https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 28 Aug 2026 09:04:27 -0400https://pop.com.sg/normalizing-eigenvectors-from-pauli-matrices.htmlhttps://pop.com.sg/normalizing-eigenvectors-from-pauli-matrices.htmlFor this example of a Pauli matrix, \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} I found that one of its eigenvectors (for $\lambda = 1$) is \begin{bmatrix} -i \\ 1 \end{bmat......Fri, 28 Aug 2026 13:00:04 -0400https://pop.com.sg/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.htmlhttps://pop.com.sg/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.html(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this su......Fri, 28 Aug 2026 12:59:08 -0400https://pop.com.sg/is-the-division-property-of-equality-just-a-special-case-of-the-multip.htmlhttps://pop.com.sg/is-the-division-property-of-equality-just-a-special-case-of-the-multip.htmlMy textbook clearly states after the lesson on Transforming Equations: Addition and Subtraction "Notice that the subtraction property of equality is just a special case of the addition property, s......Fri, 28 Aug 2026 12:55:53 -0400https://pop.com.sg/showing-a-matrix-is-not-diagonalizable.htmlhttps://pop.com.sg/showing-a-matrix-is-not-diagonalizable.htmlFor eigenvalue 1, the eigenspace I got was $span[ 1,0,0]$, and for eigenvalue 4, the eigenspace I got was $span[ 1,-3,9]$. Do they look right? The reason the matrix is not diagonalizable is becaus......Fri, 28 Aug 2026 12:55:45 -0400https://pop.com.sg/integral-of-arcsin-x-x-2.htmlhttps://pop.com.sg/integral-of-arcsin-x-x-2.htmlI did the integration by parts and got this expression, but then I am stuck on how to take it further. I tried substituting u=1-x^2, but then I had to do a partial fraction decomposition (which I d......Fri, 28 Aug 2026 12:54:00 -0400https://pop.com.sg/complex-conjugation-continuous-function.htmlhttps://pop.com.sg/complex-conjugation-continuous-function.htmlHow can I show that complex conjugation is a continuous function? I tried looking at open sets $U$ and then the preimage. Can assume preimage is not empty so that if $z$ is in preimage then $f(z) \......Fri, 28 Aug 2026 12:51:42 -0400https://pop.com.sg/how-to-calculate-the-integral-of-exponential-functions.htmlhttps://pop.com.sg/how-to-calculate-the-integral-of-exponential-functions.htmlHaving an integral like $\int_{2}^{10}{\frac{x}{\ln x}}dx$ How does this function turns to an exponential integral of the form: $ \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt.\,$ For ...Fri, 28 Aug 2026 12:48:56 -0400https://pop.com.sg/example-of-function-of-bounded-variation.htmlhttps://pop.com.sg/example-of-function-of-bounded-variation.htmlI want to show that $f(x) = x \sin(\frac{1}{\sqrt{x}})$ , with $f(0)=0$ on $[0,1]$ is of bounded variation. To do that I Don't want to use of showing this is continuously differentiable hence, BV. ......Fri, 28 Aug 2026 12:39:15 -0400https://pop.com.sg/given-this-transformation-matrix-how-do-i-decompose-it-into-translatio.htmlhttps://pop.com.sg/given-this-transformation-matrix-how-do-i-decompose-it-into-translatio.htmlI have this problem from my Graphics course. Given this transformation matrix: $$\begin{pmatrix} -2 &-1& 2\\ -2 &1& -1\\ 0 &0& 1\\ \end{pmatrix}$$ I need to extract ...Fri, 28 Aug 2026 12:36:14 -0400https://pop.com.sg/power-series-to-evaluate-a-definite-integral-of-f-x-frac1-1-x-7.htmlhttps://pop.com.sg/power-series-to-evaluate-a-definite-integral-of-f-x-frac1-1-x-7.htmlFind a power series expansion of the function $f(x)=\dfrac1{1+x^7}$ about $x=0$. Use it to evaluate $\int_0^1 \dfrac1{1+x^7} dx$. Your answer should be expressed in the form of numerical series (a ......Fri, 28 Aug 2026 12:34:32 -0400https://pop.com.sg/calculating-values-for-cubic-bezier-curve.htmlhttps://pop.com.sg/calculating-values-for-cubic-bezier-curve.htmlI'm trying to use cubic bezier curves for some non-linear animations in my iOS app. Let's say I'm animating position of some element on the screen. I'm using this curve from cubic-bezier.com for ...Fri, 28 Aug 2026 12:29:39 -0400https://pop.com.sg/find-s-where-s-sqrt-3-5-2-sqrt-13-sqrt-3-5-2-sqrt-13-why-am-i-getting-.htmlhttps://pop.com.sg/find-s-where-s-sqrt-3-5-2-sqrt-13-sqrt-3-5-2-sqrt-13-why-am-i-getting-.html$\large S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$ Multiplying by conjugate: $\large S=\dfrac {-3}{\sqrt[3] {5+2 \sqrt {13}}-\sqrt[3]{5-2 \sqrt {13}}}$ From the original: $\large S-......Fri, 28 Aug 2026 12:20:20 -0400https://pop.com.sg/how-to-show-if-a-complex-function-is-analytic.htmlhttps://pop.com.sg/how-to-show-if-a-complex-function-is-analytic.htmlJust began the study of complex analysis. Let $$ f(x,y) = x^2 - y^2 + 2 i xy - x - iy. $$ I need to determine if this function is analytic. This means I have to show the partials satisfy the Cauchy-...Fri, 28 Aug 2026 12:18:21 -0400https://pop.com.sg/understanding-vector-projection.htmlhttps://pop.com.sg/understanding-vector-projection.htmlI'm learning about vector projection. I understand how to perform it, but I still can't understand what it actually means and what it gives me. Here is a common definition: Vector projection of a ...Fri, 28 Aug 2026 12:14:31 -0400https://pop.com.sg/integral-of-1-x-2.htmlhttps://pop.com.sg/integral-of-1-x-2.htmlI know that the integral of $1/x^2$ is $(-1/x +C)$ Can we split $(1/x^2)$ as $(1/x)(1/x)$ and say that the integral of $(1/x^2)$ is $\ln x\cdot\ln x$? PS: I am new to high school. And this is my fi......Fri, 28 Aug 2026 12:02:29 -0400https://pop.com.sg/is-there-any-formula-to-find-the-nth-element-in-a-sequence-where-commo.htmlhttps://pop.com.sg/is-there-any-formula-to-find-the-nth-element-in-a-sequence-where-commo.htmlTo explain my question, here is an example. Below is an AP: 2, 6, 10, 14....n Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is ...Fri, 28 Aug 2026 11:56:56 -0400https://pop.com.sg/is-there-a-straight-line-solution-for-this-separable-differential-equa.htmlhttps://pop.com.sg/is-there-a-straight-line-solution-for-this-separable-differential-equa.htmlThis is the question: Solve $$x^2 \frac{\text{d}y}{\text{d}x} = y^2 - 3xy -5x^2 .$$ Are there any straight line solutions? If so, state them. For the case $y(1)=0$, state the solution ...Fri, 28 Aug 2026 11:56:03 -0400https://pop.com.sg/what-is-the-difference-between-probability-density-function-and-probab.htmlhttps://pop.com.sg/what-is-the-difference-between-probability-density-function-and-probab.htmlWhats the difference between probability density function and probability distribution function?...Fri, 28 Aug 2026 11:48:02 -0400https://pop.com.sg/set-intersection-the-same-as-logical-conjunction.htmlhttps://pop.com.sg/set-intersection-the-same-as-logical-conjunction.htmlI have been exploring more about set theory beyond my textbook and I have ran into something I couldn't explain. Can you use logical conjunction/disjunction on sets and are they the same as union/...Fri, 28 Aug 2026 11:45:14 -0400https://pop.com.sg/mathematical-notation-for-distinct-numbers.htmlhttps://pop.com.sg/mathematical-notation-for-distinct-numbers.htmlHow can I represent a set of distinct numbers in shorthand mathematical notation? In particular I am trying to write that there are 4 distinct integers $a,b,c,d$. For three numbers, I could say: ......Fri, 28 Aug 2026 11:17:31 -0400https://pop.com.sg/equations-for-cubic-regression.htmlhttps://pop.com.sg/equations-for-cubic-regression.htmlSo, I'm making a simple program for drawing graphs, and I'm looking at making some simple best-fit curves using some basic regression analysis. I've happily got linear and quadratic regression work......Fri, 28 Aug 2026 11:09:24 -0400https://pop.com.sg/depressed-cubic.htmlhttps://pop.com.sg/depressed-cubic.htmlI have this string of questions, but when I look up depressed cubic, I don't quite understand. I'm not asking for all of these questions to be answered explicitly, but for some explanation on depre......Fri, 28 Aug 2026 11:05:09 -0400https://pop.com.sg/equation-for-curves-of-high-density-in-bifurcation-diagram.htmlhttps://pop.com.sg/equation-for-curves-of-high-density-in-bifurcation-diagram.htmlIs there an equation for the high density curves in the chaotic regions of the bifurcation diagram for the logistic map? I'm talking about the sinusoidal-looking dark curves in the following picture. ...Fri, 28 Aug 2026 10:55:43 -0400https://pop.com.sg/how-to-compute-confidence-interval-for-variance-with-unknown-mean-from.htmlhttps://pop.com.sg/how-to-compute-confidence-interval-for-variance-with-unknown-mean-from.htmlWhen mean is known, we note that $\frac{\bar{(X-a)^2}n}{\sigma ^2}$ has a Chi-squared distribution with $n$ degrees of freedom. However, what to do when $a$ is unknown? I can't just substitute sample ...Fri, 28 Aug 2026 10:50:25 -0400https://pop.com.sg/is-there-a-symbol-for-antiparallel.htmlhttps://pop.com.sg/is-there-a-symbol-for-antiparallel.htmlI've been doing some work where I've needed to talk about vectors that are parallel and those that are antiparallel, parallel to the negative of the other vector. Is there a symbol for this? I ......Fri, 28 Aug 2026 10:24:28 -0400https://pop.com.sg/divergence-of-series-with-n-factorial.htmlhttps://pop.com.sg/divergence-of-series-with-n-factorial.htmlI need to find the convergence or divergence of $$\sum_{n=1}^\infty \frac{n!