counting for some number $n$ of len......Sat, 15 Aug 2026 09:40:05 -0400https://pop.com.sg/good-examples-of-double-induction.htmlhttps://pop.com.sg/good-examples-of-double-induction.htmlI'm looking for good examples where double induction is necessary. What I mean by double induction is induction on $\omega^2$. These are intended as examples in an "Automatas and Formal Languages" ......Sat, 15 Aug 2026 09:36:34 -0400https://pop.com.sg/surface-area-of-quarter-of-a-sphere.htmlhttps://pop.com.sg/surface-area-of-quarter-of-a-sphere.htmlA quarter sphere with a radius of $10 \text{ units}$. Please help, also remember the sides. I used the normal formula of the total surface area of a sphere and divided it by $4$, then added half ......Sat, 15 Aug 2026 09:31:44 -0400https://pop.com.sg/limit-of-1-x-n-n-when-n-tends-to-infinity-duplicate.htmlhttps://pop.com.sg/limit-of-1-x-n-n-when-n-tends-to-infinity-duplicate.htmlDoes anyone know the exact proof of this limit result? $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$...Sat, 15 Aug 2026 09:20:18 -0400https://pop.com.sg/inverse-of-sparse-matrix-is-not-generally-sparse.htmlhttps://pop.com.sg/inverse-of-sparse-matrix-is-not-generally-sparse.htmlI have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the in......Sat, 15 Aug 2026 09:18:14 -0400https://pop.com.sg/how-to-find-orthogonal-projection-of-vector-on-a-subspace.htmlhttps://pop.com.sg/how-to-find-orthogonal-projection-of-vector-on-a-subspace.htmlWell, I have this subspace: $V = \operatorname{span}\left\{ \begin{pmatrix}\frac{1}{3} \\\frac{2}{3} \\\frac{2}{3}\end{pmatrix},\begin{pmatrix}1 \\3 \\4\end{pmatrix}\right\}$ and the vector $v = \b......Sat, 15 Aug 2026 09:03:07 -0400https://pop.com.sg/show-that-every-ideal-of-the-ring-mathbb-z-is-principal.htmlhttps://pop.com.sg/show-that-every-ideal-of-the-ring-mathbb-z-is-principal.htmlLet $\mathbb Z$ be the ring of integers. The question asks to show that every ideal of $\mathbb Z$ is principal. I beg someone to help me because it is a new concept to me....Sat, 15 Aug 2026 09:02:58 -0400https://pop.com.sg/memoryless-property-and-geometric-distribution.htmlhttps://pop.com.sg/memoryless-property-and-geometric-distribution.htmlSuppose $X$ is a random variable taking values in $\mathbb N_0$ with the memoryless property,i.e., for each pair of number $s,t \in \mathbb N$, $$P(X\geq s+t\mid X>t)=P(X\geq s)$$ Show that a r......Sat, 15 Aug 2026 08:52:09 -0400https://pop.com.sg/how-to-find-position-of-point-that-is-x-unit-distant-from-ab-line-segm.htmlhttps://pop.com.sg/how-to-find-position-of-point-that-is-x-unit-distant-from-ab-line-segm.htmlI am trying to calculate coordinates of point P, which is x units distant from AB line segment and y units distant from BC line segment. Edit: I am trying to write code for general solution. As ...Sat, 15 Aug 2026 08:46:39 -0400https://pop.com.sg/which-matrices-perform-dilation.htmlhttps://pop.com.sg/which-matrices-perform-dilation.htmlFor each angle $\theta \in \mathbb{R}$, we get two corresponding matrices: $$\mathrm{Rot}^\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos\theta\end{bmatrix}, \qqu......Sat, 15 Aug 2026 08:39:22 -0400https://pop.com.sg/example-of-a-simple-pole.htmlhttps://pop.com.sg/example-of-a-simple-pole.htmlI was told that $\operatorname{sech} x$ has a simple pole. Could someone please explain what that means? I have looked up the definition but it involves too much jargon like holomorphic, etc. Is th......Sat, 15 Aug 2026 08:23:02 -0400https://pop.com.sg/rarefaction-wave-in-nonlinear-pde-u-t-u-3-2-x-0.htmlhttps://pop.com.sg/rarefaction-wave-in-nonlinear-pde-u-t-u-3-2-x-0.htmlI