https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 04 Sep 2026 04:43:55 -0400https://pop.com.sg/why-does-imread-generate-3-images-closed.htmlhttps://pop.com.sg/why-does-imread-generate-3-images-closed.htmlThe MATLAB help for imread says: A = imread(filename, fmt) The return value A is an array containing the image data. If the file contains a grayscale image, A is an M-by-N array. If the f......Fri, 04 Sep 2026 08:39:34 -0400https://pop.com.sg/how-do-i-find-percentiles-of-data-sets-even-vs-odd.htmlhttps://pop.com.sg/how-do-i-find-percentiles-of-data-sets-even-vs-odd.htmlGiven the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sam......Fri, 04 Sep 2026 08:37:48 -0400https://pop.com.sg/finding-the-height-given-the-angle-of-elevation-and-depression.htmlhttps://pop.com.sg/finding-the-height-given-the-angle-of-elevation-and-depression.htmlPlease, I need help for this problem. I'm a little confused about it :( From a point A 10ft. above the water the angle of elevation of the top of a lighthouse is 46 degrees and the angle of depr......Fri, 04 Sep 2026 08:32:04 -0400https://pop.com.sg/find-n-of-geometric-sequence-closed.htmlhttps://pop.com.sg/find-n-of-geometric-sequence-closed.htmlI need to find the $n$ of a geometric sequence, this formula is to find the $n$th term, $n$th = $ar ^{n-1}$; so I need $n$ to be on the left instead of $n$th, I have all the other variables except......Fri, 04 Sep 2026 08:26:10 -0400https://pop.com.sg/why-is-ln-cos-x-equal-to-ln-sec-x.htmlhttps://pop.com.sg/why-is-ln-cos-x-equal-to-ln-sec-x.htmlWhy does the value $-\ln(|\cos(x)|)$ become $\ln(|\sec(x)|)?$ I was doing an integral and I got my final answer as that, but I don't understand how you can just send the negative sign inside and m......Fri, 04 Sep 2026 08:23:00 -0400https://pop.com.sg/find-the-coordinates-of-a-point-on-a-circle.htmlhttps://pop.com.sg/find-the-coordinates-of-a-point-on-a-circle.htmlI have a circle like so $r$, with angle $\theta$ to the $y$-axis"> Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 ......Fri, 04 Sep 2026 08:16:14 -0400https://pop.com.sg/equation-of-a-line-as-a-determinant-equation.htmlhttps://pop.com.sg/equation-of-a-line-as-a-determinant-equation.htmlI've been looking for an answer in the forums and Google but couldn't find one. Given two different points in the plane, $(x_1,\,y_1)$ and $(x_2,\,y_2)$, we know that the equation of that line can......Fri, 04 Sep 2026 08:13:11 -0400https://pop.com.sg/how-to-express-transformations-of-exponential-functions-stretches-and-.htmlhttps://pop.com.sg/how-to-express-transformations-of-exponential-functions-stretches-and-.htmlWhen referring to the transformations of exponential functions, how do you refer to vertical "stretches" and "compressions" in terms of "a dilation of $z$ from the y-axis o......Fri, 04 Sep 2026 08:10:17 -0400https://pop.com.sg/how-to-calculate-the-moment-of-inertia-of-a-uniform-disk-about-an-axis.htmlhttps://pop.com.sg/how-to-calculate-the-moment-of-inertia-of-a-uniform-disk-about-an-axis.htmlConsider a uniform disk of mass m and radius R, and let's calculate the moment of inertia about an axis passing through a diameter of the disk. One easy way is to use the perpendicular axis theorem: ...Fri, 04 Sep 2026 08:01:57 -0400https://pop.com.sg/isomorphism-vs-equality-of-graphs.htmlhttps://pop.com.sg/isomorphism-vs-equality-of-graphs.htmlI have just started studying graph theory and having trouble with understanding the difference b/w isomorphism and equality of two graphs.According what I have studied so far, I am able to conclude......Fri, 04 Sep 2026 07:52:11 -0400https://pop.com.sg/what-is-a-co-prime.htmlhttps://pop.com.sg/what-is-a-co-prime.htmlI've never encountered this question in any of my math classes and it just shows up randomly in my comsci class with no further info about it. I've wiki'ed it, but can't even understand that. Could ...Fri, 04 Sep 2026 07:50:42 -0400https://pop.com.sg/basic-equivalence-class-discrete-math.htmlhttps://pop.com.sg/basic-equivalence-class-discrete-math.htmlI read through the textbook definition of the equivalence class, but still cannot clearly understand what an equivalence class is. Does anyone have a good example with a definition that can hit m......Fri, 04 Sep 2026 07:47:03 -0400https://pop.com.sg/what-is-the-outward-normal-vector-at-the-end-points-of-the-one-dimensi.htmlhttps://pop.com.sg/what-is-the-outward-normal-vector-at-the-end-points-of-the-one-dimensi.htmlIn the one-dimensional space, suppose the interval $[a, b]$ exists on the real line with boundary points $a$ and $b$. What is the outward normal vector at the point $a$ and at the point $b$? The an......Fri, 04 Sep 2026 07:43:35 -0400https://pop.com.sg/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-.htmlhttps://pop.com.sg/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-.htmlI was reading the wikipedia page for symmetric matrices, and I noticed this part: a real n×n matrix A is symmetric if and only if there is an orthonormal basis of Rn consisting of eigenvectors f......Fri, 04 Sep 2026 07:30:59 -0400https://pop.com.sg/how-to-pronounce-tableaux-closed.htmlhttps://pop.com.sg/how-to-pronounce-tableaux-closed.htmlHow do you pronounce Young tableaux? Does it sound just like its singular form?...Fri, 04 Sep 2026 07:23:53 -0400https://pop.com.sg/finding-cosets-and-index-in-d8.htmlhttps://pop.com.sg/finding-cosets-and-index-in-d8.htmla). Find all the left and right cosets of $<τ>$ in $D_8$. $<τ>$ is reflection and I know that there are two reflections about a diagonal. So I am wondering how to represent this idea a......Fri, 04 Sep 2026 07:10:48 -0400https://pop.com.sg/show-that-the-sum-of-2n-1-consecutive-integers-is-divisible-by-2n-1.htmlhttps://pop.com.sg/show-that-the-sum-of-2n-1-consecutive-integers-is-divisible-by-2n-1.htmlBase case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds. Let $n=k>=1$ and let $2k+1$ be true. We want to show $2......Fri, 04 Sep 2026 07:06:12 -0400https://pop.com.sg/zassenhaus-formula-exponents.htmlhttps://pop.com.sg/zassenhaus-formula-exponents.htmlfound Zassenhaus formula for noncommutative $X,Y$ $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X......Fri, 04 Sep 2026 07:05:15 -0400https://pop.com.sg/right-triangular-pyramid.htmlhttps://pop.com.sg/right-triangular-pyramid.htmlI am somehow confused with this problem. It is said that this is a right triangular pyramid, but it has an equilateral triangle base? I am confused. How can I find VA? It doesn't look like it has a......Fri, 04 Sep 2026 06:53:20 -0400https://pop.com.sg/strangest-notation-closed.htmlhttps://pop.com.sg/strangest-notation-closed.htmlWhile this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?...Fri, 04 Sep 2026 06:53:19 -0400https://pop.com.sg/calculators-using-taylor-polynomials.htmlhttps://pop.com.sg/calculators-using-taylor-polynomials.htmlI've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?...Fri, 04 Sep 2026 06:48:47 -0400https://pop.com.sg/derivative-of-y-sec-x-2-cos-x.htmlhttps://pop.com.sg/derivative-of-y-sec-x-2-cos-x.htmlAccording to wolfram the derivative of this is zero. This derivative though is part of a larger problem and I don't think it is possible for it to be zero. According to my own calculation the answe......Fri, 04 Sep 2026 06:46:47 -0400https://pop.com.sg/attempt-to-view-irrational-number-as-a-fraction.htmlhttps://pop.com.sg/attempt-to-view-irrational-number-as-a-fraction.htmlI am wondering if an irrational number can be represented as a fraction in this way: For example (to represent $\pi$): $$\pi= 3.14159265359...=\frac{314159265359...}{100000000000...}$$ In the fr......Fri, 04 Sep 2026 06:42:08 -0400https://pop.com.sg/closed-vs-compact-surface.htmlhttps://pop.com.sg/closed-vs-compact-surface.htmlWikipedia defines a surface to be a two-dimensional manifold, and a closed surface to be a surface that is compact and without boundary. Am I correct that this definition of "closed surface" is not ...Fri, 04 Sep 2026 06:40:22 -0400https://pop.com.sg/what-does-overlap-two-functions-mean.htmlhttps://pop.com.sg/what-does-overlap-two-functions-mean.htmlWhat does overlap mean in the case of two functions. Or atleast, what does overlap mean in the case of 2 lines or 2 curves. Thanks in advance. The complete sentence to clarify the context: ...Fri, 04 Sep 2026 06:38:31 -0400https://pop.com.sg/expectation-of-geometric-brownian-motion.htmlhttps://pop.com.sg/expectation-of-geometric-brownian-motion.htmlI was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t......Fri, 04 Sep 2026 06:37:24 -0400https://pop.com.sg/what-does-within-the-same-order-of-magnitude-convey.htmlhttps://pop.com.sg/what-does-within-the-same-order-of-magnitude-convey.htmlThis question originates from a quandary about the meaning of the statement that two values are within the same order of magnitude. I wonder whether there is an established usage, of (rather more ...Fri, 04 Sep 2026 06:10:17 -0400https://pop.com.sg/are-all-sets-ordered-sets.htmlhttps://pop.com.sg/are-all-sets-ordered-sets.htmlThe definition of an ordered set according to W. Rudin in his book, Principles of Mathematical Analysis is: An ordered set is a set S in which an order is defined He also defined order in his boo......Fri, 04 Sep 2026 06:08:56 -0400https://pop.com.sg/finding-formula-for-partial-sum-of-polynomial-terms.htmlhttps://pop.com.sg/finding-formula-for-partial-sum-of-polynomial-terms.htmlFor example, we know that the following is true (and can be easily derived): $\sum\limits_{x=1}^{n}x = \frac{1}{2}n(n+1)$ But, what if we wanted to find the sum of a series like this: $\sum\limi......Fri, 04 Sep 2026 06:08:18 -0400https://pop.com.sg/what-do-small-and-large-mean-for-sets-and-categories.htmlhttps://pop.com.sg/what-do-small-and-large-mean-for-sets-and-categories.htmlCategories for the Working Mathematicians says In addition to the metacategory of all sets ~ which is not a set ~ we want an actual category $Set$, the category of all small sets. We shall ...Fri, 04 Sep 2026 06:00:00 -0400https://pop.com.sg/why-do-we-assume-in-the-principle-of-strong-complete-induction.htmlhttps://pop.com.sg/why-do-we-assume-in-the-principle-of-strong-complete-induction.htmlI get induction. (i) show that the statement holds when n=1 (or some basis) (ii) show that the statement holds for a general subsequent case, $n+1$. By PMI the statement holds for all cases. Do......Fri, 04 Sep 2026 05:57:48 -0400https://pop.com.sg/vieta-039-s-formula-failed.htmlhttps://pop.com.sg/vieta-039-s-formula-failed.htmlFind the value of $p$ if $p$ and $q$ are the roots of the equation, $x^2+px+q=0, \ \ \{x,p,q\}\in\ \mathbb{R}$ By using vieta's formula for sum and product of roots, $\begin{cases} p+q=-p \......Fri, 04 Sep 2026 05:54:25 -0400https://pop.com.sg/what-does-mean-in-math.htmlhttps://pop.com.sg/what-does-mean-in-math.htmlIn a Linear Algebra textbook I am reading, the following is stated: $b\notin \operatorname{span}(A \cup \{a\})\setminus \operatorname{span}(A)$. It does so without explaining what "$\setminus$" mea......Fri, 04 Sep 2026 05:45:19 -0400https://pop.com.sg/can-we-compute-the-area-of-a-quadrilateral-with-one-right-angle-when-w.htmlhttps://pop.com.sg/can-we-compute-the-area-of-a-quadrilateral-with-one-right-angle-when-w.htmlI took an IQ test for fun recently, but I take issue with the answer to one of the questions. Here's the question: My issue is that the explanation assumes angle DC is a right angle. Given that ...Fri, 04 Sep 2026 05:39:26 -0400https://pop.com.sg/a-demonstration-of-lagrange-039-s-form-for-the-remainder-of-a-taylor-s.htmlhttps://pop.com.sg/a-demonstration-of-lagrange-039-s-form-for-the-remainder-of-a-taylor-s.htmlThe following argument for Lagrange's Form for the Remainder of a Taylor polynomial is a typical one in analysis books. Even in the case of finding the remainder when the Taylor polynomial is a li......Fri, 04 Sep 2026 05:38:49 -0400https://pop.com.sg/how-does-sinc-interpolation-work.htmlhttps://pop.com.sg/how-does-sinc-interpolation-work.htmlEvery now and then I come across mention of sinc interpolation. Trying to read up on it, I have yet to get what it's about. I have done basic DSP work, have programmed stuff using FFT (using just a ...Fri, 04 Sep 2026 05:25:31 -0400https://pop.com.sg/microeconomics-calculating-tax-revenue-and-tax-incidence.htmlhttps://pop.com.sg/microeconomics-calculating-tax-revenue-and-tax-incidence.htmlAustralian Government has imposed a tax on Beer. Assume that the tax on Beer is $20 per unit (a unit is a carton of drinks) Assume the demand and supply functions for cartons of Beers per week are:......Fri, 04 Sep 2026 05:22:46 -0400https://pop.com.sg/why-does-the-graph-not-show-the-vertical-asymptote.htmlhttps://pop.com.sg/why-does-the-graph-not-show-the-vertical-asymptote.htmlI have a rational function in which the denominator is equal to zero when $x = -1.4142135$. So: $f(x)=\dfrac{1}{x + 1.4142135}$ Then the vertical asymptote