https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 01 Sep 2026 15:44:17 -0400https://pop.com.sg/normalizing-an-eigenvector-to-have-unit-length.htmlhttps://pop.com.sg/normalizing-an-eigenvector-to-have-unit-length.htmlI worked out a problem where one of the eigenvectors is $(1+ \sqrt{2}, 1)$. How do I normalize this vector? Should I use approximate values?...Tue, 01 Sep 2026 19:42:48 -0400https://pop.com.sg/minimum-possible-number-of-friendships.htmlhttps://pop.com.sg/minimum-possible-number-of-friendships.htmlHere is the problem: There are 2000 people on a social network. Each person sends 1000 friend requests. Two people are friends if they've sent a friend request to each other. What is the minimum ...Tue, 01 Sep 2026 19:40:12 -0400https://pop.com.sg/when-will-the-ball-hit-the-ground.htmlhttps://pop.com.sg/when-will-the-ball-hit-the-ground.htmlA girl is throwing a ball. If I have an equation $h= -4.9*t^2+20t+1.2$, where $t$ is number of second after the girl has thrown the ball, and $h$ is the height above ground. I want to find out w......Tue, 01 Sep 2026 19:30:00 -0400https://pop.com.sg/volume-and-lateral-surface-area-are-equal.htmlhttps://pop.com.sg/volume-and-lateral-surface-area-are-equal.htmlI need to express the radius $r$ of the right circular cone as a function of its height $h$ given that its volume equals to its lateral surface area. I know the two equations $\pi r \times \sqrt{......Tue, 01 Sep 2026 19:27:55 -0400https://pop.com.sg/permutations-vs-combinatorial-vs-factorials-vs-exponents.htmlhttps://pop.com.sg/permutations-vs-combinatorial-vs-factorials-vs-exponents.htmlI'm currently working on a probability course, and I am constantly having trouble figuring out when to use permutations vs. combinations vs. factorials vs. exponents in order to calculate sample si......Tue, 01 Sep 2026 19:27:54 -0400https://pop.com.sg/why-can-a-5x4-matrix-never-be-surjective.htmlhttps://pop.com.sg/why-can-a-5x4-matrix-never-be-surjective.htmlMy thought is that since a matrix can only be surjective if null(T)= n-m, no matrix can have a nullity of -1 therefore a 5x4 matrix cannot be surjective. Is this a valid proof?...Tue, 01 Sep 2026 19:27:49 -0400https://pop.com.sg/rgb-to-hsv-color-conversion-algorithm.htmlhttps://pop.com.sg/rgb-to-hsv-color-conversion-algorithm.htmlI'm a programmer looking to build an RGB to HSV color converter. I found an algorithm, but I have very little mathematical background and I'm not quite sure what's going on with it. A step-by-step ...Tue, 01 Sep 2026 19:26:59 -0400https://pop.com.sg/gaussian-elimination-with-4-variables.htmlhttps://pop.com.sg/gaussian-elimination-with-4-variables.htmlAfter this step however, I end up with 2 simultanious equations but still with four unknowns, I'm not sure if my gaussian elimination is wrong or not but how will i find the exact values of x,y,z a......Tue, 01 Sep 2026 19:21:48 -0400https://pop.com.sg/why-do-we-say-dn-dt-kn.htmlhttps://pop.com.sg/why-do-we-say-dn-dt-kn.htmlSo, I just started doing differential equations and I'm trying to understand the basic idea of dN/dt = kN. I understand the whole process of solving a differential equation (the separation of varia......Tue, 01 Sep 2026 19:19:43 -0400https://pop.com.sg/when-is-a-graph-balanced-bipartite.htmlhttps://pop.com.sg/when-is-a-graph-balanced-bipartite.htmlI have a quick question: is there any sufficient condition (theorem, lemma, proposition,...) to show that a graph (vertices do not have the same degree) is balanced bipartite?...Tue, 01 Sep 2026 19:17:02 -0400https://pop.com.sg/confused-about-matrix-tracing.htmlhttps://pop.com.sg/confused-about-matrix-tracing.htmlI need help with this one question I found from my Linear Algebra textbook. While I was doing problems and checking my work from the back of my textbook, I found out one of the problems I did is wr......Tue, 01 Sep 2026 19:06:58 -0400https://pop.com.sg/how-to-find-the-inradius-of-a-triangle-with-given-side-lengths.htmlhttps://pop.com.sg/how-to-find-the-inradius-of-a-triangle-with-given-side-lengths.htmlI need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Thank you....Tue, 01 Sep 2026 19:02:53 -0400https://pop.com.sg/widest-part-of-a-sector.htmlhttps://pop.com.sg/widest-part-of-a-sector.htmlCoordinates of the center of a circle, radius, initial and final angles of a sector are given. How to find coordinates of the endpoints of a segment (A and B) connecting