https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 26 Aug 2026 08:02:05 -0400https://pop.com.sg/what-does-it-mean-to-be-functorial-in-something.htmlhttps://pop.com.sg/what-does-it-mean-to-be-functorial-in-something.htmlIs there a definition of the word functorial? Often I read that certain things are functorial and often I have an intuition of what is meant. But is there a precise definition of what is meant by t......Wed, 26 Aug 2026 11:57:56 -0400https://pop.com.sg/is-it-acceptable-to-cite-commonly-used-definitions-from-mathworld-wolf.htmlhttps://pop.com.sg/is-it-acceptable-to-cite-commonly-used-definitions-from-mathworld-wolf.htmlI am writing a document/article for the average layman, with the aim of explaining certain well known results in mathematics while glossing over the more gritty details. I'm going to include several ...Wed, 26 Aug 2026 11:53:28 -0400https://pop.com.sg/if-a-is-an-element-of-b-does-that-imply-that-a-is-a-subset-of-b.htmlhttps://pop.com.sg/if-a-is-an-element-of-b-does-that-imply-that-a-is-a-subset-of-b.htmlQuestion: Suppose that $A\in B$, does this imply that $A\subset B$? If so, why? I have tried to look at using the following example: Suppose that $A = 4$ and $B = \{4\}$. We definitely have $A \i......Wed, 26 Aug 2026 11:52:11 -0400https://pop.com.sg/find-the-least-integer-n-such-that-f-x-is-o-x-n-for-given-functions.htmlhttps://pop.com.sg/find-the-least-integer-n-such-that-f-x-is-o-x-n-for-given-functions.htmlI am to find the least integer $n$ such that $f(x)$ is $O(x^n)$ for the following functions: $\begin{equation} (a) f(x) = 2x^2 + x^7 log(x) \end{equation}$ $\begin{equation} (b) f(x) = 3x^9 + (lo......Wed, 26 Aug 2026 11:51:23 -0400https://pop.com.sg/linear-algebra-polynomial-subspace.htmlhttps://pop.com.sg/linear-algebra-polynomial-subspace.htmlLet $P_n$ be the set of polynomials of degree at most $n$ with real coefficients. Is the set of polynomials of the form $p(t) = a + t^2,$ such that '$a$' is an element of $\mathbb{R},$ a subspace o......Wed, 26 Aug 2026 11:50:14 -0400https://pop.com.sg/find-the-unique-x-in-the-interval-0-pi-with.htmlhttps://pop.com.sg/find-the-unique-x-in-the-interval-0-pi-with.htmlI don't understand what I should calculate if is said: "find the unique x", e.g. in the following exercise: "Find the unique x in the interval [0,pi] with cos(x) = 0,5 * sqrt(3)"...Wed, 26 Aug 2026 11:37:15 -0400https://pop.com.sg/dividing-a-rectangle-into-4-parts-in-the-ratio-1-2-3-4-with-only-2-lin.htmlhttps://pop.com.sg/dividing-a-rectangle-into-4-parts-in-the-ratio-1-2-3-4-with-only-2-lin.htmlI have a rectangle made up of 30 identical squares (5 tall and 6 wide). By only drawing two lines on the rectangle, split the rectangle into 4 parts where the areas are in the ratio 1:2:3:4. How ......Wed, 26 Aug 2026 11:33:48 -0400https://pop.com.sg/dihedral-angles-between-tetrahedron-faces-from-triangles-039-angles-at.htmlhttps://pop.com.sg/dihedral-angles-between-tetrahedron-faces-from-triangles-039-angles-at.htmlSay I have a general tetrahedron, or in fact only the tip of a tetrahedron, being three triangles in 3-space sharing a single vertex and sharing edges pairwise. Given that I only know the three an......Wed, 26 Aug 2026 11:31:22 -0400https://pop.com.sg/what-is-model-theory.htmlhttps://pop.com.sg/what-is-model-theory.htmlI've been reading up on some Model Theory lately. My question is simple. What is the purpose of the theory to begin with? How does it enrich mathematics? A-priori, it doesn't seem like we're doing ......Wed, 26 Aug 2026 11:21:52 -0400https://pop.com.sg/calculate-the-degree-of-convergence.htmlhttps://pop.com.sg/calculate-the-degree-of-convergence.htmlI was studying about numerical methods to find root of functions. In the False Position Method, I came to the following claim: $$\text{Degree of convergence } = P = \frac{1+\sqrt5}{2}$$ I cannot fi......Wed, 26 Aug 2026 11:20:47 -0400https://pop.com.sg/first-de-rham-cohomology-group-of-the-2-torus.htmlhttps://pop.com.sg/first-de-rham-cohomology-group-of-the-2-torus.htmlShow that for the de Rham cohomology, $H^{1}_{dR}({T^{2}})$ is isomorphic to $\mathbb{R}^{2}$ by showing that the following map: $[\alpha] \to (\int_{S^{1}}f^{*}_{1}\alpha,\int_{S^{1}}f^{*}_{1}\alp......Wed, 26 Aug 2026 11:09:29 -0400https://pop.com.sg/find-all-x-in-mathbb-z-with-x-equiv-2-mod-4-x-equiv-8-mod9-and-x-equiv.htmlhttps://pop.com.sg/find-all-x-in-mathbb-z-with-x-equiv-2-mod-4-x-equiv-8-mod9-and-x-equiv.htmlFind