https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 13 Aug 2026 07:03:38 -0400https://pop.com.sg/deriving-sin-ix-i-sinh-x.htmlhttps://pop.com.sg/deriving-sin-ix-i-sinh-x.htmlDerive $\sin{ix}=i\sinh{x}$ from $(5)$. What is $\sin{i}$? $$\cos{x}=\frac{1}{2}\left(e^{ix}+e^{-ix}\right)\quad\text{and}\quad\sin{x}=\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)\tag 5$$ We have $$\......Thu, 13 Aug 2026 11:02:33 -0400https://pop.com.sg/what-are-the-uses-of-euler-039-s-number-e.htmlhttps://pop.com.sg/what-are-the-uses-of-euler-039-s-number-e.htmlPeople make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ......Thu, 13 Aug 2026 11:01:38 -0400https://pop.com.sg/user-jialin-zhang-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-jialin-zhang-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 10:56:01 -0400https://pop.com.sg/find-the-domain-of-the-given-function.htmlhttps://pop.com.sg/find-the-domain-of-the-given-function.htmlI have the function $f(x)=\cos^{-1}(\frac{1}{2\cos(x)})$ and I have to find its domain. What I know is that the domain of $\cos^{-1}(x)$ is $[-1,1]$ so I think that $\frac{1}{2\cos(x)}$ should be at ...Thu, 13 Aug 2026 10:55:27 -0400https://pop.com.sg/what-is-the-time-complexity-of-euclid-039-s-algorithm-upper-bound-lowe.htmlhttps://pop.com.sg/what-is-the-time-complexity-of-euclid-039-s-algorithm-upper-bound-lowe.htmlI looked it up online in many sites but none give a clear answer. They all give a lot of complicated mathematical stuff which is not only hard for me to grasp but also irrelevant as I simply want t......Thu, 13 Aug 2026 10:53:21 -0400https://pop.com.sg/how-to-prove-that-n-frac-1-3-is-not-a-polynomial.htmlhttps://pop.com.sg/how-to-prove-that-n-frac-1-3-is-not-a-polynomial.htmlI'm reading Barbeau's Polynomials, there's an exercise: How to prove that $n^{\frac{1}{3}}$ is not a polynomial? I've made this question and with the first answer as an example, I guess I should ...Thu, 13 Aug 2026 10:48:38 -0400https://pop.com.sg/solve-for-a-and-b-of-this-piecewise-function-such-that-the-function-f-.htmlhttps://pop.com.sg/solve-for-a-and-b-of-this-piecewise-function-such-that-the-function-f-.htmlConsider the function defined by \begin{align*} f(x) = \begin{cases} ax^{2} + x - b, & x < 2\\ ax+b, & 2\leq x \leq 5\\ 2ax - 7, & x > 5. \end{cases} \end{align*} I was able to ......Thu, 13 Aug 2026 10:48:13 -0400https://pop.com.sg/expectation-of-absolute-difference-of-two-uniform-random-variables.htmlhttps://pop.com.sg/expectation-of-absolute-difference-of-two-uniform-random-variables.htmlLet $X \sim U(0, 1)$ and $Y \sim U(0, 2)$ be independent random variables. Find $\mathbb{E}[|X - Y|]$. I found $f_{X}(x) = 1$ and $f_{Y}(y) = 1/2$, and I took an integral to find $P(Y > X) = 3......Thu, 13 Aug 2026 10:45:45 -0400https://pop.com.sg/determine-the-fourth-roots-of-16-duplicate.htmlhttps://pop.com.sg/determine-the-fourth-roots-of-16-duplicate.htmlDetermine the fourth roots of -16 in the form $x +iy$ where $x$ and $y$ are not trigonometric functions. I do not even know what they really want from me in this question. My initial thought wass......Thu, 13 Aug 2026 10:45:44 -0400https://pop.com.sg/a-lot-of-confusion-in-the-polynomial-remainder-theorem.htmlhttps://pop.com.sg/a-lot-of-confusion-in-the-polynomial-remainder-theorem.htmlLately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynom......Thu, 13 Aug 2026 10:43:38 -0400https://pop.com.sg/lie-group-and-lie-algebra-homomorphisms.htmlhttps://pop.com.sg/lie-group-and-lie-algebra-homomorphisms.htmlI am reading Brian C Hall's Lie Groups, Lie Algebras, and Representations and trying to do one of the exercises about showing that isomorphic matrix Lie groups give rise to isomorphic Lie Algebras.......Thu, 13 Aug 2026 10:33:00 -0400https://pop.com.sg/find-closest-power-of-2-to-a-specific-number.htmlhttps://pop.com.sg/find-closest-power-of-2-to-a-specific-number.htmlThis question is weird and it's not a homework question. I couldn't come up with anything substantial, so sorry if you think that I should have posted my tried methods (I have tried but most of the......Thu, 13 Aug 2026 10:31:51 -0400https://pop.com.sg/you-can-039-t-find-an-equation-for-the-cross-sectional-shape-when-give.htmlhttps://pop.com.sg/you-can-039-t-find-an-equation-for-the-cross-sectional-shape-when-give.htmlWhy not? I'm trying to design a calculus I problem. I want them to eventually prove that the