borders of sector in its wi......Tue, 01 Sep 2026 19:01:01 -0400https://pop.com.sg/how-to-find-vector-orthogonal-to-a-given-pair-of-vectors.htmlhttps://pop.com.sg/how-to-find-vector-orthogonal-to-a-given-pair-of-vectors.html(a) $[2,-3,5], [4,-5,1]$ Solution: $[2,-3,5] \times [4,-5,1] = [22,18,2]$ No idea how they got $ [22,18,2]$...Tue, 01 Sep 2026 18:54:30 -0400https://pop.com.sg/total-average-of-averages-not-same-as-the-average-of-total-values.htmlhttps://pop.com.sg/total-average-of-averages-not-same-as-the-average-of-total-values.htmlI am having a strange problem while computing the overall percentage. Let me demonstrate my problem using an example. Assume that I am receiving multiple batches of apple from the vendor and the nu......Tue, 01 Sep 2026 18:53:54 -0400https://pop.com.sg/how-to-find-coordinates-of-reflected-point.htmlhttps://pop.com.sg/how-to-find-coordinates-of-reflected-point.htmlHow can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If P is a reflection (image) of point (3, -3) in the line $2y = ......Tue, 01 Sep 2026 18:35:36 -0400https://pop.com.sg/importance-of-rolle-039-s-and-lagrange-039-s-theorem-in-daily-life-for.htmlhttps://pop.com.sg/importance-of-rolle-039-s-and-lagrange-039-s-theorem-in-daily-life-for.htmlI am school teacher and teaching math. I know the Rolle's theorem and Lagrange's theorem. I can solve the problems numerically then and there. However, I was completely failed to explain the signif......Tue, 01 Sep 2026 18:32:49 -0400https://pop.com.sg/inscribing-square-in-circle-in-just-seven-compass-and-straightedge-ste.htmlhttps://pop.com.sg/inscribing-square-in-circle-in-just-seven-compass-and-straightedge-ste.htmlProblem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the ...Tue, 01 Sep 2026 18:30:31 -0400https://pop.com.sg/conditional-probability-involving-sum-of-bernoulli-trials.htmlhttps://pop.com.sg/conditional-probability-involving-sum-of-bernoulli-trials.htmlI have $X_i (i=1,...,n)$ which has $\text{Bernoulli}(p)$ or $\text{binomial}(1,p)$ distribution and each $X_i$ is independent with $X_i=1$ with probability $p$ and $X_i=0$ with probability $(1-p)$. I ...Tue, 01 Sep 2026 18:27:36 -0400https://pop.com.sg/evaluating-a-surface-integral-using-symmetry.htmlhttps://pop.com.sg/evaluating-a-surface-integral-using-symmetry.htmlLet $S$ denote the surface of the cylinder $x^2+y^2=4,-2\le z\le2$ and consider the surface integral $$\int_{S}(z-x^2-y^2)\,dS$$ How can one use geometry and symmetry to evaluate the integral without ...Tue, 01 Sep 2026 18:24:10 -0400https://pop.com.sg/a-question-about-the-arctangent-addition-formula.htmlhttps://pop.com.sg/a-question-about-the-arctangent-addition-formula.htmlIn the arctangent formula, we have that: $$\arctan{u}+\arctan{v}=\arctan\left(\frac{u+v}{1-uv}\right)$$ however, only for $uv<1$. My question is: where does this condition come from? The situa......Tue, 01 Sep 2026 18:22:37 -0400https://pop.com.sg/how-to-derive-a-function-that-depends-on-another-function-evaluated-at.htmlhttps://pop.com.sg/how-to-derive-a-function-that-depends-on-another-function-evaluated-at.htmlI'm coming from physics, but I figured this is a mathematical question, so here I am. I have a scalar $f(\vec{x},\vec{y})$ that is a function of two different points $\vec{x}$ and $\vec{y}$ in spac......Tue, 01 Sep 2026 18:16:07 -0400https://pop.com.sg/calculating-sqrt-28-cdot-29-cdot-30-cdot-31-1.htmlhttps://pop.com.sg/calculating-sqrt-28-cdot-29-cdot-30-cdot-31-1.htmlIs it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work....Tue, 01 Sep 2026 18:09:47 -0400https://pop.com.sg/why-is-the-empty-set-a-subset-of-every-set-duplicate.htmlhttps://pop.com.sg/why-is-the-empty-set-a-subset-of-every-set-duplicate.htmlTake for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?...Tue, 01 Sep 2026 17:55:14 -0400https://pop.com.sg/is-it-valid-to-calculate-standard-deviation-for-n-2.htmlhttps://pop.com.sg/is-it-valid-to-calculate-standard-deviation-for-n-2.htmlMy colleagues and I are having discussion whether it's valid to calculate standard deviation for $n=2$ or not? I think it's valid since I can calculate based on the equation, but higher N will give......Tue, 01 Sep 2026 17:52:39 -0400https://pop.com.sg/how-to-integrate-cos-x-e-ix-dx.htmlhttps://pop.com.sg/how-to-integrate-cos-x-e-ix-dx.htmlHow to integrate $$\int \cos(t)e^{it}dt $$ I tried integrating by parts twice, but it doesn't work because an i shows up in the end and one gets $0=….