}{3^n}$$ The first thing I did was to take the nth root. So I got: $$\lim_{n\rightarrow\infty}\big(\frac{n!}{3^n}\big)^\......Fri, 28 Aug 2026 10:12:48 -0400https://pop.com.sg/probably-an-easy-question-e-2-ln-3-without-a-calculator.htmlhttps://pop.com.sg/probably-an-easy-question-e-2-ln-3-without-a-calculator.htmlHow to evaluate $e^{2\ln(3)}$ without using a calculator? I've tried playing with $\ln$ but couldn't figure out...Fri, 28 Aug 2026 10:11:04 -0400https://pop.com.sg/angle-between-curves-at-point-of-intersection.htmlhttps://pop.com.sg/angle-between-curves-at-point-of-intersection.htmlQuestion reads- Find the acute angle between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $y=x^......Fri, 28 Aug 2026 10:08:10 -0400https://pop.com.sg/arc-length-definition.htmlhttps://pop.com.sg/arc-length-definition.htmlI've always felt that the notion of 'area under the curve' is more precise than 'length of a curve', because of the Riemann integral and the reasoning used there: First, we assume the quantity we ......Fri, 28 Aug 2026 10:07:37 -0400https://pop.com.sg/how-does-the-focus-directrix-definition-of-a-conic-section-apply-to-a-.htmlhttps://pop.com.sg/how-does-the-focus-directrix-definition-of-a-conic-section-apply-to-a-.htmlOne of the definitions of a conic section is that the conic section is a locus of points whose distance to the focus $F$ is a constant multiple of distance between them and the directrix $D$, i.e. ......Fri, 28 Aug 2026 10:06:06 -0400https://pop.com.sg/how-to-solve-x-2-3-4.htmlhttps://pop.com.sg/how-to-solve-x-2-3-4.htmlOK I know this sounds pretty stupid, but I am stuck on solving $x^{{2}/{3}}=4$. I rewrote it to $\sqrt[3]{x^2}=4$, but I don't know what to do next. Would the radical go away if I took the $\sqrt[3......Fri, 28 Aug 2026 10:06:05 -0400https://pop.com.sg/simple-theorems-that-are-instances-of-deep-mathematics.htmlhttps://pop.com.sg/simple-theorems-that-are-instances-of-deep-mathematics.htmlSo, this question asks about how useful computational tricks are to mathematics research, and several people's response was "well, computational tricks are often super cool theorems in disguise.&...Fri, 28 Aug 2026 10:04:56 -0400https://pop.com.sg/why-is-i-2-equal-to-1-duplicate.htmlhttps://pop.com.sg/why-is-i-2-equal-to-1-duplicate.htmlIn this KhanAcademy link at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why. Why is this so? What is the intuition behind it?...Fri, 28 Aug 2026 09:54:34 -0400https://pop.com.sg/block-matrix-multiplication.htmlhttps://pop.com.sg/block-matrix-multiplication.htmlIf $A,B$ are $2 \times 2$ matrices of real or complex numbers, then $$AB = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot \left[ \begin{array}{cc} b_{......Fri, 28 Aug 2026 09:54:13 -0400https://pop.com.sg/pick-5-box-play-pa-lottery-odds.htmlhttps://pop.com.sg/pick-5-box-play-pa-lottery-odds.htmlI was looking over the PA lottery odds and payout table for the Pick 5 "Quinto" game, specifically for the second row "Boxed - 5 in any order - 5 distinct digits" http://www.palottery.state.pa.us/......Fri, 28 Aug 2026 09:48:34 -0400https://pop.com.sg/how-long-does-the-ball-take-to-reach-half-of-its-terminal-velocity.htmlhttps://pop.com.sg/how-long-does-the-ball-take-to-reach-half-of-its-terminal-velocity.htmlA ball of mass m = 0.1kg falls from rest under the influence of gravity (on earth) in a medium that provides resistance that is proportional to its velocity. For a velocity of 0.2 m/s, the resistance ...Fri, 28 Aug 2026 09:44:02 -0400https://pop.com.sg/how-can-i-graph-the-derivative-of-1-4th-of-a-circle-or-a-semicircle-in.htmlhttps://pop.com.sg/how-can-i-graph-the-derivative-of-1-4th-of-a-circle-or-a-semicircle-in.htmlI'm having trouble with questions like these. In the first image, the original function is what is the two sharp lines and a semicircle in between. I understand how to find and graph the derivative......Fri, 28 Aug 2026 09:42:16 -0400https://pop.com.sg/sum-and-difference-of-tangents-in-degrees-duplicate.htmlhttps://pop.com.sg/sum-and-difference-of-tangents-in-degrees-duplicate.html$$\tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ$$ At first, I tried to break the tangent into multiple angles… $$\text{FAILED}$$ Then I proceeded by converting it into other trigonometric ...Fri, 28 Aug 2026 09:38:05 -0400https://pop.com.sg/adding-2-absolute-values-together-x-2-x-3-5-duplicate.htmlhttps://pop.com.sg/adding-2-absolute-values-together-x-2-x-3-5-duplicate.htmlI came across a very basic absolute value question $|x+2| + |x-3| =5.