all $x\in \mathbb{Z}$ with $x \equiv 2 \mod 4$ , $x \equiv 8\mod9$ and $x \equiv 1\mod5$ My attempt: $\gcd\left(9,5,4\right) = 1 = 9r+5s+4\times0$ $9 = 1\times5+4$ $5 = 1\times4+1$ so $1 ......Wed, 26 Aug 2026 11:07:01 -0400https://pop.com.sg/what-is-the-difference-between-an-unbounded-solution-and-infinite-opti.htmlhttps://pop.com.sg/what-is-the-difference-between-an-unbounded-solution-and-infinite-opti.htmlI am currently studying Linear Programming, and one of the few things that I haven't found a solution for (pun obviously intended) is the difference between a solution being either unbounded or hav......Wed, 26 Aug 2026 11:02:28 -0400https://pop.com.sg/simplify-the-expression-by-using-a-double-angle-formula-or-a-half-angl.htmlhttps://pop.com.sg/simplify-the-expression-by-using-a-double-angle-formula-or-a-half-angl.htmlSimplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) $\cos^2 \left( \cfrac{θ}{2} \right)− \sin^2 \left( \cfrac{θ}{2} \right)$ (b) $2 \sin \left( \cfrac{θ}{2} \right)......Wed, 26 Aug 2026 11:02:15 -0400https://pop.com.sg/why-are-the-coefficients-of-the-equation-of-a-plane-the-normal-vector-.htmlhttps://pop.com.sg/why-are-the-coefficients-of-the-equation-of-a-plane-the-normal-vector-.htmlWhy are the coefficients of the equation of a plane the normal vector of a plane? I borrowed the below picture from Pauls Online Calculus 3 notes: http://tutorial.math.lamar.edu/Classes/CalcIII/...Wed, 26 Aug 2026 11:01:13 -0400https://pop.com.sg/what-is-the-difference-between-zero-vector-and-null-vector-closed.htmlhttps://pop.com.sg/what-is-the-difference-between-zero-vector-and-null-vector-closed.htmlWhat is the difference between zero vector and null vector? I think it is not a duplicate....Wed, 26 Aug 2026 10:57:38 -0400https://pop.com.sg/the-difference-between-inverse-function-and-a-function-that-is-inverti.htmlhttps://pop.com.sg/the-difference-between-inverse-function-and-a-function-that-is-inverti.htmlMy discrete mathematics book says: But I read an answer, https://math.stackexchange.com/a/2415543/390226, said: ["...]There are invertible functions which are not bijective,[..."] And to the s......Wed, 26 Aug 2026 10:54:31 -0400https://pop.com.sg/is-9-1-an-improper-fraction.htmlhttps://pop.com.sg/is-9-1-an-improper-fraction.htmlMy son took a test in school. The teacher told them that they did not need to simplify improper fractions in their answers. On one question, for example, the answer of 28/3 was marked as correct.......Wed, 26 Aug 2026 10:48:28 -0400https://pop.com.sg/cauchy-integral-formula-when-f-z-is-not-analytic-everywhere-inside-c.htmlhttps://pop.com.sg/cauchy-integral-formula-when-f-z-is-not-analytic-everywhere-inside-c.htmlLet $\alpha$ be a given complex number satisfying $0<|\alpha|<1$, and let $\gamma$ be the circle $|z|=2$ oriented in the counterclockwise direction. Determine the value of $$ \int_{\gamma} \f......Wed, 26 Aug 2026 10:44:14 -0400https://pop.com.sg/determine-an-interval-in-which-the-solution-to-the-given-ivp-exists-di.htmlhttps://pop.com.sg/determine-an-interval-in-which-the-solution-to-the-given-ivp-exists-di.htmlDetermine an interval in which the solution to the given IVP exists. $$(x-3){dy\over dx}+(\ln x)y=2x; (1,2)$$ I was wondering if I needed to to separation of variables or the thing where I multipl......Wed, 26 Aug 2026 10:42:47 -0400https://pop.com.sg/what-does-arc-in-arcsin-arccos-arctan-stands-for.htmlhttps://pop.com.sg/what-does-arc-in-arcsin-arccos-arctan-stands-for.htmlI was just wondering, what does the "arc" in arcsin, arccos, arctan stands for? Is there any particular reason why it is named the way it is?...Wed, 26 Aug 2026 10:41:34 -0400https://pop.com.sg/show-that-m-le-2e-v-le-m.htmlhttps://pop.com.sg/show-that-m-le-2e-v-le-m.html$G$ be a graph with $v$ vertices and $e$ edges.Let $M$ be maximum degree of the vertices of $G$, and let $m$ be the minimum degree of the vertices $G$.Show that $$m \le 2e/v \le M$$ I am completely ...Wed, 26 Aug 2026 10:33:07 -0400https://pop.com.sg/express-solution-in-simplest-radical-form-4x-1-3x-7-5x-1-2x-3.htmlhttps://pop.com.sg/express-solution-in-simplest-radical-form-4x-1-3x-7-5x-1-2x-3.htmlI need help with the problem. I'm in grade 11 math. $(4x-1)(3x+7)=(5x-1)(2x+3)$ So far I think I have to factor it and I got $(12x^2+25x-7)=(10x^2-13x-3)$ Thanks!...Wed, 26 Aug 2026 10:29:13 -0400https://pop.com.sg/determine-the-expected-value-and-variance-of-bar-x.htmlhttps://pop.com.sg/determine-the-expected-value-and-variance-of-bar-x.htmlLet $X_1,..,X_n$ be independent, identically distributed (continuous) random variables. Let $$\bar{X} = \frac{1}{n} \cdot \left(X_1+...