https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 05 Sep 2026 06:43:59 -0400https://pop.com.sg/tetrahedral-dice-throw.htmlhttps://pop.com.sg/tetrahedral-dice-throw.htmlIf we threw $30$ tetrahedral dice and summed the outcomes, how many values in the distribution would have a non-zero probability? a) If we calculate the distribution of the sum of two thrown ...Sat, 05 Sep 2026 10:28:35 -0400https://pop.com.sg/from-the-graph-of-the-derivative-f-039-x-make-a-sketch-of-the-original.htmlhttps://pop.com.sg/from-the-graph-of-the-derivative-f-039-x-make-a-sketch-of-the-original.htmlI haven't started finding the derivatives of functions yet, so at the moment this is strictly about finding the right derivative graph to an original graph. The task was this: Look at the graph be......Sat, 05 Sep 2026 10:11:16 -0400https://pop.com.sg/rigorous-definition-of-geometric-similarity.htmlhttps://pop.com.sg/rigorous-definition-of-geometric-similarity.htmlOn wikipedia it says : " Two geometrical objects are called similar if one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotatio......Sat, 05 Sep 2026 10:10:29 -0400https://pop.com.sg/how-do-we-know-what-the-integral-of-sin-x-is.htmlhttps://pop.com.sg/how-do-we-know-what-the-integral-of-sin-x-is.htmlSince I started more or less formally learning the foundations of calculus, I naturally came across the Riemann definition of the integral. At first I considered this definition intuitive but not r......Sat, 05 Sep 2026 10:09:30 -0400https://pop.com.sg/why-two-vectors-cannot-span-bf-r-3.htmlhttps://pop.com.sg/why-two-vectors-cannot-span-bf-r-3.htmlConsider two vectors $$ u_{1} = \begin{bmatrix} -1 \\ 3 \\ 2 \\ \end{bmatrix},\quad u_{2} = \begin{bmatrix} 6 \\ 1 \\ 1 \\ \end{bmatrix} $$ I can tell they don't span $R^3$ because $R^3$ re......Sat, 05 Sep 2026 10:03:19 -0400https://pop.com.sg/how-many-sets-of-5-can-you-make-from-22-elements.htmlhttps://pop.com.sg/how-many-sets-of-5-can-you-make-from-22-elements.htmlI know you can just say $\large\frac{22!}{(22!-5!)5!}$. My question is deeper than this, however. What if the set of 5 cannot be used again? For instance, I have 22 coloured balls, my first set of 5 ...Sat, 05 Sep 2026 09:55:32 -0400https://pop.com.sg/sum-of-independent-gamma-distributions-is-a-gamma-distribution.htmlhttps://pop.com.sg/sum-of-independent-gamma-distributions-is-a-gamma-distribution.htmlIf $X\sim \Gamma(a_1,b)$ and $Y \sim \Gamma(a_2,b)$, I need to prove $X+Y\sim\Gamma(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and just tryin......Sat, 05 Sep 2026 09:53:10 -0400https://pop.com.sg/why-non-real-means-only-the-square-root-of-negative.htmlhttps://pop.com.sg/why-non-real-means-only-the-square-root-of-negative.htmlOnce in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...Sat, 05 Sep 2026 09:46:15 -0400https://pop.com.sg/explanation-circular-permutation.htmlhttps://pop.com.sg/explanation-circular-permutation.htmlWell if one looks at the formula of circular permutations $Pc = (n-1)!$ But as we come to that formula, I need a concrete example and an explanation. Another thing, I have seen that when working ......Sat, 05 Sep 2026 09:45:34 -0400https://pop.com.sg/trace-of-a-diagonalized-matrix.htmlhttps://pop.com.sg/trace-of-a-diagonalized-matrix.htmlWhy do I have: $Tr(SDS^{-1})=Tr(D)$?...Sat, 05 Sep 2026 09:42:52 -0400https://pop.com.sg/what-do-mathematicians-do.htmlhttps://pop.com.sg/what-do-mathematicians-do.htmlThe American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: Wha......Sat, 05 Sep 2026 09:41:51 -0400https://pop.com.sg/f-ax-f-x-2-1-what-is-f.htmlhttps://pop.com.sg/f-ax-f-x-2-1-what-is-f.htmlSuppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm cons......Sat, 05 Sep 2026 09:40:53 -0400https://pop.com.sg/what-are-the-interesting-applications-of-hyperbolic-geometry.htmlhttps://pop.com.sg/what-are-the-interesting-applications-of-hyperbolic-geometry.htmlI am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...Sat, 05 Sep 2026 09:35:30 -0400https://pop.com.sg/proving-a-reduction-formula-for-the-antiderivative-of-cos-n-x-duplicat.htmlhttps://pop.com.sg/proving-a-reduction-formula-for-the-antiderivative-of-cos-n-x-duplicat.htmlI want to show that for all $n\ge 2$, it holds that $$ \int \cos^n x\ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n}\int \cos^{n-2} x\ dx. $$ I'm not even getting the result for the induction......Sat, 05 Sep 2026 09:24:44 -0400https://pop.com.sg/how-far-is-the-list-of-known-primes-known-to-be-complete.htmlhttps://pop.com.sg/how-far-is-the-list-of-known-primes-known-to-be-complete.htmlSo there is always the search for the next "biggest known prime number". The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twenty million digits. Wikipedia also lists the ...Sat, 05 Sep 2026 09:10:41 -0400https://pop.com.sg/how-to-prove-the-closed-map-lemma.htmlhttps://pop.com.sg/how-to-prove-the-closed-map-lemma.htmlThe closed map lemma says that if $f : X \to Y$ is a continuous function, $X$ is compact and $Y$ is Hausdorff, then $f$ is a closed map. How can I prove this ? Here is my attempt so far: Suppose......Sat, 05 Sep 2026 08:32:35 -0400https://pop.com.sg/quaternions-why-is-the-angle-frac-theta-2-duplicate.htmlhttps://pop.com.sg/quaternions-why-is-the-angle-frac-theta-2-duplicate.htmlThe equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta 2\ri......Sat, 05 Sep 2026 08:29:28 -0400https://pop.com.sg/how-do-i-find-the-equation-of-a-parabola-given-the-max-and-two-points.htmlhttps://pop.com.sg/how-do-i-find-the-equation-of-a-parabola-given-the-max-and-two-points.htmlI was given the points (2, -1) and (10,-1) and also a max of 4. How would I go about finding the equation of the parabola given this info?...Sat, 05 Sep 2026 08:24:21 -0400https://pop.com.sg/prove-the-sum-to-product-formulas-with-complex-numbers.htmlhttps://pop.com.sg/prove-the-sum-to-product-formulas-with-complex-numbers.htmlFirst of all, I don't know if it's possible but I'm just wondering about it. I'm pretty standard when it comes to complex numbers knowledge. So my goal: $$\sin x+\sin y=2\sin\left(\frac{x+y}{2}\rig......Sat, 05 Sep 2026 08:15:22 -0400https://pop.com.sg/the-relationship-between-spectral-decomposition-eigendecomposition-and.htmlhttps://pop.com.sg/the-relationship-between-spectral-decomposition-eigendecomposition-and.htmlI am trying to clarify the relationship between the spectral decomposition / eigendecomposition of a matrix and projection operators. I understand that there is a connection between diagonalizabil......Sat, 05 Sep 2026 08:14:36 -0400https://pop.com.sg/a-detail-in-the-proof-of-auslander-buchsbaum-theorem.htmlhttps://pop.com.sg/a-detail-in-the-proof-of-auslander-buchsbaum-theorem.htmlI'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m)$, where $m$ is the maximal ideal, and a finitely generated non-zero $R$-module $M$ such ...Sat, 05 Sep 2026 07:51:32 -0400https://pop.com.sg/find-the-volume-v-of-the-described-solid-s.htmlhttps://pop.com.sg/find-the-volume-v-of-the-described-solid-s.htmlFind the volume $V$ of the described solid $S$. The base of $S$ is the triangular region with vertices $(0, 0), (4,0)$, and $(0, 4)$.Cross-sections perpendicular to the x−axis are squares. My an......Sat, 05 Sep 2026 07:46:35 -0400https://pop.com.sg/antiderivative-of-sin-x-pi-3.htmlhttps://pop.com.sg/antiderivative-of-sin-x-pi-3.htmlI used $u$-substitution to break down the problem to be $-\cos(u)$ and then got my final answer to be $-\cos(x-\pi/3)$ (re-insertion for my $u$) and it is not completely correct. What am I missing ......Sat, 05 Sep 2026 07:43:23 -0400https://pop.com.sg/why-is-this-series-of-square-root-of-twos-equal-pi.htmlhttps://pop.com.sg/why-is-this-series-of-square-root-of-twos-equal-pi.htmlWikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, ...Sat, 05 Sep 2026 07:42:54 -0400https://pop.com.sg/what-is-the-integral-of-a-cumulative-distribution-function.htmlhttps://pop.com.sg/what-is-the-integral-of-a-cumulative-distribution-function.htmlI cannot find what is the integral of a cumulative distribution function $$\int G(\xi)d\xi$$ I think it should be simple, but I have no idea where else to look for it....Sat, 05 Sep 2026 07:40:34 -0400https://pop.com.sg/log-or-antilog-tables-which-ones-are-more-useful.htmlhttps://pop.com.sg/log-or-antilog-tables-which-ones-are-more-useful.htmlI'm trying to make a Log or Antilog table small enough to fit in the back of a wallet calendar (or a business card). My intend is to build a mathematically useful gift that can be used by anybody e......Sat, 05 Sep 2026 07:36:09 -0400https://pop.com.sg/solve-x-3-3x-4-0-in-real-numbers.htmlhttps://pop.com.sg/solve-x-3-3x-4-0-in-real-numbers.htmlSolve $x^3 +3x -4 =0$ in Real Numbers. I know the answer is 1 but i want the solution. if you solve it with decomposition, explain how did you find what to do to decompose this expression. (I know......Sat, 05 Sep 