https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 22 Aug 2026 05:31:24 -0400https://pop.com.sg/proving-monge-039-s-theorem-using-menelaus-039.htmlhttps://pop.com.sg/proving-monge-039-s-theorem-using-menelaus-039.htmlMonge's theorem states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are ...Sat, 22 Aug 2026 09:28:35 -0400https://pop.com.sg/finding-a-joint-moment-generating-function.htmlhttps://pop.com.sg/finding-a-joint-moment-generating-function.htmlPlease consider the following problem and my solution to it: Problem: Let $(X,Y)$ be a continues bivariate r.v. with joint pdf \begin{eqnarray*} f_{XY}(x,y) &=& \begin{cases} e^{-(x+y)} &a......Sat, 22 Aug 2026 09:10:25 -0400https://pop.com.sg/norm-of-gradient-in-gradient-descent.htmlhttps://pop.com.sg/norm-of-gradient-in-gradient-descent.htmlThis question discusses the size of gradient in gradient descent. Some examples were pointed to show it is not necessarily the case that gradient will decrease, for example, $f(x) = \sqrt{|x|}$ or ......Sat, 22 Aug 2026 09:07:56 -0400https://pop.com.sg/how-does-the-reduced-cone-look-like.htmlhttps://pop.com.sg/how-does-the-reduced-cone-look-like.htmlI have to find in terms of standard spaces (that is interval, circle, disk, sphere, cone, etc.) the reduced cone (i.e. cone in category of pointed spaces) $\operatorname{Cone}(*)$ where $\{*\}$ is ......Sat, 22 Aug 2026 09:03:59 -0400https://pop.com.sg/binomial-distribution-word-problem.htmlhttps://pop.com.sg/binomial-distribution-word-problem.htmlA student gets at least 8 hours of sleep 45% of the nights; the sleeping schedule is independent from night to night. Let Xi indicate whether the student gets at least 8 hours of sleep during the n......Sat, 22 Aug 2026 09:00:50 -0400https://pop.com.sg/what-is-the-symbol-for-coincident-in-geometry.htmlhttps://pop.com.sg/what-is-the-symbol-for-coincident-in-geometry.htmlI am looking for a symbol to say that one geometrical figure coincides with another without writing the phrase "is coincident with." For example, the altitude $a$ of an equilateral triangle coinci......Sat, 22 Aug 2026 08:55:19 -0400https://pop.com.sg/get-polynomial-function-from-3-points.htmlhttps://pop.com.sg/get-polynomial-function-from-3-points.htmlI need to understand how to define a polynomial function from 3 given points. Everything I found on the web so far is either too complicated or the reversed way around. (how to get points with a gi......Sat, 22 Aug 2026 08:54:41 -0400https://pop.com.sg/how-does-squaring-both-sides-of-an-equation-lead-to-extraneous-solutio.htmlhttps://pop.com.sg/how-does-squaring-both-sides-of-an-equation-lead-to-extraneous-solutio.htmlLet’s say I have $x = x + 1$, which is a false statment for real $x$; why can I solve for real $x$ when I square both sides of the equation, giving $x^2=(x+1)^2$?...Sat, 22 Aug 2026 08:50:43 -0400https://pop.com.sg/convert-y-2-4-x-1-to-a-polar-equation.htmlhttps://pop.com.sg/convert-y-2-4-x-1-to-a-polar-equation.htmlI'm trying to convert the rectangular cartesian equation $$ y^2 = 4(x + 1) $$ to a polar equation. After replacing $y = r \sin \theta$ and $x = r \cos \theta$, I get $$ r^2 \sin^2 \theta = 4(r \......Sat, 22 Aug 2026 08:40:36 -0400https://pop.com.sg/what-is-the-perimeter-of-a-sector.htmlhttps://pop.com.sg/what-is-the-perimeter-of-a-sector.htmlI don't understand this. So we have: \begin{align} r &= 12 \color{gray}{\text{ (radius of circle)}} \\ d &= 24 \text{ (r}\times2) \color{gray}{\text{ (diameter of circle)}} \\ c &= 24\......Sat, 22 Aug 2026 08:37:00 -0400https://pop.com.sg/symbols-for-odd-and-even.htmlhttps://pop.com.sg/symbols-for-odd-and-even.htmlLet $A$ be a sequence of letters $\langle a,b,c,d,e,f \rangle$. I want to create two subsequences, one with the values with odd index and other with the values with even index: $A_\mathrm{odd} = \l......Sat, 22 Aug 2026 08:33:11 -0400https://pop.com.sg/context-free-grammar-for-palindrome-over-all-terminals.htmlhttps://pop.com.sg/context-free-grammar-for-palindrome-over-all-terminals.htmlI've found various examples of context-free grammars for palindromes but they all seem to hardcode the rules to be of the form "terminal Statement (same) terminal" For example, if $T \...Sat, 22 Aug 2026 08:32:49 -0400https://pop.com.sg/what-is-the-square-root-of-infinity-and-what-is-infinity-2.htmlhttps://pop.com.sg/what-is-the-square-root-of-infinity-and-what-is-infinity-2.htmlIn rigorous