https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 15 Aug 2026 02:33:33 -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 -0400https://pop.com.sg/trivial-and-non-trivial-closed.htmlhttps://pop.com.sg/trivial-and-non-trivial-closed.htmlHow do we know what to do when asked to find the solutions of given equations saying find the solution when it has non trivial solution??...Sat, 15 Aug 2026 01:58:58 -0400https://pop.com.sg/function-for-second-quadrant-in-unit-circle.htmlhttps://pop.com.sg/function-for-second-quadrant-in-unit-circle.htmlI am no "math-guy" and would really appreciate some help in writing a function for the second quadrant of the unit circle. Conditions I want to be met: Centre of the circle is moved so s......Sat, 15 Aug 2026 01:57:44 -0400https://pop.com.sg/why-does-the-imaginary-number-i-satisfy-i-times-0-0.htmlhttps://pop.com.sg/why-does-the-imaginary-number-i-satisfy-i-times-0-0.htmlWhy does the imaginary number $i$ satisfy $i\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that....Sat, 15 Aug 2026 01:54:05 -0400https://pop.com.sg/what-is-the-name-for-the-shape-formed-by-removing-a-square-from-the-co.htmlhttps://pop.com.sg/what-is-the-name-for-the-shape-formed-by-removing-a-square-from-the-co.htmlSquares of consecutive numbers differ by the sum of those numbers, so $6^2 = 5^2 + 5 + 6$. Geometrically, this is because the difference between the two squares is a pair of strips, $5$ and $6$ un......Sat, 15 Aug 2026 01:51:02 -0400https://pop.com.sg/equation-of-a-circle-tangent-to-the-y-axis.htmlhttps://pop.com.sg/equation-of-a-circle-tangent-to-the-y-axis.htmlThe problem is: Find the general equation of a circumference with center at $(-4,3)$ and tanget to the $y$ axis I know that calculating the distance between the center and any point on the ...Sat, 15 Aug 2026 01:34:12 -0400https://pop.com.sg/prove-that-every-convex-function-is-continuous.htmlhttps://pop.com.sg/prove-that-every-convex-function-is-continuous.htmlA function $f : (a,b) \to \Bbb R$ is said to be convex if $$f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$$ whenever $a < x, y < b$ and $0 < \lambda <1$. Prove that every co......Sat, 15 Aug 2026 01:31:13 -0400https://pop.com.sg/finding-integrating-factor-for-inexact-differential-equation.htmlhttps://pop.com.sg/finding-integrating-factor-for-inexact-differential-equation.htmlSuppose we have the following differential equation that is NOT exact, i.e. $M_y \ne N_x$: $2xy^3+y^4+(xy^3-2y)y'=0$ How would I find an integrating factor $μ(x,y)$ so that when I multiply this ...Sat, 15 Aug 2026 01:31:11 -0400https://pop.com.sg/finding-the-limit-of-a-piecewise-function.htmlhttps://pop.com.sg/finding-the-limit-of-a-piecewise-function.htmlI had a question about finding limits of piecewise functions through graphs. I believe, I am missing something in my fundamentals about finding limits for these functions. Firstly, I would like to ...Sat, 15 Aug 2026 01:27:49 -0400https://pop.com.sg/invertability-of-singular-2x2-matrix-with-all-same-real-values.htmlhttps://pop.com.sg/invertability-of-singular-2x2-matrix-with-all-same-real-values.htmlQuestion: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...Sat, 15 Aug 2026 01:20:20 -0400https://pop.com.sg/what-039-s-the-degree-of-a-multivariate-polynomial-in-artin-algebra.htmlhttps://pop.com.sg/what-039-s-the-degree-of-a-multivariate-polynomial-in-artin-algebra.htmlAccording to Degree of a polynomial, it is highest degree among the monomials. Where is this in Artin Algebra? In Chapter 11.9, an exercise gives a degree to an irreducible complex polynomial of two ...Sat, 15 Aug 2026 01:16:33 -0400https://pop.com.sg/how-do-i-evaluate-this-integral-given-a-table-of-values-for-f-x-and-f-.htmlhttps://pop.com.sg/how-do-i-evaluate-this-integral-given-a-table-of-values-for-f-x-and-f-.htmlThe