https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 26 Aug 2026 16:59:35 -0400https://pop.com.sg/why-can-a-matrix-without-a-full-rank-not-be-invertible.htmlhttps://pop.com.sg/why-can-a-matrix-without-a-full-rank-not-be-invertible.htmlI know you could just say because the $\det = 0$. But during the introduction of determinants the professor said, obviously if two columns of the matrix are linearly dependent the matrix can't be ...Wed, 26 Aug 2026 20:58:06 -0400https://pop.com.sg/proof-of-a-2-ab-b-2-0-a-b-both-are-not-zero-both-are-real-duplicate.htmlhttps://pop.com.sg/proof-of-a-2-ab-b-2-0-a-b-both-are-not-zero-both-are-real-duplicate.htmlShow that for any reals a,b, both of which are not zero, it holds true that a^2 + ab + b^2 >0...Wed, 26 Aug 2026 20:55:07 -0400https://pop.com.sg/what-is-the-difference-between-average-and-expected-value.htmlhttps://pop.com.sg/what-is-the-difference-between-average-and-expected-value.htmlQuestion : What is the difference between Average and Expected value? I have been going through the definition of expected value on Wikipedia beneath all that jargon it seems that the expected val......Wed, 26 Aug 2026 20:54:15 -0400https://pop.com.sg/prove-that-x-is-rational.htmlhttps://pop.com.sg/prove-that-x-is-rational.htmlLet $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the d......Wed, 26 Aug 2026 20:36:45 -0400https://pop.com.sg/differentiating-with-respect-to-1-x.htmlhttps://pop.com.sg/differentiating-with-respect-to-1-x.htmlI am fairly sure this is a silly question, but a Google search was insufficient to find a satisfactory answer. If I differentiate some function of $x$ with respect to $1-x$, what do I get compare......Wed, 26 Aug 2026 20:36:15 -0400https://pop.com.sg/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta.htmlhttps://pop.com.sg/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta.htmlLet $X_1,\dots,X_n$ be a sample from uniform distribution on $(-\theta,\theta)$ with parameter $\theta>0$. It is easy to show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic for $\the......Wed, 26 Aug 2026 20:34:22 -0400https://pop.com.sg/partial-differential-equation-u-t-3u-xx.htmlhttps://pop.com.sg/partial-differential-equation-u-t-3u-xx.htmlSolve the partial differential equation $$u_t=3u_{xx}, u(0,t)=0, u(x,0)=\cos{x}\sin{5x}$$ Attempt: Using separation of variables, let $u(x,t)=f(x)g(t)$, so $$f(x)g'(t)=3f''(x)g(t)$$ $$\frac{g'(t)}{......Wed, 26 Aug 2026 20:26:57 -0400https://pop.com.sg/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-pro.htmlhttps://pop.com.sg/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-pro.htmlI am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...Wed, 26 Aug 2026 20:26:33 -0400https://pop.com.sg/how-to-use-parametric-equations-for-line-integral.htmlhttps://pop.com.sg/how-to-use-parametric-equations-for-line-integral.htmlThe question is to find the work done in moving a particle in a force field: $$\overrightarrow F = 3x^2i+(2xy-y)j+3k$$ along the straight line from (0, 0, 0) to (2, 1, 3) So, work done $=\int \...Wed, 26 Aug 2026 20:22:25 -0400https://pop.com.sg/what-039-s-the-difference-between-theorem-lemma-and-corollary.htmlhttps://pop.com.sg/what-039-s-the-difference-between-theorem-lemma-and-corollary.htmlCan anybody explain me what is the basic difference between theorem, lemma and corollary? We have been using it for a long time but I never paid any attention. I am just curious to know....Wed, 26 Aug 2026 20:19:07 -0400https://pop.com.sg/proving-and-distributive-law-using-boolean-algebra.htmlhttps://pop.com.sg/proving-and-distributive-law-using-boolean-algebra.html$$ X(Y+Z) = (XY) + (XZ) $$ I can’t seem to derive the proper steps to prove this equation using Boolean axioms. The hint I’ve been given is