$...Tue, 01 Sep 2026 17:49:06 -0400https://pop.com.sg/given-coordinates-of-beginning-and-end-of-two-intersecting-line-segmen.htmlhttps://pop.com.sg/given-coordinates-of-beginning-and-end-of-two-intersecting-line-segmen.htmlThere are two line segments. I know for sure they intersect (so I don't have to check it). For both line segment I know coordinates of its both ends. With what formula can I find coordinates of their ...Tue, 01 Sep 2026 17:43:38 -0400https://pop.com.sg/derivative-of-y-6xe-6x-e-6x.htmlhttps://pop.com.sg/derivative-of-y-6xe-6x-e-6x.htmlCan some one walk me through how to find the derivative of $y=6xe^{6x}-e^{6x}$? I know the answer should be $36e^{6x}x$. I know I am supposed to use the product and chain rule, I just don't ...Tue, 01 Sep 2026 17:39:14 -0400https://pop.com.sg/null-space-of-a-vector.htmlhttps://pop.com.sg/null-space-of-a-vector.htmlAssuming I have a vector $v_1 \in \mathbb{R}^n $ of dimension $n$. (edit what I mean is e.g. $v_1 \in \mathbb{R}^n $) I want to describe the set of "all the vectors that are orthogonal to $v_1$"......Tue, 01 Sep 2026 17:36:58 -0400https://pop.com.sg/how-to-convert-a-vector-to-a-unit-vector-if-its-magnitude-is-smaller-t.htmlhttps://pop.com.sg/how-to-convert-a-vector-to-a-unit-vector-if-its-magnitude-is-smaller-t.htmlIn my javascript project, I need to convert a vector to a unit vector. I know that I can divide its x and y components by its magnitude. But what if the vector's magnitude is smaller than one? How to ...Tue, 01 Sep 2026 17:34:46 -0400https://pop.com.sg/questions-tagged-conic-sections.htmlhttps://pop.com.sg/questions-tagged-conic-sections.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 17:29:27 -0400https://pop.com.sg/two-numbers-two-mathematicians-puzzle.htmlhttps://pop.com.sg/two-numbers-two-mathematicians-puzzle.htmlThis is one of the most beautiful and difficult puzzles I have encountered. I have talked to several people, but I still don't know the solution - I do however know that the solution exists. Someone ...Tue, 01 Sep 2026 17:28:25 -0400https://pop.com.sg/formula-for-calculating-the-total-interest-payable-over-the-life-of-a-.htmlhttps://pop.com.sg/formula-for-calculating-the-total-interest-payable-over-the-life-of-a-.htmlwondering if someone can help a non-mathematician out. I am looking for the formula for calculating the total interest payable over the life of a loan. Given that we know: P: Principal (amount) of ...Tue, 01 Sep 2026 17:25:57 -0400https://pop.com.sg/evaluate-integral-with-fractional-exponents.htmlhttps://pop.com.sg/evaluate-integral-with-fractional-exponents.htmlCan someone please help me integrating this? $$\displaystyle \int \frac{w\mathrm dw}{(5-3w)^{2/3}}$$ I tried substituting $5-3w = u$ and $-3\mathrm dw = \mathrm du$. So $w = (u-5)/3$ and then we ......Tue, 01 Sep 2026 17:24:30 -0400https://pop.com.sg/what-is-the-039-physical-039-explanation-of-a-division-by-a-fraction.htmlhttps://pop.com.sg/what-is-the-039-physical-039-explanation-of-a-division-by-a-fraction.htmlFor example, dividing by 2, means we cut something in two. But dividing by 0.5, can only be explained with multiplying something by 2. So, is there a "physical" explanation of dividing by 0.5? Is......Tue, 01 Sep 2026 17:22:43 -0400https://pop.com.sg/setting-lineard-equation-ode-into-standard-form-proving-existence-of-u.htmlhttps://pop.com.sg/setting-lineard-equation-ode-into-standard-form-proving-existence-of-u.htmlI'm working on proving existence and uniqueness of a local solution to ODE's of the Lienard variety $$y'' + f(y)\,y' + g(y) = 0$$ I'm trying to put this into standard $y' = f(t,y)$ system So I see ...Tue, 01 Sep 2026 17:22:05 -0400https://pop.com.sg/contradiction-proof-of-the-product-of-two-irrational-numbers.htmlhttps://pop.com.sg/contradiction-proof-of-the-product-of-two-irrational-numbers.htmlI am wondering what is wrong with my contradiction proof that "The product of two irrational numbers is irrational". I understand that there are examples where this is not true: $\sqrt{2} * \frac{1}{\...Tue, 01 Sep 2026 17:16:27 -0400https://pop.com.sg/recurrence-for-number-of-words-of-length-r-over-n-with-no-three-consec.htmlhttps://pop.com.sg/recurrence-for-number-of-words-of-length-r-over-n-with-no-three-consec.htmlQuestion Let $b_{n,r}$ be the number words of length $r$ over $[n]=\{1,2,\dotsc, n\}$ with no three consecutive letters the same. Show