https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 29 Aug 2026 09:44:52 -0400https://pop.com.sg/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-p.htmlhttps://pop.com.sg/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-p.htmlIt’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an a......Sat, 29 Aug 2026 13:39:39 -0400https://pop.com.sg/analysis-determine-radius-and-interval-of-convergence-for-these-power-.htmlhttps://pop.com.sg/analysis-determine-radius-and-interval-of-convergence-for-these-power-.htmlI needed to determine the radius and interval of convergence for each of the following; $\mathbb{One:}$ $$\sum_{n=0}^{\infty}\frac{3^nx^n}{n!}$$ $\mathbb{Two:}$ $$\sum_{n=0}^{\infty}\frac{x^n}{(1......Sat, 29 Aug 2026 13:24:29 -0400https://pop.com.sg/how-to-find-the-length-of-the-line-segment-of-this-following-triangle.htmlhttps://pop.com.sg/how-to-find-the-length-of-the-line-segment-of-this-following-triangle.htmlGiven: right triangle ABC with C as a right angle. Line DE is formed, such that ED is perpendicular to AB. D at AB and E at CB. If AC=AD= 8 cm and CE=DE, what is the length of BE? My thinking(sin......Sat, 29 Aug 2026 13:18:28 -0400https://pop.com.sg/example-of-a-quantifier-elimination-procedure-for-a-simple-but-nontriv.htmlhttps://pop.com.sg/example-of-a-quantifier-elimination-procedure-for-a-simple-but-nontriv.htmlIs there a simple-but-nontrivial example of a concrete quantifier elimination procedure with a concrete theory, especially one that's a standard example of what a constructive argument for quantifier ...Sat, 29 Aug 2026 13:09:33 -0400https://pop.com.sg/spherical-triangle-given-two-angles-and-one-length-find-another-length.htmlhttps://pop.com.sg/spherical-triangle-given-two-angles-and-one-length-find-another-length.htmlLet $[u,v,w]$ be a positively oriented spherical triangle in $S^2$ If $a=\cos^{-1}(\frac13)$, $\beta = \frac{2\pi}{3}$, $\gamma = \frac{\pi}{3}$, find $b$. We have that: $a=\cos^{-1}(\frac13)\imp......Sat, 29 Aug 2026 13:09:01 -0400https://pop.com.sg/ratios-make-me-feel-like-an-idiot-help-me-mix-up-some-coca-cola.htmlhttps://pop.com.sg/ratios-make-me-feel-like-an-idiot-help-me-mix-up-some-coca-cola.htmlI may be over-complicating things, but something doesn't seem right (and I swear this isn't homework, I'm friggin 30 years old :P). I want to see what it will cost me to make a TWO liter of Coca-C......Sat, 29 Aug 2026 13:04:07 -0400https://pop.com.sg/what-is-wrong-with-this-solution-of-2x-ln-2-3y-ln-3.htmlhttps://pop.com.sg/what-is-wrong-with-this-solution-of-2x-ln-2-3y-ln-3.htmlWhat is wrong with this solution of Solve for $ (2x)^{\ln2 }= (3y)^{\ln3 }$ $3^{\ln x} = 2^{\ln y} $ Here is my solution- $ (2x)^{\ln2 }= (3y)^{\ln3 }$ $ 2^{\ln2x }= 3^{\ln3y }$ $ 2^{(\ln ......Sat, 29 Aug 2026 12:56:41 -0400https://pop.com.sg/proof-of-the-associative-law-and-the-commutative-law.htmlhttps://pop.com.sg/proof-of-the-associative-law-and-the-commutative-law.htmlThe associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily. We can prove$^2$ the ...Sat, 29 Aug 2026 12:40:53 -0400https://pop.com.sg/if-ax-2-bx-7-15-x-2-cx-14-for-all-values-of-x-and-a-b-8-what-are-the-t.htmlhttps://pop.com.sg/if-ax-2-bx-7-15-x-2-cx-14-for-all-values-of-x-and-a-b-8-what-are-the-t.htmlPlease solve the following problem and show your work: If $(ax + 2)( bx + 7) = 15 x^2 + cx + 14$ for all values of $x$, and $a + b =8$, what are the two possible values for $c$ ?...Sat, 29 Aug 2026 12:25:09 -0400https://pop.com.sg/maclaurin-series-from-sin-x-to-cos-x-using-derivative.htmlhttps://pop.com.sg/maclaurin-series-from-sin-x-to-cos-x-using-derivative.htmlI understand how to find the MacLaurin series for $\sin(x)$. $$\sum_{n=1}^\infty \frac{x^{2n-1}\cdot\!(-1)^{n-1}}{(2n-1)!