using demorgans laws proofs but I still can’t seem to fi......Wed, 26 Aug 2026 20:17:02 -0400https://pop.com.sg/finding-contrapositive-of-a-universal-statement.htmlhttps://pop.com.sg/finding-contrapositive-of-a-universal-statement.htmlFor all $x$, if $x^2$ is even, then $x$ is even. The contrapositive to this statement is: For all $x$, if $x$ is odd, then $x^2$ is odd. Why do we ignore the "For all $x$" and not say ......Wed, 26 Aug 2026 20:16:58 -0400https://pop.com.sg/soft-question-definition-of-the-real-plane.htmlhttps://pop.com.sg/soft-question-definition-of-the-real-plane.htmlI have only seen the real plane to be defined as $\mathbb R \times \mathbb R$. In Euclidean geometry, the origin does not play any central role, but in this set theoretic definition, it does. Is t......Wed, 26 Aug 2026 20:13:26 -0400https://pop.com.sg/how-to-prepare-for-integral-calculus-calculus-2.htmlhttps://pop.com.sg/how-to-prepare-for-integral-calculus-calculus-2.htmlI'm majoring in computer engineering and I have Calculus 2 coming up this semester. From what I understand, Calculus 2 is the most difficult math class in the engineering path. Over the summer, I......Wed, 26 Aug 2026 20:10:41 -0400https://pop.com.sg/probability-of-drawing-a-flush-from-a-standard-deck-of-cards.htmlhttps://pop.com.sg/probability-of-drawing-a-flush-from-a-standard-deck-of-cards.htmlThe following is the problem put to me: A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Flush (5 nonconsecutive cards e......Wed, 26 Aug 2026 19:54:52 -0400https://pop.com.sg/how-to-calculate-the-height-of-a-cone-at-particular-volume.htmlhttps://pop.com.sg/how-to-calculate-the-height-of-a-cone-at-particular-volume.htmlThe equation for the volume of a cone is $\frac{1}{3}\pi r^2h$ starting from the base. However, I wanna calculate the height of the cone for a particular volume from the tip of the cone. Could you ......Wed, 26 Aug 2026 19:31:46 -0400https://pop.com.sg/with-the-help-of-chinese-remainder-theorem-show-that-x-144-equiv-1-pmo.htmlhttps://pop.com.sg/with-the-help-of-chinese-remainder-theorem-show-that-x-144-equiv-1-pmo.htmlWith the help of Chinese remainder theorem show that $ x^{144} \equiv 1 \pmod{323}$ for all $x$ relatively prime to 323. The problem with me is that I used to use CRT when $x$ is raised to a power......Wed, 26 Aug 2026 19:25:38 -0400https://pop.com.sg/what-does-curly-curved-less-than-sign-succcurlyeq-mean.htmlhttps://pop.com.sg/what-does-curly-curved-less-than-sign-succcurlyeq-mean.htmlI am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq) $$f(x^*) \succcurlyeq \alpha$$ and curved less than or equal to (\preccu......Wed, 26 Aug 2026 19:22:58 -0400https://pop.com.sg/is-it-possible-to-reverse-a-gradient-vec-nabla-operation.htmlhttps://pop.com.sg/is-it-possible-to-reverse-a-gradient-vec-nabla-operation.htmlIn calculus, the antiderivative (indefinite integral) can be considered as the reverse operation of a derivative. A gradient yields a vector. Is there a similar way of reversing gradient, as you d......Wed, 26 Aug 2026 19:15:48 -0400https://pop.com.sg/constant-slope-in-straight-line.htmlhttps://pop.com.sg/constant-slope-in-straight-line.htmlI'm now reading about the formal definition of line as the graph of the function $f(x)=f(x_0)+m(x-x_0)$ for an arbitrary $x_0$ through which the graph passes, where the constant $m$ is called slope......Wed, 26 Aug 2026 18:57:11 -0400https://pop.com.sg/height-of-pyramid.htmlhttps://pop.com.sg/height-of-pyramid.htmlKnowing the base sides and the slant heights, do we calculate the height of the pyramid by the formula $h^2=h_b^2-\left (\frac{a}{2}\right )^2$, no matter what shape the base of the pyramid has, i.e. ...Wed, 26 Aug 2026 18:47:17 -0400https://pop.com.sg/what-is-a-cardinal-basis-spline.htmlhttps://pop.com.sg/what-is-a-cardinal-basis-spline.htmlWikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ......Wed, 26 Aug 2026 18:44:09 -0400https://pop.com.sg/finding-transient-and-steady-state-solution.htmlhttps://pop.com.sg/finding-transient-and-steady-state-solution.htmlObtain the steady periodic solutin $x_{sp}(t)=Asin(\omega t+\phi)$ and the transient equation for the solution t $x''+2x'+26x=82cos(4t)$, where $x(0)=6$ & $x'(0)=0$. So I'm not sure what's being ...Wed, 26 Aug 2026 18:42:59 -0400https://pop.com.sg/limit-as-n-goes-to-infinity.htmlhttps://pop.com.sg/limit-as-n-goes-to-infinity.htmlConsider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$ I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, $......Wed, 26 Aug 2026 18:39:58 -0400https://pop.com.sg/definition-of-left-right-exact-functors.htmlhttps://pop.com.sg/definition-of-left-right-exact-functors.htmlLet $P,Q$ be abelian categories and $F:P\to Q$ be an additive functor. Wikipedia states two definitions on left exact functors (right dually): $F$ is left exact if $0\to A\to B\to C\to 0$ is exact ...Wed, 26 Aug 2026 18:39:58 -0400https://pop.com.sg/solving-equations-with-factorials-duplicate.htmlhttps://pop.com.sg/solving-equations-with-factorials-duplicate.htmlI looked on the internet but couldn't find anything relevant, so I was hoping you could help because I have no clue where to even start with how to solve this equations: x! = 6 Obviously trial and......Wed, 26 Aug 2026 18:37:50 -0400https://pop.com.sg/how-would-you-discover-stokes-039-s-theorem.htmlhttps://pop.com.sg/how-would-you-discover-stokes-039-s-theorem.htmlLet $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$, and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$. Stokes's theorem s......Wed, 26 Aug 2026 18:13:57 -0400https://pop.com.sg/can-someone-give-me-a-deeper-understanding-of-implicit-differentiation.htmlhttps://pop.com.sg/can-someone-give-me-a-deeper-understanding-of-implicit-differentiation.htmlI'm doing calculus and I want to be an engineer so I would like to understand the essence of the logic of implicit differentials rather than just memorizing the algorithm. Yes, I could probably mem......Wed, 26 Aug 2026 18:09:26 -0400https://pop.com.sg/archimedean-spiral-symmetry-test.htmlhttps://pop.com.sg/archimedean-spiral-symmetry-test.htmlIt is usually stated in Precalculus textbooks that in polar coordinates when a relation between $r$ and $\theta$ passes a symmetry test then the curve described by that relation has that symmetry. ......Wed, 26 Aug 2026 17:58:08 -0400https://pop.com.sg/don-039-t-understand-why-this-binomial-expansion-is-not-valid-for-x-1.htmlhttps://pop.com.sg/don-039-t-understand-why-this-binomial-expansion-is-not-valid-for-x-1.htmltoday I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...Wed, 26 Aug 2026 17:57:18 -0400https://pop.com.sg/where-does-this-probability-formula-come-from.htmlhttps://pop.com.sg/where-does-this-probability-formula-come-from.htmlFrom Introduction to Probability, 2ed (Bersekas et al), chapter 8.1 p. 417: I would like to ask where the underlined formula comes from. So far, the formulas given in that chapter are: Please he......Wed, 26 Aug 2026 17:52:38 -0400https://pop.com.sg/probability-of-balls-drawn-with-replacement.htmlhttps://pop.com.sg/probability-of-balls-drawn-with-replacement.htmlWe have two bags, Bag A has 40 red balls and 15 blue balls, Bag B has 40 blue balls and 10 red balls. One of these bags is selected at random and from it five balls are drawn at random, replacing e......Wed, 26 Aug 2026 17:45:29 -0400https://pop.com.sg/why-does-the-difference-between-the-left-hand-and-right-hand-sum-get-s.htmlhttps://pop.com.sg/why-does-the-difference-between-the-left-hand-and-right-hand-sum-get-s.htmlQ) For