am dealing with the following PDE: $$u_t + \big(u^{\frac{3}{2}}\big)_{x} = 0$$ subject to: $$u(x,0) = \begin{cases} 1 & x\leq 0 \\ 4 & 0\leq x\leq 10 \\ 1 & x > ......Sat, 15 Aug 2026 08:22:27 -0400https://pop.com.sg/in-a-family-with-3-children-what-is-the-probability-that-they-have-2-b.htmlhttps://pop.com.sg/in-a-family-with-3-children-what-is-the-probability-that-they-have-2-b.htmlI'm doing some regents practice questions, and one of them asked In a family with 3 children, what is the probability that they have 2 boys and 1 girl? And the answer choices are 3/8 1/4 1......Sat, 15 Aug 2026 08:20:34 -0400https://pop.com.sg/about-plus-minus-sign.htmlhttps://pop.com.sg/about-plus-minus-sign.htmlIf, say, we know that $x=1$, is then the expression $x=\pm 1$ mathematically incorrect? I ask this because when we use $\pm$ sign in the discriminant formula we imply that any of the plus or minus ......Sat, 15 Aug 2026 08:17:39 -0400https://pop.com.sg/organic-chemistry-for-mathematicians.htmlhttps://pop.com.sg/organic-chemistry-for-mathematicians.htmlRecently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do ...Sat, 15 Aug 2026 08:11:17 -0400https://pop.com.sg/find-t-v-using-the-standard-matrix-and-the-matrix-relative-to-b-and-b-.htmlhttps://pop.com.sg/find-t-v-using-the-standard-matrix-and-the-matrix-relative-to-b-and-b-.htmlFind T(V) by using the standard matrix and the matrix relative to B and B' $T: R^3 -> R^2, T(x,y,z) = (x-y + 0, 0 + y-z), v = (2,4,6)$ $B = {(1,1,1), (1,1,0), (0,1,1)}$ $B' = {(1,1),(2,1)}$ ...Sat, 15 Aug 2026 08:07:26 -0400https://pop.com.sg/finding-limit-of-a-sequence-defined-by-another-sequence.htmlhttps://pop.com.sg/finding-limit-of-a-sequence-defined-by-another-sequence.htmlLet there be $\{a_n\}$ such that $a_{n+2}=2a_{n+1}+2a_n$ and $\{b_n\}$ such that $b_n=\frac{a_{n+1}}{a_n}$ if $\lim_{n \to \infty}b_n=L$ what is the limit $L$? I thought about if $L$ exist so I ca......Sat, 15 Aug 2026 08:07:04 -0400https://pop.com.sg/how-does-a-floor-function-work.htmlhttps://pop.com.sg/how-does-a-floor-function-work.htmlI understand what a floor function does, and got a few explanations here, but none of them had a explanation, which is what i'm after. Can someone explain to me what is going on behind the scenes ......Sat, 15 Aug 2026 08:01:40 -0400https://pop.com.sg/can-someone-prove-why-sqrt-ab-sqrt-a-sqrt-b-is-only-valid-when-a-and-b.htmlhttps://pop.com.sg/can-someone-prove-why-sqrt-ab-sqrt-a-sqrt-b-is-only-valid-when-a-and-b.htmlI have seen many people say that a and b can't be positive for example in this false proof : $$1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1} \sqrt{-1} = i^2 = -1$$ Trust me, I understand that $1\neq -1$ ......Sat, 15 Aug 2026 07:48:56 -0400https://pop.com.sg/inversion-formula-for-the-abel-transform.htmlhttps://pop.com.sg/inversion-formula-for-the-abel-transform.htmlI need an inversion formula for the Abel transform $$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$ Hint: The inversion formula found on Wikipedia appears to be incorrect. The "Verific......Sat, 15 Aug 2026 07:36:32 -0400https://pop.com.sg/is-there-a-definition-of-a-pseudo-period-for-f-x-sin-3x-sin-pi-x.htmlhttps://pop.com.sg/is-there-a-definition-of-a-pseudo-period-for-f-x-sin-3x-sin-pi-x.htmlSums of trigonometric functions may or may not be periodic functions; in particular, $\sin(ax)+\sin(bx)$ is periodic if $a/b$ is rational. If we consider the function \begin{equation} f(x) = \sin(......Sat, 15 Aug 2026 07:34:18 -0400https://pop.com.sg/derivative-of-x-mapsto-x-sin-y-x.htmlhttps://pop.com.sg/derivative-of-x-mapsto-x-sin-y-x.htmlIf