that $$ b_{n,r}=(n-1)(b_{n,r-1}+b_{n,r-2})\quad (r>2)......Tue, 01 Sep 2026 16:47:48 -0400https://pop.com.sg/showing-that-x-4-4x-3-11x-2-10x-9-is-irreducible-over-mathbb-q.htmlhttps://pop.com.sg/showing-that-x-4-4x-3-11x-2-10x-9-is-irreducible-over-mathbb-q.htmlLet $p(x) = x^4 + 4x^3 + 11x^2 + 10x + 9$. How to how that it is irreducible over $\mathbb{Q}$? The first thing I tried was Eisenstein, but that obviously doesn't work with the polynomial in that ......Tue, 01 Sep 2026 16:46:37 -0400https://pop.com.sg/volume-of-a-hyperbolic-cube-shaped-solid-in-euclidean-space.htmlhttps://pop.com.sg/volume-of-a-hyperbolic-cube-shaped-solid-in-euclidean-space.htmlI'm searching for a formula for the volume (and, if available, also other information like area, etc.) of a so-called "hyperbolic cube": I couldn't find anything in Wikipedia, and also MathWorld's "...Tue, 01 Sep 2026 16:20:59 -0400https://pop.com.sg/does-cos-pi-2-really-equal-zero.htmlhttps://pop.com.sg/does-cos-pi-2-really-equal-zero.htmlI was watching this video and at 5:30 $\cos \frac{\pi}{2}$ is substituted with $0$. Using my calculator the result I got was $-0.5$. Which one is correct?...Tue, 01 Sep 2026 16:14:47 -0400https://pop.com.sg/where-can-one-find-a-list-of-prime-numbers.htmlhttps://pop.com.sg/where-can-one-find-a-list-of-prime-numbers.htmlI am looking for the biggest list of precomputed prime numbers one can find and download. Where should I look?...Tue, 01 Sep 2026 16:10:30 -0400https://pop.com.sg/finding-the-adjoint-of-an-operator.htmlhttps://pop.com.sg/finding-the-adjoint-of-an-operator.htmlThis is from my homework, I'm totally lost as to how to proceed. Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $(Tf)(x) = \int^x_0 f(s) \ ds$ What is the adjoint of $T$? ......Tue, 01 Sep 2026 16:06:55 -0400https://pop.com.sg/since-is-a-subset-of-every-set-it-is-a-subset-of-itself-and-hence-dupl.htmlhttps://pop.com.sg/since-is-a-subset-of-every-set-it-is-a-subset-of-itself-and-hence-dupl.htmlI am bit confused with the concept of empty set here. Since {} is a subset of every set, it is a subset of itself? and hence {} = {{}}? Also, say A = {a}, but since {} is a subset of A, is it tru......Tue, 01 Sep 2026 15:59:24 -0400https://pop.com.sg/decomposition-of-a-biquadratic-polynomial.htmlhttps://pop.com.sg/decomposition-of-a-biquadratic-polynomial.htmlLet $f(t):=T^4+aT^2+b$ be a polynomial in a field $K[T]$ such that it can be decomposed in two irreducible factors of degree 2. My intuition then is screaming to me that then we neccessarily have ......Tue, 01 Sep 2026 15:54:49 -0400https://pop.com.sg/a-general-formula-for-n-distinguishable-arrangements-of-length-ell-of-.htmlhttps://pop.com.sg/a-general-formula-for-n-distinguishable-arrangements-of-length-ell-of-.htmlIs there a general formula for the number of distinguishable arrangements of length $\ell$ of a sequence of size $s$, given the repeated elements? So, For example, would there be a general formula......Tue, 01 Sep 2026 15:49:37 -0400https://pop.com.sg/what-is-the-cardinality-of-powerset-of-a-union-set.htmlhttps://pop.com.sg/what-is-the-cardinality-of-powerset-of-a-union-set.htmlIs there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), a......Tue, 01 Sep 2026 15:46:40 -0400https://pop.com.sg/31-331-3331-33331-333331-3333331-33333331-are-prime.htmlhttps://pop.com.sg/31-331-3331-33331-333331-3333331-33333331-are-prime.html31,331,3331, 33331,333331,3333331,33333331 are prime. This law can continue it? Will there emerge a composite number? Without using a computer how to judge....Tue, 01 Sep 2026 15:40:09 -0400https://pop.com.sg/2015-amc-10b-problem-21.htmlhttps://pop.com.sg/2015-amc-10b-problem-21.htmlThe problem and solutions I've attempted to solve another AMC 10 problem, and the problem is basically like this: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of......Tue, 01 Sep 2026 15:36:11 -0400https://pop.com.sg/continuum-between-addition-multiplication-and-exponentiation.htmlhttps://pop.com.sg/continuum-between-addition-multiplication-and-exponentiation.htmlI noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we f......Tue, 01 Sep 2026 15:32:32 -0400https://pop.com.sg/when-is-991n-2-1-a-perfect-square.htmlhttps://pop.com.sg/when-is-991n-2-1-a-perfect-square.htmlWhat should be the value of $n$ so that the number obtained after adding $1$ to $991$ times its square is itself a perfect square? Can you please give me a few