volume of a cone (oblique or even with irregular base) is $\frac{\pi r^2 h}{3}$, I also want them to know ...Thu, 13 Aug 2026 10:18:42 -0400https://pop.com.sg/is-a-omega-spectrum-a-connective-one.htmlhttps://pop.com.sg/is-a-omega-spectrum-a-connective-one.htmlI can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback Let $X$ be a $\Omega$-spectrum ......Thu, 13 Aug 2026 10:18:21 -0400https://pop.com.sg/what-are-we-really-doing-when-we-integrate-a-complex-exponential.htmlhttps://pop.com.sg/what-are-we-really-doing-when-we-integrate-a-complex-exponential.htmlFirstly, I have read this related question from this site, but it does not answer what I am asking here. What I like to know is why $$\int_{-n}^n e^{ix}dx \ne 0$$ for any finite $n \in \mathbb N$ ......Thu, 13 Aug 2026 10:04:47 -0400https://pop.com.sg/proper-suffix-suffix-and-empty-string.htmlhttps://pop.com.sg/proper-suffix-suffix-and-empty-string.htmlI found this definition on Wikipedia. Suffix. A suffix of a string is any substring of the string which includes its last letter, including itself. A proper suffix of a string is not equal to the ......Thu, 13 Aug 2026 10:03:14 -0400https://pop.com.sg/deriving-the-hamilton-product-of-two-quaternions-for-spatial-rotation.htmlhttps://pop.com.sg/deriving-the-hamilton-product-of-two-quaternions-for-spatial-rotation.htmlI work in computational chemistry, and I want make a simple Python program that rotates a molecular structure by some angle $\alpha$ around some axis defined by the unit vector $\vec{v}$. I define......Thu, 13 Aug 2026 09:57:41 -0400https://pop.com.sg/is-this-derivation-of-the-path-length-formula-actually-correct.htmlhttps://pop.com.sg/is-this-derivation-of-the-path-length-formula-actually-correct.htmlSaw this in a physics lecture. This all assumes we have some function, $y=f(x)$. First he defined $$ds = \sqrt{dx^2+dy^2},$$ where the professor drew a picture and seemed to be using dx and dy t......Thu, 13 Aug 2026 09:41:15 -0400https://pop.com.sg/probability-that-exactly-k-bins-are-empty-given-m-balls-and-n-bins.htmlhttps://pop.com.sg/probability-that-exactly-k-bins-are-empty-given-m-balls-and-n-bins.htmlI've searched for an understandable answer to this exact question and have failed to find it. How do you find the probability that exactly $k$ bins are empty, given $m$ balls and $n$ bins? (Each b......Thu, 13 Aug 2026 09:33:22 -0400https://pop.com.sg/definition-of-adjoint-operator-for-quantum-mechanics.htmlhttps://pop.com.sg/definition-of-adjoint-operator-for-quantum-mechanics.htmlWhile learning about adjoint operators for quantum mechanics, I encountered two definitions. The first definition is given by Shankar in The Principle of Quantum Mechanics: Given a ket $$ A\lve......Thu, 13 Aug 2026 09:28:58 -0400https://pop.com.sg/expected-value-of-e-tx-moment-generating-function.htmlhttps://pop.com.sg/expected-value-of-e-tx-moment-generating-function.htmlThis is a question from a Statistics class exam that I still am not sure on how to complete. The number of accidents in a week follows a discrete random variable X with a probability function x|......Thu, 13 Aug 2026 09:11:53 -0400https://pop.com.sg/what-does-50-choose-4-mean-in-statistics.htmlhttps://pop.com.sg/what-does-50-choose-4-mean-in-statistics.htmlI have a test tomorrow in statistics and was wondering what the following means? $$\binom{50}{4}$$ My professor along with most of my classmates have a calculator they can just plug that into. The ...Thu, 13 Aug 2026 09:06:57 -0400https://pop.com.sg/summing-digits-of-powers-of-2-to-get-1-2-4-8-7-5-pattern.htmlhttps://pop.com.sg/summing-digits-of-powers-of-2-to-get-1-2-4-8-7-5-pattern.htmlIf you add the decimal digits of multiples of 9,possibly repeatedly, you get 9. For instance, with 18 we have $$18\mapsto 1+8=9$$ and with 909 we have $$ 909\mapsto 9+0+9=18\mapsto 1+8=9.