a given function on a given interval, the difference between the left-hand sum and right-hand sum always gets smaller as the number of subdivisions gets larger. I remember that the answer i......Wed, 26 Aug 2026 17:38:47 -0400https://pop.com.sg/solving-inhomogeneous-heat-equation.htmlhttps://pop.com.sg/solving-inhomogeneous-heat-equation.htmlSolve the inhomogeneous heat equation $u_t=c^2u_{xx} +\sin(5\pi x)$ for all $0 < x < 1, t > 0$ subject to homogeneous Dirichlet boundary conditions $u(t,0)=u(t,1)=0$ and initial condition ......Wed, 26 Aug 2026 17:36:29 -0400https://pop.com.sg/simplify-boolean-product-of-sums-function.htmlhttps://pop.com.sg/simplify-boolean-product-of-sums-function.htmlI've got a product of sums expression: F=(A'+B+C')&(A+D')(C+D') I need to show it as a sum of products and then simplify it. Right now I got: F=(A'&D')+(A&B&C)+(B&D')+(C&D......Wed, 26 Aug 2026 17:36:22 -0400https://pop.com.sg/order-of-essential-singularity.htmlhttps://pop.com.sg/order-of-essential-singularity.htmlcan we say that essential singularities has order like poles ? I mean ; we know e^(1/z) has an essential singularity at z=0 then do I need to say e^(1/z^3) has essential singularity at z=0 with or......Wed, 26 Aug 2026 17:21:30 -0400https://pop.com.sg/what-exactly-is-real-number.htmlhttps://pop.com.sg/what-exactly-is-real-number.htmlThis question may sound philosophy, but it has been bothering me for a very long time, therefore I have to ask it here. The story goes back when my first time reading Apostol's Calculus, then I had ...Wed, 26 Aug 2026 17:19:17 -0400https://pop.com.sg/how-to-find-functional-square-root-of-sin-x.htmlhttps://pop.com.sg/how-to-find-functional-square-root-of-sin-x.htmlMaybe I am overlooking something, but is there some easy way to find a function $x\to f(x)$ so that $$(f\circ f)(x) = f(f(x)) = \sin(x)$$ on some interval, say $x\in [-\pi,\pi]\subset \mathbb R$ of ...Wed, 26 Aug 2026 16:51:30 -0400https://pop.com.sg/prove-that-the-product-of-a-non-zero-rational-and-irrational-number-is.htmlhttps://pop.com.sg/prove-that-the-product-of-a-non-zero-rational-and-irrational-number-is.htmlCould you please confirm if this proof is correct? Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational. Proof: Proof by contradiction, we assume that $qy$ is rational. ...Wed, 26 Aug 2026 16:44:55 -0400https://pop.com.sg/quickest-way-to-find-characteristic-polynomial-from-a-given-matrix.htmlhttps://pop.com.sg/quickest-way-to-find-characteristic-polynomial-from-a-given-matrix.htmlFind the Rational form of $$ A=\begin{pmatrix} 1&2&0&4\\ \:\:\:4&1&2&0\\ \:\:\:0&4&1&2\\ \:\:\:2&0&4&1\end{pmatrix} $$ I don't wanna the solution, ...Wed, 26 Aug 2026 16:44:26 -0400https://pop.com.sg/given-a-byte-in-2-s-complement-what-is-0x56-in-decimal-closed.htmlhttps://pop.com.sg/given-a-byte-in-2-s-complement-what-is-0x56-in-decimal-closed.htmla) -13 b) -110 c) 38 d) none of the above Im not sure on how to do this question. What I have done so far is convert the hexadecimal number into a decimal, but since it's already in 2's complement......Wed, 26 Aug 2026 16:40:46 -0400https://pop.com.sg/area-of-trapezoids-vs-area-of-parallelograms.htmlhttps://pop.com.sg/area-of-trapezoids-vs-area-of-parallelograms.htmlBy definition parallelograms are special type of trapezoids. Given a trapezoid with known sides one can calculate its area according to wikipedia. But in order to calculate the area of a parallelog......Wed, 26 Aug 2026 16:30:11 -0400https://pop.com.sg/statistics-calculating-mean-given-standard-deviation-and-percentage.htmlhttps://pop.com.sg/statistics-calculating-mean-given-standard-deviation-and-percentage.htmlThis is one of our homework problems for Statistics. I'm really stuck on this one, and