problem gives me a list of values for $f(x)$ and $f'(x)$. I tried solving the problem using integration by parts, allowing $u=\sin(2x), u'=2\cos(2x), v'=f'(\cos(2x))$, and $v=f(\cos(2x))$. I use ...Sat, 15 Aug 2026 01:13:22 -0400https://pop.com.sg/can-different-quaternions-represent-the-same-orientation.htmlhttps://pop.com.sg/can-different-quaternions-represent-the-same-orientation.htmlFor a current project I'm working on I have to use quaternions to represent the orientation of an object. The piece of code I'm working on now integrates rotation rates to the quaternion represent......Sat, 15 Aug 2026 01:09:26 -0400https://pop.com.sg/why-bbb-z-integers-is-an-integral-domain.htmlhttps://pop.com.sg/why-bbb-z-integers-is-an-integral-domain.htmlIs there a proof for $\Bbb Z$ being an integral domain? Or is it an axiom that if $a*b=0$ then $a=0$ or $b=0$ where $a$, $b$ belong to $\Bbb Z$. Is it so obvious?...Sat, 15 Aug 2026 01:01:08 -0400https://pop.com.sg/proving-holder-039-s-inequality-for-sums.htmlhttps://pop.com.sg/proving-holder-039-s-inequality-for-sums.htmlI want to prove the Holder's inequality for sums: Let $p\ge1$ be a real number. Let $(x_{k})\in l_{p}$ and $(y_{k})\in l_{q}$ . Then, $$\overset{\infty}{\underset{k=1}{\sum}}\vert x......Sat, 15 Aug 2026 00:58:52 -0400https://pop.com.sg/how-to-graph-gradient-vector.htmlhttps://pop.com.sg/how-to-graph-gradient-vector.htmlI'm working on a practice problem for my Calculus 3 course. It gives the function: $z=x^2+y^2$, and asks to graph the contours for $c=1,2,3$. Than asks to calculate the gradient at point $(2,1)$ ......Sat, 15 Aug 2026 00:57:18 -0400https://pop.com.sg/decomposition-group-and-inertia-group.htmlhttps://pop.com.sg/decomposition-group-and-inertia-group.htmlLet $L/K$ be a Galois extension with Galois group $G$. Let $O_K$ and $O_L$ be the ring of algebraic integers of $K$ and $L$ respectively. Let $P\subseteq O_K$ be a prime. Let $Q\subseteq O_L$ be a ......Sat, 15 Aug 2026 00:53:33 -0400https://pop.com.sg/euler-path-for-directed-graph.htmlhttps://pop.com.sg/euler-path-for-directed-graph.htmlHow do we find Euler path for directed graphs? I don't seem to get the algorithm below! Algorithm To find the Euclidean cycle in a digraph (enumerate the edges in the cycle), using a greedy proc......Sat, 15 Aug 2026 00:48:58 -0400https://pop.com.sg/depressed-cubic-roots.htmlhttps://pop.com.sg/depressed-cubic-roots.htmlAccording to Wikipedia, a depressed cubic has only one root and 2 imaginary roots. Is this true? Can a depressed cubic of the form $x^3+px+q=0$ have 2 or 3 real roots? Edit: Here is a screenshot o......Sat, 15 Aug 2026 00:48:11 -0400https://pop.com.sg/how-to-self-study-topology.htmlhttps://pop.com.sg/how-to-self-study-topology.htmlI'm a first year undergraduate student and I'm a math major. Currently, I'm taking an intro to analysis class and a linear algebra class. However, I often feel constrained by what I do in class and......Sat, 15 Aug 2026 00:43:29 -0400https://pop.com.sg/proof-of-the-dirichlet-dini-criterion-for-pointwise-convergence-of-fou.htmlhttps://pop.com.sg/proof-of-the-dirichlet-dini-criterion-for-pointwise-convergence-of-fou.htmlI have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (source: Wikipedia) if $f$ is $2\pi$–periodic, locally integrable and ...Sat, 15 Aug 2026 00:42:49 -0400https://pop.com.sg/do-other-trigonometric-functions-than-tan-sin-cos-and-their-derivative.htmlhttps://pop.com.sg/do-other-trigonometric-functions-than-tan-sin-cos-and-their-derivative.htmlI remember my physics teacher mentioning that other trigonometric functions exist apart from the Sin Cos and Tan, he mentioned a few and they did not sound familiar, nothing like Sec Csc and Cot. I......Sat, 15 Aug 2026 00:42:26 -0400https://pop.com.sg/definition-of-sheaf-using-equalizer.htmlhttps://pop.com.sg/definition-of-sheaf-using-equalizer.htmlWikipedia give sheaf property using equalizer diagram by saying sheaf property means for any open cover $\{U_i\}$ of $U$ $$F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\...Sat, 15 Aug 2026 00:36:42 -0400https://pop.com.sg/questions-tagged-sequences-and-series.htmlhttps://pop.com.sg/questions-tagged-sequences-and-series.