at $x = -1.4142135$ is pretty clear. ......Fri, 04 Sep 2026 05:13:12 -0400https://pop.com.sg/simple-proof-of-krein-smulian.htmlhttps://pop.com.sg/simple-proof-of-krein-smulian.htmlI have a question concerning a simple and short proof of Krein-Shmulian, the proof can be found here https://people.math.ethz.ch/~jteichma/slides_ftap.pdf on page 35/36. Here the proof: Let $X$ be a ...Fri, 04 Sep 2026 05:04:37 -0400https://pop.com.sg/if-f-039-2-0-and-f-039-039-2-4-what-can-you-say-about-f.htmlhttps://pop.com.sg/if-f-039-2-0-and-f-039-039-2-4-what-can-you-say-about-f.htmlI was doing very well in Calculus up until this point. I realize that concavity and $f'$ and $f''$ require one to really visualize what is happening with a function, but can someone please help me......Fri, 04 Sep 2026 05:04:14 -0400https://pop.com.sg/why-characteristic-zero-and-not-infinite-characteristic.htmlhttps://pop.com.sg/why-characteristic-zero-and-not-infinite-characteristic.htmlThe characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we de......Fri, 04 Sep 2026 04:57:20 -0400https://pop.com.sg/notation-for-compact-sets.htmlhttps://pop.com.sg/notation-for-compact-sets.htmlIs there some accepted notation for a set that is compact? E.g. I am currently writing "... [blah] is true if for every compact set $A \subset \mathbb{R}^n$ and ...". I could simplify my writing if ...Fri, 04 Sep 2026 04:44:39 -0400https://pop.com.sg/how-many-3-digit-positive-integers-are-odd-and-do-not-contain-digit-5.htmlhttps://pop.com.sg/how-many-3-digit-positive-integers-are-odd-and-do-not-contain-digit-5.htmlIt is a GRE question. And it has been answered here. But I still want to ask it again, just to know why I am wrong. The correct is 288. My idea is, first I get the total number of 3-digit integer......Fri, 04 Sep 2026 04:29:59 -0400https://pop.com.sg/a-specific-example-of-the-gns-construction.htmlhttps://pop.com.sg/a-specific-example-of-the-gns-construction.htmlIn an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X......Fri, 04 Sep 2026 04:27:43 -0400https://pop.com.sg/the-transitive-closure-of-a-relation-r-on-set-a-whose-relation-matrix.htmlhttps://pop.com.sg/the-transitive-closure-of-a-relation-r-on-set-a-whose-relation-matrix.htmlThe transitive closure of a relation $R$ on set $A$ whose relation matrix My attempt: If I assumed node $(1, 2, 3)$, then given is: $R=\{(1, 2), (2, 3), (3, 1)\}$ The transitive closer would be......Fri, 04 Sep 2026 04:25:00 -0400https://pop.com.sg/proving-1-frac-1-4-frac-1-9-cdots-frac-1-n-2-leq-2-frac-1-n-for-all-n-.htmlhttps://pop.com.sg/proving-1-frac-1-4-frac-1-9-cdots-frac-1-n-2-leq-2-frac-1-n-for-all-n-.htmlQuestion: Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case. Attempt at soluti......Fri, 04 Sep 2026 04:23:26 -0400https://pop.com.sg/how-to-solve-int-c-im-z-dz-where-c-is-the-unit-circle.htmlhttps://pop.com.sg/how-to-solve-int-c-im-z-dz-where-c-is-the-unit-circle.htmlI want to solve $\int_C Im(z) dz$ where $C$ is the unit circle. The first thing is to parameterize the curve, $C$: $$ z(t)=e^{it} $$ where $0\leq t \leq 2\pi$ Then $$ \frac {dz}{dt} = ie^{it} $$ ......Fri, 04 Sep 2026 04:23:02 -0400https://pop.com.sg/isometries-and-orthogonal-matrices.htmlhttps://pop.com.sg/isometries-and-orthogonal-matrices.htmlI know how to show that multiplying by an orthogonal matrix preserves the angle and distance between two vectors. I have seen everywhere that Orthogonal matrices are kind of related to rotations and ...Fri, 04 Sep 2026 04:19:00 -0400https://pop.com.sg/if-type-theories-are-all-logics.htmlhttps://pop.com.sg/if-type-theories-are-all-logics.htmlSo it sounds like Higher Order Logic (HOL) and Type Theory are equivalent. Then there is Intuitionistic Logic and Intuitionistic Type Theory, but I'm not sure of the connection there. I am just ...Fri, 04 Sep 2026 04:19:00 -0400https://pop.com.sg/universal-factoring-method-or-list-of-methods-trinomials.htmlhttps://pop.com.sg/universal-factoring-method-or-list-of-methods-trinomials.htmlI am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it seems to exceedingly hard to find anything ...Fri, 04 Sep 2026 04:11:49 -0400https://pop.com.sg/for-which-angles-we-know-the-sin-value-algebraically-exact.htmlhttps://pop.com.sg/for-which-angles-we-know-the-sin-value-algebraically-exact.htmlFor example: $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$ $\sin(30^\circ) = \frac{1}{2}$ $\sin(45^\circ) = \frac{1}{\sqrt{2}}$ $\s......Fri, 04 Sep 2026 04:07:09 -0400https://pop.com.sg/finding-linear-dependence-relation.htmlhttps://pop.com.sg/finding-linear-dependence-relation.htmlFind a linear dependence relation between the following vectors: x1 = (1, -1, 2) x2 = (-3, 2, 1) x3= (1, 2, -3) x4= (2, 3, 1) I've already created a matrix and reduced and I know how to tell w......Fri, 04 Sep 2026 04:06:42 -0400https://pop.com.sg/how-to-simplify-sin-x-y-cos-y-cos-x-y-sin-y.htmlhttps://pop.com.sg/how-to-simplify-sin-x-y-cos-y-cos-x-y-sin-y.htmlthe question How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$ my steps I tried to use trig identities on the $\sin(x-y)$ and $\cos(x-y)$ and tried to distribute the others in but it didn't wo......Fri, 04 Sep 2026 03:58:46 -0400https://pop.com.sg/use-rational-zero-theorem-to-find-real-zeros-of-2x-3-3x-2-x-1.htmlhttps://pop.com.sg/use-rational-zero-theorem-to-find-real-zeros-of-2x-3-3x-2-x-1.htmlI am to use the rational zero theorem to find the real zeros of $2x^3-3x^2-x+1$. The provided answers are $\frac{1}{2}$, $\frac{1+\sqrt{5}}{2}$ and $\frac{1-\sqrt{5}}{2}$. I was able to get $\frac{......Fri, 04 Sep 2026 03:53:55 -0400https://pop.com.sg/is-there-such-an-inequality-between-product-and-integral-for-functions.htmlhttps://pop.com.sg/is-there-such-an-inequality-between-product-and-integral-for-functions.htmlGiven a measure space $(\Omega, \mathbb{F},\mu)$ and any two measurable real-valued functions $f,g$ defined on $\Omega$, I was wondering if there is an inequality like $$ \int_\Omega |f*g| d\mu \l......Fri, 04 Sep 2026 03:42:03 -0400https://pop.com.sg/finding-the-standard-deviation-of-a-probability-distribution.htmlhttps://pop.com.sg/finding-the-standard-deviation-of-a-probability-distribution.htmlHere is the question: The time, to the nearest whole minute, that a city bus takes to go from one end of its route to the other has the probability distribution shown. As sometimes happens with ...Fri, 04 Sep 2026 03:41:02 -0400https://pop.com.sg/unordered-sampling-with-replacement.htmlhttps://pop.com.sg/unordered-sampling-with-replacement.htmlI have difficulties with understanding of how do we count this. Why can't we just divide number of ways of ordered sample with replacement $n^k$ by $k!$, i.e. to get rid of permutations since we c......Fri, 04 Sep 2026 03:38:59 -0400https://pop.com.sg/how-to-find-the-area-of-a-triangle-with-lengths-of-heights.htmlhttps://pop.com.sg/how-to-find-the-area-of-a-triangle-with-lengths-of-heights.htmlGiven the lengths of 3 heights in a triangle, I need to find its area....Fri, 04 Sep 2026 03:37:07 -0400https://pop.com.sg/change-the-cartesian-integral-into-an-equivalent-polar-integral.htmlhttps://pop.com.sg/change-the-cartesian-integral-into-an-equivalent-polar-integral.htmlI started learning double integrals. I am trying to solve one problem but its not going well :). $$ \int_{-1}^{1} \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} \ln\bigl(x^2+y^2+1\bigr)\, dx\,dy $$ then I wro......Fri, 04 Sep 2026 03:33:02 -0400https://pop.com.sg/switching-iterated-integrals-from-dxdy-to-dydx.htmlhttps://pop.com.sg/switching-iterated-integrals-from-dxdy-to-dydx.htmlI am having issues switching an iterated integral from dxdy to dydx: Switch the integral $\int_0^2 \int_y^{2y}6xy$ dx dy to a dy dx integral in the form of $\int_0^2 \int_{...}^{...}6xydydx$ + $\...Fri, 04 Sep 2026 03:28:30 -0400https://pop.com.sg/how-to-solve-differential-equation-y-039-2y.htmlhttps://pop.com.sg/how-to-solve-differential-equation-y-039-2y.htmlHow to solve differential equation $y'=2y$ when $y(7)=6$ ? The answer is $6e^{2(x-7)}$, but i don't get how come it is $(x-7)$? I know that it is from formula of $y(x)=Ce^{kx}$, where $C=6$ and $......Fri, 04 Sep 2026 03:26:13 -0400https://pop.com.sg/amazing-isomorphisms-closed.htmlhttps://pop.com.sg/amazing-isomorphisms-closed.htmlJust as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether......Fri, 04 Sep 2026 03:17:53 -0400https://pop.com.sg/what-is-the-integral-of-x-ln-x.htmlhttps://pop.com.sg/what-is-the-integral-of-x-ln-x.htmlI need calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank you !...Fri, 04 Sep 2026 03:16:08 -0400https://pop.com.sg/why-do-we-not-measure-angles-like-this.htmlhttps://pop.com.sg/why-do-we-not-measure-angles-like-this.htmlAn angle is a simple geometric figure that is obtained by taking a ray and rotating it about a point to form another ray, both of which have their starting point in common. The initial rays is call......Fri, 04 Sep 2026 03:12:46 -0400https://pop.com.sg/discrete-mathematics-relations-question.htmlhttps://pop.com.sg/discrete-mathematics-relations-question.htmlif r2 is in the set of N*N ( natural numbers) with (X,y) in the subset of r2, if and only if x+y=0 is it reflexive? is it symmetric? is it anti symmetric? is it Transitive? i said it is reflect......Fri, 04 Sep 2026 03:05:49 -0400https://pop.com.sg/find-the-maximum-radius-of-a-incircle.htmlhttps://pop.com.sg/find-the-maximum-radius-of-a-incircle.htmlI was soving geometry problem. And found this question , i tried to solve it .But i am unable to get the answer. The hypotenuse of a right triangle has length of $5$ cm. Determine its maximum pos......Fri, 04 Sep 2026 03:03:41 -0400https://pop.com.sg/how-to-prove-that-if-the-determinant-of-the-matrix-is-zero-then-at-lea.htmlhttps://pop.com.sg/how-to-prove-that-if-the-determinant-of-the-matrix-is-zero-then-at-lea.htmlFor a matrix $A$, if $\det(A)=0,$ prove and provide an example that at least one eigenvalue must be zero. At first, I tried using the identity that the product of eigenvalues is the determinant of......Fri, 04 Sep 2026 02:59:14 -0400https://pop.com.sg/simplifying-cos-arcsin-x-closed.htmlhttps://pop.com.sg/simplifying-cos-arcsin-x-closed.htmlI remember reading somewhere that we can simplify $\cos(\arcsin x)$ and $\sin(\arccos x)$ in terms of a polynomial by making the substitution $m=\arcsin x$ or $m=\arccos x$ (respectively), then ...Fri, 04 Sep 2026 02:55:21 -0400https://pop.com.sg/getting-a-mixed-probability-distribution-function-from-a-mixed-cumulat.htmlhttps://pop.com.sg/getting-a-mixed-probability-distribution-function-from-a-mixed-cumulat.htmlI am having trouble understanding how to get a complete mixed distribution function from a mixed CDF. The question is from my actuarial exam p manual. Namely, the discrete portion is confusing me. ......Fri, 04 Sep 2026 02:52:49 -0400https://pop.com.sg/integrate-over-t-if-dr-dt-sqrt-1-r-2.htmlhttps://pop.com.sg/integrate-over-t-if-dr-dt-sqrt-1-r-2.htmlI'm stuck on this step trying to solve a math puzzle. How do I integrate this?...Fri, 04 Sep 2026 02:50:30 -0400https://pop.com.sg/proof-that-the-runge-phenomenon-occurs.htmlhttps://pop.com.sg/proof-that-the-runge-phenomenon-occurs.htmlIs there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?...Fri, 04 Sep 2026 02:45:58 -0400https://pop.com.sg/fluid-force-on-semicircular-gate-partially-submerged.htmlhttps://pop.com.sg/fluid-force-on-semicircular-gate-partially-submerged.htmlWhat is the best way to approach this problem? Does the problem change since the gate is only partially submerged?...Fri, 04 Sep 2026 02:45:47 -0400https://pop.com.sg/figuring-out-the-percentage-of-a-in-relation-to-b.htmlhttps://pop.com.sg/figuring-out-the-percentage-of-a-in-relation-to-b.htmlI have two numbers 13.49 and 7.8 I want to know how to figure out how much 7.8 is of 13.49 in percentage e.g. 7.8 is 80% of 13.49 What is the best way to do this?...Fri, 04 Sep 2026 02:40:30 -0400https://pop.com.sg/divide-circle-into-9-pieces-of-equal-area.htmlhttps://pop.com.sg/divide-circle-into-9-pieces-of-equal-area.htmlI'd like to divide a unit circle disk into nine parts of equal area, using circle arcs as delimiting lines. The whole setup should be symmetric under the symmetry group of the square, i.e. 4 mirro......Fri, 04 Sep 2026 02:36:35 -0400https://pop.com.sg/sufficient-conditions-for-the-spectral-decomposition.htmlhttps://pop.com.sg/sufficient-conditions-for-the-spectral-decomposition.htmlI know that is possible to apply the spectral decomposition (diagonalization) to a matrix when the sum of the dimensions of its eigenspaces is equal to the size of the matrix. The spectral decompo......Fri, 04 Sep 2026 02:24:30 -0400https://pop.com.sg/solve-sin-3-x-cos-3-x-1.htmlhttps://pop.com.sg/solve-sin-3-x-cos-3-x-1.htmlSolve for $x\\ \sin^{3}x+\cos^{3}x=1$ $\sin^{3}x+\cos^{3}x=1\\(\sin x+\cos x)(\sin^{2}x-\sin x\cdot\cos x+\cos^{2}x)=1\\(\sin x+\cos x)(1-\sin x\cdot\cos x)=1$ What should I do