+X_n\right)$$ Determine the expected value and variance......Wed, 26 Aug 2026 10:14:15 -0400https://pop.com.sg/nullity-of-t-vs-nullity-of-t-2.htmlhttps://pop.com.sg/nullity-of-t-vs-nullity-of-t-2.htmlFor a finite dimensional vector space, $V$, and linear transformation, $T$, what is the relationship between the nullity of $T$ and the nullity of $T^2$? I read somwhere that if $nullity(T)=k$, th......Wed, 26 Aug 2026 10:07:28 -0400https://pop.com.sg/good-video-lectures-for-studying-calculus.htmlhttps://pop.com.sg/good-video-lectures-for-studying-calculus.htmlI am looking for a good online video resource to start studying Calculus. I am studying it alone, not part of any school or university. Trying to learn and enhance my mathematical skills. Thanks!...Wed, 26 Aug 2026 10:02:12 -0400https://pop.com.sg/how-to-find-what-a-given-series-sequence-converges-to.htmlhttps://pop.com.sg/how-to-find-what-a-given-series-sequence-converges-to.htmlGiven a function series, what are the ways often used to find the function that the series converges to? For example, how do you construct the function on the right hand side from the series on th......Wed, 26 Aug 2026 09:43:39 -0400https://pop.com.sg/basic-tensor-derivation.htmlhttps://pop.com.sg/basic-tensor-derivation.htmlLet $D$ be a tensor derivation on a mnaifold $M$. I have to show that if $D(\partial_i)=\sum F_i^j \partial_j$, then $D(dx^j)=-\sum F_i^j dx^i$. Any help on how to do this? Thanks in advance....Wed, 26 Aug 2026 09:43:09 -0400https://pop.com.sg/proof-of-vector-calculus-identities.htmlhttps://pop.com.sg/proof-of-vector-calculus-identities.htmlHere is the all identities : http://en.wikipedia.org/wiki/Vector_calculus_identities I need help concerning vector functions and indexing notations. Let $\overrightarrow{a}$ be a (smooth) vector ...Wed, 26 Aug 2026 09:35:32 -0400https://pop.com.sg/what-is-convex-combination-of-two-points.htmlhttps://pop.com.sg/what-is-convex-combination-of-two-points.htmlI am studying algorithms and i saw a definition like the following: Given $3$ points $p_1 = (x_1, y_1)$, $p_2 = (x_2, y_2)$ and $p_3 = (x_3, y_3)$, $p_3$ is a convex combination of $p_1$ and $p_2$......Wed, 26 Aug 2026 09:33:26 -0400https://pop.com.sg/eigenvectors-of-similar-matrices.htmlhttps://pop.com.sg/eigenvectors-of-similar-matrices.htmlIf $A$ and $B$ are similar matrices then every eigenvector of $A$ is an eigenvector of $B$. Is the above statement is true? I know that similar matrices have same eigenvalue, but I'm not sure about......Wed, 26 Aug 2026 09:32:14 -0400https://pop.com.sg/is-a-positively-homogeneous-convex-function-lower-semi-continuous.htmlhttps://pop.com.sg/is-a-positively-homogeneous-convex-function-lower-semi-continuous.htmlHere the definition of positively homogeneous function $f$ is \begin{equation} f(\lambda x) = \lambda f(x), \quad \lambda \ge 0, \end{equation} and $f$ is defined on linear space $\mathbb{R}^n$. I ...Wed, 26 Aug 2026 09:24:42 -0400https://pop.com.sg/square-roots-of-complex-number-3-4i.htmlhttps://pop.com.sg/square-roots-of-complex-number-3-4i.htmlWhat I did $z^2=3-4i$ $(a+bi)^2 = 3-4i$ $a^2-b^2+2abi = 3-4i$ Then got 2 simultaneous equations $a^2-b^2=3$ and $2ab=-4$ Solve for $a^2$ in 1st equation: $a^2=3+b^2$ Subbed into 2nd equation......Wed, 26 Aug 2026 09:19:32 -0400https://pop.com.sg/equivalence-relation-and-sets.htmlhttps://pop.com.sg/equivalence-relation-and-sets.htmlThe question: Let $X$ and $Y$ be two sets, and let $S$ be an equivalence relation on set $X$ and $T$ be an equivalence relation on set $Y$. Define a relation $R$ on $X ×Y$ by $(a,b)R(c,d)$ if and ......Wed, 26 Aug 2026 09:18:27 -0400https://pop.com.sg/what-comes-after-seconds.htmlhttps://pop.com.sg/what-comes-after-seconds.htmlAngles can be measured in different ways. For example, one can measure angles in degrees/minutes/seconds. So $1^\circ$ is divded into $60$ min. and $1$ min is divided into $60$ sec. That way a tent......Wed, 26 Aug 2026 09:17:31 -0400https://pop.com.sg/prove-that-the-square-root-of-any-irrational-number-is-irrational.htmlhttps://pop.com.sg/prove-that-the-square-root-of-any-irrational-number-is-irrational.htmlThe problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irratio......Wed, 26 Aug 2026 09:00:17 -0400https://pop.com.sg/why-does-an-integral-change-signs-when-flipping-the-boundaries.htmlhttps://pop.com.sg/why-does-an-integral-change-signs-when-flipping-the-boundaries.htmlLet us define a very simple integral: $f(x) = \int_{a}^{b}{x}$ for $a,b\ge 0$. Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$? I drew the graphs and thought about it but to me ...Wed, 26 Aug 2026 08:51:07 -0400https://pop.com.sg/intuitive-and-convincing-argument-that-functions-are-vectors.htmlhttps://pop.com.sg/intuitive-and-convincing-argument-that-functions-are-vectors.htmlBack to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge ......Wed, 26 Aug 2026 08:48:24 -0400https://pop.com.sg/define-a-binary-composition-operation-star-such-that-langle-g-star-ran.htmlhttps://pop.com.sg/define-a-binary-composition-operation-star-such-that-langle-g-star-ran.htmlLet $G$ be a group and $a\in G $ be any non identity element i.e. $a\neq e$. Define a binary composition ( operation ) $\star$ such that $\langle G,\star \rangle$ is a group with $a$ as its identit......Wed, 26 Aug 2026 08:32:26 -0400https://pop.com.sg/how-to-prove-that-the-matrix-is-not-invertible.htmlhttps://pop.com.sg/how-to-prove-that-the-matrix-is-not-invertible.html$A$ and $B$ are $3\times 3$ matrices ($A,B \in \mathcal{M}_{3 \times 3}(\mathbb{R})$). There are two equations: $$A^2+3BA=I$$ $$A^2=AB$$ I want to prove that $A$ does not have an inverse. I tri......Wed, 26 Aug 2026 08:24:24 -0400https://pop.com.sg/lower-limit-topology-properties.htmlhttps://pop.com.sg/lower-limit-topology-properties.htmlI am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...Wed, 26 Aug 2026 08:15:07 -0400https://pop.com.sg/how-many-solutions-exists-for-this-equation-duplicate.htmlhttps://pop.com.sg/how-many-solutions-exists-for-this-equation-duplicate.html$$x_1 + x_2 + x_3 + x_4 = 28$$ I tried to solve it with generating functions. Is it correct to get to the form of $${(1 + x + {x^2} + {x^3} + ....)^4}$$ and this equals to: $${(1 - x)^{ - 4}}$$......Wed, 26 Aug 2026 08:08:12 -0400https://pop.com.sg/find-center-of-rotation-after-object-rotated-by-known-angle-2d.htmlhttps://pop.com.sg/find-center-of-rotation-after-object-rotated-by-known-angle-2d.htmlI need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle. Previously I would simply find the center of the object, rotate 180 ...Wed, 26 Aug 2026 08:07:07 -0400https://pop.com.sg/negation-of-biconditional-statements.htmlhttps://pop.com.sg/negation-of-biconditional-statements.htmlLet $p$ and $q$ be two sub statements of the compound biconditional statement given as $p$⇔$q$. The negation of this biconditional statement is given as ($p$^~$q$)∨($q$^~$p$) In the above statem......Wed, 26 Aug 2026 07:58:31 -0400https://pop.com.sg/camera-rotation-draws-twoards-z-animation-script-built-from-mathformul.htmlhttps://pop.com.sg/camera-rotation-draws-twoards-z-animation-script-built-from-mathformul.htmlI have constructed a script in FME which calculates the position and rotation of the camera along a track "railroad track". I use math when I write my script so no standard functions. The software ......Wed, 26 Aug 2026 07:56:41 -0400https://pop.com.sg/proving-x-n-y-n-x-y-x-n-1-x-n-2-y-x-y-n-2-y-n-1.htmlhttps://pop.com.sg/proving-x-n-y-n-x-y-x-n-1-x-n-2-y-x-y-n-2-y-n-1.htmlIn Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've g......Wed, 26 Aug 2026 07:48:00 -0400https://pop.com.sg/how-to-perform-wedge-product.htmlhttps://pop.com.sg/how-to-perform-wedge-product.htmlI have heard all kinds of great things about Clifford/Geometric algebra, but I can't find any good resources. I have been looking EVERYWHERE for just one actual example of a wedge product being ...Wed, 26 Aug 2026 07:46:31 -0400https://pop.com.sg/how-is-the-subgradient-method-different-from-gradient-descent-in-pract.htmlhttps://pop.com.sg/how-is-the-subgradient-method-different-from-gradient-descent-in-pract.htmlI'm trying to learn about the subgradient descent optimization method. I'm having trouble understanding how it differs from basic gradient descent in a practical sense. According to this lecture,......Wed, 26 Aug 2026 07:42:50 -0400https://pop.com.sg/solving-inequalities-with-variables-in-the-denominator.htmlhttps://pop.com.sg/solving-inequalities-with-variables-in-the-denominator.htmlOk so I've got an exam in 4 hours and I can't ever figure out these problems. Here's the problem I'm struggling with now: $$\frac{x}{3x - 5} \leq \frac{2}{x - 1}.