I differentiate $x\sin(y)$ with respect to $x$ normally then what happens to the $\sin(y)$? In partial differentiation it's taken as constant thus, partially differentiating with respect to $x$ ......Sat, 15 Aug 2026 07:25:09 -0400https://pop.com.sg/a-in-b-vs-a-subset-b-for-proofs.htmlhttps://pop.com.sg/a-in-b-vs-a-subset-b-for-proofs.htmlI have to prove a few different statements. The first is if $A \subset B$ and $B \subset C$ then prove $A \subset C$. This one is fairly straight forward, but I'm stuck on how the next one differ......Sat, 15 Aug 2026 07:09:43 -0400https://pop.com.sg/int-0-infty-sin-ax-sin-bx-dx.htmlhttps://pop.com.sg/int-0-infty-sin-ax-sin-bx-dx.htmlHow can I calculate the integral: $$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx$$ ?? I got stuck.. :/ Could you give me some hint?? Do I have to use the following formula?? $\displaystyle{\sin{(......Sat, 15 Aug 2026 07:03:24 -0400https://pop.com.sg/proof-of-descartes-039-theorem.htmlhttps://pop.com.sg/proof-of-descartes-039-theorem.htmlI came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any proof.Even Wikipedia also, just states the theorem!!I want to know ......Sat, 15 Aug 2026 06:51:58 -0400https://pop.com.sg/anyone-familiar-with-a-mathematical-symbol-for-that-is-to-clarify-wher.htmlhttps://pop.com.sg/anyone-familiar-with-a-mathematical-symbol-for-that-is-to-clarify-wher.htmlMany years ago, a math teacher of mine thought us that there was a mathematical symbol for "that is" / "to clarify" / "whereby" (I'm not from an English speaking country, so I'm writing several ...Sat, 15 Aug 2026 06:49:23 -0400https://pop.com.sg/using-lagrange-multipliers-to-determine-the-point-on-a-surface-nearest.htmlhttps://pop.com.sg/using-lagrange-multipliers-to-determine-the-point-on-a-surface-nearest.htmlI'm attempting to figure this problem out. I would appreciate some guidance on how to get the answer. Thanks. Consider the surface defined as $S: x^2+y^2+z^2 = 8$. If we have a $P = (0,1,1)$, use ...Sat, 15 Aug 2026 06:48:57 -0400https://pop.com.sg/e-2x-e-x-2-can-some-one-explain-how-these-two-are-the-same-i-know-it-i.htmlhttps://pop.com.sg/e-2x-e-x-2-can-some-one-explain-how-these-two-are-the-same-i-know-it-i.html$ (e^{2x})=(e^x)^2$ Can Some one explain how these two are the same? I know it is trick, but I cannot see how it fits into the exponent rules....Sat, 15 Aug 2026 06:27:47 -0400https://pop.com.sg/derivative-of-exponent.htmlhttps://pop.com.sg/derivative-of-exponent.htmlLooking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!...Sat, 15 Aug 2026 06:18:49 -0400https://pop.com.sg/solving-sin-3-theta-1-2-on-the-interval-0-2-pi-i-don-039-t-understand-.htmlhttps://pop.com.sg/solving-sin-3-theta-1-2-on-the-interval-0-2-pi-i-don-039-t-understand-.htmlI am learning precalculus, and I understand how to obtain first two solutions, but I don't understand where did last four solutions came from: All values of $\theta$ in the interval $[0,2\pi]$ t......Sat, 15 Aug 2026 06:18:06 -0400https://pop.com.sg/problem-using-casio-fx-991ce-x-calculator-to-solve-e-i-pi-closed.htmlhttps://pop.com.sg/problem-using-casio-fx-991ce-x-calculator-to-solve-e-i-pi-closed.htmlI'm trying to solve e^(i*pi) on a Casio fx-991CE X calculator. The calculator is in Complex mode (MENU>2:Complex). I Input e^(i*pi) and press "=" and get an error message instead of -1......Sat, 15 Aug 2026 06:16:41 -0400https://pop.com.sg/finding-a-basis-for-w-perp-and-understanding-it.htmlhttps://pop.com.sg/finding-a-basis-for-w-perp-and-understanding-it.htmlGiven $$ w_1 = \begin{bmatrix} 1 \\ -1 \\ 1 \\ 1 \end{bmatrix},w_2= \begin{bmatrix} 0 \\ 1\\2\\3 \end{bmatrix} $$ let $W$ be the subspace spanned by the given vectors. Find a basis for $W^\pe......Sat, 15 Aug 2026 06:15:49 -0400https://pop.com.sg/what-is-x-tx-called-where-x-is-a-matrix.htmlhttps://pop.com.sg/what-is-x-tx-called-where-x-is-a-matrix.htmlWith $X$ being a matrix, is there a general name for the expression $X^TX$?...Sat, 15 Aug 2026 06:09:50 -0400https://pop.com.sg/limit-with-fraction-in-numerator.htmlhttps://pop.com.sg/limit-with-fraction-in-numerator.htmlI need to find the limit for $$ \lim_{x\,\rightarrow\,0} \frac {\frac{1}{x+3} - \frac{1}{3}}{x} $$ It's supposed to be $-1/9$. I've tried changing it around multiple ways, and get the $9$ but nev......Sat, 15 Aug 2026 06:06:29 -0400https://pop.com.sg/calculate-the-dimension-of-the-vector-subspace-u-text-span-left-v-1-v-.htmlhttps://pop.com.sg/calculate-the-dimension-of-the-vector-subspace-u-text-span-left-v-1-v-.htmlGiven are the vectors $v_{1}=\begin{pmatrix} 1\\ 3\\ 5 \end{pmatrix}, v_{2}=\begin{pmatrix} 4\\ 5\\ 6 \end{pmatrix}, v_{3}=\begin{pmatrix} 6\\ 4\\ 2 \end{pmatrix}$ from $\mathbb{R}^{3}$. ...Sat, 15 Aug 2026 05:52:30 -0400https://pop.com.sg/show-that-the-area-of-a-triangle-is-given-by-this-determinant.htmlhttps://pop.com.sg/show-that-the-area-of-a-triangle-is-given-by-this-determinant.htmlI'm not sure how to solve this problem. Can you guys provide some input/hints? Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$. Show that the area of $\...Sat, 15 Aug 2026 05:48:33 -0400https://pop.com.sg/continuity-correction-factor-adding-or-subtracting-0-5.htmlhttps://pop.com.sg/continuity-correction-factor-adding-or-subtracting-0-5.htmlIn this file :http://mkarefin.weebly.com/uploads/2/0/7/9/20793168/assignment_6-solution.pdf At the solution of problem number 11 on top of page 7. The author writes that $$p(x >200) =p(z>......Sat, 15 Aug 2026 05:38:11 -0400https://pop.com.sg/why-is-this-way-used-to-solve-for-base-10-to-base-16.htmlhttps://pop.com.sg/why-is-this-way-used-to-solve-for-base-10-to-base-16.htmlIn one of my courses there is a problem to convert $142$ base $10$ to base $16$, I know the answer is $8E$ base $16$ just through dividing $142$ by $16$ but the solution is show to use base $2$ lik......Sat, 15 Aug 2026 05:37:40 -0400https://pop.com.sg/calculate-the-volume-a-trough-of-length-250cm-and-cross-sectional-area.htmlhttps://pop.com.sg/calculate-the-volume-a-trough-of-length-250cm-and-cross-sectional-area.htmlI am working on volumes of prisms, using $\pi$, diameter, and radius; however, I haven't done any problems with cross-sectional areas before... Also, I'm only 14 and am finding this really hard! ......Sat, 15 Aug 2026 05:30:09 -0400https://pop.com.sg/i-i-d-what-does-this-stand-for.htmlhttps://pop.com.sg/i-i-d-what-does-this-stand-for.htmlSo almost everywhere in the book it's written "random variables are IID", what does this mean? I think it means independent and identically distributed but not sure. So by definition A and B R.V ......Sat, 15 Aug 2026 05:11:28 -0400https://pop.com.sg/can-the-word-derive-be-used-to-mean-take-the-derivative-of.htmlhttps://pop.com.sg/can-the-word-derive-be-used-to-mean-take-the-derivative-of.htmlBack when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be "different......Sat, 15 Aug 2026 05:07:00 -0400https://pop.com.sg/what-does-f-5-2-tell-me-about-f-x.htmlhttps://pop.com.sg/what-does-f-5-2-tell-me-about-f-x.htmlAs part of a math quiz I was given the following two functions: f(-5) = -2 and g(x) = -2 * f(x) The way I interpret the first function