next?...Fri, 04 Sep 2026 02:16:25 -0400https://pop.com.sg/limit-evaluation-conjugate-method-further-algebraic-manipulation.htmlhttps://pop.com.sg/limit-evaluation-conjugate-method-further-algebraic-manipulation.html$$\lim_\limits{x\to 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}$$ I have tried evaluating the above limit by multiplying either both the conjugate of numerator and the denominator with no avail in exitin......Fri, 04 Sep 2026 02:11:54 -0400https://pop.com.sg/10-bit-2-039-s-complement-of-positive-numbers.htmlhttps://pop.com.sg/10-bit-2-039-s-complement-of-positive-numbers.htmlWhen doing this for a negative number, like -213 for example, I successfully found the answer by doing $(2^9 + -327)$ => $(2^9 - 213)$ taking the result of that, and using the division by 2 method, ...Fri, 04 Sep 2026 02:09:18 -0400https://pop.com.sg/why-does-the-sequence-frac-1-n-diverge.htmlhttps://pop.com.sg/why-does-the-sequence-frac-1-n-diverge.htmlMy math teacher said that it is not a Cauchy sequence thus it isn't convergent but divergent. It is bounded and its limit exists ($=0$). Now i am really confused. I thought that things started to get ...Fri, 04 Sep 2026 02:08:04 -0400https://pop.com.sg/the-circumference-of-the-circle-is-c-what-is-the-area-of-circle-in-ter.htmlhttps://pop.com.sg/the-circumference-of-the-circle-is-c-what-is-the-area-of-circle-in-ter.htmlThe circumference of the circle is $C$, what is the area of circle in terms of $C$? a). $\dfrac {C^2}{4\pi }$ b). $2\pi C$ c). $\dfrac {4}{3} \pi C^2$ d). $2\pi C^2$ My Attempt: $$\textrm {...Fri, 04 Sep 2026 02:05:54 -0400https://pop.com.sg/derivative-with-multiplication-and-division.htmlhttps://pop.com.sg/derivative-with-multiplication-and-division.htmlSo I have the following homework. I don't want the answer, only point me in the right direction please. Thanks. I'm stuck in the product rule. Do I apply the product rule twice or just one time af......Fri, 04 Sep 2026 02:01:50 -0400https://pop.com.sg/express-this-limit-as-a-definite-integral-no-interval-given-lim-limits.htmlhttps://pop.com.sg/express-this-limit-as-a-definite-integral-no-interval-given-lim-limits.htmlI am having trouble trying to convert a limit to a definite integral. I am unsure about how to go about this. I have already tried googling this but can not find anything that is comprehensive enou......Fri, 04 Sep 2026 02:00:21 -0400https://pop.com.sg/what-does-this-039-l-039-and-upside-down-039-l-039-symbol-mean.htmlhttps://pop.com.sg/what-does-this-039-l-039-and-upside-down-039-l-039-symbol-mean.htmlI need help finding out what the following symbols are called and what they do. I searched up math symbols but couldn't find them anywhere near there. $$\lceil{-3.14}\rceil=$$ $$\lfloor{-3.14}\rf......Fri, 04 Sep 2026 01:58:18 -0400https://pop.com.sg/write-the-exact-answer-using-base-10-logarithms.htmlhttps://pop.com.sg/write-the-exact-answer-using-base-10-logarithms.htmlSolve for x. $11^{-8x} = 18^{-x+10}$ Write the exact answer using base-10 logarithms. Every online tutorial I've looked up does not have a step by step process explaining how they got it....Fri, 04 Sep 2026 01:51:42 -0400https://pop.com.sg/differential-equations-ivp-application.htmlhttps://pop.com.sg/differential-equations-ivp-application.htmlI am given $$y'=0.05y-800$$ I am asked to: (a) Find all constant solutions of the differential equation. (b) Suppose $y = M$ is your constant solution from (a). Plot two solutions of the differen......Fri, 04 Sep 2026 01:47:59 -0400https://pop.com.sg/simplify-sqrt-4-frac-162x-6-16x-4-is-frac-3-sqrt-4-2x-2-2.htmlhttps://pop.com.sg/simplify-sqrt-4-frac-162x-6-16x-4-is-frac-3-sqrt-4-2x-2-2.html(I've been posting a lot today and yesterday, not sure if too many posts are frowned upon or not. I am studying and making sincere efforts to solve on my own and only post here as a last resort) I'm ...Fri, 04 Sep 2026 01:36:13 -0400https://pop.com.sg/proof-of-drinker-paradox-duplicate.htmlhttps://pop.com.sg/proof-of-drinker-paradox-duplicate.htmlI searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...Fri, 04 Sep 2026 01:34:31 -0400https://pop.com.sg/fair-5-sided-die.htmlhttps://pop.com.sg/fair-5-sided-die.htmlCommonly used polyhedral dice all have sides of the same shape. However: is this strictly required? Take a triangular prism. A