$$ I've learned this so many tim......Wed, 26 Aug 2026 07:37:34 -0400https://pop.com.sg/why-are-natural-numbers-called-natural-numbers.htmlhttps://pop.com.sg/why-are-natural-numbers-called-natural-numbers.htmlWhen we say $1,2,3...$ are natural numbers, why don't we include rational and irrational numbers? Isn't $\pi$ something natural? Shouldn't we say all real numbers the Natural numbers? Shouldn't......Wed, 26 Aug 2026 07:18:53 -0400https://pop.com.sg/determine-whether-the-series-frac-e-frac-1-n-n-2-converges-or-diverges.htmlhttps://pop.com.sg/determine-whether-the-series-frac-e-frac-1-n-n-2-converges-or-diverges.htmlThe problem states Determine whether the series converges or diverges: $$\sum_{n=1}^\infty\frac{e^{1/n}}{n^2}$$ Immediately I always like to try the test for divergence: $$\lim_{n\to\infty} \s......Wed, 26 Aug 2026 07:17:55 -0400https://pop.com.sg/find-the-six-trigonometric-functions-given-a-point-with-y-2x.htmlhttps://pop.com.sg/find-the-six-trigonometric-functions-given-a-point-with-y-2x.htmlThe terminal side of the angle θ is standard position lies on the given line and satisfies the given condition. Determine the 6 trigonometric functions of said angle. $$y = 2x, \quad \sec θ &gt......Wed, 26 Aug 2026 07:08:44 -0400https://pop.com.sg/what-is-the-symbol-for-imaginary-numbers.htmlhttps://pop.com.sg/what-is-the-symbol-for-imaginary-numbers.html$\mathbb Q$ is used to represent rational numbers. $\mathbb R$ is used to represent real numbers. Is there an accepted symbol for imaginary numbers?...Wed, 26 Aug 2026 07:07:16 -0400https://pop.com.sg/how-to-integrate-when-integration-by-parts-never-ends.htmlhttps://pop.com.sg/how-to-integrate-when-integration-by-parts-never-ends.htmlSo I have the following integration: $\int{e^{-x}\sin{x}}$ I let u be the $e^{-x}$ and $dv = \sin{x} dx$ After doing the integration by parts I get:$(e^{-x})(-\cos{x})-\int{-e^{-x}*(-\cos{x})}$ ......Wed, 26 Aug 2026 07:04:53 -0400https://pop.com.sg/what-does-the-mathematical-symbol-with-an-underline-below-the-variable.htmlhttps://pop.com.sg/what-does-the-mathematical-symbol-with-an-underline-below-the-variable.htmlI have been reading a research paper. The author has used a variable '$x$' with an underline beneath '$x$'. I know that a variable with a line on top of it implies it's arithmetic mean. But I have ......Wed, 26 Aug 2026 07:03:55 -0400https://pop.com.sg/two-different-formulas-for-standard-error-of-difference-between-two-me.htmlhttps://pop.com.sg/two-different-formulas-for-standard-error-of-difference-between-two-me.htmlI mostly see this formula when searching for a formula for the estimate of the standard error in difference between two means, and it is also used in this video. $$\Delta=\sqrt{s_1^2/N_1+s_2^2/N_2}......Wed, 26 Aug 2026 07:00:52 -0400https://pop.com.sg/delta-x-in-limit-problem.htmlhttps://pop.com.sg/delta-x-in-limit-problem.htmlI was working on some limit homework and everything was going fine until I reached this problem: $$\lim_{\Delta x \to 0} \frac{2(x + \Delta x) - 2x}{\Delta x}.$$ I am understanding limits but the ...Wed, 26 Aug 2026 06:58:10 -0400https://pop.com.sg/calculate-x-y-z-from-two-specific-degrees-on-a-sphere.htmlhttps://pop.com.sg/calculate-x-y-z-from-two-specific-degrees-on-a-sphere.htmlI am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...Wed, 26 Aug 2026 06:56:00 -0400https://pop.com.sg/why-does-x-2-5x-6-0-which-is-the-same-as-x-3-x-2-0-represent-a-parabol.htmlhttps://pop.com.sg/why-does-x-2-5x-6-0-which-is-the-same-as-x-3-x-2-0-represent-a-parabol.htmlConsider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 ......Wed, 26 Aug 2026 06:54:40 -0400https://pop.com.sg/the-sum-of-series-with-natural-logarithm-sum-n-1-infty-ln-left-frac-n-.htmlhttps://pop.com.sg/the-sum-of-series-with-natural-logarithm-sum-n-1-infty-ln-left-frac-n-.htmlCalculate the sum of series: $$\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$$ I tried to spread this logarithm, but I'm not seeing any method for this exercise....Wed, 26 Aug 2026 06:51:24 -0400https://pop.com.sg/integral-of-x-2-e-x-2.htmlhttps://pop.com.sg/integral-of-x-2-e-x-2.htmlLike the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has le......Wed, 26 Aug 2026 