is "return -2 regardless of the input" since the right hand ......Sat, 15 Aug 2026 05:00:46 -0400https://pop.com.sg/integration-and-calculating-the-area-under-a-trig-function.htmlhttps://pop.com.sg/integration-and-calculating-the-area-under-a-trig-function.htmlSo my assignment states the following: $$\text{Let}\;\;f(x) = -\sin(x),\,\;\text{where}\;\,x \in \left[-\dfrac{\pi}{4}, \dfrac{\pi}{6}\right]$$ Ok, so I know the forum isn't here to do my homewor......Sat, 15 Aug 2026 05:00:01 -0400https://pop.com.sg/square-fibonacci-numbers.htmlhttps://pop.com.sg/square-fibonacci-numbers.htmlAre there Fibonacci numbers other than $F_0 = 0 = 0^2, F_1 = F_2 = 1 = 1^2,$ and $F_{12} = 144 = 12^2$ which are square numbers? If not, what is the proof?...Sat, 15 Aug 2026 04:47:42 -0400https://pop.com.sg/what-does-cramer-039-s-theorem-tell-us.htmlhttps://pop.com.sg/what-does-cramer-039-s-theorem-tell-us.htmlCramer's Theorem States, Let $(Y_i)_{i\geq 1}$ be a sequence of i.i.d. random variables, ${S_n=\frac{1}{n}\sum_{i=1}^n Y_i}$ be their average sum and $M_{Y_1}(u):=\mathrm{E}[e^{uY_1}]<\infty$ b......Sat, 15 Aug 2026 04:39:34 -0400https://pop.com.sg/prove-that-three-points-lie-on-a-semicircle.htmlhttps://pop.com.sg/prove-that-three-points-lie-on-a-semicircle.htmlI have three points $A(-10,-12)$, $B(6,18)$, $C(-2,-14)$. I have to prove they exist on the same semicircle. How? If I try to draw the three points to investigate I see that I can draw a right tria......Sat, 15 Aug 2026 04:38:52 -0400https://pop.com.sg/finding-the-nth-derivative-of-y-e-ax-b.htmlhttps://pop.com.sg/finding-the-nth-derivative-of-y-e-ax-b.htmlHow to find the $n$-th derivative of $y=e^{ax+b}$ Please provide an explanation of the steps. Thanks....Sat, 15 Aug 2026 04:36:25 -0400https://pop.com.sg/there-is-no-set-of-all-sets.htmlhttps://pop.com.sg/there-is-no-set-of-all-sets.htmlSo there are a few posts about this already, but they skip over the problem I have. The proof of Cantor's theorem elegantly shows that if we consider $f:A\rightarrow \mathcal{P}(A)$, the set $B=\{......Sat, 15 Aug 2026 04:36:16 -0400https://pop.com.sg/stokes-039-theorem-with-a-sphere.htmlhttps://pop.com.sg/stokes-039-theorem-with-a-sphere.htmlUse Stokes' theorem to evaluate $$ \iint_S \operatorname{curl} F \cdot \hat{n}\, dS $$ where $F =\langle xyz, x, e^{xy} \cos(z)\rangle$ $S$ is the hemisphere $x^2+y^2+z^2=25$ for $z ≥ 0$ oriented ...Sat, 15 Aug 2026 04:34:03 -0400https://pop.com.sg/what-does-the-graph-of-a-norm-look-like.htmlhttps://pop.com.sg/what-does-the-graph-of-a-norm-look-like.htmlI talking about functions of the form ||X||^p, when p>1 for different values of p. I know these are all convex functions, but I don't know how to graph them....Sat, 15 Aug 2026 04:31:49 -0400https://pop.com.sg/how-to-remember-the-trigonometric-identities.htmlhttps://pop.com.sg/how-to-remember-the-trigonometric-identities.htmlI have a test tomorrow and I am having trouble remembering those pesky trigonometrical identities (such as $1-\cos x=2\sin^2(\frac{x}{2})$ ) Do you guys have any tips on how I can remember these? ...Sat, 15 Aug 2026 04:29:18 -0400https://pop.com.sg/proof-that-trace-of-039-hat-039-matrix-in-linear-regression-is-rank-of.htmlhttps://pop.com.sg/proof-that-trace-of-039-hat-039-matrix-in-linear-regression-is-rank-of.htmlI understand that the trace of the projection matrix (also known as the "hat" matrix) X*Inv(X'X)*X' in linear regression is equal to the rank of X. How can we prove that from first principles, i.e. ...Sat, 15 Aug 2026 04:14:38 -0400https://pop.com.sg/how-to-factorise-x-2-x-1.htmlhttps://pop.com.sg/how-to-factorise-x-2-x-1.htmlI