https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 29 Aug 2026 13:44:30 -0400https://pop.com.sg/show-that-a-positive-operator-on-a-complex-hilbert-space-is-self-adjoi.htmlhttps://pop.com.sg/show-that-a-positive-operator-on-a-complex-hilbert-space-is-self-adjoi.htmlLet $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A : \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{......Sat, 29 Aug 2026 17:37:38 -0400https://pop.com.sg/what-is-the-correct-formula-for-the-washer-method.htmlhttps://pop.com.sg/what-is-the-correct-formula-for-the-washer-method.htmlAlmost everywhere I look the formula is: $$ \pi \int_b^a {\left(f(x)^2 - g(x)^2\right) dx} $$ where f(x) is the big function and g(x) is the smaller function. Though I've run into problems while ...Sat, 29 Aug 2026 17:34:19 -0400https://pop.com.sg/what-is-the-difference-between-a-kernel-and-a-function.htmlhttps://pop.com.sg/what-is-the-difference-between-a-kernel-and-a-function.htmlI have been looking around for this question, but all results I found only describe the definition and not the answer I seek. Is "kernel" basically a synonym of "function"? When should be the time......Sat, 29 Aug 2026 17:15:50 -0400https://pop.com.sg/duplication-formula-for-gamma-function.htmlhttps://pop.com.sg/duplication-formula-for-gamma-function.htmlUsing the Weierstrass definition for $\Gamma(x)$ and $\Gamma\Big(x + \frac12\Big)$, how can I prove the duplication formula? This is problem $10.7.3$ in the book Irresistible Integrals, by Boros an......Sat, 29 Aug 2026 17:11:51 -0400https://pop.com.sg/computing-the-value-of-zeta-6-and-zeta-8.htmlhttps://pop.com.sg/computing-the-value-of-zeta-6-and-zeta-8.htmlI know the closed form formula for calculating the values of Zeta function at even integers. I was able to derive it from the coefficients of the power series of $-\dfrac{\pi x}{2}\cot(\pi x)$. Now......Sat, 29 Aug 2026 17:08:01 -0400https://pop.com.sg/what-is-the-precise-definition-of-the-ambient-space.htmlhttps://pop.com.sg/what-is-the-precise-definition-of-the-ambient-space.htmlFor instance, what is the ambient space of a singleton $\{x\}$, where $x \in \mathbb{R}$? Can it be the singleton itself? $\mathbb{R}$? $\mathbb{R}^n$? or some arbitary set that happens to contain......Sat, 29 Aug 2026 17:00:10 -0400https://pop.com.sg/how-do-i-evaluate-the-integral-of-a-function-g-x-using-a-graph.htmlhttps://pop.com.sg/how-do-i-evaluate-the-integral-of-a-function-g-x-using-a-graph.htmlI attempted to solve this problem in the following manner: Part1: The area under the curve is defined by the integral below: $$ \int_0^{10} g(x)dx $$ and can be easily obtained by using the Are......Sat, 29 Aug 2026 16:59:24 -0400https://pop.com.sg/solve-initial-value-problem-ivp.htmlhttps://pop.com.sg/solve-initial-value-problem-ivp.htmlI have equations \begin{align} 7z'(x) &= x^2 + y(x)^2 + z(x)^2 \\ y''(x) &= -(y'(x) + 7y(x))\sin(z(x)) \end{align} where $$(y(0),y'(0),z(0))=(1.5,-2.6,0.5)$$ The task is to solve this pro......Sat, 29 Aug 2026 16:55:31 -0400https://pop.com.sg/constructive-intermediate-value-theorem-ivt.htmlhttps://pop.com.sg/constructive-intermediate-value-theorem-ivt.htmlI'm trying to learn a bit about intuitionistic/constructive mathematics, as I want to understand a little about topos theory and homotopy type theory (HoTT). I'm confused as to why the intermediate ...Sat, 29 Aug 2026 16:51:20 -0400https://pop.com.sg/how-to-simply-increase-probability-of-an-event.htmlhttps://pop.com.sg/how-to-simply-increase-probability-of-an-event.htmlLets say there is 40% chance that an event will happen. Than it is said that the probability of the event is increased by five times. I do realize that the resulting percentage is not 200% or even ......Sat, 29 Aug 2026 16:47:48 -0400https://pop.com.sg/is-it-possible-to-make-a-2d-net-of-a-spherical-truncated-icosahedron-a.htmlhttps://pop.com.sg/is-it-possible-to-make-a-2d-net-of-a-spherical-truncated-icosahedron-a.htmlI know that a Truncated Icosahedron is the underlying shape that makes a Football (Soccer ball). In real life, the 20 Hexagons and 12 Pentagons are cut out of fabric and a balloon inside is blown......Sat, 29 Aug 2026 16:47:23 -0400https://pop.com.sg/trigonometric-equation-involving-tangent-how-do-i-solve-it.htmlhttps://pop.com.sg/trigonometric-equation-involving-tangent-how-do-i-solve-it.htmlI would like to know how to solve this and most importantly make intuitive sense of