}$$ Now I am trying to find the MacLaurin series for $\cos(x)$ by taking the ...Sat, 29 Aug 2026 12:22:48 -0400https://pop.com.sg/is-1e-005-1e-5.htmlhttps://pop.com.sg/is-1e-005-1e-5.htmlTotally new to Matlab, I encountered the notation $1e-005$ without proper context to guess its meaning. After googling, I found that it may represent the number $0.00001$. However, I know that $1e-5$ ...Sat, 29 Aug 2026 12:13:39 -0400https://pop.com.sg/how-to-solve-x-log-x-10-6.htmlhttps://pop.com.sg/how-to-solve-x-log-x-10-6.htmlI am trying to solve $$x\log(x) = 10^6$$ but can't find an elegant solution. Any ideas ?...Sat, 29 Aug 2026 12:04:14 -0400https://pop.com.sg/is-there-a-difference-between-theorem-proof-and-theorem-demonstration.htmlhttps://pop.com.sg/is-there-a-difference-between-theorem-proof-and-theorem-demonstration.htmlWhen referring to proving that a theorem is true, I have seen both words used interchangeably. Do they mean the same thing ?...Sat, 29 Aug 2026 12:01:57 -0400https://pop.com.sg/why-is-cos135-circ-negative-when-length-is-always-positive.htmlhttps://pop.com.sg/why-is-cos135-circ-negative-when-length-is-always-positive.htmlConsider the following diagram. I am told that $\cos 45^{\circ}$ = $\frac{1}{\sqrt 2}$. I understand this. I am next told taught that $\cos 135^{\circ}$ = $\cos 45^{\circ}$ in 2nd quadrant. And f......Sat, 29 Aug 2026 11:59:11 -0400https://pop.com.sg/what-is-a-symbolic-expression-to-calculate-height-of-a-triangle-from-t.htmlhttps://pop.com.sg/what-is-a-symbolic-expression-to-calculate-height-of-a-triangle-from-t.htmlGiven one side (a), which we'll call the 'base', and it's adjacent angles (B, C), calculate the 'height' of a triangle. In this case 'height' means the distance from the opposite vertex of the give......Sat, 29 Aug 2026 11:44:54 -0400https://pop.com.sg/evaluate-the-limit-of-f-2-h-f-2-h-as-h-approaches-0-for-f-x-sin-x.htmlhttps://pop.com.sg/evaluate-the-limit-of-f-2-h-f-2-h-as-h-approaches-0-for-f-x-sin-x.htmlIf $f(x)=\sin x$, evaluate $\displaystyle\lim_{h \to 0} \frac{f(2+h) - f(2)}{h}$ to two decimal places. Have tried to answer in many different ways, but always end up getting confused along the w......Sat, 29 Aug 2026 11:36:22 -0400https://pop.com.sg/decompose-a-vector-in-a-part-parallel-to-a-plane-and-other-to-a-line.htmlhttps://pop.com.sg/decompose-a-vector-in-a-part-parallel-to-a-plane-and-other-to-a-line.htmlI have the problem: Decompose the vector $\vec v = (1,2,4)$ in two parts, one parallel to the plane $X = (1,1,0) + \lambda(1,0,1) + \gamma(0,1,-1)$ and other parallel to the line $X = (0,0,0) + h(......Sat, 29 Aug 2026 11:33:28 -0400https://pop.com.sg/in-a-triangle-does-an-angle-bisector-necessarily-bisect-the-opposite-s.htmlhttps://pop.com.sg/in-a-triangle-does-an-angle-bisector-necessarily-bisect-the-opposite-s.htmlHave a look at the triangle and tell me if $AD=DB$:...Sat, 29 Aug 2026 11:25:31 -0400https://pop.com.sg/precalculus-angular-velocity-and-linear-velocity-closed.htmlhttps://pop.com.sg/precalculus-angular-velocity-and-linear-velocity-closed.htmlCan someone help me please? I don't really understand Angular and Linear Velocity. Thank you! A back wheel on a tricycle has a radius of 8cm and rotates at a rate of 200 times per minute. Approxim......Sat, 29 Aug 2026 11:20:38 -0400https://pop.com.sg/find-the-inflection-points-in-the-graph.htmlhttps://pop.com.sg/find-the-inflection-points-in-the-graph.htmlThe question is which of the $x$-values of the given points are inflection points of the function $f(x)$ itself? I chose $C,F$ and $H$ because at this point the $f'(x)$ is zero. But my answer was......Sat, 29 Aug 2026 11:17:01 -0400https://pop.com.sg/expectation-of-an-exponential-function.htmlhttps://pop.com.sg/expectation-of-an-exponential-function.htmlWhy is the expectation of an exponential function: $$\mathbb{E}[\exp(A x)] = \exp((1/2) A^2)\,?$$ I am struggling to find references that shows this, can anyone help me please? If anyone could ...Sat, 29 Aug 2026 11:12:54 -0400https://pop.com.sg/properties-of-the-number-137.htmlhttps://pop.com.sg/properties-of-the-number-137.htmlThe number $137$ is a prime number.One of the permutation of $137$ is $173$, which is a prime number. The summation of $137$ digits is $11$ which is again a prime number! My question: Is there a ......Sat, 29 Aug 2026 11:04:36 -0400https://pop.com.sg/do-asymptotes-disprove-0-9-repeating-equal-1.htmlhttps://pop.com.sg/do-asymptotes-disprove-0-9-repeating-equal-1.htmlI am in 9th grade and taking geometry. Several of my friends taking pre-calc say that 0.9999... does not equal 1, but is just an asymptote. I have not taken that subject yet and they don't give any ...Sat, 29 Aug 2026 11:02:34 -0400https://pop.com.sg/probability-of-p-a-b-given-p-a-p-b.htmlhttps://pop.com.sg/probability-of-p-a-b-given-p-a-p-b.htmlGiven $P(A)=0.9$ and $P(B)=0.8$. How to prove that $P(A|B)\ge 0.875$ ?...Sat, 29 Aug 2026 10:55:54 -0400https://pop.com.sg/families-of-square-roots-of-identity-matrices.htmlhttps://pop.com.sg/families-of-square-roots-of-identity-matrices.htmlI