06:45:12 -0400https://pop.com.sg/which-algorithm-does-matlab-eig-use-to-diagonalize-a-complex-symmetric.htmlhttps://pop.com.sg/which-algorithm-does-matlab-eig-use-to-diagonalize-a-complex-symmetric.htmlI used MATLAB eig() to find eigenvectors and eigenvalues of a complex symmetric matrix. I searched through MATLAB online documentation to find a link to some algorithm, but failed. My curiosity is ......Wed, 26 Aug 2026 06:23:49 -0400https://pop.com.sg/convert-the-integral-from-rectangular-to-cylindrical-coordinates-and-s.htmlhttps://pop.com.sg/convert-the-integral-from-rectangular-to-cylindrical-coordinates-and-s.htmlConvert the integral from rectangular to cylindrical coordinates and solve I think I know how to do this, but I just want to double check my method. So assuming I have the below problem: $$\int^......Wed, 26 Aug 2026 06:21:33 -0400https://pop.com.sg/how-can-cyclic-groups-be-infinite.htmlhttps://pop.com.sg/how-can-cyclic-groups-be-infinite.htmlI am a little confused about how a cyclic group can be infinite. To provide an example, look at $\langle 1\rangle$ under the binary operation of addition. You can never make any negative numbers with ...Wed, 26 Aug 2026 06:19:58 -0400https://pop.com.sg/how-to-find-curves-of-intersection-between-two-surfaces.htmlhttps://pop.com.sg/how-to-find-curves-of-intersection-between-two-surfaces.htmlThe curve of the intersection of the surface $x^2+y^2-2z^2=1$ and the plane $y=2z+1$ is given by $\varphi(t)=(\sqrt2\cos t,2\sin t-1,\sin t-1), t\in [0,2\pi)$. I would like to know how to find it ......Wed, 26 Aug 2026 06:17:40 -0400https://pop.com.sg/what-is-the-difference-between-a-forest-and-a-spanning-forest.htmlhttps://pop.com.sg/what-is-the-difference-between-a-forest-and-a-spanning-forest.htmlIf a graph is labelled as a forest it does not contain any cycles, meaning it consists of all trees, which I realize can even be a single node (since that is technically a tree). If a graph is lab......Wed, 26 Aug 2026 06:07:33 -0400https://pop.com.sg/reduction-formula-for-int-cosh-n-x-dx.htmlhttps://pop.com.sg/reduction-formula-for-int-cosh-n-x-dx.htmlGiven $n\in\Bbb N$, $n > 2$, find the reduction formula for: $$ \int \cosh^n x dx $$ I've derived an answer, but the signs do not match, so my question would be: where is the mistake in my ...Wed, 26 Aug 2026 06:05:22 -0400https://pop.com.sg/integral-int-infty-infty-x-2-e-x-2-mathrm-dx.htmlhttps://pop.com.sg/integral-int-infty-infty-x-2-e-x-2-mathrm-dx.htmlI don't know how to evaluate it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates. $$\int_{-\infty}^\infty x^2 e^......Wed, 26 Aug 2026 06:01:41 -0400https://pop.com.sg/generalization-x-text-sup-q-in-mathbb-q-q-x.htmlhttps://pop.com.sg/generalization-x-text-sup-q-in-mathbb-q-q-x.htmlHow do I prove that $x=\text{sup}\{q\in \mathbb{Q}:q<x\}$? Provided that $x\in\mathbb{R}$......Wed, 26 Aug 2026 06:00:32 -0400https://pop.com.sg/random-math-questions-modular-arithmetic-notation.htmlhttps://pop.com.sg/random-math-questions-modular-arithmetic-notation.htmlI found this amazing wall clock picture on the internet but I really don't know a few things. I don't know what's $B'_L$, 3, why $2^{-1}\equiv 4[7]$ and the one with the black and white ci......Wed, 26 Aug 2026 06:00:08 -0400https://pop.com.sg/how-does-a-branch-cut-define-a-branch.htmlhttps://pop.com.sg/how-does-a-branch-cut-define-a-branch.htmlI am studying complex analysis and I have problem understanding the concept of branch cut. The lecturer draw this as some curve that starts from a point and goes on and on in some direction (for e......Wed, 26 Aug 2026 05:44:53 -0400https://pop.com.sg/what-is-a-convex-polyhedron.htmlhttps://pop.com.sg/what-is-a-convex-polyhedron.htmlI was doing a spot of light reading (crystallography), when the term "convex" polyhedron came up in a a section (very prominently) in conjunction with something else called the "Euler ...Wed, 26 Aug 2026 05:38:15 -0400https://pop.com.sg/is-a-hypersurface-really-defined-by-an-arbitrary-polynomial.htmlhttps://pop.com.sg/is-a-hypersurface-really-defined-by-an-arbitrary-polynomial.htmlIn An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...Wed, 26 Aug 2026 05:32:27 -0400https://pop.com.sg/why-is-euler-039-s-number-2-71828-and-not-anything-else-closed.htmlhttps://pop.com.sg/why-is-euler-039-s-number-2-71828-and-not-anything-else-closed.htmlWhy is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$? I