mathematics, how can one formalize the notion of a "square root of infinity", as well as that of "infinity squared"?...Sat, 22 Aug 2026 08:27:23 -0400https://pop.com.sg/what-is-the-best-calculus-book-for-a-strong-introdutory-course.htmlhttps://pop.com.sg/what-is-the-best-calculus-book-for-a-strong-introdutory-course.htmlI'm teaching for a small group of future undergraduate science students. I'm from Brazil, so here, we don't see any calculus in school, Just definition of function and some properties of most known ...Sat, 22 Aug 2026 08:26:03 -0400https://pop.com.sg/coin-toss-probability-using-bayes-theorem.htmlhttps://pop.com.sg/coin-toss-probability-using-bayes-theorem.htmlIn a box, there are the same number of two kinds of coins: the fair coins (50% chance for head) and the biased coins (70% chance for head). One person randomly selected a coin and tosses it twice. ......Sat, 22 Aug 2026 08:23:09 -0400https://pop.com.sg/2-easy-gre-questions.htmlhttps://pop.com.sg/2-easy-gre-questions.htmlI've been having trouble with these two questions. The first is simple interest, the second is rate. I'm sure they're easy but I can't focus on getting the solution because I'm terrible at focusing......Sat, 22 Aug 2026 08:19:15 -0400https://pop.com.sg/does-sum-limits-x-1-infty-sin-x-converge.htmlhttps://pop.com.sg/does-sum-limits-x-1-infty-sin-x-converge.htmlI received a task to find out whether the following series converges: $$\sum_{x=1}^\infty\sin(x)$$ On first look it seems simple, but as I keep thinking about it, there's not a single lemma or ...Sat, 22 Aug 2026 08:19:04 -0400https://pop.com.sg/what-is-the-minimum-possible-exterior-surface-area-of-an-open-glassed-.htmlhttps://pop.com.sg/what-is-the-minimum-possible-exterior-surface-area-of-an-open-glassed-.htmlThis is a question from Khan Academy I do not understand. An open-topped glass aquarium with a square base is designed to hold $32$ space cubic feet of water. What is the minimum possible exterior ...Sat, 22 Aug 2026 08:13:46 -0400https://pop.com.sg/finding-integer-solutions-to-y-2-x-3-2.htmlhttps://pop.com.sg/finding-integer-solutions-to-y-2-x-3-2.htmlI have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a ...Sat, 22 Aug 2026 08:11:45 -0400https://pop.com.sg/terminal-value-formula-in-the-discounted-cash-flow-dcf-valuation.htmlhttps://pop.com.sg/terminal-value-formula-in-the-discounted-cash-flow-dcf-valuation.htmlI have just started with Finance and have a physics background. However, I am struggling to understand the formula for the terminal value in a discounted cash flow valuation (DCF valuation) accordi......Sat, 22 Aug 2026 08:06:34 -0400https://pop.com.sg/disconnected-connected-graphs.htmlhttps://pop.com.sg/disconnected-connected-graphs.htmlDetermine whether the statements below are true or false. If the statement is true, then prove it; and if it is false, give a counterexample. (a) Every disconnected graph has a vertex of degree 0.......Sat, 22 Aug 2026 08:06:19 -0400https://pop.com.sg/finding-the-number-of-surfaces-containing-a-curve-given-its-parametriz.htmlhttps://pop.com.sg/finding-the-number-of-surfaces-containing-a-curve-given-its-parametriz.htmlIf there is a parametrized curve: $$r(t) = t^2\,i + ln(t)\,j + \frac{1}{t}\,k$$ then as I understand it, to find surfaces that contain the curve, you can solve for $t$ and find three equations, each ...Sat, 22 Aug 2026 08:00:20 -0400https://pop.com.sg/is-there-something-like-a-normal-distribution-model-for-discrete-proba.htmlhttps://pop.com.sg/is-there-something-like-a-normal-distribution-model-for-discrete-proba.htmlIf one wants to find the probability that a continuous random variable will fall within a range of $a \leq X \leq b$, based on a mean value $\mu$, and a deviation of $\sigma$, he would integrate the ...Sat, 22 Aug 2026 07:59:51 -0400https://pop.com.sg/finding-moment-of-inertia-rotational-inertia-i-using-integration.htmlhttps://pop.com.sg/finding-moment-of-inertia-rotational-inertia-i-using-integration.htmlI just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was Rotational Inertia, or Moment of Inertia. It's basically the equivalent of......Sat, 22 Aug 2026 07:50:01 -0400https://pop.com.sg/conditional-introduction-rule.htmlhttps://pop.com.sg/conditional-introduction-rule.htmlIn the derivation (the image below) the author shows that given the premise $\neg S \land \neg J$, the conclusion is $S \implies J$. All these deductive