$$ This is ...Thu, 13 Aug 2026 08:59:23 -0400https://pop.com.sg/why-does-operatorname-var-x-e-x-2-e-x-2.htmlhttps://pop.com.sg/why-does-operatorname-var-x-e-x-2-e-x-2.html$ \operatorname{Var}(X) = E[X^2] - (E[X])^2 $ I have seen and understand (mathematically) the proof for this. What I want to understand is: intuitively, why is this true? What does this formula te......Thu, 13 Aug 2026 08:55:21 -0400https://pop.com.sg/covariance-of-normal-distribution.htmlhttps://pop.com.sg/covariance-of-normal-distribution.htmlSuppose we have two normal distribution $\mathcal{N}(0,s)$, $\mathcal{N}(0,t)$. how can I find the covariance if I don't have the distribution function $f(x,y)$ or anything, not given the correla......Thu, 13 Aug 2026 08:55:15 -0400https://pop.com.sg/how-to-transform-fraction-powers-between-numerator-and-denominator.htmlhttps://pop.com.sg/how-to-transform-fraction-powers-between-numerator-and-denominator.htmlI apologize in advance if this question is trivial to most of you but I'd like to verify if my understanding is correct. I want to verify the following rules: If we have $\dfrac{1}{\sqrt{x}}$, the......Thu, 13 Aug 2026 08:32:30 -0400https://pop.com.sg/how-to-integrate-a-4th-power-of-sine-and-cosine.htmlhttps://pop.com.sg/how-to-integrate-a-4th-power-of-sine-and-cosine.htmlI'm having some trouble figuring out the right substitutions to make to integrate $$\int \sin^4(\theta)d\theta$$ and $$\int \cos^4(\theta)d\theta$$ Any hints or suggestions are welcome. Thanks,...Thu, 13 Aug 2026 08:31:49 -0400https://pop.com.sg/discriminant-of-multivariable-function.htmlhttps://pop.com.sg/discriminant-of-multivariable-function.htmlHey I'm doing an online course and I just can't figure out what I'm doing wrong on one of the questions. The question is to find the discriminant of the function $$f(x,y) = 5x^2y^2 + 8x^2 + 9y^2$$ ......Thu, 13 Aug 2026 08:28:22 -0400https://pop.com.sg/finding-a-logarithmic-function-from-a-graph.htmlhttps://pop.com.sg/finding-a-logarithmic-function-from-a-graph.htmlHere's the graph. When I use the points $(-1,1)$ or $(-3,2)$ to use in the equation $a\log(-x-1)+k$, I can't find a finite value for k. Any ideas?...Thu, 13 Aug 2026 08:25:19 -0400https://pop.com.sg/writing-an-objective-function-such-that-the-problem-has-an-optimal-sol.htmlhttps://pop.com.sg/writing-an-objective-function-such-that-the-problem-has-an-optimal-sol.htmli know it's basic but I'm struggling with how to find the objective function of a linear program. How do i write an objective function given only two constraints of a linear program in three variab......Thu, 13 Aug 2026 08:24:18 -0400https://pop.com.sg/what-are-the-formulas-for-topological-transformations-how-to-obtain-th.htmlhttps://pop.com.sg/what-are-the-formulas-for-topological-transformations-how-to-obtain-th.htmlI'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi......Thu, 13 Aug 2026 08:23:24 -0400https://pop.com.sg/what-is-the-probability-of-selecting-a-red-marble-then-a-blue-marble-i.htmlhttps://pop.com.sg/what-is-the-probability-of-selecting-a-red-marble-then-a-blue-marble-i.htmlA bag contains 4 green marbles, 9 red marbles, 10 orange marbles, 9 brown marbles, and 11 blue marbles. You choose a marble, replace it, and choose again. What is P (red, then blue) This is my que......Thu, 13 Aug 2026 08:20:23 -0400https://pop.com.sg/when-to-use-normal-approximation.htmlhttps://pop.com.sg/when-to-use-normal-approximation.htmlDo we use normal approximation when discrete distributions are hard to solve? For example, $P(X\ge 7000)$ where is $X\sim\operatorname{Binomial}(13000, 0.7)$. Obviously, summing each case manually ...Thu, 13 Aug 2026 08:14:27 -0400https://pop.com.sg/finding-c-in-a-definite-integral-with-no-reference-point.htmlhttps://pop.com.sg/finding-c-in-a-definite-integral-with-no-reference-point.htmlHow to find C in this equation? $$\int_0^x f(t)~dt=\int_x^1t^2f(t)~dt + x^2/4 + x^4/8 +C$$ I attempted to move over $\int_0^x f(t)~dt$ to the right side, so that I could solve a definite integral......Thu, 13 Aug 2026 07:51:39 -0400https://pop.com.sg/problem-comprehending-the-representation-of-complex-numbers-by-a-unit-.htmlhttps://pop.com.sg/problem-comprehending-the-representation-of-complex-numbers-by-a-unit-.htmlI'm having a very basic problem with understanding the representation of complex numbers using the unit circle. An example of what I'm struggling with is shown in the initial diagram on this page ......Thu, 13 Aug 2026 07:45:34 -0400https://pop.com.sg/how-do-i-evaluate-the-following-derivative.htmlhttps://pop.com.sg/how-do-i-evaluate-the-following-derivative.htmlHow can I evaluate the derivative $$\left. \frac{d}{dx} \left( x \, \ln(x^5) \right)\right|_{x=1} \, ?