know that $e$ is the base of natural logarithms. I know about areas on hyperbola xy=1 and I know its formula: $$e =\......Wed, 26 Aug 2026 05:23:36 -0400https://pop.com.sg/why-the-slope-of-a-line-is-change-in-y-divided-by-change-in-x-duplicat.htmlhttps://pop.com.sg/why-the-slope-of-a-line-is-change-in-y-divided-by-change-in-x-duplicat.htmlWhy the slope of a line isn't the change in x divided by the change in y?...Wed, 26 Aug 2026 05:17:42 -0400https://pop.com.sg/product-of-two-independent-poisson-variables.htmlhttps://pop.com.sg/product-of-two-independent-poisson-variables.htmlSuppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ \operatorname{Poisson}(\lambda_1)$ and $X_2 ∼ \operatorname{Poisson}(\lambda_2)$. What will be the probability ...Wed, 26 Aug 2026 05:14:53 -0400https://pop.com.sg/if-my-average-speed-is-50-mph-is-it-better-to-go-10-faster-or-5-mph-fa.htmlhttps://pop.com.sg/if-my-average-speed-is-50-mph-is-it-better-to-go-10-faster-or-5-mph-fa.htmlBACKGROUND Suppose we have a journey between $A$ and $B$ where different sections of roads have different speed limits and our average speed is $50$ mph. If when I repeat the journey I travel $10......Wed, 26 Aug 2026 05:10:26 -0400https://pop.com.sg/show-marsaglia-polar-method-produce-the-standard-normal-distribution.htmlhttps://pop.com.sg/show-marsaglia-polar-method-produce-the-standard-normal-distribution.htmlSuppose $X, Y$ are two independent variables that uniformly distributed on the interval $[-1,1]$. If $s = x^2 + y^2 < 1$, show the result of $\frac{x}{\sqrt{s}} \sqrt{-2\ln(s)}$ comes from a sta......Wed, 26 Aug 2026 05:03:07 -0400https://pop.com.sg/how-is-the-graph-coloring-problem-np-complete.htmlhttps://pop.com.sg/how-is-the-graph-coloring-problem-np-complete.htmlAn NP-Complete problem can be checked efficiently, but has no known way of being solved in polynomial time. Then, how is the graph coloring problem (http://en.wikipedia.org/wiki/Graph_coloring_pro......Wed, 26 Aug 2026 05:01:31 -0400https://pop.com.sg/show-that-the-perimeter-of-an-octagon-is-8-sqrt-2-1.htmlhttps://pop.com.sg/show-that-the-perimeter-of-an-octagon-is-8-sqrt-2-1.htmlMy question is as follows: The top of a table is made in the shape of a regular octagon by cutting the congruent isosceles triangles from the corners of a $1$ m square piece of wood. Show that the ...Wed, 26 Aug 2026 04:47:44 -0400https://pop.com.sg/finding-the-limit-when-denominator-0.htmlhttps://pop.com.sg/finding-the-limit-when-denominator-0.htmlI'm having trouble solving this limit: $$\lim_{x \to -2^-} \frac{1}{(x + 2)^2}$$ I can't find a way to rationalize the denominator. Also, is there a way to do it without plugging in -2.001 and st......Wed, 26 Aug 2026 04:47:01 -0400https://pop.com.sg/prove-that-if-a-and-b-are-mutually-exclusive-then-p-a-leq-p-b-c.htmlhttps://pop.com.sg/prove-that-if-a-and-b-are-mutually-exclusive-then-p-a-leq-p-b-c.htmlSuppose $\textbf{P}(A\cap B) = 0$. Prove $\textbf{P}(A) \leq \textbf{P}(B^{c})$ using the axioms of probability. I know the axioms are: $\textbf{P}(A)\geq 0$. $\textbf{P}(\Omega) = 1$. If A and......Wed, 26 Aug 2026 04:41:00 -0400https://pop.com.sg/find-the-composition-of-a-piecewise-function.htmlhttps://pop.com.sg/find-the-composition-of-a-piecewise-function.htmlLet $$f(x)= \begin{cases} 5 &\text{, x $\geq$ 0}\\ x^2 &\text{, x < 0} \end{cases}$$ and $$g(x)= \begin{cases} \sqrt{x} &\text{, x > 3}\\ 4 &\text{, x $<$ 3} \end{cases}$$ ......Wed, 26 Aug 2026 04:37:40 -0400https://pop.com.sg/show-that-the-dimention-of-the-intersection-of-projective-linear-sub-s.htmlhttps://pop.com.sg/show-that-the-dimention-of-the-intersection-of-projective-linear-sub-s.htmlProposition: Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap M)\geq d_1+d_2-n$ (we consider the dimenti......Wed, 26 Aug 2026 04:36:48 -0400https://pop.com.sg/difference-between-imaginary-part-and-imaginary-number.htmlhttps://pop.com.sg/difference-between-imaginary-part-and-imaginary-number.htmlIs there a difference between imaginary part and imaginary number? Because our teacher says there is. I mean in $5+3i$ , is it right to say the imaginary part is $3i$ and imaginary number is $3$?......Wed, 26 Aug 2026 04:34:58 -0400https://pop.com.sg/find-a-circle-perpendicular-to-two-other-circles.htmlhttps://pop.com.sg/find-a-circle-perpendicular-to-two-other-circles.htmlHere are two circles $C_1 = [ x^2 + y^2 = 