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 15 Aug 2026 00:25:17 -0400https://pop.com.sg/solving-the-congruence-7x-3-1-mod-31.htmlhttps://pop.com.sg/solving-the-congruence-7x-3-1-mod-31.htmlI am having a problem when the LHS has an addition function; if the question is just a multiple of $x$ it's fine. But when I have questions like $3x+3$ or $4x+7$, I don't seem to get the right ans......Sat, 15 Aug 2026 00:21:55 -0400https://pop.com.sg/finding-an-orthogonal-vector-by-inspection.htmlhttps://pop.com.sg/finding-an-orthogonal-vector-by-inspection.htmlIf I have a vector say: $$ \begin{bmatrix} -1 \\ -\frac{1}{2} \\ 1 \end{bmatrix} $$ How can I identify a vector orthogonal to this by inspection? For example if I have this vector: $ \begin{bmatr......Sat, 15 Aug 2026 00:16:44 -0400https://pop.com.sg/will-this-be-equivalent-to-d-f-x-dx.htmlhttps://pop.com.sg/will-this-be-equivalent-to-d-f-x-dx.htmlJust wondering if $$d(f(e^x))/d(e^x)$$ would be equivalent to $$d(f(x))/d(x)$$ and if not what could you do to it to find $$d(f(x))/d(x)$$...Sat, 15 Aug 2026 00:12:32 -0400https://pop.com.sg/in-the-definition-of-k-partite-graph-is-it-necessary-that-k-should-be-.htmlhttps://pop.com.sg/in-the-definition-of-k-partite-graph-is-it-necessary-that-k-should-be-.htmlIt is well known that a graph is said to be $k$-partite if its vertex set can be partitioned in to k non empty sets such that no two vertices in the same set are adjacent. My question is whether ......Sat, 15 Aug 2026 00:12:03 -0400https://pop.com.sg/definition-of-overlapping-sets.htmlhttps://pop.com.sg/definition-of-overlapping-sets.htmlIs this definition of overlapping sets correct? If the intersection of two sets is non-empty set but neither is a subset of the other, the sets are called overlapping sets It is written in ......Fri, 14 Aug 2026 23:50:30 -0400https://pop.com.sg/what-039-s-the-chance-of-a-random-whole-number-between-1-999-having-th.htmlhttps://pop.com.sg/what-039-s-the-chance-of-a-random-whole-number-between-1-999-having-th.htmlI've thought about this problem the following way. If this was represented by 3 dials which have number 0-9 in them ([0-9][0-9][0-9]) then the chance of hitting 9 in one of them is 1/10. Since ther......Fri, 14 Aug 2026 23:38:26 -0400https://pop.com.sg/using-exchange-argument-in-proving-greedy-algorithm.htmlhttps://pop.com.sg/using-exchange-argument-in-proving-greedy-algorithm.htmlHere's a problem solvable by greedy algorithm: You are a company and you have list of tasks that still need to be done (but you're late with them already). For a given task we have information about ...Fri, 14 Aug 2026 23:38:26 -0400https://pop.com.sg/negative-exponents-in-binomial-theorem.htmlhttps://pop.com.sg/negative-exponents-in-binomial-theorem.htmlI'm looking at extensions of the binomial formula to negative powers. I've figured out how to do $n \choose k$ when $n < 0 $ and $k \geq 0$: $${n \choose k} = (-1)^k {-n + k - 1 \choose k}$$ So......Fri, 14 Aug 2026 23:30:10 -0400https://pop.com.sg/super-hard-system-of-equations.htmlhttps://pop.com.sg/super-hard-system-of-equations.htmlSolve the system of equation for real numbers \begin{split} (a+b) &(c+d) &= 