https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 31 Aug 2026 23:50:09 -0400https://pop.com.sg/user-ikshu-neithalath-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-ikshu-neithalath-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 03:26:27 -0400https://pop.com.sg/range-of-sin-x-6-cos-x-6.htmlhttps://pop.com.sg/range-of-sin-x-6-cos-x-6.htmlI need to find the range of the function $y = (\sin x)^6+ (\cos x)^6$ I did find the answer but working in a crude way rather than a methodical step by step approach. I give below the steps I used , ...Tue, 01 Sep 2026 03:07:14 -0400https://pop.com.sg/on-the-horizontal-and-slanted-asymptotes-of-rational-functions.htmlhttps://pop.com.sg/on-the-horizontal-and-slanted-asymptotes-of-rational-functions.htmlI am really confused about the horizontal and slant asymptotes of a function. My textbook says that given a rational function: \begin{equation} y=f(x)=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+\cdot\cdot\cd......Tue, 01 Sep 2026 03:07:06 -0400https://pop.com.sg/what-does-the-derivative-of-a-function-at-a-point-describe-duplicate.htmlhttps://pop.com.sg/what-does-the-derivative-of-a-function-at-a-point-describe-duplicate.htmlI understand that the derivative of a function $f$ at a point $x=x_{0}$ is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where $\Delta x......Tue, 01 Sep 2026 03:06:45 -0400https://pop.com.sg/how-does-one-derive-the-centroid-formula-for-multiple-shapes.htmlhttps://pop.com.sg/how-does-one-derive-the-centroid-formula-for-multiple-shapes.htmlI know how to derive the formula for the centroid of n sets of finite points, each with $m_k$ points and centroid $C_k$. The formula is: $$C=\frac{\sum_{k=1}^{n}C_km_k}{\sum_{k=1}^{n}m_k}$$ However......Tue, 01 Sep 2026 02:57:38 -0400https://pop.com.sg/find-the-sum-or-value-of-following-expression-closed.htmlhttps://pop.com.sg/find-the-sum-or-value-of-following-expression-closed.htmlIn the problem I am not able to derive difference term. I know for solving summation we have to make difference term. How to proceed with this problem? $$\frac{{\displaystyle \sum_{n=1}^{99}} \s......Tue, 01 Sep 2026 02:48:57 -0400https://pop.com.sg/how-to-obtain-more-than-two-abcissas-for-one-ordinate.htmlhttps://pop.com.sg/how-to-obtain-more-than-two-abcissas-for-one-ordinate.htmlI'm reading a book on trigonometry and the author shown that some of the examples contain one abcissa for one ordinate and he showed that a parabola has two abscissas for every positive ordinates, ......Tue, 01 Sep 2026 02:48:20 -0400https://pop.com.sg/inverse-function-of-tanh-x.htmlhttps://pop.com.sg/inverse-function-of-tanh-x.htmlI have a problem while calculating inverse function of tanh(x). I know it is y = sinh(x)/cosh(x) and then I should express x, but I am stuck with that. Will you help me with this? thx a lot...Tue, 01 Sep 2026 02:34:16 -0400https://pop.com.sg/how-to-factor-4x-2-2x-1.htmlhttps://pop.com.sg/how-to-factor-4x-2-2x-1.htmlI want to know how to factor $4x^2 + 2x + 1$? I found the roots using quadratic equation and got $-1 + \sqrt{-3}$ and $-1 - \sqrt{-3}$, so I thought the factors would be $(x - (-1 + \sqrt{-3}))$ an......Tue, 01 Sep 2026 02:28:43 -0400https://pop.com.sg/why-does-the-function-r-theta-graph-a-spiral.htmlhttps://pop.com.sg/why-does-the-function-r-theta-graph-a-spiral.htmlIf $\theta$ denotes an angle (in radians I assume), and $r$ means the distance from the origin, then why does $r = \theta$ make a perfect spiral? I'm not that advanced in math (only in geometry and ...Tue, 01 Sep 2026 02:23:55 -0400https://pop.com.sg/calculus-finding-the-minimum-vertical-distance-between-graphs.htmlhttps://pop.com.sg/calculus-finding-the-minimum-vertical-distance-between-graphs.htmlQuestion:Find the minimum vertical distance between the graphs of $2+\sin x$ and $\cos x$? In order to find out the required distance, what should I do? It seems that there is a problem if I ...Tue, 01 Sep 2026 02:21:06 -0400https://pop.com.sg/how-to-calculate-the-number-of-pieces-in-the-border-of-a-puzzle.htmlhttps://pop.com.sg/how-to-calculate-the-number-of-pieces-in-the-border-of-a-puzzle.htmlIs there any way to calculate how many border-pieces a puzzle has, without knowing its width-height ratio? I guess it's not even possible, but I am trying to be sure about it. Thanks for your help......Tue, 01 Sep 2026 02:14:33 -0400https://pop.com.sg/linear-function.htmlhttps://pop.com.sg/linear-function.htmlI don't get this, need some help, examples and information The linear function $f$ is given by $$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$ Enter the independent variable and the dependent ...Tue, 01 Sep 2026 02:09:19 -0400https://pop.com.sg/lagrange-remainder-is-t-contained-within-an-open-or-closed-interval.htmlhttps://pop.com.sg/lagrange-remainder-is-t-contained-within-an-open-or-closed-interval.htmlAbove is the taylor series centred at a including the lagrange remainder. Can t be equal to a or b?...Tue, 01 Sep 2026 01:45:39 -0400https://pop.com.sg/expected-value-of-an-expected-value.htmlhttps://pop.com.sg/expected-value-of-an-expected-value.htmlI am looking at a proof that $\text{Var}(X)= E((X - EX)^2) = E(X^2) - (E(X))^2$ $E((X - EX)^2) =$ $E(X^2 - 2XE(X) + (E(X))^2) =$ $E(X^2) - 2E(X)E(X) + (E(X))^2)$ I can't see how the second line......Tue, 01 Sep 2026 01:41:55 -0400https://pop.com.sg/how-can-i-get-the-fourier-transform-of-this-generalized-function.htmlhttps://pop.com.sg/how-can-i-get-the-fourier-transform-of-this-generalized-function.htmlwhen function f(x) of the generalized function is $f(x) = x+2$, the fourier transform of the generalized function will be like ($F$ is the Fourier transform) $$\int_{-∞}^{∞}(x+2)(F[k(x)])dx$$ whe......Tue, 01 Sep 2026 01:40:15 -0400https://pop.com.sg/understanding-all-sub-sigma-algebras-of-a-sigma-algebra.htmlhttps://pop.com.sg/understanding-all-sub-sigma-algebras-of-a-sigma-algebra.htmlSuppose I have a set $\Omega$, and I partition $\Omega$ into $4$ sets $A_{1}, A_{2}, A_{3}, A_{4}$. Clearly, the $A_{i}$'s are pairwise disjoint and their union equals $\Omega$. Now, it's also cl......Tue, 01 Sep 2026 01:10:30 -0400https://pop.com.sg/finding-global-max-min.htmlhttps://pop.com.sg/finding-global-max-min.htmlmy task is to figure out the critical points of $f(x,y)=e^y(x^4-x^2+y)$, $\ $$\mathbb{R}^2 \rightarrow \mathbb{R}$, and show which of them is a maximum or minimum. As far as I got, I've shown that ......Tue, 01 Sep 2026 01:09:03 -0400https://pop.com.sg/how-to-compute-the-limit-of-a-rational-function-at-infinity.htmlhttps://pop.com.sg/how-to-compute-the-limit-of-a-rational-function-at-infinity.htmlI am unable to compute the limit, anyone can help ? I only understand some basic knowledge of limit ....Tue, 01 Sep 2026 01:07:05 -0400https://pop.com.sg/units-for-spring-constant-closed.htmlhttps://pop.com.sg/units-for-spring-constant-closed.htmlI have the question: A baby of mass $9.0\;$kg bounces with a period of $1.2\;$s in a baby bouncer. What is the spring constant of the bouncer? I rearranged the equation $T = 2\pi$ $\sqrt{\frac{......Tue, 01 Sep 2026 01:04:36 -0400https://pop.com.sg/name-and-layperson-039-s-explanation-for-an-e8-group-diagram.htmlhttps://pop.com.sg/name-and-layperson-039-s-explanation-for-an-e8-group-diagram.htmlI am taking a risk here, but hoping it will not ignite wrath in the reader. In trying to get an intuition of Lie theory this diagram is all but impossible to ignore: Unfortunately, there are many ...Tue, 01 Sep 2026 01:02:13 -0400https://pop.com.sg/area-of-a-rectangle-by-its-perimeter.htmlhttps://pop.com.sg/area-of-a-rectangle-by-its-perimeter.htmlIf I have a rectangle, with its perimiter 100m how do I show that the area A is given by the quadratic function A = 50x - $x^2$ Where x is the length of one side...Tue, 01 Sep 2026 01:01:46 -0400https://pop.com.sg/definition-of-a-region.htmlhttps://pop.com.sg/definition-of-a-region.htmlDefinition given - Region: Open set with none, some, or all of its boundary points. This seems quite unimportant... and it seems that almost all sets are regions (I can only think of regions, I ......Tue, 01 Sep 2026 00:38:58 -0400https://pop.com.sg/evaluating-frac-1-log-2-100-cdots-frac-1-log-100-100.htmlhttps://pop.com.sg/evaluating-frac-1-log-2-100-cdots-frac-1-log-100-100.htmlFirstly, I am completely aware that logarithms can be rewritten in exponential form. So, I do know at least something about logs. :) However, given the task to simplify and rewrite the expression,......Tue, 01 Sep 2026 00:38:29 -0400https://pop.com.sg/study-materials-to-help-understand-the-generalized-stokes-039-theorem-.htmlhttps://pop.com.sg/study-materials-to-help-understand-the-generalized-stokes-039-theorem-.htmlDear MSE: My goal