it would be great if someone could show me how to solve this, or at least set up the equation. The weights of ...Wed, 26 Aug 2026 16:27:12 -0400https://pop.com.sg/how-can-i-construct-the-centroid-of-a-quadrilateral.htmlhttps://pop.com.sg/how-can-i-construct-the-centroid-of-a-quadrilateral.htmlHow can I construct the centroid of a quadrilateral? I suppose that it is the intersection between the lines that join the middles of opposite sides....Wed, 26 Aug 2026 16:25:59 -0400https://pop.com.sg/calculate-inverse-discrete-time-fourier-transform.htmlhttps://pop.com.sg/calculate-inverse-discrete-time-fourier-transform.htmlCalculate Inverse Discrete Time Fourier Transform of the following where $|a| < 1$: $$ X(e^{j\omega}) = \frac{1-a^2}{(1-ae^{-j\omega})(1-ae^{j\omega})} $$ Plugging this directly into the IDTFT ...Wed, 26 Aug 2026 16:09:18 -0400https://pop.com.sg/what-comes-before-precalculus-closed.htmlhttps://pop.com.sg/what-comes-before-precalculus-closed.htmlI studied in Europe and followed mostly European academic ways. I'm tutoring a young person , and I will to know what math class comes before pre calculus. Because soon we will be starting some ...Wed, 26 Aug 2026 16:05:56 -0400https://pop.com.sg/baby-papa-mama-big-rudin.htmlhttps://pop.com.sg/baby-papa-mama-big-rudin.htmlRecently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that ...Wed, 26 Aug 2026 16:01:51 -0400https://pop.com.sg/relationships-between-det-a-b-and-a-b.htmlhttps://pop.com.sg/relationships-between-det-a-b-and-a-b.htmlWhen computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$. However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?...Wed, 26 Aug 2026 16:01:05 -0400https://pop.com.sg/what-is-the-largest-positive-n-for-which-n-3-100-is-divisible-by-n-10.htmlhttps://pop.com.sg/what-is-the-largest-positive-n-for-which-n-3-100-is-divisible-by-n-10.htmlWhat is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$ I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$....Wed, 26 Aug 2026 15:55:43 -0400https://pop.com.sg/prove-the-principle-of-strong-induction.htmlhttps://pop.com.sg/prove-the-principle-of-strong-induction.htmlThe following is from Analysis with an Introduction to Proof by Steven Lay Prove the principle of strong induction: Let $P(n)$ be a statement that is either true or false for each $n \in \mathbb{N}$ ...Wed, 26 Aug 2026 15:48:46 -0400https://pop.com.sg/what-is-meant-by-the-double-vertical-line-notation-here.htmlhttps://pop.com.sg/what-is-meant-by-the-double-vertical-line-notation-here.htmlWhat do the double vertical lines around $\vec i$ and $\vec j$ in this equation actually mean? $$ sim(i,j) = cos(\vec i, \vec j) = \frac{\vec i \cdot \vec j}{\lVert \vec i\rVert^2 * \lVert\vec j\r......Wed, 26 Aug 2026 15:46:52 -0400https://pop.com.sg/conditions-for-a-unique-root-of-a-fifth-degree-polynomial.htmlhttps://pop.com.sg/conditions-for-a-unique-root-of-a-fifth-degree-polynomial.htmlFifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient ...Wed, 26 Aug 2026 15:43:00 -0400https://pop.com.sg/symbolic-operation-on-ti-84-calculator-closed.htmlhttps://pop.com.sg/symbolic-operation-on-ti-84-calculator-closed.htmlI want the calculator to answer with the parameter answer, not with a numeric value, I have tried to clear "M" variable with no success. Any idea? http://imgur.com/cHjfLln ( sorry I don't have ......Wed, 26 Aug 2026 15:35:36 -0400https://pop.com.sg/x-x-y-how-to-solve-for-x-duplicate.htmlhttps://pop.com.sg/x-x-y-how-to-solve-for-x-duplicate.htmlI tried looking for ways to solve this equation and came across something like Lambert's W function, which, by the way, I did not understand a bit, because I've never learned it nor do I have a dec......Wed, 26 Aug 2026 15:17:02 -0400https://pop.com.sg/possible-polyhedra-from-euler-039-s-formula.htmlhttps://pop.com.sg/possible-polyhedra-from-euler-039-s-formula.htmlI'm not very clear with the euler's formula, and I couldn't find it anywhere. I'm sorry if it is a double post. F + V - E = 2 Is the euler's formula. If the equation balances, is it polyhedra al......Wed, 26 Aug 2026 15:03:32 -0400https://pop.com.sg/the-chain-rule-and-squared-trig-functions.htmlhttps://pop.com.sg/the-chain-rule-and-squared-trig-functions.htmlI am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to......Wed, 26 Aug 2026 14:59:04 -0400https://pop.com.sg/moment-of-inertia-around-z-axis.htmlhttps://pop.com.sg/moment-of-inertia-around-z-axis.htmlHello I am having difficulty with the following; I am wanting to find I, the moment of inertia about the z axis of the region that is bounded by the paraboloid $z=x^{2}+y^{2}$ and the $z=1$ plane,......Wed, 26 Aug 2026 14:55:40 -0400https://pop.com.sg/use-the-given-graph-f-over-the-interval-0-7-to-find-the-following.htmlhttps://pop.com.sg/use-the-given-graph-f-over-the-interval-0-7-to-find-the-following.html(a) The open intervals on which $f$ is increasing. I answered $(0, 1), (3, 5), (5,7)$ (b) The open intervals on which $f$ is concave upward. I answered $(1, 4)$ (c) The open intervals on which ......Wed, 26 Aug 2026 14:54:39 -0400https://pop.com.sg/using-l-039-hopital-039-s-rule-with-the-indeterminate-form-of-infinity.htmlhttps://pop.com.sg/using-l-039-hopital-039-s-rule-with-the-indeterminate-form-of-infinity.htmlI'm aware that with L'Hopital's rule, we're dealing with indeterminate forms of $ \lim_{x\to a} \frac{f(x)}{g(x)} $, and that includes re-writing $\lim_{x\to a} \infty - \infty$ into that format. ...Wed, 26 Aug 2026 14:51:16 -0400https://pop.com.sg/how-to-graph-the-parametric-without-calculator-x-sin-t-y-t-2.htmlhttps://pop.com.sg/how-to-graph-the-parametric-without-calculator-x-sin-t-y-t-2.htmlQuestion How to graph the parametric without calculator: $x=\sin t$ , $y=t^2$ I first make t by itself by doing the following: $\sqrt{y}=t$ and plugging into the other function to get $x=\sin(\sq......Wed, 26 Aug 2026 14:43:45 -0400https://pop.com.sg/prove-that-there-exists-a-shortest-distance-between-two-skew-lines.htmlhttps://pop.com.sg/prove-that-there-exists-a-shortest-distance-between-two-skew-lines.htmlIn every geometry textbook it states that the shortest distance between two skew lines (lines that are not co-planar) is given by the unique line that runs perpendicular to both skew lines. This is ...Wed, 26 Aug 2026 14:41:41 -0400https://pop.com.sg/finding-the-angles-of-a-triangle-using-algebraic-expressions.htmlhttps://pop.com.sg/finding-the-angles-of-a-triangle-using-algebraic-expressions.htmlI think I have solved this question but want to check if I am correct with my formula I used the theorem (rule) : the measure of the exterior angle of a triangle is equal to the sum of the remote (...Wed, 26 Aug 2026 14:40:57 -0400https://pop.com.sg/positive-definite-and-block-matrix.htmlhttps://pop.com.sg/positive-definite-and-block-matrix.htmlSuppose a $n \times n$ symmetric matrix $\mathbf{M}$ is positive definite. Its block matrix form is written as follow. \begin{align} \mathbf{M} \; = \; \begin{pmatrix} \mathbf{A} &am......Wed, 26 Aug 2026 14:40:17 -0400https://pop.com.sg/difference-between-spe-and-spne-in-game-theory.htmlhttps://pop.com.sg/difference-between-spe-and-spne-in-game-theory.htmlI am slightly confused as to what the difference between Subgame Perfect Equilibrium and Subgame Perfect Nash Equilibrium is. I was told there is a difference, but now I found many lecture notes t......Wed, 26 Aug 2026 14:34:22 -0400https://pop.com.sg/definition-singular-manifold.htmlhttps://pop.com.sg/definition-singular-manifold.htmlI'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ex......Wed, 