1 & \qquad (1)\\ (a^2+b^2)&(c^2+d^2) &= 9 & \qquad (2)\\ (a^3+b^3)&(c^3+d^3) &= 7......Fri, 14 Aug 2026 23:29:49 -0400https://pop.com.sg/partial-fraction-decomp-why-repeated-factors-and-why-the-cx-d-for-quad.htmlhttps://pop.com.sg/partial-fraction-decomp-why-repeated-factors-and-why-the-cx-d-for-quad.html1) First, why are powers of linear factors repeated? For example, $(x+1)^2$ gets $\frac{A}{(x+1)}$ and $\frac{B}{(x+1)^2}$ 2) Why does a quad factor like $x^2+1$ get a term $\frac{Cx+d}{x^2+1}$ ...Fri, 14 Aug 2026 23:15:20 -0400https://pop.com.sg/how-to-use-the-lagrange-039-s-remainder-to-prove-that-log-1-x-sum.htmlhttps://pop.com.sg/how-to-use-the-lagrange-039-s-remainder-to-prove-that-log-1-x-sum.htmlUsing Lagrange's remainder, I have to prove that: $\log(1+x) = \sum\limits_{n=1}^\infty (-1)^{n+1} \cdot \frac{x^n}{n}, \; \forall |x| < 1$ I am not quite sure how to do this. I started with ......Fri, 14 Aug 2026 23:14:26 -0400https://pop.com.sg/number-of-ways-to-divide-n-people-into-k-groups-with-at-least-2-people.htmlhttps://pop.com.sg/number-of-ways-to-divide-n-people-into-k-groups-with-at-least-2-people.htmlI'm trying to figure out the number of ways to divide n people into k groups with at least 2 people in each group. Should I first decide a recurrence relation of the number? I don't know how I could ...Fri, 14 Aug 2026 23:07:34 -0400https://pop.com.sg/do-two-right-triangles-with-the-same-length-hypotenuse-have-the-same-a.htmlhttps://pop.com.sg/do-two-right-triangles-with-the-same-length-hypotenuse-have-the-same-a.htmlI watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with......Fri, 14 Aug 2026 22:57:45 -0400https://pop.com.sg/does-0-chance-mean-impossible-duplicate.htmlhttps://pop.com.sg/does-0-chance-mean-impossible-duplicate.htmlSuppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in......Fri, 14 Aug 2026 22:46:07 -0400https://pop.com.sg/why-do-we-multiply-by-100-for-finding-percentage.htmlhttps://pop.com.sg/why-do-we-multiply-by-100-for-finding-percentage.htmlFor example, if there are 50 boys in a school and the total number of students is $200$ then, for finding percentage of boys why do we do like this, $\left(\, 50/200\, \right)100\ \% = 25\ \%$&nbsp......Fri, 14 Aug 2026 22:38:35 -0400https://pop.com.sg/proof-of-the-dkw-inequality.htmlhttps://pop.com.sg/proof-of-the-dkw-inequality.htmlMy goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= \frac{1}{......Fri, 14 Aug 2026 22:32:44 -0400https://pop.com.sg/how-to-solve-x-6-6x-5-0.htmlhttps://pop.com.sg/how-to-solve-x-6-6x-5-0.htmlIs there any elementary way to solve (i.e. not involving derivatives etc.) $x^6 - 6x + 5 > 0$ ? I can divide it two times by $(x - 1)$ to get $x^6 - 6x + 5 = (x - 1)^2 (x^4 + 2 x^3 + 3 x^2 + 4 x......Fri, 14 Aug 2026 22:31:39 -0400https://pop.com.sg/how-to-understand-intuitively-the-stolz-cesaro-theorem-for-sequences.htmlhttps://pop.com.sg/how-to-understand-intuitively-the-stolz-cesaro-theorem-for-sequences.htmlI have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the proof and verified that, yes, step B does indeed follow logically from step A, etc. and ...Fri, 14 Aug 2026 22:27:15 -0400https://pop.com.sg/check-whether-a-point-is-within-a-3d-triangle.htmlhttps://pop.com.sg/check-whether-a-point-is-within-a-3d-triangle.htmlI have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on and inside the 3D triangle? So, for example, if I have a plane defined by $......Fri, 14 Aug 2026 22:20:47 -0400https://pop.com.sg/trying-to-understand-what-a-polynomial-ring-is.htmlhttps://pop.com.sg/trying-to-understand-what-a-polynomial-ring-is.htmlI don' really understand what a polynomial ring is, maybe because the lack of examples. Consider for example the polynomial ring $\mathbb{Z}[x]$. Can you please tell me how this polynomial ring (its ...Fri, 14 Aug 2026 22:20:35 -0400https://pop.com.sg/what-are-autonomous-and-non-autonomous-systems.htmlhttps://pop.com.sg/what-are-autonomous-and-non-autonomous-systems.htmlWhat are autonomous and non-autonomous systems and how are they different from each other. What are the differences between the types of systems they are describing? Do both autonomous and non-auto......Fri, 14 Aug 2026 22:11:06 -0400https://pop.com.sg/maximum-area-perimeter-ratio.htmlhttps://pop.com.sg/maximum-area-perimeter-ratio.htmlWith no limitation, to achieve the maximum area with a fixed perimeter, the shape is a circle, and the area / perimeter ratio would be $\frac{L}{4\pi}$ where $L$ is the perimeter lenght. However, ......Fri, 14 Aug 2026 22:10:37 -0400https://pop.com.sg/position-of-a-particle-moving-along-the-x-axis-question.htmlhttps://pop.com.sg/position-of-a-particle-moving-along-the-x-axis-question.htmlI am in need of help solving this question. The position of a particle moving along the $x$-axis is given by the function $s(t) = t^3 -6t^2 +9t$ where $s$ is in meters and $t$ is in seconds. When......Fri, 14 Aug 2026 22:06:24 -0400https://pop.com.sg/visualizing-why-rotations-preserve-orientation.htmlhttps://pop.com.sg/visualizing-why-rotations-preserve-orientation.htmlIt's clear geometrically that if you have two vectors in $\mathbb{R}^3$ a rotation will preserve their lengths and the angle between them. But how do you visualize that a rotation preserves orienta......Fri, 14 Aug 2026 22:02:00 -0400https://pop.com.sg/solving-x-2-2x-3-0.htmlhttps://pop.com.sg/solving-x-2-2x-3-0.htmlIf i have to solve $x^2-2x-3<0$ I would do $$x+1 < 0, \quad x-3<0$$ and end up getting $x<-1$ and $x<3$. Why is it wrong to use the same inequality sign? Shouldn’t both of the li......Fri, 14 Aug 2026 22:01:19 -0400https://pop.com.sg/functional-minimization-with-constraints.htmlhttps://pop.com.sg/functional-minimization-with-constraints.htmlSuppose you have a multi-variable functional given as $$ \int_{t_0}^{t_1} L(t,x,x', x'' ..., y, y', y'', ... ) \ dt $$ That you wish to to optimize [i.e find an $x$ and $y$ that maximize] but subje......Fri, 14 Aug 2026 21:55:07 -0400https://pop.com.sg/why-can-039-t-i-use-the-chain-rule-to-solve-this-trigonometric-integra.htmlhttps://pop.com.sg/why-can-039-t-i-use-the-chain-rule-to-solve-this-trigonometric-integra.htmlThe question is: what is the indefinite integral: $\int \sin^2(kx) \, \mathrm dx$? I get the correct answer using trig identities to change the $(\sin(kx))^2$ into $\dfrac{1}{2} - \dfrac{(\cos(2kx......Fri, 14 Aug 2026 21:46:43 -0400https://pop.com.sg/different-definitions-for-semisimple-lie-group.htmlhttps://pop.com.sg/different-definitions-for-semisimple-lie-group.htmlI am confused about two definitions for the notion of a semisimple Lie group i found. Lets say for simplicity i am only interested in matrix groups. In this case, do the following two object-classes ...Fri, 14 Aug 2026 21:46:22 -0400https://pop.com.sg/there-is-a-formal-definition-of-mathematical-space.htmlhttps://pop.com.sg/there-is-a-formal-definition-of-mathematical-space.htmlYesterday I was searching for a formal definition of mathematical space, then I found it wikipedia article, where the definition rely in the concept of mathematical structure. But I didnt found a ......Fri, 14 Aug 2026 21:45:46 -0400https://pop.com.sg/how-to-determine-and-explain-the-sum-of-angles-without-measuring.htmlhttps://pop.com.sg/how-to-determine-and-explain-the-sum-of-angles-without-measuring.htmlBelow is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f. I understand that angles are formed my intersecting ...Fri, 14 Aug 2026 21:40:23 -0400