just analysed this equation for real matrices $$ A^2=\begin{pmatrix}a&b\\c&d\end{pmatrix}^2=I $$ From the main diagonal of $A^2$ we must have $a^2+bc=bc+d^2=1$ showing that $d=\pm a$. CA......Sat, 29 Aug 2026 10:49:12 -0400https://pop.com.sg/find-o-b-if-aba-1-b-2-and-given-that-a-5-e.htmlhttps://pop.com.sg/find-o-b-if-aba-1-b-2-and-given-that-a-5-e.htmlIf in a group $G$, $a^5=e$, $aba^{-1}=b^2$ for some $a,b\in{G}$. Find $o(b)$. I wrote $aba^{-1}=b^2$ as $ab=b^2a$. Then $(ab)^5=(b^2a)^5$ but then I am stucked up....Sat, 29 Aug 2026 10:42:43 -0400https://pop.com.sg/laplace-transform-of-f-t-t-closed.htmlhttps://pop.com.sg/laplace-transform-of-f-t-t-closed.htmlFor a function $f(t)$ Laplace transform is defined as $F(s)=\int_0^{\infty} f(t)e^{-st}dt$. I have to show the property that the Laplace transform of $f(t)\over t$ is $\int _s^\infty F(s')ds'$. ......Sat, 29 Aug 2026 10:36:05 -0400https://pop.com.sg/real-life-usage-of-benford-039-s-law.htmlhttps://pop.com.sg/real-life-usage-of-benford-039-s-law.htmlI recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great....Sat, 29 Aug 2026 10:33:47 -0400https://pop.com.sg/is-sin-x-sin-x-1.htmlhttps://pop.com.sg/is-sin-x-sin-x-1.htmlI tested this in python using: import numpy as np import matplotlib.pyplot as plt x = np.linspace(0, 10*2*np.pi, 10000) y = np.sin(x) plt.plot(y/y) plt.plot(y) Which produces: The blue line ...Sat, 29 Aug 2026 10:33:26 -0400https://pop.com.sg/how-do-you-find-the-range-of-a-given-function-is-there-a-process-to-fo.htmlhttps://pop.com.sg/how-do-you-find-the-range-of-a-given-function-is-there-a-process-to-fo.htmlI have never quite understood how to evaluate domains and ranges of functions. I specifically have trouble with ranges. For simple ones, like quadratic equations, I can usually find the minimum poi......Sat, 29 Aug 2026 10:30:15 -0400https://pop.com.sg/simple-sum-of-finite-exponential-series.htmlhttps://pop.com.sg/simple-sum-of-finite-exponential-series.htmlImagine I have a series such as: $$(1 + n)^0 + (1 + n)^1 + (1 + n)^2 + (1 + n)^3 + ... (1+n)^x$$ Is there a simple way to summarize this as a function of $n$ and $x$?...Sat, 29 Aug 2026 10:25:40 -0400https://pop.com.sg/find-area-of-a-seven-point-star-inscribed-in-a-circle-of-radius-1-clos.htmlhttps://pop.com.sg/find-area-of-a-seven-point-star-inscribed-in-a-circle-of-radius-1-clos.htmlI need to know area of the 7 point star and radius of the circle is 1....Sat, 29 Aug 2026 10:20:11 -0400https://pop.com.sg/constrained-parameters-in-least-square-curve-fitting.htmlhttps://pop.com.sg/constrained-parameters-in-least-square-curve-fitting.htmlI have some data points that need to be fit to the curve defined by $$y(x)=\frac{k}{(x+a)^2} - b$$ I have considered that it can be done by the least squares method. However, the analytical solut......Sat, 29 Aug 2026 10:14:12 -0400https://pop.com.sg/what-is-the-closest-fraction-that-isn-039-t-something-like-31415-1000-.htmlhttps://pop.com.sg/what-is-the-closest-fraction-that-isn-039-t-something-like-31415-1000-.htmlI'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For ex......Sat, 29 Aug 2026 10:13:15 -0400https://pop.com.sg/linear-transformation-of-normal-distribution.htmlhttps://pop.com.sg/linear-transformation-of-normal-distribution.htmlNot sure if "linear transformation" is the correct terminology, but... Let $X$ be a random variable with a normal distribution $f(x)$ with mean $\mu_{X}$ and standard deviation $\sigma_{X}$: $$f(x......Sat, 29 Aug 2026 10:08:39 -0400https://pop.com.sg/approximations-involving-exponential-functions.htmlhttps://pop.com.sg/approximations-involving-exponential-functions.htmlI am reading a text and I am curious to know how certain approximations were reached. The first function approximations is: $$ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)......Sat, 29 Aug 2026 10:04:21 -0400https://pop.com.sg/why-only-compass-and-straightedge.htmlhttps://pop.com.sg/why-only-compass-and-straightedge.htmlI've read and watched some lectures on euclidean geometry - not so advanced but I've seen the focus on constructions. Two instruments are used, compass and straightedge, I had the following doubts:......Sat, 29 Aug 2026 10:03:22 -0400https://pop.com.sg/a-prize-of-27-000-is-to-be-divided-among-three-people-in-the-ratio-3-5.htmlhttps://pop.com.sg/a-prize-of-27-000-is-to-be-divided-among-three-people-in-the-ratio-3-5.htmlThis