is to understand the generalized Stokes' theorem both intuitively and rigorously. Could someone give advice or recommend study materials to help understand the generalized Stokes' ...Tue, 01 Sep 2026 00:27:09 -0400https://pop.com.sg/why-is-a-sphere-in-an-n-dimensional-space-called-n-1-sphere.htmlhttps://pop.com.sg/why-is-a-sphere-in-an-n-dimensional-space-called-n-1-sphere.htmlWhy is a sphere in an $n $-dimensional space called $(n-1) $-sphere? Isn't it natural to call a sphere in 3D a 3-sphere, a sphere in 2D (i.e. a circle) a 2-sphere, etc?...Tue, 01 Sep 2026 00:04:31 -0400https://pop.com.sg/what-is-the-concept-of-region-of-convergence-of-z-transform.htmlhttps://pop.com.sg/what-is-the-concept-of-region-of-convergence-of-z-transform.htmlI have been trying to learn Z transform. But I haven't found any good source that will clear my concept about the region of convergence. I have found some keywords like unit circle, but I don't hav......Mon, 31 Aug 2026 23:58:16 -0400https://pop.com.sg/calculating-the-oblique-ness-of-a-line-that-intersects-two-non-paralle.htmlhttps://pop.com.sg/calculating-the-oblique-ness-of-a-line-that-intersects-two-non-paralle.htmlI have a 3d space containing two planes that may have any non-orthogonal relative orientation. In addition I have a line the intersects both planes. I am trying to compute an index of the "oblique-......Mon, 31 Aug 2026 23:49:08 -0400https://pop.com.sg/relation-between-image-and-pre-image-of-a-function.htmlhttps://pop.com.sg/relation-between-image-and-pre-image-of-a-function.htmlWe know that given some function, each element of its domain has only one image in its codomain. But is it possible for an image of some element to have two preimages? For example, the function $f......Mon, 31 Aug 2026 23:45:34 -0400https://pop.com.sg/calculate-periodicity-of-f-x-cos-sin-x.htmlhttps://pop.com.sg/calculate-periodicity-of-f-x-cos-sin-x.htmlGiven the function: $$f(x) = \cos(\sin(x))$$ Calculate the periodicity of $f(x)$. I understand it equals $\pi$ but how would a solid answer look like? Thank you....Mon, 31 Aug 2026 23:36:00 -0400https://pop.com.sg/how-to-solve-a-quartic-equation.htmlhttps://pop.com.sg/how-to-solve-a-quartic-equation.htmlCould someone please explain how to solve this : $x^4 - 10x^3 + 21x^2 + 40x - 100 = 0$ - not the answer only, but a step-by-step solution. I tried to solve it, with the help of khanacademy, but st......Mon, 31 Aug 2026 23:33:02 -0400https://pop.com.sg/how-is-the-phrase-n-bit-number-related-to-the-big-o-notation.htmlhttps://pop.com.sg/how-is-the-phrase-n-bit-number-related-to-the-big-o-notation.htmlWhen it comes to algorithms, you frequently have to evaluate problems like this: Let $x$ be an $n$-bit integer. For each of the following questions, give your answer as a function of $n$. Or a ...Mon, 31 Aug 2026 23:30:41 -0400https://pop.com.sg/proving-trig-identities-complex-numbers.htmlhttps://pop.com.sg/proving-trig-identities-complex-numbers.htmlQuestion: Prove that if $z = \cos (\theta) + i\sin(\theta)$, then $$ z^n + {1\over z^n} = 2\cos(n\theta) $$ Hence prove that $\cos^6(\theta)$ $=$ $\frac 1{32}$$(\cos(6\theta)$ + ......Mon, 31 Aug 2026 23:23:57 -0400https://pop.com.sg/what-is-the-differentiation-operator.htmlhttps://pop.com.sg/what-is-the-differentiation-operator.htmlOn Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator. The link that underlies the words "differentiation operator" in fact gives......Mon, 31 Aug 2026 23:21:59 -0400https://pop.com.sg/proving-that-n-over-2-n-converges-to-0-duplicate.htmlhttps://pop.com.sg/proving-that-n-over-2-n-converges-to-0-duplicate.htmlI'm completely clueless on this one. I can easily calculate the limit using L'Hopital's rule, but proving that the series is converging to 0 is far more tricky. $$a_n= {n \over 2^n}$$ Any help?...Mon, 31 Aug 2026 23:15:25 -0400https://pop.com.sg/checking-whether-a-graph-is-planar.htmlhttps://pop.com.sg/checking-whether-a-graph-is-planar.htmlI have to check whether a graph is planar. The given type is $$ e ≤ 3v − 6 .