maneuver for concluding implications I find ...Sat, 22 Aug 2026 07:48:29 -0400https://pop.com.sg/how-can-we-make-11-non-isomorphic-graphs-on-4-vertices.htmlhttps://pop.com.sg/how-can-we-make-11-non-isomorphic-graphs-on-4-vertices.htmlHow can we draw all the non-isomorphic graphs on $4$ vertices ? But it is mentioned that $ 11 $ graphs are possible....Sat, 22 Aug 2026 07:47:34 -0400https://pop.com.sg/how-are-asymptotes-actually-defined-in-rigorous-mathematics.htmlhttps://pop.com.sg/how-are-asymptotes-actually-defined-in-rigorous-mathematics.htmlThis question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x \m......Sat, 22 Aug 2026 07:29:24 -0400https://pop.com.sg/find-the-value-of-the-expression.htmlhttps://pop.com.sg/find-the-value-of-the-expression.htmlThe expression $ax^2 + bx + 1$ takes the values $1$ and $4$ when $x$ takes the values $2$ and $3$ respectively. How can we find the value of the expression when $x$ takes the value of $4$?...Sat, 22 Aug 2026 07:16:04 -0400https://pop.com.sg/determining-whether-cartesian-first-order-tensor.htmlhttps://pop.com.sg/determining-whether-cartesian-first-order-tensor.htmlI'm struggeling to understand when we call a certain tuple a vector or first order Cartesian tensor $\mathbf v$. Riley, Hobson and Riley (3rd) states that when you apply a rotation, the new compone......Sat, 22 Aug 2026 07:13:08 -0400https://pop.com.sg/why-is-the-determinant-of-a-symplectic-matrix-1.htmlhttps://pop.com.sg/why-is-the-determinant-of-a-symplectic-matrix-1.htmlSuppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 & E_n\\ -E_n&0 \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J.$$ How to figure out $$\det......Sat, 22 Aug 2026 07:09:18 -0400https://pop.com.sg/the-interior-of-a-connected-set-in-mathbb-r-k.htmlhttps://pop.com.sg/the-interior-of-a-connected-set-in-mathbb-r-k.htmlIs the interior of a connected set in $\mathbb R^k$ connected?...Sat, 22 Aug 2026 07:02:22 -0400https://pop.com.sg/intuitively-what-is-the-difference-between-eigendecomposition-and-sing.htmlhttps://pop.com.sg/intuitively-what-is-the-difference-between-eigendecomposition-and-sing.htmlI'm trying to intuitively understand the difference between SVD and eigendecomposition. From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three b......Sat, 22 Aug 2026 07:02:16 -0400https://pop.com.sg/i-have-a-d3-three-sided-die-and-i-m-trying-to-get-the-highest-possible.htmlhttps://pop.com.sg/i-have-a-d3-three-sided-die-and-i-m-trying-to-get-the-highest-possible.htmlAs the title says, if I roll a 2 on a d3 and am allowed to reroll it, is there an advantage to re-rolling? It seems like the choice doesn’t matter, or is purely preferential but what is wrong with ...Sat, 22 Aug 2026 07:01:38 -0400https://pop.com.sg/empty-and-trivial-graph-diestel-039-s-book.htmlhttps://pop.com.sg/empty-and-trivial-graph-diestel-039-s-book.htmlThe number of vertices of a graph $G$ is its order, written as $|G|$; For the empty graph $(\varnothing, \varnothing)$ we simply write $\varnothing$. A graph of order $0$ or $1$ is called trivial. ......Sat, 22 Aug 2026 06:54:17 -0400https://pop.com.sg/closure-of-integers.htmlhttps://pop.com.sg/closure-of-integers.htmlFellow Mathers, is the fact that the Integers are closed under addition and multiplication... ∀ a,b ∈ Z , a + b = c where c ∈ Z ∀ a,b ∈ Z , ab = c where c ∈ Z ...universal axioms, theorems, depen......Sat, 22 Aug 2026 06:50:13 -0400https://pop.com.sg/dense-sequence-definition.htmlhttps://pop.com.sg/dense-sequence-definition.htmlA subset $A$ of a space $X$ is called dense if every point $x$ in $X$ either belongs to $A$ or is a limit point of $A$ ; that is, the closure of $A$ constitutes the whole set $X$. That the definiti......Sat, 22 Aug 2026 06:45:56 -0400https://pop.com.sg/2-norm-vs-operator-norm.htmlhttps://pop.com.sg/2-norm-vs-operator-norm.htmlI have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the......Sat, 22 Aug 2026 06:33:32 -0400https://pop.com.sg/riemann-sum-given-partition-and-sample-points.htmlhttps://pop.com.sg/riemann-sum-given-partition-and-sample-points.htmlCalculate the Riemann sum $R(f,p,c)$ for the function $f(x)=3x^2+2x$. Partition $p={0,1,2.5,3.2,5}$ and sample points $c={0.5,2,3,4.5}$. Am able to find a Riemann sum whereby partitions have been ...Sat, 22 Aug 2026 06:33:27 -0400https://pop.com.sg/what-is-the-difference-between-cluster-point-and-limit-point.htmlhttps://pop.com.sg/what-is-the-difference-between-cluster-point-and-limit-point.htmlIn