1]$ and $C_2 = [ (x-2)^2 + y^2 = 3 ]$. The radii are $1$ and $3$ and the two circles are orthogonal to each other $C_1 \cdot C_2 = 0$. What is the equati......Wed, 26 Aug 2026 04:34:09 -0400https://pop.com.sg/finding-the-local-trivilization-of-canonical-line-bundle.htmlhttps://pop.com.sg/finding-the-local-trivilization-of-canonical-line-bundle.htmlI am going through Nakahara's "Geometry, topology and physics" by myself and I got stuck on a calculation. I have the total space $$L=\{(p,v)\in\mathbb{C}P^n\times\mathbb{C}^{n+1}|v=ap,a\in\math......Wed, 26 Aug 2026 04:32:13 -0400https://pop.com.sg/express-cos-tan-1-x-in-terms-of-x.htmlhttps://pop.com.sg/express-cos-tan-1-x-in-terms-of-x.htmlExpress $\cos(\tan^{-1}(x))$ in terms of $x$. My Attempt: Let $\tan^{-1}(x)=A$. $$x=\tan (A)$$ Then, $$\cos(\tan^{-1}(x))=\cos (A)$$...Wed, 26 Aug 2026 04:30:10 -0400https://pop.com.sg/composition-of-function-with-itself.htmlhttps://pop.com.sg/composition-of-function-with-itself.htmlFor the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider the following statements: $A$ : For ...Wed, 26 Aug 2026 04:18:25 -0400https://pop.com.sg/boolean-algebra-sum-of-products-form.htmlhttps://pop.com.sg/boolean-algebra-sum-of-products-form.htmlI have a logic circuit where the output can be represented with the following boolean expression (1)$\overline {xy}$ + x $\bar y z$ + $\overline {\bar x + z} $ + y Using truth tables I found the ...Wed, 26 Aug 2026 04:05:49 -0400https://pop.com.sg/simultaneous-diagonalization-of-commuting-linear-transformations.htmlhttps://pop.com.sg/simultaneous-diagonalization-of-commuting-linear-transformations.htmlLet $V$ be a vector space of finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that if $TS=ST$ every eigenspace $V_\lambda$ of $S$ is $T$-inv......Wed, 26 Aug 2026 04:04:31 -0400https://pop.com.sg/derived-categories-of-filtered-modules.htmlhttps://pop.com.sg/derived-categories-of-filtered-modules.htmlFor a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which decl......Wed, 26 Aug 2026 04:02:36 -0400https://pop.com.sg/sum-of-sets-of-measure-zero.htmlhttps://pop.com.sg/sum-of-sets-of-measure-zero.htmlLet $A$ and $B$ be two subsets of $\Bbb R$ of measure zero. Is it true that the Minkowski sum $A+B = \{ a + b \mid a \in A, b \in B \}$ has measure zero as well? I think so but I can't prove it. The ...Wed, 26 Aug 2026 03:59:58 -0400https://pop.com.sg/how-to-evaluate-the-integral-int-e-2x-sin-2x-dx-closed.htmlhttps://pop.com.sg/how-to-evaluate-the-integral-int-e-2x-sin-2x-dx-closed.htmlHow can I evaluate the following integral? $$\int e^{2x} \sin^2x\ dx$$...Wed, 26 Aug 2026 03:58:37 -0400https://pop.com.sg/universal-hashing-family.htmlhttps://pop.com.sg/universal-hashing-family.htmlI'm not really sure this question is proper for MathExchange but I have found it more suitable than StackOverFlow. So, I'm taking the course of Data Structures and Algorithms and we got some examp......Wed, 26 Aug 2026 03:53:12 -0400https://pop.com.sg/proof-of-intersection-and-union-of-set-a-with-empty-set.htmlhttps://pop.com.sg/proof-of-intersection-and-union-of-set-a-with-empty-set.htmlI need to prove the following: Prove that $A\cup \!\, \varnothing \!\,=A$ and $A\cap \!\, \varnothing \!\,=\varnothing \!\,$ It's my understanding that to prove equality, I must prove that both are ...Wed, 26 Aug 2026 03:45:43 -0400https://pop.com.sg/calculating-the-square-root-of-2.htmlhttps://pop.com.sg/calculating-the-square-root-of-2.htmlSince $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so th......Wed, 26 Aug 2026 03:32:14 -0400https://pop.com.sg/convergent-or-divergent-integral.htmlhttps://pop.com.sg/convergent-or-divergent-integral.htmlConvergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do......Wed, 26 Aug 2026 03:29:23 -0400https://pop.com.sg/differentiation-using-d-or-delta.htmlhttps://pop.com.sg/differentiation-using-d-or-delta.htmlAre the symbols $d$ and $\delta$ equivalent in expressions like $dy/dx$? Or do they mean something different? Thanks...Wed, 26 Aug 2026 03:27:24 -0400https://pop.com.sg/limacon-cardioid-orientation-confusion.htmlhttps://pop.com.sg/limacon-cardioid-orientation-confusion.htmlI'm learning about polar equations and am trying to graph the curve r = 3 + 3 sin(theta), with theta between 0 and 2pi. Based on what I've learned and the image below, since a = b = 3, it would see......Wed, 26 Aug 2026 03:26:52 -0400