$$ From Wikipedia: Note that these theorems provide necessary conditions for planarity that are not sufficient ...Mon, 31 Aug 2026 23:10:57 -0400https://pop.com.sg/positive-constant-scalar-definition.htmlhttps://pop.com.sg/positive-constant-scalar-definition.htmlIn French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one expl......Mon, 31 Aug 2026 23:07:33 -0400https://pop.com.sg/what-039-s-the-connection-between-the-indefinite-integral-and-the-defi.htmlhttps://pop.com.sg/what-039-s-the-connection-between-the-indefinite-integral-and-the-defi.htmlI want to understand the connection between the primitive function or antiderivative and the definite integral. My problem with this is the independent variable called t in the formula for the first ...Mon, 31 Aug 2026 23:01:08 -0400https://pop.com.sg/basic-open-sets-in-the-zariski-topology-are-also-compact.htmlhttps://pop.com.sg/basic-open-sets-in-the-zariski-topology-are-also-compact.htmlLet $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complement......Mon, 31 Aug 2026 22:58:43 -0400https://pop.com.sg/what-is-an-inductive-step.htmlhttps://pop.com.sg/what-is-an-inductive-step.htmlI have a set of problems I am trying to complete. They are all proofs by induction. I have made attempts at most of them and always get stuck doing the inductive step. Can someone explain in laym......Mon, 31 Aug 2026 22:56:12 -0400https://pop.com.sg/lists-versus-sets-in-linear-algebra.htmlhttps://pop.com.sg/lists-versus-sets-in-linear-algebra.htmlI’m currently learning linear algebra from “Linear Algebra Done Right” by Sheldon Axler. The author, in his proofs, makes use of lists of vectors, as opposed to the more conventional usage of sets of ...Mon, 31 Aug 2026 22:42:58 -0400https://pop.com.sg/what-exactly-are-eigen-things.htmlhttps://pop.com.sg/what-exactly-are-eigen-things.htmlWikipedia defines an eigenvector like this: An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at mos......Mon, 31 Aug 2026 22:28:50 -0400https://pop.com.sg/when-is-the-order-of-convergence-of-newton-raphson-greater-than-2.htmlhttps://pop.com.sg/when-is-the-order-of-convergence-of-newton-raphson-greater-than-2.htmlIf some function $f$ has a simple root at $x_*$ then I know that the order of convergence of the Newton-Raphson iteration is at least 2. But when is this order strictly greater than 2?...Mon, 31 Aug 2026 22:26:46 -0400https://pop.com.sg/theorem-1-8-viii-proof-of-mathematical-statistics-jun-shao.htmlhttps://pop.com.sg/theorem-1-8-viii-proof-of-mathematical-statistics-jun-shao.htmlI have some questions about the proof of Theorem 1.8(viii) of Jun Shao's Mathematical Statistics. The theorem is stating that If $X_n$ converges to $X$ in distribution, then for any $r > 0$, $li......Mon, 31 Aug 2026 22:21:26 -0400https://pop.com.sg/prove-s-is-a-supremum-iff-for-all-epsilon-0-there-exists-a-in-a-such-t.htmlhttps://pop.com.sg/prove-s-is-a-supremum-iff-for-all-epsilon-0-there-exists-a-in-a-such-t.htmlLet $A\subseteq\Bbb{R}$ is a nonempty set and $s\in \Bbb{R}$ is an upper bound. Prove $s$ is the supremum iff for all $\epsilon>0$ there exists $a\in A$ such that $a>s-\epsilon$. This is impo......Mon, 31 Aug 2026 22:19:29 -0400https://pop.com.sg/the-membership-table-for-a-u-b-c-a-c-u-b-c.htmlhttps://pop.com.sg/the-membership-table-for-a-u-b-c-a-c-u-b-c.htmlI have attached the image of my membership table. I have supposedly mistook three values I marked with green question marks. Can someone please tell me why they are wrong? the correct values are ones ...Mon, 31 Aug 2026 22:10:59 -0400https://pop.com.sg/understanding-quotient-groups.htmlhttps://pop.com.sg/understanding-quotient-groups.htmlAdmittedly this will be probably be a naive question, but here it goes: Is it possible to flesh out in simple terms, for someone with little background in group theory, what it means to take the ...Mon, 31 Aug 2026 22:08:53 -0400https://pop.com.sg/prove-8-cos-x-cos-2x-cos-3x-1-dfrac-sin-7x-sin-x.htmlhttps://pop.com.sg/prove-8-cos-x-cos-2x-cos-3x-1-dfrac-sin-7x-sin-x.htmlHow do you prove that $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?...Mon, 31 Aug 2026 21:43:23 -0400https://pop.com.sg/formula-for-the-diameter-of-an-emerging-beam.htmlhttps://pop.com.sg/formula-for-the-diameter-of-an-emerging-beam.htmlI have the following problem: A thin beam of white light with negligible radius is incident at $50.0^\circ$ on a $10.0 cm$ thick slab of clear plastic. The index of refraction for red light in t......Mon, 31 Aug 2026 21:38:59 -0400https://pop.com.sg/is-the-euclidean-plane-equal-to-mathbb-r-2.htmlhttps://pop.com.sg/is-the-euclidean-plane-equal-to-mathbb-r-2.htmlI understand what the Cartesian plane is and how one can