the book of real mathematical analysis written by Pugh, there is a sentence about these two and condensation: Thus, S limits at p, clusters at p, or condenses at p according to whether each Mr(p......Sat, 22 Aug 2026 06:30:31 -0400https://pop.com.sg/making-change-for-a-dollar-and-other-number-partitioning-problems.htmlhttps://pop.com.sg/making-change-for-a-dollar-and-other-number-partitioning-problems.htmlI was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quo......Sat, 22 Aug 2026 06:24:03 -0400https://pop.com.sg/what-does-lg-x-mean-is-it-log-2-x-or-log-10-x-in-binary-trees.htmlhttps://pop.com.sg/what-does-lg-x-mean-is-it-log-2-x-or-log-10-x-in-binary-trees.htmlI'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = \lg x$ but some people say $\lg = \log$. So what does $\lg$ really stand for? specifically when t......Sat, 22 Aug 2026 06:14:56 -0400https://pop.com.sg/definition-of-the-ordered-triple-a-b-c-according-to-kuratowski-039-s-s.htmlhttps://pop.com.sg/definition-of-the-ordered-triple-a-b-c-according-to-kuratowski-039-s-s.htmlCan someone give Kuratowski's definition of the ordered triple $(a,b,c)$ assuming $A \times B \times C$ is rewritten as $(A \times B) \times C$, please? I noticed there is already an answered quest......Sat, 22 Aug 2026 06:03:49 -0400https://pop.com.sg/fourier-transform-of-sinc-function-confused-about-operatorname-sinc-pi.htmlhttps://pop.com.sg/fourier-transform-of-sinc-function-confused-about-operatorname-sinc-pi.htmlI am confused about the fourier transform of the $\operatorname{sinc}$ function. First I don't know if $$\operatorname{sinc} (x) = \frac{\sin(\pi x)}{(\pi x)}$$ or $$\operatorname{sinc} (\pi x......Sat, 22 Aug 2026 06:03:39 -0400https://pop.com.sg/order-of-operations-for-multiplying-three-matrices.htmlhttps://pop.com.sg/order-of-operations-for-multiplying-three-matrices.htmlIf I have a $1\times 2$ matrix $A$, a $2\times 2$ matrix $B$, and a $2\times 2$ matrix $C$, and am asked to calculate ABC, is there a specific order in which I have to carry out the multiplication? ...Sat, 22 Aug 2026 05:52:56 -0400https://pop.com.sg/find-the-area-of-a-segment-of-a-circle-given-radius-and-central-angle.htmlhttps://pop.com.sg/find-the-area-of-a-segment-of-a-circle-given-radius-and-central-angle.htmlFind the Area of a segment of a circle if the central angle of the segment is $105^\circ$ degrees and the radius is $70$. Formulas I have: Area of a non-right angle triangle= $\frac{1}{2}a b \sin......Sat, 22 Aug 2026 05:39:35 -0400https://pop.com.sg/a-semicircle-has-a-radius-of-2-m-determine-the-dimensions-of-a-rectang.htmlhttps://pop.com.sg/a-semicircle-has-a-radius-of-2-m-determine-the-dimensions-of-a-rectang.html$y^2 + x^2 = 4$ $A(x) = 2xy$ (make base of semicircle = $2x$) plug it in: $A(x) = (\space 2\sqrt{(4-y^2)}\space)\cdot y$ Final derivative: $$\begin{align} A'(x) & = \frac{-2y^2}{\sqrt{4-y^2......Sat, 22 Aug 2026 05:29:42 -0400https://pop.com.sg/understanding-when-a-graph-represents-a-function.htmlhttps://pop.com.sg/understanding-when-a-graph-represents-a-function.htmlWhich of the following is the graph of a function where $y = f(x)$? I got the answer for the above question is A function has one output for each input in its domain. Each $x$-value on this graph ...Sat, 22 Aug 2026 05:29:25 -0400https://pop.com.sg/deriving-the-cosine-formula-using-vectors.htmlhttps://pop.com.sg/deriving-the-cosine-formula-using-vectors.htmlHow did they go from 2b⋅c to -2bccosA? Where did they get the negative sign from?...Sat, 22 Aug 2026 05:21:12 -0400https://pop.com.sg/what-really-is-diagonalization.htmlhttps://pop.com.sg/what-really-is-diagonalization.htmlIf I have a square matrix $A$ representing a linear transformation $T:V\rightarrow V$ w.r.t the basis, $B=\{v_1,v_2..,v_n\}$ and $A$ is Hermitian. So we have $Av_n=\lambda_{n}v_n$ where $\{v_1,v_......Sat, 22 Aug 2026 05:16:43 -0400https://pop.com.sg/how-to-make-continued-fractions-of-any-number.htmlhttps://pop.com.sg/how-to-make-continued-fractions-of-any-number.htmlI recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...Sat, 22 Aug 2026 05:10:54 -0400https://pop.com.sg/if-ab-i-then-ba-i.htmlhttps://pop.com.sg/if-ab-i-then-ba-i.htmlIf $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...Sat, 22 Aug 2026 05:01:34 -0400https://pop.com.sg/why-does-the-area-of-a-rhombus-with-same-lengths-as-a-square-has-a-dif.htmlhttps://pop.com.sg/why-does-the-area-of-a-rhombus-with-same-lengths-as-a-square-has-a-dif.htmla