2026 07:34:49 -0400https://pop.com.sg/find-the-smallest-distance-between-point-and-ellipsoid.htmlhttps://pop.com.sg/find-the-smallest-distance-between-point-and-ellipsoid.htmlFind the smallest distance between the points on the ellipsoid $x^2+2y^2+z^2=16$ and the point $(0,0,1)$. Answer: Let $(x,y,z)$ be the closest point of the ellipsoid. The distance is $$\sqrt{x^2+......Sat, 05 Sep 2026 07:34:48 -0400https://pop.com.sg/definition-of-identical-in-distribution.htmlhttps://pop.com.sg/definition-of-identical-in-distribution.htmlWhile one can find various online references (and in books) for the definition of "convergence in distribution", I have several times today been stumped (as a non-probabilist) by the notion of "ide......Sat, 05 Sep 2026 07:22:03 -0400https://pop.com.sg/surface-area-and-volume-of-cone.htmlhttps://pop.com.sg/surface-area-and-volume-of-cone.htmlConsider a cone with height $H$ and radius $R$. Let $dh$ and $dr$ be the respective changes in $H$ and $R$. Therefore $$\text{Volume}=\int H \pi r^2 dh$$ Now $r$ is a function of $h$, so $$h=H−H/R*......Sat, 05 Sep 2026 07:16:05 -0400https://pop.com.sg/which-is-bigger-a-googolplex-or-10-100.htmlhttps://pop.com.sg/which-is-bigger-a-googolplex-or-10-100.htmlA googol is defined as $ 10^{100}$ Let x = $10^{100}$ A googolplex is defined as $10^{x}$ Which is bigger: a googolplex or $10^{100!}$ I only know that: $100! = 1×2×3×...×98×99×100$ $10^{100} = 1......Sat, 05 Sep 2026 07:11:33 -0400https://pop.com.sg/an-ill-conditioned-matrix.htmlhttps://pop.com.sg/an-ill-conditioned-matrix.htmlIf C is an ill-conditioned matrix and I want to get the inverse, one way is to take a pseudo-inverse of some sort. Instead, is the following, which uses the (normal) inverse, also a way to deal wit......Sat, 05 Sep 2026 07:11:04 -0400https://pop.com.sg/axiom-of-double-induction.htmlhttps://pop.com.sg/axiom-of-double-induction.htmlWhat would the set-theoretical axiom of induction look like for double induction* when stated in the mathematical language of first- or second-order logic? *References as to What Double Inductio......Sat, 05 Sep 2026 07:10:04 -0400https://pop.com.sg/extending-039-guess-2-3-of-the-average-039-game.htmlhttps://pop.com.sg/extending-039-guess-2-3-of-the-average-039-game.htmlIn game theory, 'Guess $\frac{2}3$ of the Average' is a game where $n$ people are asked to choose a real number between $0$ and $100$ inclusive. The person with the closest answer to $\frac{2}3$ of......Sat, 05 Sep 2026 07:04:56 -0400https://pop.com.sg/ogf-for-partitions-of-integer-r-n-into-even-parts-less-than-or-equal-t.htmlhttps://pop.com.sg/ogf-for-partitions-of-integer-r-n-into-even-parts-less-than-or-equal-t.htmlLet $a_r$ be the number of partitions of the integer r-N into even parts, the largest of which is less than or equal to 2m, where N is some integer that may depend on m. Find the ordinary generative ...Sat, 05 Sep 2026 06:55:52 -0400https://pop.com.sg/cauchy-039-s-residue-theorem-on-a-singularity-outside-a-contour.htmlhttps://pop.com.sg/cauchy-039-s-residue-theorem-on-a-singularity-outside-a-contour.htmlI recently ran into the following exercise: Evaluate $$\oint_\Gamma\frac{\cos z}{(z-\pi)^2}dz,$$where $\Gamma$ is a complete circuit of the circle $|z|=1$. Clearly, the singularity lies outsi......Sat, 05 Sep 2026 06:55:20 -0400https://pop.com.sg/minimum-number-of-attempts-to-guess-a-pin-code-given-constraints.htmlhttps://pop.com.sg/minimum-number-of-attempts-to-guess-a-pin-code-given-constraints.htmlI'm playing a video game at the moment called Sleeping Dogs, in which some of the mini-missions are to 'hack' a security camera, by guessing a four-digit PIN code. Here are the rules: 1) You are ...Sat, 05 Sep 2026 06:50:42 -0400https://pop.com.sg/a-consequence-of-the-mean-value-theorem-for-integrals.htmlhttps://pop.com.sg/a-consequence-of-the-mean-value-theorem-for-integrals.htmlSuppose that $f,g: [a,b] \rightarrow R$ are continuous functions. Its true that exists $a \leq c \leq b$, $$ \int _a^b f(x)g(x) dx = f(c) \int_a^b g(x) dx ? $$ And if $g \geq 0$ what happens? Thi......Sat, 05 Sep 2026 06:42:21 -0400https://pop.com.sg/proving-the-splitting-lemma-on-pg-147-in-at.htmlhttps://pop.com.sg/proving-the-splitting-lemma-on-pg-147-in-at.htmlThe Lemma and a sketch of its proof are given below: My questions are: 1- why the diagram that must be commutative looks specifically like this? My professor actually drew a diagram without the ...Sat, 