triangular prism with small height compared to the base edge would al......Fri, 04 Sep 2026 01:33:09 -0400https://pop.com.sg/compute-the-double-sum-sum-n-m-0-n-neq-m-frac-1-n-left-m-2-n-2-right-f.htmlhttps://pop.com.sg/compute-the-double-sum-sum-n-m-0-n-neq-m-frac-1-n-left-m-2-n-2-right-f.htmlI am trying to compute the following double sum $$\boxed{\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)}$$ I proceeded as following $$\sum_{n=1}^{\infty}\sum_{m......Fri, 04 Sep 2026 01:25:06 -0400https://pop.com.sg/definition-of-good-reduction.htmlhttps://pop.com.sg/definition-of-good-reduction.htmlLet $C/\mathbb{Q}$ be a curve. I am trying to understand the difference between (1) There is a model $\mathcal{C}/\mathbb{Z}$ of $C$ such that the fiber $\mathcal{C}_p$ is regular and (2) ......Fri, 04 Sep 2026 01:24:26 -0400https://pop.com.sg/symbol-for-the-set-of-integers-from-1-to-n-duplicate.htmlhttps://pop.com.sg/symbol-for-the-set-of-integers-from-1-to-n-duplicate.htmlIs there a special symbol for the set: $$ \{1, 2, 3, \dots, n\}$$, or $$\{x| 1\leq x\leq n, n \in \mathbb{Z} \} $$?...Fri, 04 Sep 2026 01:14:06 -0400https://pop.com.sg/an-open-ball-is-an-open-set.htmlhttps://pop.com.sg/an-open-ball-is-an-open-set.htmlProve that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open. My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an $r_1\in\mathbb{R^......Fri, 04 Sep 2026 01:12:47 -0400https://pop.com.sg/finding-plane-equation-given-two-lines.htmlhttps://pop.com.sg/finding-plane-equation-given-two-lines.htmlDetermine an equation of the plane containing the lines $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-5}{6}$; $r=\lt1,-1,5\gt+t\lt1,1,-3\gt$. I calculated the cross product between the directional vector......Fri, 04 Sep 2026 01:09:26 -0400https://pop.com.sg/could-someone-explain-me-what-is-a-clopen-set.htmlhttps://pop.com.sg/could-someone-explain-me-what-is-a-clopen-set.htmlI came across this term while studying for a test and got curious as to what this actually is. could someone please the concept in a relatively easy language?...Fri, 04 Sep 2026 01:00:20 -0400https://pop.com.sg/what-is-the-point-of-non-commutative-probability.htmlhttps://pop.com.sg/what-is-the-point-of-non-commutative-probability.htmlI read this article by Terry Tao about non-commutative probability. I honestly don't really get it. Apparently, we can generalize Kolmogorov's system of probability theory, in such a way that it i......Fri, 04 Sep 2026 01:00:13 -0400https://pop.com.sg/solving-cos4-theta-cos3-theta-to-find-the-roots-of-8x-3-4x-2-4x-1-0.htmlhttps://pop.com.sg/solving-cos4-theta-cos3-theta-to-find-the-roots-of-8x-3-4x-2-4x-1-0.htmlFind all the solution to the equation $\cos 4\theta=\cos 3\theta$. Hence or otherwise, show that the roots of the equation $$8x^3+4x^2-4x-1=0$$ are $$\cos \frac{2\pi}{7},\cos \frac{4\pi}{7},\cos \f......Fri, 04 Sep 2026 00:59:49 -0400https://pop.com.sg/what-are-the-common-abbreviation-for-minimum-in-equations.htmlhttps://pop.com.sg/what-are-the-common-abbreviation-for-minimum-in-equations.htmlI'm searching for some symbol representing minimum that is commonly used in math equations....Fri, 04 Sep 2026 00:59:48 -0400https://pop.com.sg/is-the-laplacian-a-vector-or-a-scalar.htmlhttps://pop.com.sg/is-the-laplacian-a-vector-or-a-scalar.htmlNeed to prove $\operatorname{div}(\nabla u)=\nabla ^2 u$ where $u=g(x,y,z)$ The RHS is the Lapacian which we were told is a vector. But $\nabla u=(g_x,g_y,g_z)$ and the divergence of that is $g_{x......Fri, 04 Sep 2026 00:56:13 -0400https://pop.com.sg/how-to-check-if-5-is-a-square-mod-701-without-a-calculator.htmlhttps://pop.com.sg/how-to-check-if-5-is-a-square-mod-701-without-a-calculator.htmlThe Legendre symbol tells us to calculate $5^{350} \mod 701$, but this question was on an exam where no calculators are allowed, so I wasn't able to do this question. How can you find if $5$ is a s......Fri, 04 Sep 2026 00:55:34 -0400https://pop.com.sg/difference-between-polynomial-functions-and-polynomials-and-why-these-.htmlhttps://pop.com.sg/difference-between-polynomial-functions-and-polynomials-and-why-these-.htmlHere's an excerpt from abstract algebra book that I'm reading and my question is given later: The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Giv......Fri, 04 Sep 2026 00:50:39 -0400