rhombus can be created by dragging the four sides of the square to a certain angle. so how does the area of the new formed rhombus differ from the square.it can be proven mathematically but how d......Sat, 22 Aug 2026 05:01:08 -0400https://pop.com.sg/inductive-definition-of-a-set.htmlhttps://pop.com.sg/inductive-definition-of-a-set.htmlI'm a beginner in set theory and I have doubt regarding mathematical induction. I came across the following examples. Example 1: Find the set given by the following definition: 1) $ 3 \in P $ 2......Sat, 22 Aug 2026 04:55:46 -0400https://pop.com.sg/what-is-the-scientific-notation-of-zero.htmlhttps://pop.com.sg/what-is-the-scientific-notation-of-zero.htmlThis question was asked here, where the answer uses this description. The last line reads: "The special case of $0$ does not have a unique representation in scientific notation, i.e., $0=0×10^0=......Sat, 22 Aug 2026 04:50:32 -0400https://pop.com.sg/how-to-draw-hasse-diagram.htmlhttps://pop.com.sg/how-to-draw-hasse-diagram.htmlHow would you draw a Hasse diagram of the divisibility relation? when A = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} Any help would be appreciated, thank you....Sat, 22 Aug 2026 04:48:43 -0400https://pop.com.sg/what-is-the-mathematical-formula-for-square-root.htmlhttps://pop.com.sg/what-is-the-mathematical-formula-for-square-root.htmlI wasn't at school when we were learning this, and I've forgot how to calculate a square root on paper using a formula? Can anyone please help me? What is the formula? I need this to write an alg......Sat, 22 Aug 2026 04:48:20 -0400https://pop.com.sg/can-all-maths-be-represented-in-geometry.htmlhttps://pop.com.sg/can-all-maths-be-represented-in-geometry.htmlI'm aware that many maths problems can be expressed and understood in geometry, for example, complex numbers can be expressed in 2 dimensions, that can be useful for some problems. Maths started in ...Sat, 22 Aug 2026 04:32:58 -0400https://pop.com.sg/how-to-use-eigenvalues-and-eigenvector-to-solve-systems-of-linear-equa.htmlhttps://pop.com.sg/how-to-use-eigenvalues-and-eigenvector-to-solve-systems-of-linear-equa.htmlA simple system of algebraic (not differential) equations such as: $$ 3 x + 4 y - 8 z = 23 $$ $$ -2 x + 8 y - 11 z = -32 $$ $$ -4 x + 9 y - 32 z = - 8 $$ can be written as: $$\mathbf{A} \vec x = ......Sat, 22 Aug 2026 04:32:31 -0400https://pop.com.sg/eulerian-graph-with-odd-even-vertices-edges.htmlhttps://pop.com.sg/eulerian-graph-with-odd-even-vertices-edges.htmlFind an Eulerian graph with an even/odd number of vertices and an even/odd number of edges or prove that there is no such graph (for each of the four cases). I came up with the graphs shown below ......Sat, 22 Aug 2026 04:31:34 -0400https://pop.com.sg/second-derivative-of-xy-with-respect-to-x.htmlhttps://pop.com.sg/second-derivative-of-xy-with-respect-to-x.html$$\frac{d^2}{dx^2}xy$$ I know it equals zero but I don't know the in-between steps. I'm using it to prove that Newton's Laws work in any frame of reference. So, say, two guys start from the same......Sat, 22 Aug 2026 04:31:18 -0400https://pop.com.sg/consensus-theorem-legal-to-use-redundant-terms-to-find-more-redundant-.htmlhttps://pop.com.sg/consensus-theorem-legal-to-use-redundant-terms-to-find-more-redundant-.htmlWhen using the Consensus Theorem in Boolean algebra to minimize an expression, is it a legal move to find and add a redundant term to the expression and then use that term to find more redundant te......Sat, 22 Aug 2026 04:15:25 -0400https://pop.com.sg/what-is-a-pullback-of-a-metric-and-how-does-it-work.htmlhttps://pop.com.sg/what-is-a-pullback-of-a-metric-and-how-does-it-work.htmlThe term "metric" is familiar, but not the idea of a pullback on it. I have tried to find intuitive, beginner-friendly explanations of this concept without success. Your attempts would be appreciat......Sat, 22 Aug 2026 04:14:50 -0400https://pop.com.sg/limit-of-x-ln-x.htmlhttps://pop.com.sg/limit-of-x-ln-x.htmlI am trying to solve $$\lim \limits_{x \to 0}x\ln{x}$$ which according to WolframAlpha (and Wikipedia) equals $0$. I managed to solve it by substituting such that $y = \dfrac{1}{x}$ and then using L'...Sat, 22 Aug 2026 04:02:32 -0400https://pop.com.sg/solving-x-3-y-3-xy-61-in-integers.htmlhttps://pop.com.sg/solving-x-3-y-3-xy-61-in-integers.htmlI'm