it. Thanks in advance! Equation: $\tan(x) = -1.5$ (using a calculator with $\arctan(-1.5)$) My Results: $x = -0......Sat, 29 Aug 2026 16:42:54 -0400https://pop.com.sg/prove-2-boolean-expressions-are-equal-by-multiplying-out-and-simplifyi.htmlhttps://pop.com.sg/prove-2-boolean-expressions-are-equal-by-multiplying-out-and-simplifyi.htmlI was asked to prove that the following is TRUE by "multiplying it out" and simplifying: $(A+BC)(A+DE)=A+BCDE$ I am already familiar with the theorem $A+BC=(A+B)(A+C)$ which would be enough pr......Sat, 29 Aug 2026 16:41:54 -0400https://pop.com.sg/two-linearly-independent-vectors-perpendicular-to-vector-u.htmlhttps://pop.com.sg/two-linearly-independent-vectors-perpendicular-to-vector-u.htmlI'm having trouble with these types of questions. I have the following vector $u = (4, 7, -9)$ and it wants me to find 2 vectors that are perpendicular to this one. I know that $(4,7,-9)\cdot (x,y......Sat, 29 Aug 2026 16:38:50 -0400https://pop.com.sg/help-with-coding-class-exercise-find-the-half-of-a-number-without-divi.htmlhttps://pop.com.sg/help-with-coding-class-exercise-find-the-half-of-a-number-without-divi.htmlThe teacher explained more or less what we have to do. My question is about the maths part not the coding part. The exercise is: Create a function that given a positive integer will return the int......Sat, 29 Aug 2026 16:23:43 -0400https://pop.com.sg/the-derivative-of-constant-raised-to-the-power-of-n-which-is-raised-to.htmlhttps://pop.com.sg/the-derivative-of-constant-raised-to-the-power-of-n-which-is-raised-to.htmlI need a reminder of how to go about taking the derivative of $c^{n^x}$ where $x$ and $c$ are constants. An example would be $$\large 4^{n^2}$$ Thanks in advance....Sat, 29 Aug 2026 16:09:09 -0400https://pop.com.sg/abel-differential-equation-of-second-kind.htmlhttps://pop.com.sg/abel-differential-equation-of-second-kind.htmlConsider this Abel differential equation: $$4\left[ y(z)+ z^2\right] y'(z)-(1+6z)y(z)=0$$ Is there a way to separate variables? I just tried some change of variables, but with no good results....Sat, 29 Aug 2026 16:06:01 -0400https://pop.com.sg/proving-difference-of-two-powers-using-induction.htmlhttps://pop.com.sg/proving-difference-of-two-powers-using-induction.htmlProve, that $\forall n \in \mathbb{N}$ the following identity holds: $$a^n - b^n = (a-b)\sum_{k=0}^{n-1} a^k b^{n-k-1}.$$ For $n = 1$ we get $a^1 - b^1 = a - b$ on LHS, and on RHS we get $(a-b)\su......Sat, 29 Aug 2026 16:00:02 -0400https://pop.com.sg/maximum-value-of-function-on-unit-circle.htmlhttps://pop.com.sg/maximum-value-of-function-on-unit-circle.htmlFind the maximum value of the function $f(x,y) = (x+2y)^2$ on the unit circle $x^2+y^2=1$. I defined a new function $g(x)= x^2 +4x\sqrt{1-x^2} +4(1-x^2) $ and let its derivative equal to zero. So......Sat, 29 Aug 2026 15:56:52 -0400https://pop.com.sg/what-does-it-mean-to-call-something-naive-in-mathematics.htmlhttps://pop.com.sg/what-does-it-mean-to-call-something-naive-in-mathematics.htmlI apologize if this is a less-than appropriate place for the question, but I've seen this term come up repeatedly in textbooks and lectures without any knowledge of what it indicates. There is alwa......Sat, 29 Aug 2026 15:52:15 -0400https://pop.com.sg/archimedes-039-derivation-of-the-spherical-cap-area-formula.htmlhttps://pop.com.sg/archimedes-039-derivation-of-the-spherical-cap-area-formula.htmlArchimedes derived a formula for the area of a spherical cap. so Archimedes says that the curved surface area of a spherical cap is equal to the area of a circle with radius equal to the distance ...Sat, 29 Aug 2026 15:46:45 -0400https://pop.com.sg/find-the-perimeter-of-any-triangle-given-the-three-altitude-lengths.htmlhttps://pop.com.sg/find-the-perimeter-of-any-triangle-given-the-three-altitude-lengths.htmlThe altitude lengths are 12, 15 and 20. I would like a process rather than just a single solution....Sat, 29 Aug 2026 15:21:29 -0400https://pop.com.sg/user-akhil-mathew-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-akhil-mathew-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 29 Aug 2026 15:12:42 -0400https://pop.com.sg/how-to-find-the-eigenvalues-of-tridiagonal-toeplitz-matrix.htmlhttps://pop.com.sg/how-to-find-the-eigenvalues-of-tridiagonal-toeplitz-matrix.htmlAssume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & &\\ b & a & c &am......Sat, 29 Aug 2026 15:04:20 -0400https://pop.com.sg/parallel-surface.htmlhttps://pop.com.sg/parallel-surface.htmlFor a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$ How could I show the ...Sat, 29 Aug 2026 14:57:27 -0400https://pop.com.sg/expectation-of-hitting-time-for-simple-symmetric-random-walk.htmlhttps://pop.com.sg/expectation-of-hitting-time-for-simple-symmetric-random-walk.htmlAssume there is a simple symmetric random walk $$S_n=X_1+...