is not homework; I was just reviewing some old math flash cards and I came across this one I couldn't solve. I'm not interested in the solution so much as the reasoning. Thanks...Sat, 29 Aug 2026 10:01:15 -0400https://pop.com.sg/domain-of-parametric-equation.htmlhttps://pop.com.sg/domain-of-parametric-equation.htmlIf there is a parametric equation $x=2\cos{2t}$ and $y=6\sin{t}$, $0\le t \le \frac{\pi}{2}$ the Cartesian equation is $y=3\sqrt{2-x}$. How do I find the domain of the Cartesian equation? I tried:......Sat, 29 Aug 2026 09:58:24 -0400https://pop.com.sg/to-what-extent-is-pi-non-repeating-closed.htmlhttps://pop.com.sg/to-what-extent-is-pi-non-repeating-closed.htmlI've been told that pi is an irrational (infinite and non-repeating) number. But to what extent is it non-repeating? It obviously repeats individual numbers, and I find it hard to believe that it......Sat, 29 Aug 2026 09:57:01 -0400https://pop.com.sg/do-all-graphs-have-interval-representations.htmlhttps://pop.com.sg/do-all-graphs-have-interval-representations.htmlThe interval representation of a graph is the idea of depicting graph as layers of sequences of intervals. Two intervals intersect iff there exist an edge in the initial graph between corresponding ...Sat, 29 Aug 2026 09:54:00 -0400https://pop.com.sg/how-many-different-sums-of-the-dots-can-one-obtain-if-three-ordinary-d.htmlhttps://pop.com.sg/how-many-different-sums-of-the-dots-can-one-obtain-if-three-ordinary-d.htmlHow many different sums of the dots can one obtain if three ordinary dice are thrown at the same time? (an ordinary standard die is a regular cube with its six sides numbered with dots from 1 t......Sat, 29 Aug 2026 09:52:30 -0400https://pop.com.sg/does-lim-frac-xy-x-y-exist-at-0-0.htmlhttps://pop.com.sg/does-lim-frac-xy-x-y-exist-at-0-0.htmlGiven the function $f(x,y) = \frac{xy}{x+y}$, after my analysis I concluded that the limit at $(0,0)$ does not exists. In short, if we approach to $(0,0)$ through the parabola $y = -x^2 -x$ and $y......Sat, 29 Aug 2026 09:48:36 -0400https://pop.com.sg/conditional-probability-mass-function-independent-binomial-what-is-the.htmlhttps://pop.com.sg/conditional-probability-mass-function-independent-binomial-what-is-the.htmlSuppose $X$ and $Y$ are independent. $X$ is binomial based on $m$ trials and success probability $\theta$. $Y$ is binomial based on n trials and success probability (i) What is the conditional ...Sat, 29 Aug 2026 09:44:54 -0400https://pop.com.sg/how-does-dv-2-pi-r-2t-sin-theta-d-theta-come.htmlhttps://pop.com.sg/how-does-dv-2-pi-r-2t-sin-theta-d-theta-come.htmlHow does $dV=2\pi R^2 t \sin \theta\, d\theta$ come? From the figure shown in the screenshot, How do I prove $dV=2\pi R^2 t \sin \theta \,d\theta$? My Attempt...Sat, 29 Aug 2026 09:43:44 -0400https://pop.com.sg/simplification-of-terms-2-1-ln2.htmlhttps://pop.com.sg/simplification-of-terms-2-1-ln2.htmlThe final answer is $\mathrm e$. Can anyone show steps? $ 2^{(1/\ln2)}$ I tried log laws without any result. I'm pretty sure I'm stuck somewhere really easy....Sat, 29 Aug 2026 09:38:45 -0400https://pop.com.sg/how-do-unitary-matrices-preserve-the-magnitude-of-unit-vectors.htmlhttps://pop.com.sg/how-do-unitary-matrices-preserve-the-magnitude-of-unit-vectors.htmlIn quantum mechanics and quantum computing, quantum particles evolve in a unitary manner. That is to say at any point in time the particle/system (represented as a vector) has a magnitude of 1, mea......Sat, 29 Aug 2026 09:36:50 -0400https://pop.com.sg/finding-an-average-value-of-a-function.htmlhttps://pop.com.sg/finding-an-average-value-of-a-function.htmlThis is a problem that someone presented to me and I tried to do. My work is shown below the problem. I got my answer (or I thought I did), but it is not in agreement with my TI-84's average value ......Sat, 29 Aug 2026 09:33:41 -0400https://pop.com.sg/how-to-write-special-set-notation-by-hand.htmlhttps://pop.com.sg/how-to-write-special-set-notation-by-hand.htmlDoes anyone know a good resource (preferably pictures) that illustrates a conventional way to write the special sets symbols, i.e. $\mathbb{N,Z,Q,R,C}$ etc., by hand?...Sat, 29 Aug 2026 09:24:06 -0400https://pop.com.sg/how-many-perfect-shuffles-are-needed-to-go-back-to-initial-state.htmlhttps://pop.com.sg/how-many-perfect-shuffles-are-needed-to-go-back-to-initial-state.htmlThe other day, in a popular-science book, I