trying to solve the following equation: $$x^3-y^3=xy+61$$ I got: $$(x-y)(x^2+xy+y^2)=xy+61$$ But I can't go any further. I am looking for a solution in integers. I need some hints to proce......Sat, 22 Aug 2026 03:59:19 -0400https://pop.com.sg/how-is-the-derivative-of-the-cdf-of-a-random-variable-x-its-pdf.htmlhttps://pop.com.sg/how-is-the-derivative-of-the-cdf-of-a-random-variable-x-its-pdf.htmlFor a continuos random variable $X$, if we have its p.d.f. $f(x)$, then the cumulative densitity function (c.d.f.) of $X$, $F(x)$, is $$F(x) = \int_{-\infty}^{x}f(t)dt$$ We also have $$F'(x) = \fra......Sat, 22 Aug 2026 03:51:03 -0400https://pop.com.sg/what-is-wrong-with-the-proof-for-ln-2-frac-1-2-ln-2.htmlhttps://pop.com.sg/what-is-wrong-with-the-proof-for-ln-2-frac-1-2-ln-2.htmlI have got a question which is as follows: Is $\ln(2)=\frac{1}{2}\ln(2)$?? The following argument seems suggesting that the answer is yes: We have the series $1-\frac{1}{2}+\frac{1}{3}-\fra......Sat, 22 Aug 2026 03:51:01 -0400https://pop.com.sg/variance-of-negative-binomial-distribution-without-moment-generating-s.htmlhttps://pop.com.sg/variance-of-negative-binomial-distribution-without-moment-generating-s.htmlGiven the discrete probability distribution for the negative binomial distribution in the form $$P(X = r) = \sum_{n\geq r} {n-1\choose r-1} (1-p)^{n-r}p^r$$ It appears there are no derivations o......Sat, 22 Aug 2026 03:50:23 -0400https://pop.com.sg/reducible-and-irreducible-polynomials-are-confusing-me.htmlhttps://pop.com.sg/reducible-and-irreducible-polynomials-are-confusing-me.htmlThe definition claims that a polynomial in a field of positive degree is a reducible polynomial when it can be written as the product of $2$ polynomials in the field with positive degrees. Other wi......Sat, 22 Aug 2026 03:38:17 -0400https://pop.com.sg/logic-behind-pisano-period.htmlhttps://pop.com.sg/logic-behind-pisano-period.htmlI just came to know about Pisano periods, and it's amazing application in computing modulus of large Fibonacci numbers. I know that a Pisano period periodically starts at $0,1$, but I haven't been ......Sat, 22 Aug 2026 03:16:12 -0400https://pop.com.sg/find-the-three-consecutive-numbers.htmlhttps://pop.com.sg/find-the-three-consecutive-numbers.htmlI have my problem here. I can't solve it on my own. I need a solution to prove my answer. Does anybody know this? thanks "Find the three consecutive numbers that have the property that the square ......Sat, 22 Aug 2026 03:15:31 -0400https://pop.com.sg/how-to-prove-that-sin-180-circ-theta-sin-theta.htmlhttps://pop.com.sg/how-to-prove-that-sin-180-circ-theta-sin-theta.htmlMi question is: How to prove $$\sin(180^\circ-\theta)=\sin\theta$$ ? Here, sine is defined for any angle such as 'alpha' This is the question mi college teacher asked me to derive it but i could ......Sat, 22 Aug 2026 03:14:25 -0400https://pop.com.sg/finding-taking-off-speed-help.htmlhttps://pop.com.sg/finding-taking-off-speed-help.htmlAn athlete jumps a horizontal distance of $7.18$m. He was airborne for $2.29$ seconds. The acceleration due to gravity is taken as $9.81$ms$^{-2}$. Assume that air resistance is negligible. Calcula......Sat, 22 Aug 2026 03:04:53 -0400https://pop.com.sg/how-to-integrate-frac-1-sqrt-1-x-2-using-substitution.htmlhttps://pop.com.sg/how-to-integrate-frac-1-sqrt-1-x-2-using-substitution.htmlHow you integrate $\frac{1}{\sqrt{1+x^2}}$ using following substitution? $1+x^2=t$ $\Rightarrow$ $x=\sqrt{t-1} \Rightarrow dx = \frac{dt}{2\sqrt{t-1}}dt$... Now I'm stuck. I don't know how to proceed ...Sat, 22 Aug 2026 02:30:50 -0400https://pop.com.sg/to-optimize-fenced-area-in-a-semi-ellipse-what-a-b-should-i-choose.htmlhttps://pop.com.sg/to-optimize-fenced-area-in-a-semi-ellipse-what-a-b-should-i-choose.htmlA fence must enclose an area that borders a straight river. We have a fixed amount of fence, say 100 meters. I want to make the shape a semi-ellipse (half an ellipse). What a/b ratio should I cho......Sat, 22 Aug 2026 02:30:37 -0400https://pop.com.sg/why-study-quadrilaterals.htmlhttps://pop.com.sg/why-study-quadrilaterals.htmlMy niece is in the 10th grade, and they have to do lot of theorems related to quadrilaterals. And, I was surprised to know that they have to learn by rote some theorems. This has made her feel that......Sat, 22 Aug 2026 02:19:46 -0400https://pop.com.sg/solving-arccos-x-arccos-2x-frac-3-pi-4.htmlhttps://pop.com.sg/solving-arccos-x-arccos-2x-frac-3-pi-4.html$\arccos (x) +\arccos (2x)=\frac{3\pi}{4}$ $\implies \arccos (x)=\frac{3\pi}{4}-\arccos (2x)$ $\implies \cos(\arccos (x))=\cos(\frac{3\pi}{4}-\arccos (2x))$ $\implies x=\cos(\frac{3\pi}{4})\cos(\...Sat, 22 Aug 2026 02:17:37 -0400https://pop.com.sg/calculating-ln-1-sqrt3.htmlhttps://pop.com.sg/calculating-ln-1-sqrt3.htmlI distributed the natural logarithm and got $(0 + 0.549)$ [placing the values in a calculator]. However, the answer key states that the answer is $1.0051$. Where did I go wrong?...Sat, 22 Aug 2026 02:17:21 -0400https://pop.com.sg/use-logarithmic-differentiation-to-find-frac-dy-dx-of-y-ln-x-ln-x.htmlhttps://pop.com.sg/use-logarithmic-differentiation-to-find-frac-dy-dx-of-y-ln-x-ln-x.htmlI've been asked to find the derivative of $\ln x^{\ln x}$ using a method called "logarithmic differentiation". I'm not familiar with this method of differentiation, and a search on Wikipedia doesn'......Sat, 22 Aug 2026 02:03:03 -0400https://pop.com.sg/is-the-derivative-of-a-straight-line-the-same-of-a-tangent-line.htmlhttps://pop.com.sg/is-the-derivative-of-a-straight-line-the-same-of-a-tangent-line.htmlSorry if I'm being too specific and for not showing an example but If you had a derivative of a straight line would the slope of the tangent line be the same as the straight line?...Sat, 22 Aug 2026 02:02:11 -0400https://pop.com.sg/why-do-we-need-topology-and-what-are-examples-of-real-life-application.htmlhttps://pop.com.sg/why-do-we-need-topology-and-what-are-examples-of-real-life-application.htmlI am very new to topology. I want to know why we need topology in mathematics. What is its significance? Without topology which mathematics concept cannot be explained. What are its real-life ...Sat, 22 Aug 2026 01:57:03 -0400https://pop.com.sg/please-help-me-understand-the-definition-of-straight-line-given-by-euc.htmlhttps://pop.com.sg/please-help-me-understand-the-definition-of-straight-line-given-by-euc.html"A straight-line is (any) one which lies evenly with points on itself." This is how Euclid defines a straight line but I don't know what it really means. Is this saying that any point picked on the ...Sat, 22 Aug 2026 01:54:22 -0400https://pop.com.sg/prove-3-is-prime-in-the-ring-bbb-z-i.htmlhttps://pop.com.sg/prove-3-is-prime-in-the-ring-bbb-z-i.htmlI am not sure what is the right phrase for primes in some ring. The definition I was gave is that in a ring $A$, $p\in A$ is prime if $\forall x,y \in A$ such that $p\mid xy$, $p\mid x$ or $p\mid y......Sat, 22 Aug 2026 01:53:45 -0400https://pop.com.sg/can-all-real-polynomials-be-factored-into-quadratic-and-linear-factors.htmlhttps://pop.com.sg/can-all-real-polynomials-be-factored-into-quadratic-and-linear-factors.htmlSo I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens ......Sat, 22 Aug 2026 01:53:22 -0400https://pop.com.sg/examples-of-one-to-one-and-onto-functions-f-mathbb-n-to-mathbb-n.htmlhttps://pop.com.sg/examples-of-one-to-one-and-onto-functions-f-mathbb-n-to-mathbb-n.htmlGive examples of functions from $\mathbb{N}$ to $\mathbb{N}$ with the following properties: i. one-to-one but not onto ii. onto but not one-to-one iii. both onto and one-t......Sat, 22 Aug 2026 01:46:00 -0400https://pop.com.sg/nested-convex-curves-inside-curve-strictly-shorter.htmlhttps://pop.com.sg/nested-convex-curves-inside-curve-strictly-shorter.htmlNote: This question is very similar but different to a previous question [1] of mine. I decided to ask a new question for the sake of clarity. In order to complete a proof, I need to show the foll......Sat, 22 Aug 2026 01:44:56 -0400https://pop.com.sg/finding-the-9th-derivative-of-frac-cos-5-x-2-1-x-3.htmlhttps://pop.com.sg/finding-the-9th-derivative-of-frac-cos-5-x-2-1-x-3.htmlHow do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series......Sat, 22 Aug 2026 01:28:12 -0400https://pop.com.sg/what-do-the-d-in-sde-notation-mean.htmlhttps://pop.com.sg/what-do-the-d-in-sde-notation-mean.htmlAn SDE is often written in the form $ dX_t=\mu dt + \sigma dW_t $. What is the meaning of this equation in English? If I had to construct an SDE, I would write something like $ \frac{dX_t}{dt} = \m......Sat, 22 Aug 2026 01:28:11 -0400https://pop.com.sg/w-such-that-it-contains-at-least-3-ones-is-my-approach-to-the-cfg-righ.htmlhttps://pop.com.sg/w-such-that-it-contains-at-least-3-ones-is-my-approach-to-the-cfg-righ.htmlSo I was trying to solve