hints on this topic with a few speci......Tue, 01 Sep 2026 15:28:15 -0400https://pop.com.sg/number-of-edges-in-n-cube.htmlhttps://pop.com.sg/number-of-edges-in-n-cube.htmlIf the number of edges in an $n$-cube (also denoted as $Q_n$) is $k$, what is the number of edges in an $(n+1)$-cube (denoted as $Q_{n+1}$)? Express your answer in terms of $k$ and $n$. How can I ......Tue, 01 Sep 2026 15:18:48 -0400https://pop.com.sg/how-to-show-that-a-cauchy-sequence-of-sequences-converges.htmlhttps://pop.com.sg/how-to-show-that-a-cauchy-sequence-of-sequences-converges.htmlLet $c_0$ be the space of real-valued sequences $\{x_n\}$ which converge to zero, equipped with the metric $d(\{x_n\}, \{y_n\}) = sup_n |x_n − y_n|$. I want to show that the metric space $(c_0, d)$......Tue, 01 Sep 2026 15:09:48 -0400https://pop.com.sg/sequence-with-no-accumulation-point.htmlhttps://pop.com.sg/sequence-with-no-accumulation-point.htmlConstruct an example of a set of real numbers E that has no points of accumulation and yet has the property that for every ε > 0 there exist points x, y ∈ E so that 0 < |x − y| < ε. so i kno......Tue, 01 Sep 2026 15:05:04 -0400https://pop.com.sg/radical-and-radical-ideal.htmlhttps://pop.com.sg/radical-and-radical-ideal.htmlI am trying to get used to the terminology of commutative algebra. Let $R$ be a commutative ring with unity. Then I know radical $\sqrt{I} := \{ a\in R| a^n \in I, \textrm{for some $n \in \mathbb{Z......Tue, 01 Sep 2026 15:03:57 -0400https://pop.com.sg/in-percent-change-why-do-we-divide-the-difference-by-the-original-numb.htmlhttps://pop.com.sg/in-percent-change-why-do-we-divide-the-difference-by-the-original-numb.htmlIf $a = 90$ and $b = 100$, and we are calculating the percent increase going from $a$ to $b$, we calculate the actual numerical difference $b - a = 10$. But why divide the difference into the star......Tue, 01 Sep 2026 15:02:13 -0400https://pop.com.sg/laplace-transform-of-the-square-of-first-derivative.htmlhttps://pop.com.sg/laplace-transform-of-the-square-of-first-derivative.htmlI want to compute the laplace transformation of this equation: $$4 -0.1{\operatorname{d}\!x(t)\over\operatorname{d}\!t} - 0.01\left({\operatorname{d}\!x(t)\over\operatorname{d}\!t}\right)^2 = 0.9{\...Tue, 01 Sep 2026 15:01:59 -0400https://pop.com.sg/how-to-calculate-a-b-bmod-n-and-b-bmod-n-closed.htmlhttps://pop.com.sg/how-to-calculate-a-b-bmod-n-and-b-bmod-n-closed.htmlConsider the following expression: (a - b) mod N Which of the following is equivalent to the above expression? 1) ((a mod N) + (-b mod N)) mod N 2) ((a mod N) - (b mod N)) mod N Also, how is (......Tue, 01 Sep 2026 14:59:43 -0400https://pop.com.sg/computing-volume-by-cross-section-method.htmlhttps://pop.com.sg/computing-volume-by-cross-section-method.htmlThe base of the solid below is the region in the $xy$-plane bounded by the $x$-axis,the graph of y = $\sqrt{x}$ and the line $ x = 3 $. Find the volume of the solid. Each cross-section of S ...Tue, 01 Sep 2026 14:56:17 -0400https://pop.com.sg/how-do-i-calculate-the-area-of-a-curved-irregular-plane-such-as-an-aer.htmlhttps://pop.com.sg/how-do-i-calculate-the-area-of-a-curved-irregular-plane-such-as-an-aer.htmlSo I need to calculate the wing area of an airbus a350 and all I have is this blueprints. Thought about searching functions that fit the wing from the top perspective and then integrate, but I dont......Tue, 01 Sep 2026 14:54:13 -0400https://pop.com.sg/how-long-will-it-take-to-learn-algebra-1-2-geometry-and-pre-calculus-c.htmlhttps://pop.com.sg/how-long-will-it-take-to-learn-algebra-1-2-geometry-and-pre-calculus-c.htmlI've previously learned algebra (wasn't the best, wasn't the worst) and have since forgotten most (basically all) of it. I'm currently homeschooled and I would like to go to college in two years. D......Tue, 01 Sep 2026 14:50:58 -0400https://pop.com.sg/how-the-jacobian-of-the-transformation-can-be-shown-to-not-depend-on-x.htmlhttps://pop.com.sg/how-the-jacobian-of-the-transformation-can-be-shown-to-not-depend-on-x.htmlTransform the random variables, $X_i$, $i=1,2,\ldots,n$ to $$ \begin{align} Y_1 & =\bar X \\ Y_2 & =X_2-\bar X \\ Y_3 & = X_3-\bar X \\ & {}\ \vdots \\ Y_n & =X_n-\bar X \end{......Tue, 01 Sep 2026 14:32:46 -0400https://pop.com.sg/probability-of-winning-twice-on-a-wheel-of-fortune.htmlhttps://pop.com.sg/probability-of-winning-twice-on-a-wheel-of-fortune.htmlA wheel of fortune is divided into 