saw: Given 32 cards ($c_i$ for $i=0..31$), cut the deck into two parts and shuffle them the American way (riffle shuffling). The deck now looks like: $c......Sat, 29 Aug 2026 09:19:49 -0400https://pop.com.sg/value-of-shape-operator-in-direction-where-the-curvature-is-zero.htmlhttps://pop.com.sg/value-of-shape-operator-in-direction-where-the-curvature-is-zero.htmlIn a problem in a differential geometry (where I prove that a surface such that all of its geodesics are planar, is either contained in a plane or in a sphere) I need to consider the shape operator......Sat, 29 Aug 2026 09:05:59 -0400https://pop.com.sg/the-definition-of-negation.htmlhttps://pop.com.sg/the-definition-of-negation.htmlWhat is the definition of the negation of a statement? I know that if a statement is true then its negation is not true and also that either the statement is true or its negation is true but they ......Sat, 29 Aug 2026 09:00:54 -0400https://pop.com.sg/find-the-slope-of-the-secant-line-with-two-points.htmlhttps://pop.com.sg/find-the-slope-of-the-secant-line-with-two-points.htmlI can't seem to figure out this problem. The point P(1,0) lies on the curve $y=\sin(10\pi/x)$. If Q is the point $(x,\sin(\frac{10\pi}{x}))$, find the slope of the secant line PQ (correct to ......Sat, 29 Aug 2026 08:59:29 -0400https://pop.com.sg/how-to-compute-the-nth-number-of-a-general-fibonacci-sequence-with-mat.htmlhttps://pop.com.sg/how-to-compute-the-nth-number-of-a-general-fibonacci-sequence-with-mat.htmlIf we want to compute the nth Fibonacci number we just power the matrix $M = \left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right]$ $n$ times and we get $M =\left[ \begin{array}{cc} F_......Sat, 29 Aug 2026 08:48:21 -0400https://pop.com.sg/factorial-of-7-2-duplicate.htmlhttps://pop.com.sg/factorial-of-7-2-duplicate.htmlIt's been many years since I studied maths, and I'm trying to figure out the half factorials $(7/2)!$ without a calculator. I did $(7/2) \times (5/2) \times (3/2) \times (1/2) = (105/16) ^ \pi = 1......Sat, 29 Aug 2026 08:41:44 -0400https://pop.com.sg/sum-of-consecutive-odd-numbers.htmlhttps://pop.com.sg/sum-of-consecutive-odd-numbers.htmlIs there a general formula for the sum of consecutive odd numbers. I don't mean from 1 to $n$ being $n^2$, I mean the way that the sum of one consecutive odd number is any odd number, 2 consecutiv......Sat, 29 Aug 2026 08:37:14 -0400https://pop.com.sg/how-to-re-write-completing-the-square-x-2-x-1.htmlhttps://pop.com.sg/how-to-re-write-completing-the-square-x-2-x-1.htmlThe exercise asks to "Re-write completing the square": $$x^2+x+1$$ The answer is: $$\left(x+\frac{1}{2}\right)^2+\frac{3}{4}$$ I don't even understand what it means with "Re-write completing the ......Sat, 29 Aug 2026 08:20:18 -0400https://pop.com.sg/what-039-s-the-definition-of-coherent-topology-in-munkres-topology.htmlhttps://pop.com.sg/what-039-s-the-definition-of-coherent-topology-in-munkres-topology.htmlMunkres Topology. In Section 71, coherent is "if" but on Wikipedia (Coherent topology), it's "if and only if" This can be seen in Example 1 of Section 83 We suppose $D \cap A_{\alpha}$ is closed ......Sat, 29 Aug 2026 08:15:32 -0400https://pop.com.sg/what-is-the-area-of-the-shaded-region-in-this-rectangle.htmlhttps://pop.com.sg/what-is-the-area-of-the-shaded-region-in-this-rectangle.htmlWhat is the area of the shaded region below? I think the solution requires subtracting the area of each of the four triangles from the area of the rectangle. I can calculate the area of triangles ......Sat, 29 Aug 2026 07:59:55 -0400https://pop.com.sg/prove-sin-x-tan-x-x-2-for-x-in-0-frac-pi-2.htmlhttps://pop.com.sg/prove-sin-x-tan-x-x-2-for-x-in-0-frac-pi-2.htmlProve $\sin(x)\tan(x) > x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$ So I did the following: Let $f(x) = \sin(x)\tan(x) - x^2$. Then of course $f(0) = 0$ and I want to show that for $x \in ( \,0,......Sat, 29 Aug 2026 07:59:47 -0400https://pop.com.sg/percent-or-percentage-point-when-rounding.htmlhttps://pop.com.sg/percent-or-percentage-point-when-rounding.htmlWhen rounding a percentage, is it preferable to use "to the nearest percent" or "to the nearest percentage point"? Does this change if we want to round to the nearest, say, 10 percent/percentage po......Sat, 29 Aug 2026 07:58:11 -0400https://pop.com.sg/decomposition-of-polynomial-into-irreducible-polynomials.htmlhttps://pop.com.sg/decomposition-of-polynomial-into-irreducible-polynomials.htmlI'm preparing to my algebra exam. And I have problem and I have no idea how to solve it. Given polynomial $$x^4+4x^3+4x^2+1.