$$ I have this math problem, and the biggest thing I do not understand, is which derivative ru......Thu, 13 Aug 2026 07:23:48 -0400https://pop.com.sg/describe-the-galois-group-of-a-field-extension.htmlhttps://pop.com.sg/describe-the-galois-group-of-a-field-extension.htmlI'm struggling to understand the basics of Galois theory. One of the things I don't understand is how to actually derive automorphisms of a field extension. Let's say you had a simple problem: $x^......Thu, 13 Aug 2026 07:17:45 -0400https://pop.com.sg/linear-transformations-and-polynomials.htmlhttps://pop.com.sg/linear-transformations-and-polynomials.htmlI've been having a really hard time recently understanding linear transformations. I would really like to know conceptually what exactly is going on. Let $T: P_3 \rightarrow P_2$ be defined by ......Thu, 13 Aug 2026 07:16:02 -0400https://pop.com.sg/user-christopher-d-long-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-christopher-d-long-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 07:14:02 -0400https://pop.com.sg/definition-of-finite-elements-ciarlet.htmlhttps://pop.com.sg/definition-of-finite-elements-ciarlet.htmlI dont understand this definition of finite elements: The triple $(K,P,\Sigma)$ with $K \subset \mathbb{R}^n$ connected lipschitz domain with non-empty interior $P$ finite dimensional vector spa......Thu, 13 Aug 2026 07:03:01 -0400https://pop.com.sg/what-is-the-approximate-area-of-the-shaded-region-of-the-given-figure.htmlhttps://pop.com.sg/what-is-the-approximate-area-of-the-shaded-region-of-the-given-figure.htmlHow do i find the area of the black shaded portion of the circle? I noticed the 4 so i think that's the radius. The formula to find the area $$A=πr^2$$ so I thought of using that to find the area ......Thu, 13 Aug 2026 07:00:12 -0400https://pop.com.sg/calculus-i-optimization-problem-maximization-of-profit.htmlhttps://pop.com.sg/calculus-i-optimization-problem-maximization-of-profit.htmlCalculus problem that I've been trying to get my head around. Problem: A company can sell 20 products if it charges $40 per product. For each dollar decrease or increase in the price, the compan......Thu, 13 Aug 2026 06:54:32 -0400https://pop.com.sg/integral-test-proof.htmlhttps://pop.com.sg/integral-test-proof.htmlCould someone explain what does it mean by "(and hence $\sum a_n$)"? Isn't $a_{n+1} \leq a_n$ for all $n$? How does the convergence of $\sum a_{n+1}$ imply $\sum a_n$ converges when we have $\sum ......Thu, 13 Aug 2026 06:48:48 -0400https://pop.com.sg/why-is-the-gradient-always-perpendicular-to-level-curves.htmlhttps://pop.com.sg/why-is-the-gradient-always-perpendicular-to-level-curves.htmlLet $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points: $$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$ Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when ...Thu, 13 Aug 2026 06:47:46 -0400https://pop.com.sg/why-is-the-area-of-an-equilateral-triangle-frac-sqrt-3-4-b-2.htmlhttps://pop.com.sg/why-is-the-area-of-an-equilateral-triangle-frac-sqrt-3-4-b-2.htmlWhy is the area of an equilateral triangle $$\frac{\sqrt{3}}{4}b^2$$ and not $$\frac12b^2$$ or $$bh$$...Thu, 13 Aug 2026 06:40:40 -0400https://pop.com.sg/question-on-applying-jensen-inequality-on-logarithm-of-a-sum.htmlhttps://pop.com.sg/question-on-applying-jensen-inequality-on-logarithm-of-a-sum.htmlI am confused by the meaning of the $t \in (0,1)$ parameter in Jensen's inequality $$ f( tx_1+(1-t)x_2) \le tf(x_1)+(1-t)f(x_2) $$ When I apply this to the logarithm $$ \log( tx_1+(1-t)x_2) \ge t \......Thu, 13 Aug 2026 06:39:30 -0400https://pop.com.sg/the-range-of-arc-cotangent-function-arccot-1.htmlhttps://pop.com.sg/the-range-of-arc-cotangent-function-arccot-1.htmlWe know that the range of arc-cotangent function is $(0,π)$ and we I calculate the value of $cot^{-1}(-1)$ by a calculator, I get ($-π/4$) Which is clearly not included in the range !! Why isn't......Thu, 13 Aug 2026 06:37:21 -0400https://pop.com.sg/explanation-of-proof-of-logarithm-power-rule.htmlhttps://pop.com.sg/explanation-of-proof-of-logarithm-power-rule.htmlCan someone explain how to get from steps $(1)$ to $(2)$ in this proof of the logarithm power rule? How is the "equivalent exponential expression found? The power rule: For any positive $x$, ......Thu, 13 Aug 2026 06:35:08 -0400https://pop.com.sg/imaginary-and-complex-dimensions.htmlhttps://pop.com.sg/imaginary-and-complex-dimensions.htmlSo I saw a really intriguing question here, and I thought, "Why stop at integers? Why stop at rationals? Why even stop at real numbers?" So I'm asking this: Is there such a thing as imaginary and c......Thu, 13 Aug 2026 06:33:04 -0400https://pop.com.sg/to-evaluate-the-limit-as-h-approaches-0-how-do-i-rewrite-the-limit-in-.htmlhttps://pop.com.sg/to-evaluate-the-limit-as-h-approaches-0-how-do-i-rewrite-the-limit-in-.htmlI found a Question from the textbook which is not well explained...or not at all explained. And it would be interesting to know how to solve it. Or perhaps I am simply that bad at understanding squ......Thu, 13 Aug 2026 06:32:40 -0400https://pop.com.sg/is-v-tav-diagonal-in-general.htmlhttps://pop.com.sg/is-v-tav-diagonal-in-general.htmlI want to know if the following is a true statement: If $A$ is real a symmetric and $V$ is orthogonal then $V^TAV=D$ where $D$ is a diagonal matrix. To prove it I can say $$(V^TAV)^T=V^TA^T......Thu, 13 Aug 2026 06:32:25 -0400https://pop.com.sg/how-do-i-plot-the-curve-of-a-conical-spiral-on-an-unwrapped-2d-conic-s.htmlhttps://pop.com.sg/how-do-i-plot-the-curve-of-a-conical-spiral-on-an-unwrapped-2d-conic-s.htmlExample images I need a real world solution to a reconstructive surgery problem! A fairly commonly encountered clinical scenario results in the need to create a tube (from skin) in the shape of an ...Thu, 13 Aug 2026 06:30:00 -0400https://pop.com.sg/is-a-single-line-line-segment-or-ray-a-valid-angle.htmlhttps://pop.com.sg/is-a-single-line-line-segment-or-ray-a-valid-angle.htmlAfter online research and several definitions of the term angle it seems they all fit a short set of restrictions for what defines an angle, "a shape formed by two rays that intersect at a point ca......Thu, 13 Aug 2026 06:16:05 -0400https://pop.com.sg/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-.htmlhttps://pop.com.sg/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-.htmlI've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...Thu, 13 Aug 2026 05:59:32 -0400https://pop.com.sg/work-done-pulling-a-leaky-bucket.htmlhttps://pop.com.sg/work-done-pulling-a-leaky-bucket.htmlA bucket that weighs $4.6$ pounds and a rope of negligible weight are used to draw water from a well that is $80$ feet deep. The bucket is filled with $35$ pounds of water and is pulled up at a rat......Thu, 13 Aug 2026 05:59:22 -0400https://pop.com.sg/why-does-10-35-2-7.htmlhttps://pop.com.sg/why-does-10-35-2-7.htmlMy question is not meant to be dumb. I'm working on a much more complicated problem, but I forgot some of the basics. Often we perform steps in math without really knowing why we can do a certain......Thu, 13 Aug 2026 05:53:53 -0400https://pop.com.sg/let-x-and-y-be-two-vectors-in-mathbb-r-d-with-mid-x-mid-mid-y-mid-find.htmlhttps://pop.com.sg/let-x-and-y-be-two-vectors-in-mathbb-r-d-with-mid-x-mid-mid-y-mid-find.htmlLet $u \in \mathbb{R}^d = V$ be a unit vector and set $W = \text{span}(u) ^{\bot}$ (with respect to the dot product). The reflector across W is $P_u = I_d - 2uu^T$. Let $x$ and $y$ be two vectors ......Thu, 13 Aug 2026 05:52:55 -0400https://pop.com.sg/why-is-a-maclaurin-series-the-function-itself-for-a-polynomial-f-x.htmlhttps://pop.com.sg/why-is-a-maclaurin-series-the-function-itself-for-a-polynomial-f-x.htmlIt's been about two years since I last took Calc 2, and I'm trying to brush up on some of the old material using Khanacademy. I watched a couple intro videos on the subject and understand Taylor se......Thu, 13 Aug 2026 05:27:08 -0400https://pop.com.sg/inscribed-angle-theorem-to-prove-an-angle-inscribed-in-a-semicircle-is.htmlhttps://pop.com.sg/inscribed-angle-theorem-to-prove-an-angle-inscribed-in-a-semicircle-is.htmlI came across a question in my HW book: Prove that an angle inscribed in a semicircle is a right angle. My proof was relatively simple: Proof: As the measure of an inscribed angle is equal to ......Thu, 13 Aug 2026 05:24:34 -0400https://pop.com.sg/homer-simpson-039-s-fermat-equation.htmlhttps://pop.com.sg/homer-simpson-039-s-fermat-equation.htmlDisprove the following equation $3987^{12} + 4365^{12} = 4472^{12}$ First, since both the two numbers on the LHS were odd and the RHS was even, I tried dividing by 3 and found $3 \mid 3987......Thu, 13 Aug 2026 05:20:40 -0400https://pop.com.sg/using-first-principles-find-derivative-of-ln-sec-x.htmlhttps://pop.com.sg/using-first-principles-find-derivative-of-ln-sec-x.htmlThe question is to use first principles only. Thus I started with the same and got $$ y = \ln(\sec(x)) $$ $$ \frac{dy}{dx} = \lim_{h\to 0} \frac{\ln(\sec(x+h)) - \ln(\sec(x))}{h} $$ after this I do......Thu, 13 Aug 2026 05:20:36 -0400https://pop.com.sg/questions-tagged-multinomial-theorem.htmlhttps://pop.com.sg/questions-tagged-multinomial-theorem.