identify the Euclidean plane and $\mathbb{R}^2$ in many different ways. My question comes down to this though: The Euclidean plane (as defi......Mon, 31 Aug 2026 21:38:11 -0400https://pop.com.sg/subordinate-matrix-norm-of-1-norm.htmlhttps://pop.com.sg/subordinate-matrix-norm-of-1-norm.htmlShow that for the vector norm $$\|x\|_1=\sum_{i=1}^n |x_i|$$ the subordinate matrix norm is $$ \|A\|_1=\max_{1\leq j\leq n} \sum_{i=1}^n |a_{ij}| $$ I've managed to narrow it down doing the followi......Mon, 31 Aug 2026 21:32:08 -0400https://pop.com.sg/what-does-secant-mean-in-mathematics.htmlhttps://pop.com.sg/what-does-secant-mean-in-mathematics.htmlI need your help. pleas I need some one to explain for me what does secant mean in mathematics? Clear example with images would be appreciated....Mon, 31 Aug 2026 21:25:10 -0400https://pop.com.sg/what-is-the-relationship-between-sets-sequences-and-functions.htmlhttps://pop.com.sg/what-is-the-relationship-between-sets-sequences-and-functions.htmlThere's a practice problem I've been working on that's been bothering me. I've been searching for answers here and elsewhere online but I can't the answer I'm looking for. The question is as follow......Mon, 31 Aug 2026 21:19:47 -0400https://pop.com.sg/what-does-the-sign-propto-mean.htmlhttps://pop.com.sg/what-does-the-sign-propto-mean.htmlI am a computer scientist, and one of my professors today used the symbol $\propto$. I tried to search that using google, but it returns no results, and I do not even know its name. So, I would li......Mon, 31 Aug 2026 21:19:19 -0400https://pop.com.sg/algebraic-and-geometric-multiplicities-of-eigenvalues-of-a-3-times-3-m.htmlhttps://pop.com.sg/algebraic-and-geometric-multiplicities-of-eigenvalues-of-a-3-times-3-m.html"Let A = \begin{pmatrix}-1&-1&-2\\ \:2&2&1\\ \:6&2&6\end{pmatrix} Find the characteristic polynomial of A. Find the distinct eigenvalues of A and their respective algebra......Mon, 31 Aug 2026 21:15:28 -0400https://pop.com.sg/find-a-function-of-depth-with-respect-to-time-when-filling-a-cone.htmlhttps://pop.com.sg/find-a-function-of-depth-with-respect-to-time-when-filling-a-cone.htmlThe task is about filling containers of different shape (cone, sphere, etc) – to find a function of depth $h(t)$ when filling a container with a constant rate of liquid supply. The problem is that......Mon, 31 Aug 2026 21:14:52 -0400https://pop.com.sg/burnside-039-s-lemma.htmlhttps://pop.com.sg/burnside-039-s-lemma.htmlI've been trying to understand what Burnside's Lemma is, and how to apply it, but the wiki page is confusing me. The problem I am trying to solve is: You have 4 red, 4 white, and 4 blue identical ...Mon, 31 Aug 2026 21:14:47 -0400https://pop.com.sg/what-is-the-meaning-of-orthogonal-projection-of-vectors.htmlhttps://pop.com.sg/what-is-the-meaning-of-orthogonal-projection-of-vectors.htmlThe vector $\dfrac{x\cdot y}{y\cdot y}y$ is called the projection of $x$ along $y$. I do not understand what it means, geometrically. Can anyone give me n specific example of what it means?...Mon, 31 Aug 2026 21:08:35 -0400https://pop.com.sg/what-is-an-algebraic-variety.htmlhttps://pop.com.sg/what-is-an-algebraic-variety.htmlI'm studying abelian varieties from Milne's book, but I'm having difficulty juggling different conventions and definitions of basic concepts, like those of algebraic and projective varieties. First......Mon, 31 Aug 2026 21:08:04 -0400https://pop.com.sg/derivatives-by-complex-number-and-conjugate.htmlhttps://pop.com.sg/derivatives-by-complex-number-and-conjugate.htmlSuppose we write a complex number $z=x+iy$. Then we have $\overline{z}=x-iy$, and therefore $x=\dfrac12(z+\overline{z})$, $y=-\dfrac12i(z-\overline{z})$. If the rules of calculus were applicabl......Mon, 31 Aug 2026 21:05:56 -0400https://pop.com.sg/does-a-bi-3-x-yi-3-implies-a-bi-x-yi.htmlhttps://pop.com.sg/does-a-bi-3-x-yi-3-implies-a-bi-x-yi.htmlDoes $(a + bi)^3 = (x+yi)^3 \implies a + bi = x + yi$, where $a,b,x,y \in \mathbb{R}?