$$ The task is find expansion of the polynomial as a product of ...Sat, 29 Aug 2026 07:55:46 -0400https://pop.com.sg/transforming-a-single-gaussian-into-a-mixture-of-gaussians.htmlhttps://pop.com.sg/transforming-a-single-gaussian-into-a-mixture-of-gaussians.htmlGiven a standard normal random variable: $X\sim \mathcal{N}(0,1)$ and given two real positive numbers $\mu$ and $\sigma$, I am interested in finding a (non-random) function $g:\mathbb{R}\rightar......Sat, 29 Aug 2026 07:49:56 -0400https://pop.com.sg/square-root-of-a-sum-bound.htmlhttps://pop.com.sg/square-root-of-a-sum-bound.htmlIs there any upper bound for an expression like: $$\left( a_1 + a_2 + \cdots + a_n\right)^{1/2} ?$$ I need it for $n=3$. I know Hardy's inequality but it is for exponent greater than 1. Is there ...Sat, 29 Aug 2026 07:33:21 -0400https://pop.com.sg/homogeneous-or-non-homogeneous-ode.htmlhttps://pop.com.sg/homogeneous-or-non-homogeneous-ode.htmlI've just gotten back a corrected homework about differential equations, and now I need your help: Why is the ODE $u''(x)=u(x)\sqrt{x}$ homogeneous, but the PDE $u_{xx}(x,y)+u_{yy}(x,y)e^{\......Sat, 29 Aug 2026 07:32:49 -0400https://pop.com.sg/how-do-you-show-sin-180-circ-2x-sin-2x.htmlhttps://pop.com.sg/how-do-you-show-sin-180-circ-2x-sin-2x.htmlMy approach would be let $2x$ become $q$ and $\sin(180-q)=\sin(q)$ therefore $\sin(q)=\sin(2x)$ but I think there is a better proof....Sat, 29 Aug 2026 07:14:04 -0400https://pop.com.sg/one-point-perspective-formula.htmlhttps://pop.com.sg/one-point-perspective-formula.htmlA column of cubes In the image, the length of the side of each cube is one unit. The dotted black line represents the negative side of the z axis. Given a vanishing point P(x, y) on the image plan......Sat, 29 Aug 2026 06:59:34 -0400https://pop.com.sg/what-is-a-false-premise.htmlhttps://pop.com.sg/what-is-a-false-premise.htmlI don't understand what a sound argument is. And what does it mean for a premise to be false? Why does case 3 (A is false, B is true) not apply in the real world? Here the author says that the first ...Sat, 29 Aug 2026 06:56:15 -0400https://pop.com.sg/why-union-find-algorithm-runs-in-amortized-time.htmlhttps://pop.com.sg/why-union-find-algorithm-runs-in-amortized-time.htmlI have an exercise that is stated: What's amortized time of series on $n$ operations where all union operations precede find operations? I know it should be $n$ for series but I struggle with ...Sat, 29 Aug 2026 06:53:22 -0400https://pop.com.sg/formula-for-working-out-the-number-of-dice-combinations-resulting-in-a.htmlhttps://pop.com.sg/formula-for-working-out-the-number-of-dice-combinations-resulting-in-a.htmlIt's been about nine years since I last undertook any formal mathematical training, and I am wrestling with generating probability curves in R. What I want to know is, for a dicepool of an arbitrary ...Sat, 29 Aug 2026 06:51:37 -0400https://pop.com.sg/finding-the-second-derivative-of-a-polar-equation.htmlhttps://pop.com.sg/finding-the-second-derivative-of-a-polar-equation.html$$\dfrac{dy}{dx} =\dfrac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\dfrac{f(\theta) \cos \theta +f'(\theta) \sin \theta}{-f(\theta) \sin \theta +f'(\theta) \cos \theta} \\ r=f(\theta), ~~y=f(\theta)\......Sat, 29 Aug 2026 06:50:57 -0400https://pop.com.sg/what-is-edge-maximal-graph.htmlhttps://pop.com.sg/what-is-edge-maximal-graph.htmlGoing through Graph theory , i found We call Graph $G$ edge Maximal if with a given graph property if $G$ itself has the property but no graph $G$+$xy$ does for non adjacent vertices $x,y$ $\...Sat, 29 Aug 2026 06:45:05 -0400https://pop.com.sg/vertex-cover-reduction.htmlhttps://pop.com.sg/vertex-cover-reduction.htmlI'm having a hard time figuring out how to reduce a problem to prove that it is NP-complete. Basically, my problem is this: Given an undirected graph, find a set $S$ such that for all nodes in $G \...Sat, 29 Aug 2026 06:44:39 -0400https://pop.com.sg/fourier-series-for-cos-x.htmlhttps://pop.com.sg/fourier-series-for-cos-x.htmlI'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx =......Sat, 29 Aug 2026 06:41:41 -0400https://pop.com.sg/need-help-to-find-remainder-when-5-25-is-divided-by-31.htmlhttps://pop.com.sg/need-help-to-find-remainder-when-5-25-is-divided-by-31.htmlI'm really confused about congruence. I tried hard, but I kept failing :( $$30! \equiv -1 \pmod{31} \text{ by Wilson's Theorem}$$ $$ \Longleftrightarrow 30.29.28.27.26.25! \equiv -1 \pmod{3......Sat, 29 Aug 2026 06:38:05 -0400https://pop.com.sg/how-to-integrate-frac-4x-x-3-x-2-x-1.htmlhttps://pop.com.sg/how-to-integrate-frac-4x-x-3-x-2-x-1.htmlI'd like to know how to integrate $$\frac{4x}{x^3+x^2+x+1}$$ Please could anyone help me Thanks all...Sat, 29 Aug 2026 06:31:20 -0400https://pop.com.sg/geometrical-interpretation-of-function-phi-n-duplicate.htmlhttps://pop.com.sg/geometrical-interpretation-of-function-phi-n-duplicate.htmlWe know that the $\phi(n)$ function denotes the total numbers which are co-prime to n , but like most of the mathematical concepts , can it be geometrically interpreted ? Like in terms of graph? ......Sat, 29 Aug 2026 06:27:37 -0400https://pop.com.sg/circular-definition.htmlhttps://pop.com.sg/circular-definition.htmlI am new to logic and studying it by Enderton's A Mathematical Introduction to Logic (2nd edition). For question 3.3.1 in the books, which states Show that in the structure $(\mathbb{N}; \cdot, ......Sat, 29 Aug 2026 06:13:18 -0400https://pop.com.sg/basis-for-the-solution-space.htmlhttps://pop.com.sg/basis-for-the-solution-space.htmlFind the basis for the solution space of the system and describe all solutions: $3x_1 - x_2 + x_4 = 0$ $x_1 + x_2 + x_3 + x_4 = 0$ I row reduce the matrix: $\begin{pmatrix} 1&1&1&1\......Sat, 29 Aug 2026 06:07:20 -0400https://pop.com.sg/proof-of-pinsker-039-s-inequality.htmlhttps://pop.com.sg/proof-of-pinsker-039-s-inequality.htmlHow to prove the following known (Pinsker's) inequality? For two strictly positive sequences $(p_i)^n_{i=l}$ and $(q_i)^n_{i=l}$ with $\sum_{i=1}^np_i=\sum_{i=1}^nq_i=1$ one has $$\sum_{i=1}^np......Sat, 29 Aug 2026 06:07:01 -0400https://pop.com.sg/range-of-a-piecewise-function.htmlhttps://pop.com.sg/range-of-a-piecewise-function.htmlApologies, this is a simple question but I seem to have some sort of brain freeze. I'm looking for the range of this piecewise function: $$f(x)=\begin{cases}x+9&\text{ if } x<-3\\ -2x&\......Sat, 29 Aug 2026 06:03:23 -0400https://pop.com.sg/shortest-path-between-three-nodes-in-a-graph.htmlhttps://pop.com.sg/shortest-path-between-three-nodes-in-a-graph.htmlI know Dijkstra's algorithm to find the shortest way between 2 nodes, but is there a way to find the shortest path between 3 nodes among $n$ nodes? Here are the details: I have $n$ nodes, some of ......Sat, 29 Aug 2026 06:02:07 -0400https://pop.com.sg/mean-value-theorem-find-c.htmlhttps://pop.com.sg/mean-value-theorem-find-c.htmlThe derivative of $\frac1x$ is $\frac{-1}{x^2}$. How do I find the $c$ if there is no zero in the derivative of the function? I started with $-1/x^2= -0.0625$ but I'm confused from here on....Sat, 29 Aug 2026 05:55:58 -0400https://pop.com.sg/why-is-the-ratio-of-circumference-to-diameter-pi-and-not-any-other-num.htmlhttps://pop.com.sg/why-is-the-ratio-of-circumference-to-diameter-pi-and-not-any-other-num.htmlHow do we specifically get to 3.14 from axioms through logic to that particular transcendental number being the case? I'm guessing it's the sum of an infinite series. But how do we go from "there'......Sat, 29 Aug 2026 05:53:46 -0400https://pop.com.sg/what-is-n-when-n-0-duplicate.htmlhttps://pop.com.sg/what-is-n-when-n-0-duplicate.htmlPossible Duplicate: Prove $0! = 1$ from first principles Why does 0! = 1? If I'm right, factorial $!$ means: $$n!=1 \cdot 2 \cdot 3 \cdot 4 \cdots n $$ so: $$ \begin{align} 5!&=1\cdot2\...Sat, 29 Aug 2026 05:44:39 -0400https://pop.com.sg/are-the-real-product-rule-and-quotient-rule-for-integration-already-kn.htmlhttps://pop.com.sg/are-the-real-product-rule-and-quotient-rule-for-integration-already-kn.htmlIn "A Quotient Rule Integration by Parts Formula", the authoress integrates the product rule of differentiation and gets the known formula for integration by parts: \begin{equation}\int f(x)g'(x)dx......Sat, 29 Aug 2026 05:41:55 -0400https://pop.com.sg/why-does-the-euclidean-algorithm-for-finding-gcd-work.htmlhttps://pop.com.sg/why-does-the-euclidean-algorithm-for-finding-gcd-work.htmlI am having trouble understanding why the Euclidean algorithm for finding the GCD of two numbers always works? I found some resources