$ Initially, I thought the answer was $x=-2$ and $x=3$ because I let each absolute values be either positive and negative and ......Fri, 28 Aug 2026 09:36:55 -0400https://pop.com.sg/rectangular-box-optimization-problem.htmlhttps://pop.com.sg/rectangular-box-optimization-problem.htmlA rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs \$0.31 per square foot, the material for the sides costs $0.05 per square foot, and the materia......Fri, 28 Aug 2026 09:27:34 -0400https://pop.com.sg/create-an-isosceles-triangle-from-only-equilateral-triangles-possible-.htmlhttps://pop.com.sg/create-an-isosceles-triangle-from-only-equilateral-triangles-possible-.htmlMy friend recently asked me a question. And I think I have a nice solution but wanted to share this question here to see how you approach this problem and hopefully with a solid proof. The question......Fri, 28 Aug 2026 09:26:15 -0400https://pop.com.sg/expected-waiting-time-for-bus.htmlhttps://pop.com.sg/expected-waiting-time-for-bus.htmlSuppose that the inter-arrival time between consecutive buses is 15 minutes with probability 0.5 and 30 minutes with probability 0.5. You just arrived at a bus stop. What is your expected waiting t......Fri, 28 Aug 2026 09:19:59 -0400https://pop.com.sg/conditions-for-the-mean-value-theorem.htmlhttps://pop.com.sg/conditions-for-the-mean-value-theorem.htmlthe mean value theorem which most of us know starts with the conditions that $f$ is continuous on the closed interval $[a,b]$ and differentiable on the opened interval $(a,b)$, then there exists a ......Fri, 28 Aug 2026 09:13:34 -0400https://pop.com.sg/verification-of-proof-for-when-a-number-is-divisible-by-4.htmlhttps://pop.com.sg/verification-of-proof-for-when-a-number-is-divisible-by-4.htmlI have never taken a number theory course and so am only going off of the first few chapters in an introductory number theory book. The divisibility property I wish to prove is the following: Def......Fri, 28 Aug 2026 09:09:57 -0400https://pop.com.sg/root-test-applied-to-power-series.htmlhttps://pop.com.sg/root-test-applied-to-power-series.htmlWhen I apply the root test to power series I know that $$R= \frac{1}{\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_n|}}}$$ So if I get $1$ as the result of the limit the radius of convergence of the p......Fri, 28 Aug 2026 08:57:29 -0400https://pop.com.sg/subtract-binary-numbers-with-1-s-complement.htmlhttps://pop.com.sg/subtract-binary-numbers-with-1-s-complement.htmlI'm trying to figure out how to subtract two binary numbers with a complementary one or two, When I need to address carrier and when not? Do I need to solve the problem in decimal numbers? And ...Fri, 28 Aug 2026 08:56:09 -0400https://pop.com.sg/product-of-banded-matrices.htmlhttps://pop.com.sg/product-of-banded-matrices.htmlHow can one show that the product of two banded matrices is a banded matrix with upper and lower bandwidths equal to the sum of the upper and lower bandwidths (respectively) of the multiplicands ? ......Fri, 28 Aug 2026 08:52:25 -0400https://pop.com.sg/norm-of-multiplication-and-multiplication-of-norms.htmlhttps://pop.com.sg/norm-of-multiplication-and-multiplication-of-norms.htmlIt is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$. Is the following true for all $p \in \mathbb{R}$: $$\|u \cdot v \|_p \le \|u \|_p \cdo......Fri, 28 Aug 2026 08:51:24 -0400https://pop.com.sg/difference-between-magnitude-of-gradient-vs-directional-derivative-of-.htmlhttps://pop.com.sg/difference-between-magnitude-of-gradient-vs-directional-derivative-of-.htmlI've read that the directional derivative is the rate of change of a function $f$ in a given direction $\mathbf{v}$, given as $\nabla f\cdot \mathbf{v}$. I've also read (perhaps incorrectly) that the ...Fri, 28 Aug 2026 08:50:31 -0400https://pop.com.sg/how-does-the-second-derivative-imply-concavity.htmlhttps://pop.com.sg/how-does-the-second-derivative-imply-concavity.htmlconsider this equation, $$y=x(400-x)$$the second derivative of this equation is $$y''=-2$$ As far as I know, a negative sign in the second derivative indicates the curve will concave down. As it is a ...Fri, 28 Aug 2026 08:49:33 -0400https://pop.com.sg/optimal-strategy-for-deal-or-no-deal.htmlhttps://pop.com.sg/optimal-strategy-for-deal-or-no-deal.htmlWhen I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy t......Fri, 28 Aug 2026 08:46:23 -0400https://pop.com.sg/finding-the-exact-value-of-tan-pi-5.htmlhttps://pop.com.sg/finding-the-exact-value-of-tan-pi-5.htmlHi, I realise there has been a question already asked regarding this particular exact value, but this question requires for it to be done under different conditions, which is the part I require hel......Fri, 28 Aug 2026 08:33:23 -0400https://pop.com.sg/puzzle-123456789-100-with-three-operations.htmlhttps://pop.com.sg/puzzle-123456789-100-with-three-operations.htmlGiven the sequence 123456789: You can insert three operations ($+$,$-$,$\times$,$/$) into this sequence to make the equation = 100. My question is: is there a way to solve this without brute forc......Fri, 28 Aug 2026 08:30:30 -0400https://pop.com.sg/actually-playable-games-based-on-graphs.htmlhttps://pop.com.sg/actually-playable-games-based-on-graphs.htmlIn computer science lessons, we have recently got the task to program something using graphs. Due to my enthusiasm for computer games, i would really prefer to implement a concept for a game. The ...Fri, 28 Aug 2026 08:29:01 -0400https://pop.com.sg/how-to-show-c-n-k-c-n-n-k.htmlhttps://pop.com.sg/how-to-show-c-n-k-c-n-n-k.htmlI am doing a question from the textbook "Calculus and Statistics" by Gemigani. If $n = 2m$ and $k= 1,2, \ldots ,m$ Prove that $C(n,k)= C(n,n-k)$ Ok so my approach begins with writing out the for......Fri, 28 Aug 2026 08:21:55 -0400https://pop.com.sg/find-a-second-degree-polynomial-that-goes-through-3-points.htmlhttps://pop.com.sg/find-a-second-degree-polynomial-that-goes-through-3-points.htmlI am having trouble calculating the quadratic curve $f(x)$ that goes through 3 points; for example a curve that goes through $A(1,3), B(-1,-5), and C(-2,12)$. I can only guess that the curve is upw......Fri, 28 Aug 2026 08:17:37 -0400https://pop.com.sg/why-is-there-no-online-record-of-pentagram-area-formula-closed.htmlhttps://pop.com.sg/why-is-there-no-online-record-of-pentagram-area-formula-closed.htmlI'm talking about the short version of the area of a pentagram inscribed in a circle when the radius is given and so Area $=1.123r^2$. I only got this formula from a book but I researched about it......Fri, 28 Aug 2026 08:14:30 -0400https://pop.com.sg/how-to-find-the-roots-of-x-4-1.htmlhttps://pop.com.sg/how-to-find-the-roots-of-x-4-1.htmlI'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a......Fri, 28 Aug 2026 08:14:18 -0400https://pop.com.sg/differential-geometry-and-transition-maps-as-communative-diagrams.htmlhttps://pop.com.sg/differential-geometry-and-transition-maps-as-communative-diagrams.htmlConsider the picture below. I know pictures are usually looked down upon on this site but I have no experience with TikZ. Anyway, I am learning about regular surfaces and transition maps in differ......Fri, 28 Aug 2026 08:10:19 -0400https://pop.com.sg/the-derivative-of-log-2-n-5.htmlhttps://pop.com.sg/the-derivative-of-log-2-n-5.htmlThe derivative of $ (\log_2 n)^5$? (log base 2) Hey everyone, I am not sure how to go about this question because I am not sure what to do with the power $5$ ( or any other power ) in the log? Shou......Fri, 28 Aug 2026 08:07:22 -0400https://pop.com.sg/finding-all-orthogonal-matrices-of-a-given-form.htmlhttps://pop.com.sg/finding-all-orthogonal-matrices-of-a-given-form.htmlI am given this problem. I need to find all $a,b,c,d \in \mathbb{R}$ for which the matrix $A$ is orthogonal. $$A:=\frac{1}{3} \begin{pmatrix} -1 & 2 & a \\ 2 & 2 & b \\ 2 & c ......Fri, 28 Aug 2026 08:03:27 -0400https://pop.com.sg/lusin-039-s-theorem-proof-verification-folland-chap-2-exercise-44.htmlhttps://pop.com.sg/lusin-039-s-theorem-proof-verification-folland-chap-2-exercise-44.htmlConisder Lusin's Theorem (Folland Chap 2 Ex 44): If $f: [a,b] \rightarrow \mathbb{C}$ is Lebesgue measurable