the CFG, $$\{w \in (0,1)^* \mid w \text{ contains at least three 1's}\}$$ My approach: I decided that a string can begin with a $0$, end with a $0$, it may begin with a ......Sat, 22 Aug 2026 01:27:16 -0400https://pop.com.sg/can-an-irrational-number-raised-to-an-irrational-power-be-rational.htmlhttps://pop.com.sg/can-an-irrational-number-raised-to-an-irrational-power-be-rational.htmlCan an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?...Sat, 22 Aug 2026 01:25:21 -0400https://pop.com.sg/dual-vs-reciprocal-basis.htmlhttps://pop.com.sg/dual-vs-reciprocal-basis.htmlI am confused as to what the differences between the two are. I have just learnt reciprocal vectors, and learnt that they are defined as $u_i v_j = \delta_{ij}$. However, I read that the dual basis......Sat, 22 Aug 2026 01:21:02 -0400https://pop.com.sg/the-graphs-of-f-f-039-and-f-039-039.htmlhttps://pop.com.sg/the-graphs-of-f-f-039-and-f-039-039.htmlRecently I have been struggling to solve the following problem: The figure shows the graphs of $f$, $f'$, and $f''$. Identify each curve, and explain your choices. I know the answer, but I have n......Sat, 22 Aug 2026 01:09:32 -0400https://pop.com.sg/what-is-the-payoff-function-for-games-with-more-than-two-players.htmlhttps://pop.com.sg/what-is-the-payoff-function-for-games-with-more-than-two-players.htmlFor two player games, with payoff matrices $(A,B)$, let $x \in \Delta_x$ denote the mixed strategy of player $1$, and $y \in \Delta_y$ denote the mixed strategy of player $2$. Then the payoff func......Sat, 22 Aug 2026 01:03:09 -0400https://pop.com.sg/what-is-the-meaning-of-hermitian.htmlhttps://pop.com.sg/what-is-the-meaning-of-hermitian.htmlGoogle search-bar gives the definition of Hermitian as: Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal ...Sat, 22 Aug 2026 00:58:18 -0400https://pop.com.sg/expected-number-of-unpecked-chicks-nyt-article.htmlhttps://pop.com.sg/expected-number-of-unpecked-chicks-nyt-article.htmlIn this article, the winner of the math competition answered this question correctly: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately......Sat, 22 Aug 2026 00:55:55 -0400https://pop.com.sg/every-polynomial-with-real-coefficients-is-the-sum-of-cubes-of-three-p.htmlhttps://pop.com.sg/every-polynomial-with-real-coefficients-is-the-sum-of-cubes-of-three-p.htmlHow to prove that every polynomial with real coefficients is the sum of three polynomials raised to the 3rd degree? Formally the statement is: $\forall f\in\mathbb{R}[x]\quad \exists g,h,p\in\ma......Sat, 22 Aug 2026 00:55:25 -0400https://pop.com.sg/is-the-mle-of-an-exponential-distribution-complete-and-sufficient.htmlhttps://pop.com.sg/is-the-mle-of-an-exponential-distribution-complete-and-sufficient.htmlI've just prove that the MLE of an exponential distribution is: $$ \hat \theta = \frac{n}{\sum x_i} $$ So I'm trying to show that $\hat \theta $ is complete and sufficient. For sufficiency I've ......Sat, 22 Aug 2026 00:31:27 -0400https://pop.com.sg/how-you-could-you-change-the-surface-area-formula-for-a-cylinder-to-ca.htmlhttps://pop.com.sg/how-you-could-you-change-the-surface-area-formula-for-a-cylinder-to-ca.htmlSkateboarders use half pipes for doing tricks. A half-pipe is a half cylinder. Explain how you could manipulate the surface area formula for a cylinder to calculate the curved surface area of the h......Sat, 22 Aug 2026 00:22:34 -0400https://pop.com.sg/euclidean-distance-proof.htmlhttps://pop.com.sg/euclidean-distance-proof.htmlHow can I show that the Euclidean distance satisfies the triangle inequality? Where the Euclidean distance is given by: $$d(p,q) = \sqrt{(p_1-q_1)^2 + \cdots + (p_n-q_n)^2}$$ Triangle Inequality......Sat, 22 Aug 2026 00:18:44 -0400https://pop.com.sg/shortest-distance-from-point-to-a-plane.htmlhttps://pop.com.sg/shortest-distance-from-point-to-a-plane.htmlQ: Find the shortest distance from the point $A(1,1,1)$ to the plane $2x+3y+4z=5$. I understand that we need to pick a point P on the plane such that we can form a vector PA. Can someone explain ......Sat, 22 Aug 2026 00:18:08 -0400https://pop.com.sg/how-can-i-prove-ax-bx-for-every-n-times1-column-matrix-x-implies-a-b.htmlhttps://pop.com.sg/how-can-i-prove-ax-bx-for-every-n-times1-column-matrix-x-implies-a-b.htmlLet $A$ and $B$ be matrices $n\times n$. Suppose $AX=BX$ for every $n\times 1$ column matrix $X$. How can I prove this implies $A=B$?...Sat, 22 Aug 2026 00:16:12 -0400