05 Sep 2026 06:37:55 -0400https://pop.com.sg/is-calculus-necessary-for-computer-science-student-closed.htmlhttps://pop.com.sg/is-calculus-necessary-for-computer-science-student-closed.htmlI'm a freshman in university and I'm studying Computer science and engineering. This will be my second year of studying. We don't have Calculus as a mandatory class but I can take it from elective ...Sat, 05 Sep 2026 06:35:52 -0400https://pop.com.sg/after-adding-10-cups-of-water-to-a-container-1-8-full-it-becomes-3-4-f.htmlhttps://pop.com.sg/after-adding-10-cups-of-water-to-a-container-1-8-full-it-becomes-3-4-f.htmlA container is $1/8$ full of water. After $10$ cups of water are added, the container is $3/4$ full. What is the volume of the container, in cups? Ok, I wrote out an equation: $$\frac{1}{8}V + 10C......Sat, 05 Sep 2026 06:29:53 -0400https://pop.com.sg/on-the-jacobian-determinant-for-conversion-to-cylindrical-coordinates.htmlhttps://pop.com.sg/on-the-jacobian-determinant-for-conversion-to-cylindrical-coordinates.htmlI have an exercise that asks to compute the triple integral over a region $E$ (a paraboloid with a horizontal slice), which I know to be $$\iiint_E x\,dV=\int_{-1}^1\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^......Sat, 05 Sep 2026 06:27:22 -0400https://pop.com.sg/what-is-the-interpretation-of-the-fundamental-theorem-of-line-integral.htmlhttps://pop.com.sg/what-is-the-interpretation-of-the-fundamental-theorem-of-line-integral.htmlThe fundamental theorem of line integrals is: $$\int_{a}^{b} \nabla F \cdot \vec{dr} = F(r(b)) - F(r(a))$$ for some curve traced by $r$. What is the intuition for why this is true? The proof is ...Sat, 05 Sep 2026 06:15:25 -0400https://pop.com.sg/describe-the-span-of-the-given-vectors-geometrically-and-algebraically.htmlhttps://pop.com.sg/describe-the-span-of-the-given-vectors-geometrically-and-algebraically.htmlDescribe the span of the given vectors geometrically and algebraically: $\pmatrix{1\\0\\-1}$, $\pmatrix{-1\\1\\0}$, $\pmatrix{0\\-1\\1}$. I have figured out that these vectors are linearly depend......Sat, 05 Sep 2026 05:54:58 -0400https://pop.com.sg/precedence-between-implication-and-bi-implication.htmlhttps://pop.com.sg/precedence-between-implication-and-bi-implication.htmlI came across this question: Let p, q, and r be the propositions p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these ...Sat, 05 Sep 2026 05:33:48 -0400https://pop.com.sg/what-are-the-rules-for-a-tetartoid-pentagon.htmlhttps://pop.com.sg/what-are-the-rules-for-a-tetartoid-pentagon.htmlThe tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry. It has twelve identical ...Sat, 05 Sep 2026 05:28:46 -0400https://pop.com.sg/cartesian-coordinates-of-lemniscate.htmlhttps://pop.com.sg/cartesian-coordinates-of-lemniscate.htmlA problem from Spivak's Calculus: Sketch the graph of the lemniscate $$r^2 = 2a^2\cos 2 \theta.$$ Find an equation for its cartesian coordinates. Show that it is the collection of ......Sat, 05 Sep 2026 05:26:26 -0400https://pop.com.sg/determining-the-area-of-a-right-triangle-perimeter-given-hypotenuse-va.htmlhttps://pop.com.sg/determining-the-area-of-a-right-triangle-perimeter-given-hypotenuse-va.htmlThe problem states: Right Triangle- perimeter of $84$, and the hypotenuse is $2$ greater than the other leg. Find the area of this triangle. I have tried different methods of solving this pro......Sat, 05 Sep 2026 05:26:24 -0400https://pop.com.sg/find-the-partial-fraction-for-frac-1-x-2-1-2-x-1.htmlhttps://pop.com.sg/find-the-partial-fraction-for-frac-1-x-2-1-2-x-1.htmlI want to find the partial fraction of: $$\frac{1}{(x^2+1)^2(x-1)}$$ Although I want this expressed through the format of finding the complex number that goes into $x$ so that I can get this equati......Sat, 05 Sep 2026 05:18:07 -0400https://pop.com.sg/a-box-contains-10-balls-numbered-from-1-to-10.htmlhttps://pop.com.sg/a-box-contains-10-balls-numbered-from-1-to-10.htmlAn urn contains balls numbered $1, \ldots, 10$. Five balls are drawn without replacement. What is the probability that the second largest of the five numbers drawn will be 8? I believe the number of ...Sat, 05 Sep 2026 05:12:31 -0400https://pop.com.sg/why-can-039-t-you-flatten-a-sphere.htmlhttps://pop.com.sg/why-can-039-t-you-flatten-a-sphere.htmlIt's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old? As soon as an explanation starts using terms like "Gauss......Sat, 05 Sep 2026 05:11:40 -0400https://pop.com.sg/extended-euclidean-algorithm-for-modular-inverse.htmlhttps://pop.com.sg/extended-euclidean-algorithm-for-modular-inverse.htmlI'm currently learning how to find the inverse of a modulo with the Extended Euclid Algorithm and I stumbled upon a problem when finding an inverse when the $m>p$ as for $m \equiv 1 \pmod{p}$ For ...Sat, 05 Sep 2026 05:10:37 -0400https://pop.com.sg/converting-augmented-matrix-to-reduced-row-echelon-form.htmlhttps://pop.com.sg/converting-augmented-matrix-to-reduced-row-echelon-form.htmlI am struggling with reducing an augmented matrix to reduced row echelon form. The given linear system is \begin{cases} 3x-2z=-3 \\ -2x+z=-2 \\ -z=2 \\ \end{cases} I can write this as an augmen......Sat, 05 Sep 2026 04:57:52 -0400https://pop.com.sg/find-the-indicated-probability.htmlhttps://pop.com.sg/find-the-indicated-probability.htmlQuestion: On one tropical island, hurricanes occur with a mean of 2.74 per year. Assuming that the number of hurricanes can be modeled by a Poisson distribution, find the probability that during th......Sat, 05 Sep 2026 04:54:31 -0400https://pop.com.sg/find-the-arclength-of-the-curve-defined-by-r-t-i-9t-2j-t-3k-for-0-leq-.htmlhttps://pop.com.sg/find-the-arclength-of-the-curve-defined-by-r-t-i-9t-2j-t-3k-for-0-leq-.htmlFirst I found $r'(t)=\langle 1,18t^2,3t^2\rangle$ and so the magnitude of $r'(t)= \sqrt{1+(18t)^2+(3t^2)^2}$ thus the integral from $0$ to $\sqrt{28}$ of $\sqrt{1+324t^2+9t^4} dt$. When I plugged $......Sat, 05 Sep 2026 04:54:23 -0400https://pop.com.sg/classifying-map.htmlhttps://pop.com.sg/classifying-map.htmlLet $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f......Sat, 05 Sep 2026 04:48:51 -0400https://pop.com.sg/sum-of-multiplication-of-two-binomial-coefficients.htmlhttps://pop.com.sg/sum-of-multiplication-of-two-binomial-coefficients.htmlIs there any formula for calculating $\sum_{k=0}^n {n\choose k} {2n\choose 2k}$ ? One possible way is to use Stirling's approximation, but couldn't reach a reasonable answer....Sat, 05 Sep 2026 04:47:35 -0400https://pop.com.sg/why-do-divisions-like-1-98-and-1-998-give-us-numbers-continuously-bein.htmlhttps://pop.com.sg/why-do-divisions-like-1-98-and-1-998-give-us-numbers-continuously-bein.htmlFor example, when I divided $1$ by $98$, I got an amazing result of $0.0102040816326530612244897...$ where so many numbers get multiplied by two every time in the right pattern with some carrying. ......Sat, 05 Sep 2026 04:42:17 -0400https://pop.com.sg/positive-and-negative-complex-numbers.htmlhttps://pop.com.sg/positive-and-negative-complex-numbers.htmlCan there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number w......Sat, 05 Sep 2026 04:40:15 -0400https://pop.com.sg/relative-interpretations-alla-kunen.htmlhttps://pop.com.sg/relative-interpretations-alla-kunen.htmlat the moment I try to figure out some details of Kunen's "Relative Interpretation" Definition (within the 2013 Edition of his "Set Theory", p. 99 to 100): Definition If $\Lambda$ is some axio......Sat, 05 Sep 2026 04:38:14 -0400https://pop.com.sg/formula-for-the-sequence-1-1-2-2-3-3-closed.htmlhttps://pop.com.sg/formula-for-the-sequence-1-1-2-2-3-3-closed.htmlI am trying to solve a programming problem where i encounter this pattern 1 1 2 2 3 3 4 4 . What is the formula for computing it for Nth term ?...Sat, 05 Sep 2026 04:38:06 -0400https://pop.com.sg/understanding-the-earth-mover-s-distance-emd-from-the-matlab.htmlhttps://pop.com.sg/understanding-the-earth-mover-s-distance-emd-from-the-matlab.htmlI found difficulty in understanding the Earth Mover’s Distance (EMD) from the MATLAB implementation. The method is actually implemented in this paper. Anyone help me to where EMD is implemented in ......Sat, 05 Sep 2026 04:35:30 -0400https://pop.com.sg/is-there-any-significant-consequence-if-p-does-not-equal-np-complete.htmlhttps://pop.com.sg/is-there-any-significant-consequence-if-p-does-not-equal-np-complete.htmlI hear a lot on these forums about how if P=NP-complete how our lives would be better, is there any significant consequence if we found P to be not equal to NP-Complete?...Sat, 05 Sep 2026 04:30:32 -0400https://pop.com.sg/calculating-inverse-trig-expressions-like-cos-arctan-2.htmlhttps://pop.com.sg/calculating-inverse-trig-expressions-like-cos-arctan-2.htmlI have some problems "connecting dots". All feedback is welcomed and really, really helpful! :) Task 1: calculate $\quad \tan{(\arcsin{(-\frac{3}{4}}))}$ Solution: $\tan{(\arcsin{-\frac{3}{4}})......Sat, 05 Sep 2026 04:25:31 -0400https://pop.com.sg/proof-of-schur-039-s-lemma.htmlhttps://pop.com.sg/proof-of-schur-039-s-lemma.htmlCan someone give me a simplified proof of Schur's lemma in group theory. Sorry if the question looks a standard textbook proof. But I find the proof complicated in books. It would be helpful if som......Sat, 05 Sep 2026 04:24:35 -0400https://pop.com.sg/is-7-5040-the-largest-highly-composite-factorial.htmlhttps://pop.com.sg/is-7-5040-the-largest-highly-composite-factorial.htmlA highly composite number is a natural number $\ n\ge 1\ $ that has more divisors than any smaller natural number $\ m\ge 1$. Checking the $\ 1000\ $ entries in the OEIS-sequence, I noticed that o......Sat, 05 Sep 2026 04:22:49 -0400https://pop.com.sg/extremas-using-lagrange-or-other-methods-with-exponential-functions.htmlhttps://pop.com.sg/extremas-using-lagrange-or-other-methods-with-exponential-functions.html$ \overline{B((0,0),1))}\mapsto \mathbb{R}$ $f(x,y)= e^{(x-1)y}$ How do you find the extremas for this functions? I tried using Lagrange method but I have trouble solving the system of equations ......Sat, 05 Sep 2026 04:18:23 -0400https://pop.com.sg/giving-two-equal-line-segments-ac-and-bd-so-that-they-bisect-each-othe.htmlhttps://pop.com.sg/giving-two-equal-line-segments-ac-and-bd-so-that-they-bisect-each-othe.htmlGiven $AC=BD$ so $\angle AED= \angle CEB$ by Prop I.15 $AE+EC$ and $BE=DE$ by definition of bisect So by $SAS$, $\triangle AED$ is congruent to $\triangle BEC$ (postulate 12) Therefor $AD=BC$ ......Sat, 05 Sep 2026 04:15:46 -0400https://pop.com.sg/complex-space-why-unit-disc.htmlhttps://pop.com.sg/complex-space-why-unit-disc.htmlIn complex space for simplicity the properties of the functions on curves are sometimes considered on the unit circle on complex plane, with the center (0, 0) . So basically I would like to know why ...Sat, 05 Sep 2026 04:11:04 -0400https://pop.com.sg/complete-graph-k-n-can-be-expressed-as-the-union-of-k-bipartite-graphs.htmlhttps://pop.com.sg/complete-graph-k-n-can-be-expressed-as-the-union-of-k-bipartite-graphs.htmlI was going through the text "Introduction to Graph Theory" by Douglas West where I came across the following theorem. Theorem. The complete graph $K_n$ can be expressed as the union of ......Sat, 05 Sep 2026 03:59:08 -0400https://pop.com.sg/simplest-way-to-count-distinct-combinations-2.htmlhttps://pop.com.sg/simplest-way-to-count-distinct-combinations-2.htmlFollowing my previous problem (which was solved) Simplest way to count distinct combinations (Reading this topic is not mandatory). Consider I have these letters: A, B, C, D, E. n is the number......Sat, 05 Sep 2026 03:51:08 -0400https://pop.com.sg/making-sense-of-a-cross-product-of-three-vectors.htmlhttps://pop.com.sg/making-sense-of-a-cross-product-of-three-vectors.htmlBecause of the cross product of two vectors being another vector I can calculate $\vec a\times(\vec b\times\vec c)$ as well as $(\vec a\times\vec b)\times\vec c$. I know that the cross product is not ...Sat, 05 Sep 2026 03:47:49 -0400https://pop.com.sg/why-do-we-disregard-the-remainder-while-finding-an-oblique-asymptote.htmlhttps://pop.com.sg/why-do-we-disregard-the-remainder-while-finding-an-oblique-asymptote.htmlI have to find the oblique asymptote for the following equation: $$y=\frac{2x^{2}-4x-1}{x-3}$$ I applied long division to bring it into the form of $Q + \frac{R}{D}$ leaving me with: $$y=2x+2+\fra......Sat, 05 Sep 2026 03:25:22 -0400https://pop.com.sg/does-the-definition-of-the-angle-between-two-vectors-require-that-they.htmlhttps://pop.com.sg/does-the-definition-of-the-angle-between-two-vectors-require-that-they.htmlI am thinking specifically about $\mathbb{R}^2$ so I can visualize things. By "origin" I mean that they start at the same point. When we graphically represnt vectors we don't care where the start......Sat, 05 Sep 2026 03:22:13 -0400https://pop.com.sg/quotient-groups-of-d10.htmlhttps://pop.com.sg/quotient-groups-of-d10.htmlHow do I find the groups D10/N where N is the normal subgroups to D10? I know that the definition is aN for all a $\in$ D10. But I am unsure what the group, for example, D10/ D10 looks like? Any h......Sat, 05 Sep 2026 03:20:08 -0400https://pop.com.sg/definition-of-measurability.htmlhttps://pop.com.sg/definition-of-measurability.htmlThe alternative definition of measurability is: A subset $E \subset \mathbb{R}^n$ is measurable if for every $\epsilon > 0$ there is a closed set $F \subset E$ such that $m^*(E\setminus F)<\...Sat, 05 Sep 2026 03:10:53 -0400https://pop.com.sg/the-volume-of-a-torus.htmlhttps://pop.com.sg/the-volume-of-a-torus.htmlA torus $\mathscr{T}$ with the equation $$z^2 + \left( \sqrt{x^2 + y^2} - 2 \right)^2 = 1.