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 05:08:55 -0400https://pop.com.sg/the-derivative-of-a-power-series-has-the-same-radius-of-convergence.htmlhttps://pop.com.sg/the-derivative-of-a-power-series-has-the-same-radius-of-convergence.htmlI am trying to understand a proof regarding the derivative of a power function. Let ${ \sum_{k=0}^{ \infty }{a_k(x-x_0)^k} }$ be a real power series with a positive radius of convergence ${ R \le \......Thu, 13 Aug 2026 05:08:43 -0400https://pop.com.sg/use-of-trivially-true-in-math.htmlhttps://pop.com.sg/use-of-trivially-true-in-math.htmlTrivial, according to Google dictionary is an adjective with the meaning "of little value or importance." Some synonyms include: unimportant, insignificant, inconsequential, etc. It appea......Thu, 13 Aug 2026 05:05:18 -0400https://pop.com.sg/rsa-what-is-k-in-the-formula-to-calculate-d-closed.htmlhttps://pop.com.sg/rsa-what-is-k-in-the-formula-to-calculate-d-closed.htmlIn RSA, to calculate $d$, when given $\phi(n)$ and $e$, I stumbled upon this formula: $$d = \dfrac{k \phi(n) + 1}{e}$$ But what does $k$ stand for? How to obtain the value for $k$? Thank you, Chris...Thu, 13 Aug 2026 04:57:58 -0400https://pop.com.sg/what-does-a-space-mean.htmlhttps://pop.com.sg/what-does-a-space-mean.htmlIn Wikipedia, they say that a space is a set with some added structure. But what do they mean by "with some added structure"?...Thu, 13 Aug 2026 04:53:18 -0400https://pop.com.sg/a-circle-graph-is-not-a-function.htmlhttps://pop.com.sg/a-circle-graph-is-not-a-function.htmlI'm a little confused by the rule: If you draw a vertical line that intersects the graph at more than 1 point then it is not a function. Because then a circle like $y^2 + x^2 = 1$ is not a functio......Thu, 13 Aug 2026 04:42:52 -0400https://pop.com.sg/can-t-we-use-vector-product-to-find-the-angle-between-two-vectors.htmlhttps://pop.com.sg/can-t-we-use-vector-product-to-find-the-angle-between-two-vectors.htmlThere are two vectors : $A = (\hat i + j + k)$ and $B = (\hat i - \hat j - \hat k)$, where $\hat i$, $\hat j$, and $\hat k$ are unit vectors along $x$, $y$, and $z$ axis respectively. We have to fi......Thu, 13 Aug 2026 04:26:51 -0400https://pop.com.sg/wolfram-alpha-step-by-step-for-systems-of-differential-equation.htmlhttps://pop.com.sg/wolfram-alpha-step-by-step-for-systems-of-differential-equation.htmlDoes anyone know if wolfram alpha has step by step solutions for systems of differential equations? When I input them, it comes up with an answer but it does not give me the step by step solution. I ...Thu, 13 Aug 2026 04:12:51 -0400https://pop.com.sg/why-is-the-cross-product-of-two-vectors-orthogonal.htmlhttps://pop.com.sg/why-is-the-cross-product-of-two-vectors-orthogonal.htmlWhat is the intuition behind the fact that the cross product of two vectors is orthogonal? Every video I've seen just says it is orthogonal but they do not explain why. Since I have terrible memory......Thu, 13 Aug 2026 04:07:04 -0400https://pop.com.sg/how-to-calculate-matrix-raised-to-a-high-power.htmlhttps://pop.com.sg/how-to-calculate-matrix-raised-to-a-high-power.htmlHow would I go about calculating: $ \left( \begin{array}{cc} 3 & -\sqrt{3} \\ \sqrt{3} & -1 \end{array} \right)^{13}$ I have already attempted to find eigenvalues/eigenvectors, but I beli......Thu, 13 Aug 2026 04:01:35 -0400https://pop.com.sg/how-do-we-deal-with-arctan-or-other-inverse-functions-of-undefined-val.htmlhttps://pop.com.sg/how-do-we-deal-with-arctan-or-other-inverse-functions-of-undefined-val.htmlWhen converting coordinates from rectangular to cylindrical, to spherical, etc. we will eventually come across having to use $ArcTan(y/x)$, and/or $ArcCos(z/\rho)$ to derive $\theta$ and/or $\phi$. ...Thu, 13 Aug 2026 03:51:32 -0400https://pop.com.sg/how-do-you-pronounce-hat-n.htmlhttps://pop.com.sg/how-do-you-pronounce-hat-n.htmlI've found a symbol $\hat{n}$ in an equation, and I have no idea how to pronounce it, and how to write it phonetic. Edit: I've found it here: https://en.wikipedia.org/wiki/Quaternion#Exponential....Thu, 13 Aug 2026 03:46:08 -0400https://pop.com.sg/transitive-property-of-equality-and-the-fundamental-nature-of-algrebra.htmlhttps://pop.com.sg/transitive-property-of-equality-and-the-fundamental-nature-of-algrebra.htmlThe fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we ...Thu, 13 Aug 2026 03:43:43 -0400https://pop.com.sg/height-of-vertical-ropes-of-golden-gate-bridge.htmlhttps://pop.com.sg/height-of-vertical-ropes-of-golden-gate-bridge.htmlThe prompt is to find the parabola equation for the main span of Golden Gate bridge. Based on the $3$ points on the parabola: $(0,227)$; vertex $(640,75)$; and $(1280,227)$ I found the equation: $......Thu, 13 Aug 2026 03:37:27 -0400https://pop.com.sg/user-john-d-cook-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-john-d-cook-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 03:36:41 -0400https://pop.com.sg/what-is-the-best-book-for-pre-calculus.htmlhttps://pop.com.sg/what-is-the-best-book-for-pre-calculus.htmli have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus bo......Thu, 13 Aug 2026 03:35:58 -0400https://pop.com.sg/exterior-algebra-as-quotient.htmlhttps://pop.com.sg/exterior-algebra-as-quotient.htmlGiven a vector space $W$, I understand what the tensor algebra $T(W)$ is, and I understand that the exterior algebra $\bigwedge W$ is defined as $\bigwedge W := T(W)/N$ where $N$ is the two-sided a......Thu, 13 Aug 2026 03:32:42 -0400https://pop.com.sg/two-integrating-factors-same-solution-to-differential-equation.htmlhttps://pop.com.sg/two-integrating-factors-same-solution-to-differential-equation.htmlIf we have to valid integrating factors to a differential equation we must get the same general solution using each of them although we get two new equations by multiplying the original for each of......Thu, 13 Aug 2026 03:23:39 -0400https://pop.com.sg/intuitive-understanding-of-the-operator-norm.htmlhttps://pop.com.sg/intuitive-understanding-of-the-operator-norm.htmlI understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?...Thu, 13 Aug 2026 03:20:15 -0400https://pop.com.sg/using-right-hand-rule-for-a-current-running-in-a-loop.htmlhttps://pop.com.sg/using-right-hand-rule-for-a-current-running-in-a-loop.htmlHello. Using the right hand rule, shouldn't the magnetic force be going into the paper, since velocity is to the left and the force caused by centripetal acceleration is downward? The answer key says ...Thu, 13 Aug 2026 03:19:14 -0400https://pop.com.sg/how-do-i-write-this-basic-recursive-formula-into-desmos.htmlhttps://pop.com.sg/how-do-i-write-this-basic-recursive-formula-into-desmos.htmlI want to graph a simple equation $f(x)$ which begins at $(0,1)$, then for every increasing $x$ integer increment, $f(x) = f(x-1) - (c * f(x-1))$. So in other words each time you go up by one $x$ ...Thu, 13 Aug 2026 03:18:26 -0400https://pop.com.sg/graph-of-the-function-2x-2-2-x-2-1.htmlhttps://pop.com.sg/graph-of-the-function-2x-2-2-x-2-1.htmlI am learning precalculus and my precalculus book gives this equation: for this graph: But when I enter that equation into some online graph tool like Symbolab ( https://www.symbolab.com/graphing-...Thu, 13 Aug 2026 03:09:19 -0400https://pop.com.sg/scalar-multiplication-of-vectors-on-left-or-right.htmlhttps://pop.com.sg/scalar-multiplication-of-vectors-on-left-or-right.htmlVectors are often written column-wise as if they were $n\times 1$ matrices: $$ \mathbf{v} := \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} $$ This notation implicitly identifies the vector $\mathbf{v......Thu, 13 Aug 2026 03:04:09 -0400https://pop.com.sg/multiplication-of-ordered-pairs.htmlhttps://pop.com.sg/multiplication-of-ordered-pairs.htmlFrom what I've seen, ordered pair multiplication is $(a,b) \times (c,d) = (ac-bd,ad+bc)$. Also, $(0,1)$ represents $i$. So if I wanted to take $(0,-1) \times (0.-1)$, that would be $= (-1,0)$, rig......Thu, 13 Aug 2026 02:53:31 -0400https://pop.com.sg/find-all-solutions-to-x-9-equiv-25-pmod-29.htmlhttps://pop.com.sg/find-all-solutions-to-x-9-equiv-25-pmod-29.htmlFind all solutions to $x^9 \equiv 25 \pmod {29}$ I referred to a table of indices in the back of my book for help but am still a bit confused, any help/hints are greatly appreciated....Thu, 13 Aug 2026 02:47:36 -0400https://pop.com.sg/approximation-of-sqrt2-using-euler-039-s-method.htmlhttps://pop.com.sg/approximation-of-sqrt2-using-euler-039-s-method.htmlConsider the differential equation $$\frac{dx}{dt}=\frac1{2x}.