$ Please provide a proof/counterexample. I've tried brute force and ended up with: $$(a - x)(a^2 + ax + x^2) +......Mon, 31 Aug 2026 21:05:16 -0400https://pop.com.sg/when-can-we-say-that-a-mathrm-t-b-b-mathrm-t-a.htmlhttps://pop.com.sg/when-can-we-say-that-a-mathrm-t-b-b-mathrm-t-a.htmlI was looking at the derivation of the normal equation from here. Now, the author has used the fact that $A^{\mathrm T} B = B^{\mathrm T} A$ to reach the step shown in the below image. Can anyone ...Mon, 31 Aug 2026 20:58:58 -0400https://pop.com.sg/finding-cartesian-equation-from-parametric-trigonometric-equations.htmlhttps://pop.com.sg/finding-cartesian-equation-from-parametric-trigonometric-equations.htmlI'm trying to find the cartesian equation of the curve which is defined parametrically by: $$ x = 2\sin\theta, y = \cos^2\theta $$ Both approaches I take result in the same answer: $$ y = 1 - \s......Mon, 31 Aug 2026 20:47:17 -0400https://pop.com.sg/cantor-set-in-base-3.htmlhttps://pop.com.sg/cantor-set-in-base-3.htmlI'm trying to prove the cantor set $C$ is equivalent to the set of all numbers with ternary expansion of $2$'s and $0$'s. That is: Let $A_0=[0,1]$. $A_n$ is defined to equal $A_{n-1}$ with it's mi......Mon, 31 Aug 2026 20:45:58 -0400https://pop.com.sg/show-that-a-matrix-a-is-singular-if-and-only-if-0-is-an-eigenvalue.htmlhttps://pop.com.sg/show-that-a-matrix-a-is-singular-if-and-only-if-0-is-an-eigenvalue.htmlI can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition: A matrix $A$ is singular if and only if $0$ is an eigenvalue. Could anyone shed s......Mon, 31 Aug 2026 20:45:25 -0400https://pop.com.sg/rotating-rectangle-90-degrees-clockwise.htmlhttps://pop.com.sg/rotating-rectangle-90-degrees-clockwise.htmlI have a rectangle in the cartesian plane defined by the top left and Bottom right as $(3, 5),(5, 3)$ We rotate this around the origin clockwise by 90 degrees, what is the new top left and bottom ......Mon, 31 Aug 2026 20:40:05 -0400https://pop.com.sg/let-t-x-2y-xy-3-2-where-x-r-cos-theta-y-r-sin-theta-find-frac-partial-.htmlhttps://pop.com.sg/let-t-x-2y-xy-3-2-where-x-r-cos-theta-y-r-sin-theta-find-frac-partial-.htmlLet $T = x^2y-xy^3+2$ ; where $x = r\cos\theta, y=r\sin\theta$. Find $\dfrac{\partial T}{\partial r}$ and $\dfrac{\partial T}{\partial \theta}$. My answer: For $\dfrac{\partial T}{\partial r}$ I g......Mon, 31 Aug 2026 20:36:56 -0400https://pop.com.sg/how-to-integrate-int-x-n-e-x-dx.htmlhttps://pop.com.sg/how-to-integrate-int-x-n-e-x-dx.htmlHow can I solve this indefinite integral for an arbitrary integer $n>0$? $$ \int{x^n e^x dx}$$ I could partially integrate it for small $n$, but that's not really a solution. Edit: (TB) This ...Mon, 31 Aug 2026 20:35:32 -0400https://pop.com.sg/prob-15-chap-6-in-baby-rudin-if-f-is-a-real-continuously-differentiabl.htmlhttps://pop.com.sg/prob-15-chap-6-in-baby-rudin-if-f-is-a-real-continuously-differentiabl.htmlHere is Prob. 15, Chap. 6, in the book Principles of Mathematical Analysis, by Walter Rudin, 3rd edition: Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) =......Mon, 31 Aug 2026 20:33:14 -0400https://pop.com.sg/linear-transformation-from-r-3-to-r-2.htmlhttps://pop.com.sg/linear-transformation-from-r-3-to-r-2.htmlFind the matrix of the linear transformation $T\colon {\Bbb R}^3 \to {\Bbb R}^2$ such that $T(1,1,1) = (1,1)$, $T(1,2,3) = (1,2)$, $T(1,2,4) = (1,4)$. So far, I have only dealt with transformati......Mon, 31 Aug 2026 20:28:33 -0400https://pop.com.sg/infinity-plus-infinity.htmlhttps://pop.com.sg/infinity-plus-infinity.htmlLet $a \in \mathbb{C}$. Ahlfors says we let $a + \infty = \infty$ and $a \cdot \infty = \infty$. But we cannot define $\infty + \infty$ without violating the laws of arithimetic (i.e. field axioms)......Mon, 31 Aug 2026 20:20:12 -0400https://pop.com.sg/upper-bound-on-the-number-of-edges-in-the-farthest-point-voronoi-diagr.htmlhttps://pop.com.sg/upper-bound-on-the-number-of-edges-in-the-farthest-point-voronoi-diagr.htmlI'm trying to prove that the number of edges in the farthest-point voronoi diagram of $n$ points is upper-bounded by $2n-3$ (as claimed by Computational Geometry: Algorithms and Applications). I co......Mon, 31 Aug 2026 20:19:03 -0400https://pop.com.sg/diagonalizing-the-matrix-if-possible.htmlhttps://pop.com.sg/diagonalizing-the-matrix-if-possible.htmlDiagonalize the matrix $\begin{bmatrix}0&-4&-6\\-1&0&-3\\1&2&5\end{bmatrix}$ if possible So I know that I can check to see if this is diagonalizable by doing $A = PDP^{-1}$......Mon, 31 Aug 2026 20:17:20 -0400https://pop.com.sg/calculating-surface-area-of-a-sphere-with-cylindrical-coordinates.htmlhttps://pop.com.sg/calculating-surface-area-of-a-sphere-with-cylindrical-coordinates.htmlCalculating the surface area of a sphere of radius $a$ with spherical coordinates is quite easy ($4\pi a^2$). I'm trying to do the same with cylindrical coordinates, $\rho$, $\theta$, $z$, (just f......Mon, 31 Aug 2026 20:15:48 -0400https://pop.com.sg/does-zero-divide-zero.htmlhttps://pop.com.sg/does-zero-divide-zero.htmlI wanted to know is zero divisible by zero? I've read that division by zero is not allowed in mathematics, but for instance in Apostol's Introduction to analytic number theory, it states that only ......Mon, 31 Aug 2026 20:14:41 -0400https://pop.com.sg/how-is-ln-pronounced-by-english-speakers.htmlhttps://pop.com.sg/how-is-ln-pronounced-by-english-speakers.htmlI have always heard an expression like $\ln (x^2)$ pronounced aloud as "ell-enn ex squared". That is, the name of the function $\ln$ is read aloud as a two-letter abbreviation. However, I recentl......Mon, 31 Aug 2026 20:04:42 -0400https://pop.com.sg/length-of-a-parabolic-curve.htmlhttps://pop.com.sg/length-of-a-parabolic-curve.htmlI just wanted to know how I can find the length of a curve given by $f(x) = x^2$ from $x=0$ to $x=1$. For appproximation, the length is a bit larger than the hypotenuse of isosceles right triangle......Mon, 31 Aug 2026 20:00:50 -0400https://pop.com.sg/epsilon-delta-derivative-proof-of-x-n.htmlhttps://pop.com.sg/epsilon-delta-derivative-proof-of-x-n.htmlI'm currently trying to prove the power rule using the epsilon-delta definition of a derivative. I've already done it for the basic limit definition, but I thought it might be a helpful exercise to......Mon, 31 Aug 2026 19:50:15 -0400https://pop.com.sg/average-path-length-of-unconnected-graphs.htmlhttps://pop.com.sg/average-path-length-of-unconnected-graphs.htmlI'm reading graph theory and I found a concept named average path length. Basically, it is the average of all the shortest lengths between any two nodes of a graph. In case two nodes are not connec......Mon, 31 Aug 2026 19:48:24 -0400https://pop.com.sg/cos-2-alpha-cos-2-beta-cos-2-gamma-1-duplicate.htmlhttps://pop.com.sg/cos-2-alpha-cos-2-beta-cos-2-gamma-1-duplicate.htmlLet be $\alpha, \beta, \gamma$ the angles between a generic direction in 3D and the axes $x,y,z$, respectively. Prove that $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =1$. PS: the 2D case is tr......Mon, 31 Aug 2026 19:42:28 -0400https://pop.com.sg/definition-of-an-affine-subspace.htmlhttps://pop.com.sg/definition-of-an-affine-subspace.htmlI am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me. I cite: A subset $B$ of a $\mathbb{R}$-affine space $A$ ...Mon, 31 Aug 2026 19:40:02 -0400https://pop.com.sg/integral-int-0-1-frac-arctan-x-x-2-x-1-dx.htmlhttps://pop.com.sg/integral-int-0-1-frac-arctan-x-x-2-x-1-dx.htmlAfter seeing this integral I've decided to give a try to calculate: $$I=\int_0^1 \frac{\arctan x}{x^2-x-1}dx$$ That is because it's common for many integrals to have a combination of a polynomial i......Mon, 31 Aug 2026 19:29:03 -0400https://pop.com.sg/contour-integral-for-x-3-e-x-1.htmlhttps://pop.com.sg/contour-integral-for-x-3-e-x-1.htmlWhat contour and integrand do we use to evaluate $$ \int_0^\infty \frac{x^3}{e^x-1} dx $$ Or is this going to need some other technique?...Mon, 31 Aug 2026 19:13:01 -0400https://pop.com.sg/find-diagonal-of-inverse-matrix.htmlhttps://pop.com.sg/find-diagonal-of-inverse-matrix.htmlI have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the chole......Mon, 31 Aug 2026 19:12:07 -0400https://pop.com.sg/how-to-simplify-log-base-2-and-log-base-4.htmlhttps://pop.com.sg/how-to-simplify-log-base-2-and-log-base-4.htmlHow do I simplify the following expression: $$\log_2(2x+1) - 5\log_4(x^2) + 4\log_2(x)$$ That's it, please help me ok?...Mon, 31 Aug 2026 19:04:55 -0400https://pop.com.sg/simplify-sin-1-tan-x.htmlhttps://pop.com.sg/simplify-sin-1-tan-x.htmlI'm aware of how we can simplify functions which have $Arc$ as an argument . For example $\sin(\cos^{-1}(x)) = \sqrt{1-x^2}$ but what about cases which $Arc$ is out of the parentheses ? For instance ...Mon, 31 Aug 2026 18:50:01 -0400https://pop.com.sg/construct-a-bijective-map-and-prove-it-is-bijective.htmlhttps://pop.com.sg/construct-a-bijective-map-and-prove-it-is-bijective.htmlS is a finite set with more than 1 element. Construct a bijective map $f:S\times \mathbb{Z}\rightarrow\mathbb{Z}$ and prove it is bijective. ($\mathbb{Z}$ is the set of all integers) I'm at a