$$ (a) Give an equation with a close line in the plane $Oxz$ where $\mathscr{T}$ is a surface of ...Sat, 05 Sep 2026 03:07:51 -0400https://pop.com.sg/what-does-ir-mean-in-linear-algebra.htmlhttps://pop.com.sg/what-does-ir-mean-in-linear-algebra.htmlI am new to linear algebra and I was wondering if I could get some help for this question. I understand if it was something like this IR^2 -> IR. I have no idea what IR(=IR^1) means. Could someone ...Sat, 05 Sep 2026 03:02:51 -0400https://pop.com.sg/determine-whether-this-statement-is-true-or-false-forall-x-in-z-exists.htmlhttps://pop.com.sg/determine-whether-this-statement-is-true-or-false-forall-x-in-z-exists.htmlJust beginning in Discrete math stumbled into the symbolic logic section and need some selp on this question, any input is appreciated. $$ \forall x \in {Z}, \exists y , z \in {Z} [x = 5y + 7z] $$ ...Sat, 05 Sep 2026 02:50:06 -0400https://pop.com.sg/why-is-it-important-for-a-matrix-to-be-square.htmlhttps://pop.com.sg/why-is-it-important-for-a-matrix-to-be-square.htmlI am currently trying to self-study linear algebra. I've noticed that a lot of the definitions for terms (like eigenvectors, characteristic polynomials, determinants, and so on) require a square ma......Sat, 05 Sep 2026 02:44:01 -0400https://pop.com.sg/constructing-a-square-from-the-difference-of-diagonal-and-side.htmlhttps://pop.com.sg/constructing-a-square-from-the-difference-of-diagonal-and-side.htmlI have figured it out how to construct a square given a segment which is the difference of diagonal and side: construct an equal sided right triangle where the leg is the given segment and then add......Sat, 05 Sep 2026 02:41:29 -0400https://pop.com.sg/what-are-the-differences-between-rings-groups-and-fields.htmlhttps://pop.com.sg/what-are-the-differences-between-rings-groups-and-fields.htmlRings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?...Sat, 05 Sep 2026 02:37:39 -0400https://pop.com.sg/turn-partial-differential-equation-into-ordinary-differential-equation.htmlhttps://pop.com.sg/turn-partial-differential-equation-into-ordinary-differential-equation.html(I looked for a post similar but couldn't find one. I apologize in advance if there's a post similar to this. Apologizes to the moderators for a weak post.) I'm taking an ordinary differential equ......Sat, 05 Sep 2026 02:37:00 -0400https://pop.com.sg/square-root-of-sqrt-1-4-sqrt-3-i.htmlhttps://pop.com.sg/square-root-of-sqrt-1-4-sqrt-3-i.htmlHow can we find square root of the complex number $$\sqrt{1-4\sqrt{3}i}?$$ Now here if I assume square root to be $a+ib$ i.e. $a+ib=\sqrt{\sqrt{1-4\sqrt{3}i}}$, then after squaring both sides, ho......Sat, 05 Sep 2026 02:30:41 -0400https://pop.com.sg/how-to-use-double-angle-identities-to-find-length-039-d-039.htmlhttps://pop.com.sg/how-to-use-double-angle-identities-to-find-length-039-d-039.htmlEdit: Ok the question has now been amended by my tutor as I highlighted it was not possible to solve. A pole is tensioned at point A as shown below, if the angles are to be kept the same, then usi......Sat, 05 Sep 2026 02:17:25 -0400https://pop.com.sg/how-to-prove-distance-from-foci-on-an-ellipse-is-equal-to-twice-the-se.htmlhttps://pop.com.sg/how-to-prove-distance-from-foci-on-an-ellipse-is-equal-to-twice-the-se.htmlProve that for any point (x,y) on the conic, the sum of the distances to the two foci is always twice the semi-major axis. I know that this can be proven in general for all ellipses but the practice ...Sat, 05 Sep 2026 02:17:20 -0400https://pop.com.sg/linear-approximation-with-two-variables.htmlhttps://pop.com.sg/linear-approximation-with-two-variables.htmlThe problem I have is this: Use suitable linear approximation to find the approximate values for given functions at the points indicated: $f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$ I know how to do ...Sat, 05 Sep 2026 02:09:52 -0400https://pop.com.sg/definition-of-an-angle-vs-measurement-of-an-angle.htmlhttps://pop.com.sg/definition-of-an-angle-vs-measurement-of-an-angle.htmlThe definition of an angle in Euclidean geometry is given as follows: Angle. Definition: A shape, formed by two lines or rays diverging from a common point (the vertex). However, the definition ...Sat, 05 Sep 2026 02:01:11 -0400https://pop.com.sg/how-could-a-square-root-of-fraction-have-a-negative-root.htmlhttps://pop.com.sg/how-could-a-square-root-of-fraction-have-a-negative-root.htmlWhat I know If I'm not mistaken: $\pm \sqrt{\frac{x}{y}}=\frac{\pm \sqrt{x}}{\pm \sqrt{y}}$ a repeated $\pm$ sign in an equation means make every "$\pm$" sign a plus or make every one of them a m......Sat, 05 Sep 2026 02:00:56 -0400https://pop.com.sg/piecewise-function-plot-in-matlab.htmlhttps://pop.com.sg/piecewise-function-plot-in-matlab.htmlIs it possible in Matlab to plot an even piecewise function like: $ f(x) = \begin{cases} 3t , 0 < t < \pi \\ -3t , -\pi \le t \le 0 \end{cases}$ which has a period of $2\pi$. I ......Sat, 05 Sep 2026 01:59:36 -0400https://pop.com.sg/how-to-solve-this-limit-lim-limits-x-to0-frac-1-x-1-x-e-x-duplicate.htmlhttps://pop.com.sg/how-to-solve-this-limit-lim-limits-x-to0-frac-1-x-1-x-e-x-duplicate.htmlI tried to evaluate $$\lim_{x\to0}\frac{(1+x)^{1/x}-e}x$$but all my efforts were in vain . L'Hospital rule becomes very complicated....Sat, 05 Sep 2026 01:50:26 -0400https://pop.com.sg/even-and-odd-cubic-polynomials.htmlhttps://pop.com.sg/even-and-odd-cubic-polynomials.htmlBook says approximating even function $P_3$ must be of form $ax^2+b\ $so they neglected $x,x^3\ $. This function approximate even function $|x|^3$ very nicely in the book after some tweaking. I am ...Sat, 05 Sep 2026 01:32:58 -0400https://pop.com.sg/volume-of-a-parabola-rotated-around-a-line.htmlhttps://pop.com.sg/volume-of-a-parabola-rotated-around-a-line.htmlThe question is as follows: Find the volume of the solid generated by revolving the region bounded by the parabola $y = x^2$ and the line $y = 1$ about the line $y = -1$. My attempt: Using the was......Sat, 05 Sep 2026 01:32:13 -0400https://pop.com.sg/evaluate-the-indefinite-integral-int-1-x-2-3-2-dx.htmlhttps://pop.com.sg/evaluate-the-indefinite-integral-int-1-x-2-3-2-dx.htmlEvaluate the indefinite integral $$\int \frac {dx}{(1-x^2)^{3/2}} .$$ My answer: I have taken $u = 1-x^2$ but I arrived at the integral of $\frac{-1}{(u^3 - u^4)^{1/2}}$. What should I do next?...Sat, 05 Sep 2026 01:26:57 -0400https://pop.com.sg/riesz-functional-calculus-vs-holomorphic-functional-calculus.htmlhttps://pop.com.sg/riesz-functional-calculus-vs-holomorphic-functional-calculus.html"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...Sat, 05 Sep 2026 01:26:56 -0400https://pop.com.sg/find-the-matrix-s-of-stretch-by-a-factor-of-3.htmlhttps://pop.com.sg/find-the-matrix-s-of-stretch-by-a-factor-of-3.htmlAll mappings are from $\mathbb{R}^2$ to $\mathbb{R}^2$. Find the matrix $S$ of stretch by a factor of $3$ in the $y$-direction and the matrix $S^{-1}$. So the matrix $S$ is a $2\times2$ matrix. ......Sat, 05 Sep 2026 01:21:04 -0400https://pop.com.sg/deriving-the-midpoint-formula-in-3d-using-the-distance-formula.htmlhttps://pop.com.sg/deriving-the-midpoint-formula-in-3d-using-the-distance-formula.htmlI want to derive the midpoint of the line segment connecting the points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ using only the distance formula (since this is all that has been taught to students......Sat, 05 Sep 2026 01:17:16 -0400https://pop.com.sg/cosine-and-sine-of-the-angle-multiplied-by-a-scalar.htmlhttps://pop.com.sg/cosine-and-sine-of-the-angle-multiplied-by-a-scalar.htmlCan you write $\cos \alpha x$ in terms of $\cos x$ ? similarly for $\sin \alpha x$. where $\alpha$ is a scalar in $\mathbb{R}$. (not necessarily integer)....Sat, 05 Sep 2026 01:02:29 -0400https://pop.com.sg/what-039-s-the-formula-to-solve-summation-of-logarithms.htmlhttps://pop.com.sg/what-039-s-the-formula-to-solve-summation-of-logarithms.htmlI'm studying summation. Everything I know so far is that: $\sum_{i=1}^n\ k = \frac{n(n+1)}{2}\ $ $\sum_{i=1}^{n}\ k^2 = \frac{n(n+1)(2n+1)}{6}\ $ $\sum_{i=1}^{n}\ k^3 = \frac{n^2(n+1)^2}{4}\ $ ...Sat, 05 Sep 2026 00:59:25 -0400https://pop.com.sg/how-to-get-the-exact-value-of-sin-x-if-sin-2x-frac-24-25.htmlhttps://pop.com.sg/how-to-get-the-exact-value-of-sin-x-if-sin-2x-frac-24-25.htmlHow to get the exact value of $\sin(x)$ if $\sin(2x) = \frac{24}{25}$ ? I checked various trigonometric identities, but I am unable to derive $\sin(x)$ based on the given information. For instanc......Sat, 05 Sep 2026 00:46:46 -0400