$$ This is a separable O.D.E. so we know how to find all of its solutions: they are of the form $$x(t)=\sqrt{t+C}$$ where $C$ is a ...Thu, 13 Aug 2026 02:39:44 -0400https://pop.com.sg/calculating-the-mean-of-a-folded-normal-distribution.htmlhttps://pop.com.sg/calculating-the-mean-of-a-folded-normal-distribution.htmlI have some data that I know is distributed according to the folded normal distribution. I am looking to calculate the mean and variance of the underyling "unfolded" normal distribution. Let's sa......Thu, 13 Aug 2026 02:38:15 -0400https://pop.com.sg/give-the-bayesian-posterior-mode.htmlhttps://pop.com.sg/give-the-bayesian-posterior-mode.htmlSuppose that $X_1, X_2, \ldots, X_n$ are IID Bernoulli random variables with success probability equal to an unknown parameter $\theta \in [0,1]$. Let $A$ and $B$ be nonnegative constants. If we i......Thu, 13 Aug 2026 02:36:11 -0400https://pop.com.sg/is-p-vs-np-problem-solved.htmlhttps://pop.com.sg/is-p-vs-np-problem-solved.htmlMany people have tried to solve the very famous problem "P vs NP" and a lot of solutions are proposed. (e.g. A. D. Plotnikov, On the Relationship between Classes P and NP). But I couldn't find any ...Thu, 13 Aug 2026 02:36:01 -0400https://pop.com.sg/factors-of-zero.htmlhttps://pop.com.sg/factors-of-zero.htmlHow many integer factors of 0 are there, and what are they? I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?...Thu, 13 Aug 2026 02:35:23 -0400https://pop.com.sg/how-to-prove-that-lim-x-to-pi-2-tan-x-tan2x-1.htmlhttps://pop.com.sg/how-to-prove-that-lim-x-to-pi-2-tan-x-tan2x-1.htmlHow do I prove that the limit of the following function is $1$? $\displaystyle\lim_{x\to \pi/2} (\tan x)^{\tan2x}$ I shouldn't use l'Hôpital's rule. I have proven that limit is greater than $......Thu, 13 Aug 2026 02:25:25 -0400https://pop.com.sg/what-are-the-4-fourth-roots-of-16-complex-numbers-included-closed.htmlhttps://pop.com.sg/what-are-the-4-fourth-roots-of-16-complex-numbers-included-closed.htmlI'm trying to find the 4 fourth roots of 16, so far I have -2 and 2, but I have no idea how to find the other 2 complex numbers....Thu, 13 Aug 2026 02:23:28 -0400https://pop.com.sg/rate-of-change-with-f-x-4x-2-7-on-1-b.htmlhttps://pop.com.sg/rate-of-change-with-f-x-4x-2-7-on-1-b.htmlAs part of a textbook exercise I am to find the rate of change of $f(x)=4x^2-7$ on inputs $[1,b]$. The solution provided is $4(b+1)$ and I am unable to arrive at this solution. Tried: $f(x_2)=4b......Thu, 13 Aug 2026 02:21:13 -0400https://pop.com.sg/checking-if-a-vector-field-is-conservative.htmlhttps://pop.com.sg/checking-if-a-vector-field-is-conservative.htmlI have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ......Thu, 13 Aug 2026 02:04:20 -0400https://pop.com.sg/good-software-for-linear-integer-programming.htmlhttps://pop.com.sg/good-software-for-linear-integer-programming.htmlI never did any linear/integer programming so I am wondering the following two things What are some efficient free linear programming solvers? What are some efficient commercial linear programming ...Thu, 13 Aug 2026 01:59:29 -0400https://pop.com.sg/conditional-probability-all-heads-given-at-least-one-heads.htmlhttps://pop.com.sg/conditional-probability-all-heads-given-at-least-one-heads.htmlThree coins are flipped. If at least one of them comes up heads, then what is the probability that they all come up heads? The answer to this is 1/7, as it is the #(ways to have all heads)/#(ways t......Thu, 13 Aug 2026 01:44:59 -0400https://pop.com.sg/show-that-lim-2n-frac-1-n-1.htmlhttps://pop.com.sg/show-that-lim-2n-frac-1-n-1.htmlProve that $\lim ((2n)^\frac{1}{n}) = 1$. I have obtained the following: $$(2n)^\frac{1}{n} = 1 + k_{n}; n > 1$$ $$(2n) = (1 + k_{n})^n$$ By the Binomial Theorem: $$(1 + k_{n})^n = 1 + k_{n} + \...Thu, 13 Aug 2026 01:36:49 -0400https://pop.com.sg/why-is-the-slope-of-a-line-defined-as-the-change-in-y-over-the-change-.htmlhttps://pop.com.sg/why-is-the-slope-of-a-line-defined-as-the-change-in-y-over-the-change-.htmlI understand that this is a reasonable definition and it shows how "fast" $y$ values change corresponding to $x$ values because it's a ratio, but what I'm asking is, couldn't the slope have been de......Thu, 13 Aug 2026 01:22:53 -0400https://pop.com.sg/taylor-series-with-point-of-expansion.htmlhttps://pop.com.sg/taylor-series-with-point-of-expansion.htmlWrite the Taylor series for the function $f(x)= x^2-3x+1$ using $x=2$ as the point of expansion , i.e. write a formula for $f(2+h)$ Even though I know what Taylor Expansion is I am really confused......Thu, 13 Aug 2026 01:16:14 -0400