26 Aug 2026 14:20:58 -0400https://pop.com.sg/why-do-we-draw-the-xyz-coordinate-system-like-this.htmlhttps://pop.com.sg/why-do-we-draw-the-xyz-coordinate-system-like-this.htmlUsually people (including, for instance, Calculus teachers) draw the $xyz$ coordinate system in such a way that the $y$ and $z$ axes are perpendicular to each other: xyz http://pad1.whstatic.com/im......Wed, 26 Aug 2026 14:17:54 -0400https://pop.com.sg/derivation-of-normal-equation-for-linear-least-squares-in-matrix-form.htmlhttps://pop.com.sg/derivation-of-normal-equation-for-linear-least-squares-in-matrix-form.htmlThe derivation can be found on wikipedia but it's not clear how each step follows. We have $y=X\beta+\epsilon$, and want to minimize $\epsilon^2$. We write objective function as $S(\beta)=||y-X\b......Wed, 26 Aug 2026 14:15:43 -0400https://pop.com.sg/simplifying-division-of-integrals.htmlhttps://pop.com.sg/simplifying-division-of-integrals.htmlThe $\overline{x}$ coordinate of the center of mass of a plane region is calculated as $$ \overline{x} = \frac{M_y}{M} = \frac{\int_a^b xf(x) \, \textrm{d}x}{\int_a^b f(x) \, \textrm{d}x} $$ And ......Wed, 26 Aug 2026 14:11:43 -0400https://pop.com.sg/what-is-a-number.htmlhttps://pop.com.sg/what-is-a-number.htmlA dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and nu......Wed, 26 Aug 2026 13:43:03 -0400https://pop.com.sg/how-to-find-the-number-of-x-per-second-given-the-time-elapsed.htmlhttps://pop.com.sg/how-to-find-the-number-of-x-per-second-given-the-time-elapsed.htmlI'm benchmarking a websocket server and I am very poor at math so please forgive me. I am recording the amount of messages sent, and the elapsed time: Players Connected: 2. Messages Sent: 3528, E......Wed, 26 Aug 2026 13:39:42 -0400https://pop.com.sg/justifying-why-0-0-is-indeterminate-and-1-0-is-undefined.htmlhttps://pop.com.sg/justifying-why-0-0-is-indeterminate-and-1-0-is-undefined.html$\dfrac 00=x$ $0x=0$ $x$ can be any value, therefore $\dfrac 00$ can be any value, and is indeterminate. $\dfrac 10=x$ $0x=1$ There is no such $x$ that satisfies the above, therefore $\dfrac 10$......Wed, 26 Aug 2026 13:30:14 -0400https://pop.com.sg/find-the-absolute-maximum-and-minimum-of-f-x-frac-1-x-frac-2-x-2-on-2-.htmlhttps://pop.com.sg/find-the-absolute-maximum-and-minimum-of-f-x-frac-1-x-frac-2-x-2-on-2-.htmlNote: Turns out this is a pretty tricky problem where we need to use techniques not developed in the chapter. We will go over this problem in class on Tuesday. Find the absolute maximum and minimu......Wed, 26 Aug 2026 13:30:11 -0400https://pop.com.sg/methods-of-solving-a-problem-regarding-withdrawing-and-depositing-from.htmlhttps://pop.com.sg/methods-of-solving-a-problem-regarding-withdrawing-and-depositing-from.htmlI have an imaginary bank account with an initial balance $x$. I can only withdraw $0\% < w\% < 100\%$ (of the current balance) or deposit $0\% < d\% < 100\%$ (of the current balance) at a ...Wed, 26 Aug 2026 13:30:02 -0400https://pop.com.sg/double-integral-of-graph.htmlhttps://pop.com.sg/double-integral-of-graph.htmlI am currently trying to do this question but I am having trouble getting the correct answer of 25. I am also having trouble understanding whether my limits are right and whether my method of doin......Wed, 26 Aug 2026 13:15:30 -0400https://pop.com.sg/explicitly-calculate-shape-operator-for-graph-of-f-x-y-xy.htmlhttps://pop.com.sg/explicitly-calculate-shape-operator-for-graph-of-f-x-y-xy.htmlThis seems trivial but I am stuck. To gain intuition for a bigger problem, I am trying to compute the shape operator of the graph of $f(x,y)=xy$ at the point $p=(0,0,0)$, call this surface $\Sigma$......Wed, 26 Aug 2026 12:51:37 -0400https://pop.com.sg/strong-induction-example-using-all-of-p-1-and-and-p-k-1-and-p-k-to-pro.htmlhttps://pop.com.sg/strong-induction-example-using-all-of-p-1-and-and-p-k-1-and-p-k-to-pro.htmlI'm trying to understand how to do "real" strong induction, but my textbook seems to be of no help. It defines strong induction as follows: Let $P(n)$ be a property that is defined for integers $n......Wed, 26 Aug 2026 12:45:13 -0400https://pop.com.sg/nonsingular-z-matrix-nonsingular-m-matrix.htmlhttps://pop.com.sg/nonsingular-z-matrix-nonsingular-m-matrix.htmlHere I'm considering only M-matrices that are also Z-matrices, so all the off diagonal elements are negative in all matrices I consider. If I have a Z-Matrix with real, positive eigenvalues, is it......Wed, 26 Aug 2026 12:45:11 -0400https://pop.com.sg/how-can-i-know-that-the-graph-is-symmetric-with-respect-to-the-x-axis-.htmlhttps://pop.com.sg/how-can-i-know-that-the-graph-is-symmetric-with-respect-to-the-x-axis-.htmlHow can I know that the graph is symmetric with respect to the x axis or y axis? $y=x^4-2x^2-21$ $9x^2+4y^2=36$ $y^2=x-4$...Wed, 26 Aug 2026 12:38:59 -0400https://pop.com.sg/a-4-regular-graph-with-7-vertices-is-non-planar.htmlhttps://pop.com.sg/a-4-regular-graph-with-7-vertices-is-non-planar.htmlA Graph with 7 vertices each having degree 4 cannot be planar. Any hints on the proof?...Wed, 26 Aug 2026 12:38:24 -0400https://pop.com.sg/proof-of-derivative-of-an-exponential-function.htmlhttps://pop.com.sg/proof-of-derivative-of-an-exponential-function.htmlI was told to assume that $$\ln b=\lim_{h\to 0} \frac{\left(b^h-1\right)}{h}$$ where b is a positive, real, base. Unfortunately, being told to assume something isn't good enough. When using L'...Wed, 26 Aug 2026 12:38:03 -0400https://pop.com.sg/equation-of-chord-of-a-parabola-whose-midpoint-is-given.htmlhttps://pop.com.sg/equation-of-chord-of-a-parabola-whose-midpoint-is-given.htmlHow to prove $$ T = S1 $$ $$ i.e \qquad yy_1 - 2a(x+x_1) = y_1^2 - 4ax_1=0$$ as the equation of chord for a parabola y$^2$ = 4ax whose midpoint (x$_1,y_1$) is given. $$$$ I couldn't understand ......Wed, 26 Aug 2026 12:36:49 -0400https://pop.com.sg/how-do-you-show-that-the-three-incidence-axioms-are-independent-of-eac.htmlhttps://pop.com.sg/how-do-you-show-that-the-three-incidence-axioms-are-independent-of-eac.htmlHow do you show that the three incidence axioms: Incidence Axiom 1. For every pair of distinct points P and Q there is exactly one line l such that P and Q lie on l. Incidence Axiom 2. For every ......Wed, 26 Aug 2026 12:35:05 -0400https://pop.com.sg/what-is-multiplication-by-juxtaposition.htmlhttps://pop.com.sg/what-is-multiplication-by-juxtaposition.htmlI was reading http://www.purplemath.com/modules/orderops2.htm it shows = 16 ÷ 2[2] + 1 (**) ... = 5 The general consensus among math people is that "multiplication by juxtaposition" (......Wed, 26 Aug 2026 12:23:53 -0400https://pop.com.sg/generalizing-sherman-morrison-woodbury-formula-to-more-than-2-matrices.htmlhttps://pop.com.sg/generalizing-sherman-morrison-woodbury-formula-to-more-than-2-matrices.htmlIs there a generalization of Sherman–Morrison–Woodbury formula to find inverses of sums of more than 2 matrices? https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula seems to be for 2 mat......Wed, 26 Aug 2026 12:19:54 -0400https://pop.com.sg/does-fermat-039-s-last-theorem-imply-the-modularity-theorem.htmlhttps://pop.com.sg/does-fermat-039-s-last-theorem-imply-the-modularity-theorem.htmlThe Wikipedia article on the proof of Fermat's Last Theorem has this sentence If the link identified by Frey could be proven, then in turn, it would mean that a proof or disproof of either of Fe......Wed, 26 Aug 2026 12:16:25 -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 -0400