40 sectors, numbered from 1 to 40. Tickets are sold representing each sector. Tickets are \$1 each. All 40 tickets must be sold before the wheel can be spun. Only......Tue, 01 Sep 2026 14:30:29 -0400https://pop.com.sg/probability-question-with-black-and-white-marbles.htmlhttps://pop.com.sg/probability-question-with-black-and-white-marbles.htmlJust looking for some help with this question. The answer seems obvious but I am questioning myself as it seems a little too obvious. Max has a bag containing 2 black marbles and 2 white marbles.......Tue, 01 Sep 2026 14:25:39 -0400https://pop.com.sg/why-are-rings-called-rings.htmlhttps://pop.com.sg/why-are-rings-called-rings.htmlI've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks....Tue, 01 Sep 2026 14:22:22 -0400https://pop.com.sg/how-to-solve-a-quintic-polynomial-equation.htmlhttps://pop.com.sg/how-to-solve-a-quintic-polynomial-equation.htmlI know that not all quintics are solvable. But how do I identify the class of solvable ones?...Tue, 01 Sep 2026 14:22:00 -0400https://pop.com.sg/guess-the-value-of-the-limit.htmlhttps://pop.com.sg/guess-the-value-of-the-limit.htmlSimplify and guess the value of $\lim_{x\to 2}\frac {x^3-8}{x-2}$. I'm very confused on how to do this. Please help!...Tue, 01 Sep 2026 14:18:31 -0400https://pop.com.sg/what-does-the-phrase-except-possibly-mean.htmlhttps://pop.com.sg/what-does-the-phrase-except-possibly-mean.htmlThis term was used by my math professor when he was teaching us limits. This term also appears online like this "Let f be a function which is defined on some open interval containing $a$ except pos......Tue, 01 Sep 2026 14:11:41 -0400https://pop.com.sg/finding-angle-in-an-equilateral-triangular-pyramid.htmlhttps://pop.com.sg/finding-angle-in-an-equilateral-triangular-pyramid.htmlGiven an equilateral triangular pyramid (refer the below diagram) $\Delta ABC$ & $P$ is any point inside the triangle such that ${PA}^{2}={PB}^{2}+{PC}^{2}$, then $\angle BPC$ is - I am unable......Tue, 01 Sep 2026 14:00:36 -0400https://pop.com.sg/how-to-tell-if-3-connected-points-are-connected-clockwise-or-counter-c.htmlhttps://pop.com.sg/how-to-tell-if-3-connected-points-are-connected-clockwise-or-counter-c.htmlI have three points: p1(x1,y1) p2(x2,y2) and p3(x3,y3). I am connecting p1 to p2 to p3. how can I tell if the triangle was drawn clockwise or counter-clockwise? How can I generalize this for n po......Tue, 01 Sep 2026 13:49:49 -0400https://pop.com.sg/finately-supported-sequences-are-dense-in-l-p.htmlhttps://pop.com.sg/finately-supported-sequences-are-dense-in-l-p.htmlI want to prove that the set of finitely supported sequences $$s=\{(x_1,x_2,\dots):\exists N>0 \;st\; x_{n\geq N}=0\}$$ is dense in $l^p$, $$l^p = \{ (x_1,x_2,\dots) : \;\;x_i\in\mathbb{K} \;an......Tue, 01 Sep 2026 13:48:15 -0400https://pop.com.sg/cube-roots-of-complex-numbers-closed.htmlhttps://pop.com.sg/cube-roots-of-complex-numbers-closed.htmlI need help with finding the cube roots of the complex number 27... I know that the obvious answer is three, but what is the less simple method to solving this?...Tue, 01 Sep 2026 13:42:29 -0400https://pop.com.sg/method-of-isoclines.htmlhttps://pop.com.sg/method-of-isoclines.htmlI have this exercise and I do not know how to solve it. By using the method of isoclines represent the integrals of equation corbes nonautonomous $x'=x^2-t$. There are some indications: Let $P ......Tue, 01 Sep 2026 13:34:29 -0400https://pop.com.sg/finding-x-intercepts-and-coordinates-of-the-vertex.htmlhttps://pop.com.sg/finding-x-intercepts-and-coordinates-of-the-vertex.htmlThe following diagram shows part of graph of f, where $f(x) = x^2 - x - 2$. Find both x-intercepts. For this, $f(x)$ means $y$, should I set $y$ to 0 and solve for the $x$ values? $0 = x^2 - x ......Tue, 01 Sep 2026 13:28:01 -0400https://pop.com.sg/when-to-use-cumulative-moving-average-vs-a-simple-moving-average.htmlhttps://pop.com.sg/when-to-use-cumulative-moving-average-vs-a-simple-moving-average.htmlAfter reviewing the Wikipedia page on moving averages, the difference between the simple moving average and cumulative moving average are clear: 1) Simple moving average only considers the last n ...Tue, 01 Sep 2026 13:22:42 -0400https://pop.com.sg/coefficients-of-quadratic-function.htmlhttps://pop.com.sg/coefficients-of-quadratic-function.htmlIn a quadratic function: coefficient $a$ controls the speed of increase/decrease from the