and $\epsilon > 0$, there is a compact subset $E \in [a,b]$ s.t. $\mu(E^C) < \...Fri, 28 Aug 2026 08:00:38 -0400https://pop.com.sg/can-i-break-up-log-a-b.htmlhttps://pop.com.sg/can-i-break-up-log-a-b.htmlFor constants $a$ and $b$, I know that I can break up $\log(a/b)$ into $\log(a) - \log(b)$. Can I conveniently break up $\log(a - b)$ somehow into several terms?...Fri, 28 Aug 2026 07:52:45 -0400https://pop.com.sg/how-to-find-whether-equation-of-angle-bisector-represents-the-obtuse-o.htmlhttps://pop.com.sg/how-to-find-whether-equation-of-angle-bisector-represents-the-obtuse-o.htmlTwo lines: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are given. I know that the equation of its bisectors is ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\...Fri, 28 Aug 2026 07:48:41 -0400https://pop.com.sg/chance-on-throwing-a-six-with-6-dice.htmlhttps://pop.com.sg/chance-on-throwing-a-six-with-6-dice.htmlThe chance to throw a 6 with one die is 1/6 And 6 times 1/6 = 1 So, if I throw with 6 dice, the chance to throw at least 1 six should be 1. But when I throw 6 dice, I sometimes don't throw any 6......Fri, 28 Aug 2026 07:47:09 -0400https://pop.com.sg/why-is-gradient-the-direction-of-steepest-ascent.htmlhttps://pop.com.sg/why-is-gradient-the-direction-of-steepest-ascent.html$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a ......Fri, 28 Aug 2026 07:42:25 -0400https://pop.com.sg/sequence-series-convergence-of-2-2n-3-1-n.htmlhttps://pop.com.sg/sequence-series-convergence-of-2-2n-3-1-n.htmlthis step in the proof is confusing me: $$\sum_1^\infty {\frac{4^{n}}{3^{n-1}}}\qquad \longrightarrow \qquad\sum_1^\infty 4\left(\frac43\right)^{n-1}$$ please explain how/why this happened? che......Fri, 28 Aug 2026 07:40:14 -0400https://pop.com.sg/vertex-and-edge-connectivity-of-a-graph.htmlhttps://pop.com.sg/vertex-and-edge-connectivity-of-a-graph.htmlI want to draw a graph of 8 vertices and 16 edges with maximum vertex connectivity and maximum edge connectivity and also draw a graph with minimum vertex connectivity and minimum edge ...Fri, 28 Aug 2026 07:40:00 -0400https://pop.com.sg/general-formula-for-a-diagonal-parabola.htmlhttps://pop.com.sg/general-formula-for-a-diagonal-parabola.htmlWhat is the general formula for a diagonal parabola facing a given direction with a given vertex (x,y)?...Fri, 28 Aug 2026 07:33:18 -0400https://pop.com.sg/natural-log-of-pi-to-sixty-digits-a-reference-needed.htmlhttps://pop.com.sg/natural-log-of-pi-to-sixty-digits-a-reference-needed.htmlI have found a table of the logs of gamma functions at basic fractions, accurate to 60 decimal digits. It omits $\ln(\Gamma(1/2)) = \ln(\pi)/2$ and I want that number. Where is a cit-able sourc......Fri, 28 Aug 2026 07:29:49 -0400https://pop.com.sg/function-with-an-asymptote-at-y-1-and-y-1-closed.htmlhttps://pop.com.sg/function-with-an-asymptote-at-y-1-and-y-1-closed.htmlI'm looking for a function that has two asymptotes parallel to the x-axis. Preferably it should also only cross the x-axis at (0,0) and be built without using any trigonometric functions. Mind you,......Fri, 28 Aug 2026 07:26:38 -0400https://pop.com.sg/calculating-linear-velocity.htmlhttps://pop.com.sg/calculating-linear-velocity.htmlI have the question Use the answer to question 1 to calculate the linear velocity of a point on the surface of the Earth (radius of Earth = $6.4 \times 10^6$ [m]). So the answer for question 1 ......Fri, 28 Aug 2026 07:25:03 -0400https://pop.com.sg/normalization-condition-and-phase-constant.htmlhttps://pop.com.sg/normalization-condition-and-phase-constant.htmlI have been given a wave function and been tasked with verifying it has been normalized. $$\psi(x, 0) = \left(\frac{2\alpha}{\pi}\right)^{1/4} e^{-ikx} e^{-\alpha x^2}$$. I've normalized......Fri, 28 Aug 2026 07:24:16 -0400https://pop.com.sg/extended-3-square-problem.htmlhttps://pop.com.sg/extended-3-square-problem.htmlWe know about the famous 3-square problem. Draw three adjacent squares with same size and any two adjacent square has a common side. This looks like a rectangle. Name all of the vertices of squ......Fri, 28 Aug 2026 07:21:56 -0400https://pop.com.sg/negation-of-the-definition-of-limit-of-a-function.htmlhttps://pop.com.sg/negation-of-the-definition-of-limit-of-a-function.htmlQuestion Suppose we are dealing with real valued functions $f(x)$ of one real variable $x$. We say that the limit of $f(x)$ at point $a$ exists and equals to $L$ if and only if $$\exists L:\lef......Fri, 28 Aug 2026 07:11:44 -0400https://pop.com.sg/analyticity-and-removable-singularity.htmlhttps://pop.com.sg/analyticity-and-removable-singularity.htmlIn Churchill's book of Complex Analysis there are two statements that I can't match them to be consistent: In one place it says that a function must be analytic at a removable singular point : why......Fri, 28 Aug 2026 07:09:32 -0400https://pop.com.sg/how-should-i-study-the-matrix-cookbook.htmlhttps://pop.com.sg/how-should-i-study-the-matrix-cookbook.htmlI use The Matrix Cookbook by Kaare Brandt Petersen and Michael Syskind Pedersen to solve many problems (mostly matrix derivatives). In most cases, I just map the problem to one of the formula and s......Fri, 28 Aug 2026 06:53:25 -0400https://pop.com.sg/area-of-a-circular-segment.htmlhttps://pop.com.sg/area-of-a-circular-segment.htmlSee the picture below: How can I calculate the area in black, using no handy formulas which will give me the answer if I plug in the right values? I had the idea to take $\displaystyle \int_{0.5r}......Fri, 28 Aug 2026 06:52:00 -0400https://pop.com.sg/proof-of-the-deduction-theorem-explanation.htmlhttps://pop.com.sg/proof-of-the-deduction-theorem-explanation.htmlI'm reading through this proof of the deduction theorem, and there are a few things I don't understand. The basic idea is to show that if $\Gamma\cup \{A\}\vdash B$, then we have a proof of $B$ wi......Fri, 28 Aug 2026 06:33:19 -0400https://pop.com.sg/how-are-c-0-c-1-norms-defined.htmlhttps://pop.com.sg/how-are-c-0-c-1-norms-defined.htmlHow are $C^0,C^1$ norms defined? I know $L_p,L_\infty$ norms but are the former defined....Fri, 28 Aug 2026 06:26:24 -0400https://pop.com.sg/can-you-disprove-this-rule-pedmsa-division-before-multiplication-subtr.htmlhttps://pop.com.sg/can-you-disprove-this-rule-pedmsa-division-before-multiplication-subtr.htmlI know the proper teaching of PEMDAS/PEDMAS/BODMAS/BOMDAS.. is that multiplication and division have equal priority and are evaluated going through the arithmetic expression left to right (So, doing ...Fri, 28 Aug 2026 06:25:58 -0400https://pop.com.sg/what-is-approximation-theory.htmlhttps://pop.com.sg/what-is-approximation-theory.htmlWhat exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis a......Fri, 28 Aug 2026 06:23:41 -0400https://pop.com.sg/what-is-an-imaginary-line.htmlhttps://pop.com.sg/what-is-an-imaginary-line.htmlAn imaginary point is defined as an ordered pair of values, at least one of which is complex. My text says that if $h^2 < ab$ in the equation : $$ ax^2 + 2hxy + by^2 $$ it represents two imagi......Fri, 28 Aug 2026 06:22:13 -0400https://pop.com.sg/nice-examples-of-groups-which-are-not-obviously-groups.htmlhttps://pop.com.sg/nice-examples-of-groups-which-are-not-obviously-groups.htmlI am searching for some groups, where it is not so obvious that they are groups. In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't ...Fri, 28 Aug 2026 06:11:50 -0400https://pop.com.sg/formula-for-continuously-dividing-by-2.htmlhttps://pop.com.sg/formula-for-continuously-dividing-by-2.htmlI was helping my daughter with her physics homework. The problem had a bee flying back and forth between two bikes moving toward each other. The question was to see how far the bee traveled. Turned......Fri, 28 Aug 2026 06:02:34 -0400https://pop.com.sg/dividing-matrices.htmlhttps://pop.com.sg/dividing-matrices.htmlHI Im learning about matrices in school and am curious about how to divide them so i can find out different identity matrices for refecting shapes over linear functions...Fri, 28 Aug 2026 05:55:20 -0400https://pop.com.sg/prove-that-a-x-b-y-0-where-x-sqrt-log-a-b-y-sqrt-log-b-a-a-0-b-0-a-b-n.htmlhttps://pop.com.sg/prove-that-a-x-b-y-0-where-x-sqrt-log-a-b-y-sqrt-log-b-a-a-0-b-0-a-b-n.htmlI'm solving logarithm questions. I got stuck in this question. Prove that $a^x - b^y = 0$ where $x = \sqrt{\log_a b}$ & $y = \sqrt{\log_b a}$ , $a > 0$, $b > 0$ & $a, b \ne1$ I've ......Fri, 28 Aug 2026 05:47:28 -0400https://pop.com.sg/eigenvalues-of-product-matrix.htmlhttps://pop.com.sg/eigenvalues-of-product-matrix.htmlI have two matrices, both positive definite, real symmetric and one is diagonal. What can I say about lower and upper bound of the eigenvalues of the product matrix in terms the of lower and upper ......Fri, 28 Aug 2026 05:46:21 -0400https://pop.com.sg/what-is-the-purpose-of-the-limit.htmlhttps://pop.com.sg/what-is-the-purpose-of-the-limit.htmlI haven't taken Calculas yet, but I see the use of limit (approaching zero or infinity) in other classes such as physics. I just wanted some intuitive explanation of the limit. I was thinking t......Fri, 28 Aug 2026 05:39:00 -0400https://pop.com.sg/prove-that-two-equations-are-equivalent.htmlhttps://pop.com.sg/prove-that-two-equations-are-equivalent.htmlEDIT: Missed something very important! Sorry! We have $x^4+1=2(2x-1)^{1/4}$ not $x^4+1=2\sqrt{2x-1}$. One friend of mine told me that the equation $x^4+1=2(2x-1)^{1/4}$, where $x\geq \frac{1}{2}$ is ...Fri, 28 Aug 2026 05:34:59 -0400https://pop.com.sg/finding-d-in-rsa.htmlhttps://pop.com.sg/finding-d-in-rsa.htmlSuppose your RSA modulus is $55 = 5 * 11$ and your encryption exponent is $e = 3$. Find the decryption modulus d. I know $d = 40-13 = 27$ However, I get $1$. $$40 = (P_1-1)(P_2-1)$$ extended ...Fri, 28 Aug 2026 05:33:27 -0400https://pop.com.sg/cartesian-product-with-three-sets.htmlhttps://pop.com.sg/cartesian-product-with-three-sets.htmlI was given a question that asks "Let A={1,2} ,B={x,3,y},C={2,y}. Find A x B x C and C x B x C " I have an example that I could go by but i'm not sure what they were multiplying together to get it.......Fri, 28 Aug 2026 05:28:43 -0400https://pop.com.sg/integration-with-respect-to-dx-dy-and-dz-more-than-one-variable.htmlhttps://pop.com.sg/integration-with-respect-to-dx-dy-and-dz-more-than-one-variable.htmlSorry if my title was vague but i was not entirely sure what its called. Anyways i was solving some work and energy problems and encountered this integration: $$\int_{2,1,4}^{2,-3,3} 2x\sin^2y \......Fri, 28 Aug 2026 05:22:33 -0400https://pop.com.sg/numbers-in-a-list-which-are-perfect-squares-and-perfect-cubes-of-numbe.htmlhttps://pop.com.sg/numbers-in-a-list-which-are-perfect-squares-and-perfect-cubes-of-numbe.htmlHow many numbers in the list $$1,2,3,...,2001$$ are perfect squares and perfect cubes of whole numbers? My progress: Well I do know $$1,4,9,16,25,36,...$$ are perfect squares and $$1,8,27,64,...$$......Fri, 28 Aug 2026 05:14:59 -0400https://pop.com.sg/how-do-i-find-the-value-of-a-partial-sum-without-calculator.htmlhttps://pop.com.sg/how-do-i-find-the-value-of-a-partial-sum-without-calculator.htmlWithout using a calculator, how do I find: $$\sum_{k=0}^7 \left(\tfrac{2}{3}\right)^k $$ I know about the formula for the sum of a geometric series $\frac{a\cdot(1-r^n)}{(1-r)}$ but im not sure h......Fri, 28 Aug 2026 04:53:57 -0400https://pop.com.sg/does-anyone-know-a-simple-proof-for-the-fundamental-theorem-of-finitel.htmlhttps://pop.com.sg/does-anyone-know-a-simple-proof-for-the-fundamental-theorem-of-finitel.htmlI am looking for a simple/short proof for the fundamental theorem of finitely generated abelian group. Let $(G,\ +)$ be an abelian group. Assume $G$ is finitely generated. Prove that there exist ......Fri, 28 Aug 2026 04:25:46 -0400https://pop.com.sg/wedge-sum-example.htmlhttps://pop.com.sg/wedge-sum-example.htmlI am trying to understand the concept of wedge sum in the context of Algebraic Topology. Say I am given two sets, $A=[1, 2, 3]$ and $B=[4, 5]$ Now, disjoint union $A \sqcup B$ = $[(1,0), (2,0), (......Fri, 28 Aug 2026 04:23:11 -0400https://pop.com.sg/using-fsolve-to-solve-non-linear-equations-in-matlab.htmlhttps://pop.com.sg/using-fsolve-to-solve-non-linear-equations-in-matlab.htmlI'm trying to solve this system equations in matlab syms y z i x=[29.6,29.4,34.4,35.1,34.3,36.1,31.6,32.9,31.5,28.3,37.1,28.4,27.3,33.1,31.3,33.1] n = 16 eqns = sym(zeros(1, 2 * n)); for i = 1:n......Fri, 28 Aug 2026 04:21:10 -0400https://pop.com.sg/is-this-the-correct-notation.htmlhttps://pop.com.sg/is-this-the-correct-notation.html$\left [ \left \{ 1,2,3,4 \right \}! \right]^{-1}$Is this the correct notation if one wanted to obtain the factorial for each number in a sequence and then take the sequence and inverse each number......Fri, 28 Aug 2026 04:20:49 -0400https://pop.com.sg/why-does-the-sum-of-inverse-squares-equal-pi-2-6-duplicate.htmlhttps://pop.com.sg/why-does-the-sum-of-inverse-squares-equal-pi-2-6-duplicate.htmlI've heard that $1+1/4+1/9+1/16+1/25+...$ converges to $\pi^2/6$. This was very surprising to me and I was wondering if there was a reason that it converges to this number? I am also confused why......Fri, 28 Aug 2026 04:00:35 -0400