was asked to prepare a set of notes for year 12 students. These notes are going to be used by these students to prepare for the examinations. Naturally, I searched for problems to include so the ...Sat, 15 Aug 2026 04:12:17 -0400https://pop.com.sg/real-life-examples-for-eigenvalues-eigenvectors.htmlhttps://pop.com.sg/real-life-examples-for-eigenvalues-eigenvectors.htmlThere are already good answers about importance of eigenvalues / eigenvectors, such as this question and some others, as well as this Wikipedia article. I know the theory and these examples, but n......Sat, 15 Aug 2026 04:05:20 -0400https://pop.com.sg/is-there-an-operator-for-adding-the-numerator-and-denominator-of-a-fra.htmlhttps://pop.com.sg/is-there-an-operator-for-adding-the-numerator-and-denominator-of-a-fra.htmlNumbers in the Farey sequence are expressed as fractions e.g $F_5$: $0\over1$ $1\over5$ $1\over4$ $1\over3$ $2\over5$ $1\over2$ $3\over5$ $2\over3$ $3\over4$ $4\over5$ $1\over1$ All of the $n\ove......Sat, 15 Aug 2026 03:55:26 -0400https://pop.com.sg/why-multiply-first.htmlhttps://pop.com.sg/why-multiply-first.htmlWhy do we multiply/divide first, and then add/subtract later? I mean, what I'm curious about is that is this a universal rule, or a man-decided rule? Also how would you decide on which to operate ......Sat, 15 Aug 2026 03:52:12 -0400https://pop.com.sg/how-to-find-the-volume-of-oblique-cone.htmlhttps://pop.com.sg/how-to-find-the-volume-of-oblique-cone.htmlIt is probably dublicated but couldnt find. Let radius=r and height=h, find the volume of oblique cone with use of integral if it was right cone I'd use $y=h-\frac{xh}{r}$ so $\pi\int_0^h(\fra......Sat, 15 Aug 2026 03:48:42 -0400https://pop.com.sg/magnitude-of-a-vector-in-spherical-and-cylindrical-coordinates.htmlhttps://pop.com.sg/magnitude-of-a-vector-in-spherical-and-cylindrical-coordinates.htmlThe magnitude of a vector in Cartesian coordinates is just the length of the vector. But what about a vector in spherical coordinates and cylindrical coordinates? Do we just take the root of its dot ...Sat, 15 Aug 2026 03:46:39 -0400https://pop.com.sg/can-functions-have-multiple-inputs.htmlhttps://pop.com.sg/can-functions-have-multiple-inputs.htmlNow bear with me here, I'm not the best at math. I'm just trying to find out something that I never really learned. I was wondering, can a function have multiple inputs such as this one below? $$f......Sat, 15 Aug 2026 03:40:56 -0400https://pop.com.sg/nth-term-of-series-with-distinct-difference.htmlhttps://pop.com.sg/nth-term-of-series-with-distinct-difference.htmlI am learning some series as a naive and I came across this series where I want to find the nth term. 2, 6, 14, 30, 62, ... and so on. I am facing difficulties as the difference between elements......Sat, 15 Aug 2026 03:40:10 -0400https://pop.com.sg/how-do-irrational-numbers-lie-on-the-number-line.htmlhttps://pop.com.sg/how-do-irrational-numbers-lie-on-the-number-line.htmlIf we construct a square with side length 1, take its diagonal length : $\sqrt{2}$ However I still don't understand HOW it can lie on the number line. Imagine another irrational number $\pi = 3....Sat, 15 Aug 2026 03:35:38 -0400https://pop.com.sg/determine-the-number-of-eigenvalues-in-mathbb-r.htmlhttps://pop.com.sg/determine-the-number-of-eigenvalues-in-mathbb-r.htmlIn this problem using the characteristics equation it comes out that there are four complex roots two identical each . So according to question there is no eigenvalue on $\mathbb{R}$ but in $\mathb......Sat, 15 Aug 2026 03:31:04 -0400https://pop.com.sg/what-does-number-of-carries-mean.htmlhttps://pop.com.sg/what-does-number-of-carries-mean.htmlOn this lecture note on page no 4 Corollary 14 the term number of carries has been used. Can anyone explain what does it mean?...Sat, 15 Aug 2026 03:22:25 -0400https://pop.com.sg/how-does-cos-pi-1.htmlhttps://pop.com.sg/how-does-cos-pi-1.htmlI know this is a very elementary question, but how does $\cos(\pi) = -1$? I thought the cosine function required a minimum of 2 numbers, the adjacent side and hypotenuse of a triangle?...Sat, 15 Aug 2026 02:56:38 -0400https://pop.com.sg/normal-curvature-along-a-line-of-curvature.htmlhttps://pop.com.sg/normal-curvature-along-a-line-of-curvature.htmlI have come across the following exercise (the context is curves and surfaces in $\mathbb{R}^3$ and the Gauss map): If $C=\alpha(I)$ is a line of curvature, and $k$ is its curvature at $p$, then......Sat, 15 Aug 2026 02:49:49 -0400https://pop.com.sg/what-does-the-exclamation-mark-do.htmlhttps://pop.com.sg/what-does-the-exclamation-mark-do.htmlI've seen this but never knew what it does. Can any one let me in on the details? Thanks....Sat, 15 Aug 2026 02:39:18 -0400https://pop.com.sg/cyclic-subgroup-of-s-n.htmlhttps://pop.com.sg/cyclic-subgroup-of-s-n.htmlFind the smallest natural number n ∈ ℕ such that $S_n$ contains a cyclic subgroup of order 101. Proof: We seek the smallest n such that Sn contains a permutation of order 101. Permutation can be w......Sat, 15 Aug 2026 02:38:41 -0400https://pop.com.sg/questions-tagged-number-theory.htmlhttps://pop.com.sg/questions-tagged-number-theory.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 15 Aug 2026 02:38:20 -0400https://pop.com.sg/are-one-by-one-matrices-equivalent-to-scalars.htmlhttps://pop.com.sg/are-one-by-one-matrices-equivalent-to-scalars.htmlI am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are of......Sat, 15 Aug 2026 02:29:50 -0400https://pop.com.sg/how-do-i-represent-an-equilateral-triangle-in-cartesian-coordinates-ce.htmlhttps://pop.com.sg/how-do-i-represent-an-equilateral-triangle-in-cartesian-coordinates-ce.htmlI am trying to figure out how to find the Cartesian coordinates of an equilateral triangle centered around (0,0)....Sat, 15 Aug 2026 02:14:23 -0400https://pop.com.sg/cartesian-products-and-sets.htmlhttps://pop.com.sg/cartesian-products-and-sets.htmlI'm really confused on how to approach this question: Recall that the Cartesian product $A \times A$ is defined as the set $\{(x, y) : x \in A \land y \in A \}$. Thus if for example $A = \{1, 2,......Sat, 15 Aug 2026 02:12:59 -0400https://pop.com.sg/how-to-find-complex-eigenvectors-from-complex-eigenvalues.htmlhttps://pop.com.sg/how-to-find-complex-eigenvectors-from-complex-eigenvalues.htmlI have this matrix that represents a $2 \times 2$ linear system and I am supposed to solve to find what $x(t)$ and $y(t)$ are. $\left[ {\begin{array}{cc} 1 & 5 \\ -1 & -3 \\ \end{......Sat, 15 Aug 2026 02:08:02 -0400https://pop.com.sg/how-to-express-an-angle-of-90-degrees-between-two-lines.htmlhttps://pop.com.sg/how-to-express-an-angle-of-90-degrees-between-two-lines.htmlIf I would extend two lines $l_1$ and $l_2$ they would intersect with an angle of 90 degrees. How should I write with math terms that there would be a 90 degree angle. I assume $l_1 \perp l_2$ is w......Sat, 15 Aug 2026 02:06:06 -0400https://pop.com.sg/deriving-mean-and-variance-of-laplace-distribution.htmlhttps://pop.com.sg/deriving-mean-and-variance-of-laplace-distribution.htmlIt has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 \sigma}\,e^{-\Large\......Sat, 15 Aug 2026 02:06:03 -0400