+X_n,\quad S_0=0$$ where $\mathbb P(X_i=\pm 1)=\frac{1}{2}$. Define $T=\inf\{n:S_n=1\}$. How to compute $\mathbb E(T)$? My idea: if $\m......Sat, 29 Aug 2026 14:43:14 -0400https://pop.com.sg/right-english-wording-for-counterexamples-to-a-theorem.htmlhttps://pop.com.sg/right-english-wording-for-counterexamples-to-a-theorem.htmlThis question is about the right English wording. I give here what I call "counterexamples to Banach fixed-point theorem". What I do, is that I look to what happen if some hypothesis of the theore......Sat, 29 Aug 2026 14:34:06 -0400https://pop.com.sg/deriving-surface-area-of-a-sphere-from-the-circumference.htmlhttps://pop.com.sg/deriving-surface-area-of-a-sphere-from-the-circumference.htmlgiven the circumference of a circle, which is 2πr, how many times do I have to add it to itself to cover a whole surface of a sphere and deriving 4πr^2?...Sat, 29 Aug 2026 14:32:29 -0400https://pop.com.sg/best-strategies-for-039-squid-game-039-episode-6-game-4-marble-game-pl.htmlhttps://pop.com.sg/best-strategies-for-039-squid-game-039-episode-6-game-4-marble-game-pl.htmlTwo players each get $n=10$ marbles. Every alternating turn, one player (say, the player whose turn it is) hides an amount of own marbles in fist, and, the other player must guess if hidden amount ......Sat, 29 Aug 2026 14:29:13 -0400https://pop.com.sg/multiplying-two-inequalities.htmlhttps://pop.com.sg/multiplying-two-inequalities.htmlSuppose we have two inequalities $$a\leq x\leq b\tag{1}$$ $$c\leq y\leq d\tag{2},$$ where $a,b,c,d>0$. Then can I conclude that $$ac\leq xy\leq bd\quad ?$$ My attempt: Since $a,b,c,d>0$ and $\...Sat, 29 Aug 2026 14:16:22 -0400https://pop.com.sg/find-the-remainder-of-a-number-with-a-large-exponent-duplicate.htmlhttps://pop.com.sg/find-the-remainder-of-a-number-with-a-large-exponent-duplicate.htmlI have to find the remainder of $10^{115}$ divided by 7. I was following the way my book did it in an example but then I got confused. So far I have, $\overline{10}^{115}$=$\overline{10}^{7*73+4}......Sat, 29 Aug 2026 14:13:51 -0400https://pop.com.sg/how-to-represent-the-ceiling-function-using-mathematical-notation.htmlhttps://pop.com.sg/how-to-represent-the-ceiling-function-using-mathematical-notation.htmlHow to represent the ceiling function using mathematical notation? I need an equation to input in a program because it doesn't except the ceiling function so it has to be some sort of mathematical ...Sat, 29 Aug 2026 14:11:06 -0400https://pop.com.sg/slopes-of-inverse-functions.htmlhttps://pop.com.sg/slopes-of-inverse-functions.htmlI have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ......Sat, 29 Aug 2026 14:07:53 -0400https://pop.com.sg/generate-integer-solutions-from-a-hyperbolic-equation.htmlhttps://pop.com.sg/generate-integer-solutions-from-a-hyperbolic-equation.htmlI want to generate any integer solution from the equation $\sqrt{8r^2+1} = n$. I know there are integer solutions to this, but I have no idea how to approach this. (When working with whole numbers)......Sat, 29 Aug 2026 14:07:43 -0400https://pop.com.sg/how-to-find-the-angle-of-a-rectangle-vertex.htmlhttps://pop.com.sg/how-to-find-the-angle-of-a-rectangle-vertex.htmlbefore I go further I want you to know that I'm developing a collision detection system in a programming language (Javascript). I'm not used to math terms (it was like 10 years ago when I was in ......Sat, 29 Aug 2026 14:01:11 -0400https://pop.com.sg/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-p.htmlhttps://pop.com.sg/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-p.htmlIt’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an a......Sat, 29 Aug 2026 13:39:39 -0400https://pop.com.sg/analysis-determine-radius-and-interval-of-convergence-for-these-power-.htmlhttps://pop.com.sg/analysis-determine-radius-and-interval-of-convergence-for-these-power-.htmlI needed to determine the radius and interval of convergence for each of the following; $\mathbb{One:}$ $$\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 -0400