vertex. coefficient $b$ controls the downward slope as the function crosses the y-axis. I don't real......Tue, 01 Sep 2026 13:07:46 -0400https://pop.com.sg/prove-that-this-integral-operator-is-compact.htmlhttps://pop.com.sg/prove-that-this-integral-operator-is-compact.htmlLet $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $......Tue, 01 Sep 2026 13:05:21 -0400https://pop.com.sg/radius-of-convergence-of-power-series-or-geometric-series.htmlhttps://pop.com.sg/radius-of-convergence-of-power-series-or-geometric-series.htmlTo find radius of convergence of geometric series $$\sum_{n=1}^\infty a_n$$ I need to use ratio/root test to find $|L|&lt1$ To find radius of convergence of power series $$\sum_{n=1}^\infty c_n (x......Tue, 01 Sep 2026 13:00:09 -0400https://pop.com.sg/is-there-a-deeper-meaning-when-a-number-is-squared-closed.htmlhttps://pop.com.sg/is-there-a-deeper-meaning-when-a-number-is-squared-closed.htmlIn my opinion, math is about more than just memorizing equations, it's about numbers that are built in a way that represents our understanding of something. So I ask this, what does it mean intuit......Tue, 01 Sep 2026 12:58:31 -0400https://pop.com.sg/maclaurin-series-for-arctan-x-by-successive-differentiation.htmlhttps://pop.com.sg/maclaurin-series-for-arctan-x-by-successive-differentiation.htmlI am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...Tue, 01 Sep 2026 12:56:04 -0400https://pop.com.sg/the-size-of-the-biggest-square-that-can-be-inscribed-within-a-circle.htmlhttps://pop.com.sg/the-size-of-the-biggest-square-that-can-be-inscribed-within-a-circle.html(a) Is the size the biggest square that can be fit inside a circle always $2r^2$? (b) How do you do when you show that is indeed the case? I want to compare my boyfriend's proofs with that of o......Tue, 01 Sep 2026 12:48:59 -0400https://pop.com.sg/what-is-the-affine-space-and-what-is-it-for.htmlhttps://pop.com.sg/what-is-the-affine-space-and-what-is-it-for.htmlThese two topics already exist: (preface: got in contact with affine space through computer graphics subject in university) What are affine spaces for? What are differences between affine space ......Tue, 01 Sep 2026 12:34:36 -0400https://pop.com.sg/x-axis-of-a-linearized-polynomial.htmlhttps://pop.com.sg/x-axis-of-a-linearized-polynomial.htmlThe other day in my Physics class we had some sample data that we wanted to linearize. The graph resembled a root curve. So to linearize it, we took the square root of all the x data and replotte......Tue, 01 Sep 2026 12:28:58 -0400https://pop.com.sg/how-to-expand-cos-nx-with-cos-x.htmlhttps://pop.com.sg/how-to-expand-cos-nx-with-cos-x.htmlMultiple Angle Identities: How to expand $\cos nx$ with $\cos x$, such as $$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$ See a list of trigonometric ...Tue, 01 Sep 2026 12:18:06 -0400https://pop.com.sg/similar-matrices-how-do-you-find-them.htmlhttps://pop.com.sg/similar-matrices-how-do-you-find-them.htmlMy lecture notes write that two matrices are said to be similar if there exists an $n \times n$ invertible matrix $P$ such that $B = P^{-1} AP$. But suppose I am given two matrices, $A$ and $B$, ......Tue, 01 Sep 2026 12:03:29 -0400https://pop.com.sg/taking-partial-derivative-via-fourier-transform-using-matlab.htmlhttps://pop.com.sg/taking-partial-derivative-via-fourier-transform-using-matlab.htmlFor a task for university, I need to calculate in matlab a partial derivative using the 2D Fourier transform. Using equation (13) on pg 3 in http://www2.ph.ed.ac.uk/~wjh/teaching/Fourier/documents/...Tue, 01 Sep 2026 12:02:25 -0400https://pop.com.sg/why-can-natural-logarithm-receive-negative-x-in-the-context-of-derivat.htmlhttps://pop.com.sg/why-can-natural-logarithm-receive-negative-x-in-the-context-of-derivat.htmlIn our class we had this formula for $x<0$: $$ \ln(-x)' = \frac{1}{-x} \cdot (-x)' = \frac{1}{x} $$ The conclusion is that: $$ \int\frac{1}{x}dx = \ln|x| + c, x \neq 0 $$ I don't understand how......Tue, 01 Sep 2026 11:47:20 -0400https://pop.com.sg/formula-and-percentages.htmlhttps://pop.com.sg/formula-and-percentages.htmlI need help making a formula. Here is the basic concept: I have 1 twenty-sided die (from now on 1d20) and another 16-sided die (1d16). I roll each, and the die which displays the lowest number "win......Tue, 01 Sep 2026 11:32:29 -0400https://pop.com.sg/finding-non-closed-paths-so-that-a-line-integral-equals-certain-values.htmlhttps://pop.com.sg/finding-non-closed-paths-so-that-a-line-integral-equals-certain-values.htmlSo I'm required to find a non-closed path $C$ so that the line integral $$\displaystyle \int_C \mathbf{F} \cdot \mathrm d \mathbf{r}$$ equals $0$, and another non-closed path $C$ so that the line ...Tue, 01 Sep 2026 11:24:12 -0400https://pop.com.sg/what-are-the-length-of-the-longest-element-in-a-coexter-group-for-ever.htmlhttps://pop.com.sg/what-are-the-length-of-the-longest-element-in-a-coexter-group-for-ever.htmlWhat are the length of the longest element in a Coxeter group for every type? Thank you very much....Tue, 01 Sep 2026 11:14:02 -0400https://pop.com.sg/has-anyone-ever-attempted-to-find-all-splits-of-a-rectangle-into-small.htmlhttps://pop.com.sg/has-anyone-ever-attempted-to-find-all-splits-of-a-rectangle-into-small.htmlI'm enamored by the games Unproportional and Unproportional2. The only problem is that they have way too few levels. :) But, I'm a computer programmer, so I could make my own! With blackjack and ho......Tue, 01 Sep 2026 11:13:37 -0400https://pop.com.sg/composition-of-even-and-odd-functions-and-their-outcome.htmlhttps://pop.com.sg/composition-of-even-and-odd-functions-and-their-outcome.htmlGive an example of an even function. Give an example of an odd function. If f(x) is odd and g(x) is even, must f(g(x)) be even? Must g(f(x)) be even? I've tried generic functions like f(x) = x^3......Tue, 01 Sep 2026 11:06:20 -0400https://pop.com.sg/determining-whether-matrix-is-self-adjoint.htmlhttps://pop.com.sg/determining-whether-matrix-is-self-adjoint.htmlI have three matrices and I am trying to determine if they represent a self-adjoint linear transformation in some basis on an inner product space. The matrices are: $$A_1 = \left( \begin{array}{c......Tue, 01 Sep 2026 11:04:02 -0400https://pop.com.sg/half-full-cone-of-water.htmlhttps://pop.com.sg/half-full-cone-of-water.htmlI have been battling with this for a little while now, but can't really get my head around it. The question goes: how high must you pour water into a cone before it is half-full (by volume)? No ...Tue, 01 Sep 2026 11:02:42 -0400https://pop.com.sg/how-to-compute-these-riemann-stieltjes-integrals.htmlhttps://pop.com.sg/how-to-compute-these-riemann-stieltjes-integrals.htmlI have some doubt on this exercise which I have to solve: Compute The Integral $\displaystyle \int_0^4 F(x) \ \mathrm{d}G(x)$ for 1.) $F(x) = x$ for $x<2$, $1$ otherwise; $G(x)=x^2$ 2.) $F(x)......Tue, 01 Sep 2026 10:57:46 -0400https://pop.com.sg/finding-prime-implicants-and-essential-prime-implicants-for-boolean-fu.htmlhttps://pop.com.sg/finding-prime-implicants-and-essential-prime-implicants-for-boolean-fu.htmlI am trying to solve a EE problem and am unsure whether I doing it correctly. The problem is: Find all the prime implicants for the following Boolean functions, and determine which are essential: ......Tue, 01 Sep 2026 10:55:20 -0400https://pop.com.sg/what-does-f-039-and-f-039-039-mean.htmlhttps://pop.com.sg/what-does-f-039-and-f-039-039-mean.htmlI'm trying to learn what a Taylor series is, This is the equation I'm looking at and I know 0 calculus. I have been told that $F'(x)$ is a derivative but what does $F''(x)$ mean?...Tue, 01 Sep 2026 10:38:18 -0400https://pop.com.sg/proving-aoc-axiom-of-completeness-using-nip-nested-interval-property.htmlhttps://pop.com.sg/proving-aoc-axiom-of-completeness-using-nip-nested-interval-property.htmlNIP $\implies$ AoC. $\newcommand{\absval}[1]{\left\lvert #1 \right\rvert}$ Exercise 2.5.4 from Understanding Analysis by Abott. The Bolzanno-Weierstrass theorem is extremely important, and so is the ...Tue, 01 Sep 2026 10:38:06 -0400https://pop.com.sg/why-do-2-log-x-and-log-x-2-look-different-when-graphing-duplicate.htmlhttps://pop.com.sg/why-do-2-log-x-and-log-x-2-look-different-when-graphing-duplicate.htmlI understand how both graphs are drawn, but I do not understand why you cannot just convert one into another. It feels natural to me to just convert $2\log x$ to $\log x^2$ and not have to worry ab......Tue, 01 Sep 2026 10:37:02 -0400https://pop.com.sg/second-order-sde-solutions.htmlhttps://pop.com.sg/second-order-sde-solutions.htmlI have the following stochastic process: $$\ddot x(t)=a(t)+\nu(t)$$ where $a(t)$ is a deterministic function of the time and $\nu(t)$ is a random variable: $$\nu(t)\in\mathcal{N}(0,\sigma^2)$$ Obvi......Tue, 01 Sep 2026 10:31:35 -0400