here (http://www.cut-the-knot.org/blue/Euclid.shtml), and her......Sat, 29 Aug 2026 05:39:26 -0400https://pop.com.sg/is-the-fibonacci-sequence-exponential.htmlhttps://pop.com.sg/is-the-fibonacci-sequence-exponential.htmlI could not find any information on this online so I thought I'd make a question about this. If we take the Fibonacci sequence $F_n = F_{n-1} + F_{n-2}$, is this growing exponentially? Or perhaps ......Sat, 29 Aug 2026 05:22:51 -0400https://pop.com.sg/adapting-a-picture-proof-of-trig-039-s-angle-addition-identities-to-an.htmlhttps://pop.com.sg/adapting-a-picture-proof-of-trig-039-s-angle-addition-identities-to-an.htmlI am currently learning about the Trig addition formula. The book I am reading has a proof for it. The proof is attached here: However, I have a question about it. What if the values of theta and......Sat, 29 Aug 2026 05:21:35 -0400https://pop.com.sg/category-of-all-categories-vs-set-of-all-sets.htmlhttps://pop.com.sg/category-of-all-categories-vs-set-of-all-sets.htmlIn naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...Sat, 29 Aug 2026 05:21:14 -0400https://pop.com.sg/the-determinant-of-adjugate-matrix.htmlhttps://pop.com.sg/the-determinant-of-adjugate-matrix.htmlI have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1......Sat, 29 Aug 2026 05:13:24 -0400https://pop.com.sg/on-the-behavior-of-transcendental-functions.htmlhttps://pop.com.sg/on-the-behavior-of-transcendental-functions.htmlGiven that single-variable algebraic functions take on algebraic values when the input is algebraic, and take on transcendental values when the input is transcendental, and knowing that transcenden......Sat, 29 Aug 2026 05:12:42 -0400https://pop.com.sg/finding-bases-for-row-space-and-null-space-of-matrix.htmlhttps://pop.com.sg/finding-bases-for-row-space-and-null-space-of-matrix.htmlMy problem is: For the matrix $$A = \begin{bmatrix} 1& 4& 5& 6& 9\\ 3& −2& 1& 4& −1\\ −1& 0& −1& −2& −1\\ 2& 3&......Sat, 29 Aug 2026 05:00:32 -0400https://pop.com.sg/trouble-understanding-sum-of-subspaces.htmlhttps://pop.com.sg/trouble-understanding-sum-of-subspaces.htmlI started reviewing linear algebra, from a different textbook (Axler's), after taking a fast paced summer class. Unfortunately, I've become confused with a concept that is introduced at the end of ...Sat, 29 Aug 2026 04:52:37 -0400https://pop.com.sg/painting-cube-with-3-different-colours-closed.htmlhttps://pop.com.sg/painting-cube-with-3-different-colours-closed.htmlPainting cube with 3 colours (using each colour at least one time) Find no. of ways to colour the cube. (Rotation don't change the colouring.) Solve it for class 9 student in a easier way. Please...Sat, 29 Aug 2026 04:49:03 -0400https://pop.com.sg/what-is-cot-pi-2.htmlhttps://pop.com.sg/what-is-cot-pi-2.htmlBase on the unit circle, I know $ \begin{align} &\cot\left(\frac{\pi}{2}\right) \\ =&\frac{0}{1}\\ =&0 \end{align} $ But it is also $ \begin{align} &\cot\left(\frac{\pi}{2}\righ......Sat, 29 Aug 2026 04:40:37 -0400https://pop.com.sg/an-explanation-of-the-kalman-filter.htmlhttps://pop.com.sg/an-explanation-of-the-kalman-filter.htmlIn the past 3 months, I've been trying to understand the Kalman Filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, ...Sat, 29 Aug 2026 04:37:52 -0400https://pop.com.sg/how-to-factorize-polynomials-to-the-5th-degree.htmlhttps://pop.com.sg/how-to-factorize-polynomials-to-the-5th-degree.htmlI have the polynomial: $$2x^5-x^4+10x^3-5x^2+8x-4$$ and I know that the final result is: $$(2x-1)(x^4+5x^2+4) = (2x-1)(x^2+1)(x^2+4)$$ But how would you do it step by step? I've seen a couple of vi......Sat, 29 Aug 2026 04:33:59 -0400https://pop.com.sg/quick-way-to-find-eigenvalues-of-anti-diagonal-matrix.htmlhttps://pop.com.sg/quick-way-to-find-eigenvalues-of-anti-diagonal-matrix.htmlIf $A \in M_n(\mathbb{R})$ is an anti-diagonal $n \times n$ matrix, is there a quick way to find its eigenvalues in a way similar to finding the eigenvalues of a diagonal matrix? The standard way for ...Sat, 29 Aug 2026 04:26:48 -0400https://pop.com.sg/how-to-determine-if-a-large-number-is-prime.htmlhttps://pop.com.sg/how-to-determine-if-a-large-number-is-prime.htmlFor a very large number n, how many divisibility tests are required to establish if its prime? I know this has something to do with the Golden Number, but I can't figure out what. I did try search......Sat, 29 Aug 2026 04:24:37 -0400