los......Mon, 31 Aug 2026 18:45:45 -0400https://pop.com.sg/integral-int-limits-0-pi-sin-x-cos-x-dx.htmlhttps://pop.com.sg/integral-int-limits-0-pi-sin-x-cos-x-dx.html$$\int \limits _0 ^\pi |\sin x + \cos x|\; dx$$ If I divide integral in two parts $\int\limits_0^{\frac{\pi}{2}}{(\sin x + \cos x)\,dx}$ and $\int\limits_{\frac{\pi}{2}}^\pi{(\sin x - \cos x)\,dx......Mon, 31 Aug 2026 18:32:37 -0400https://pop.com.sg/finding-general-formula-for-a-recursion-function.htmlhttps://pop.com.sg/finding-general-formula-for-a-recursion-function.htmlIf we have a recursion relation defined as $a_n = 3a_{n-1}+1$ with $a_1=1$ then find the general formula for $a_n$ in terms of $n$ with a(1) = 1. So far I have: $a_n = 3a_{n-2}+1+1 = 3a_{n-3}+1+1......Mon, 31 Aug 2026 18:27:07 -0400https://pop.com.sg/proving-the-fundamental-period-of-tangent.htmlhttps://pop.com.sg/proving-the-fundamental-period-of-tangent.htmlI'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that $\pi$ is a fundamental period of the tangent function. I ......Mon, 31 Aug 2026 18:20:27 -0400https://pop.com.sg/probability-of-drawing-5-cards-from-a-deck-of-52-that-will-have-the-sa.htmlhttps://pop.com.sg/probability-of-drawing-5-cards-from-a-deck-of-52-that-will-have-the-sa.htmlA standard deck of cards has 52 members consisting of 4 suits each with 13 members. Five cards are dealt from the randomly mixed deck. What is the probability that all cards are the same suit? E......Mon, 31 Aug 2026 18:13:17 -0400https://pop.com.sg/prove-convergent-sequences-are-bounded.htmlhttps://pop.com.sg/prove-convergent-sequences-are-bounded.htmlI don't understand this one part in the proof for convergent sequences are bounded. Proof: Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have: $n > N \...Mon, 31 Aug 2026 18:00:45 -0400https://pop.com.sg/isomorphism-classes-of-abelian-groups-of-certain-order.htmlhttps://pop.com.sg/isomorphism-classes-of-abelian-groups-of-certain-order.htmlworking on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space......Mon, 31 Aug 2026 18:00:02 -0400https://pop.com.sg/independence-of-a-random-variable-x-from-itself.htmlhttps://pop.com.sg/independence-of-a-random-variable-x-from-itself.htmlIn our lecture on probability, my professor made the comment that "a random variable X is not independent from itself." (Here he was specifically talking about discrete random variables.) I asked h......Mon, 31 Aug 2026 17:44:58 -0400https://pop.com.sg/what-is-the-difference-between-p-a-b-and-p-a-b.htmlhttps://pop.com.sg/what-is-the-difference-between-p-a-b-and-p-a-b.htmlI feel that $p(a,b)$ = the probability that event $a$ and $b$ happen at the same time. $p(a|b)$ = the probability that event $a$ happens due to the event $b$ happens. For me, I think the meanin......Mon, 31 Aug 2026 17:42:39 -0400https://pop.com.sg/asymptotic-approximation-of-the-arctangent.htmlhttps://pop.com.sg/asymptotic-approximation-of-the-arctangent.htmlThat is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x))......Mon, 31 Aug 2026 17:39:52 -0400https://pop.com.sg/use-cylindrical-coordinates-to-find-the-volume-of-a-solid.htmlhttps://pop.com.sg/use-cylindrical-coordinates-to-find-the-volume-of-a-solid.htmlTo Find: use cylindrical coordinates to find the volume of a solid that is enclosed between the surface $r^2+z^2=20$ and the surface $z=r^2$: $$\iiint_R f(x,y,z)\,dv=\iiint_R f(r\cos\theta,r\sin\t......Mon, 31 Aug 2026 17:38:13 -0400https://pop.com.sg/definition-of-maximal-and-minimal-duplicate.htmlhttps://pop.com.sg/definition-of-maximal-and-minimal-duplicate.htmlPossible Duplicate: difference between maximal element and greatest element When I first encountered the terms maximal and minimal, I confused them with maximum and minimum. Many of my classma......Mon, 31 Aug 2026 17:36:44 -0400https://pop.com.sg/difference-between-basis-and-subbasis-in-a-topology.htmlhttps://pop.com.sg/difference-between-basis-and-subbasis-in-a-topology.htmlI was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?...Mon, 31 Aug 2026 17:29:02 -0400https://pop.com.sg/definition-of-an-induced-matrix-norm.htmlhttps://pop.com.sg/definition-of-an-induced-matrix-norm.htmlCould someone explain the second equality in the definition of a induced matrix norm to me? Let $\| \cdot \|$ be a norm for $\cdot \in \mathbb{R}^n$, then the induced matrixnorm for $A\in \ma......Mon, 31 Aug 2026 17:27:02 -0400