https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 20 Aug 2026 10:31:00 -0400https://pop.com.sg/what-is-the-difference-between-a-scalar-and-a-vector-field.htmlhttps://pop.com.sg/what-is-the-difference-between-a-scalar-and-a-vector-field.htmlCould someone please indicate precisely the difference between a scalar and a vector field? I find no matter how many times I try to understand, but I always am confused in the end. So what exactly......Thu, 20 Aug 2026 14:17:03 -0400https://pop.com.sg/in-the-proof-of-irrationality-of-sqrt-2-or-sqrt-7-why-do-the-numerator.htmlhttps://pop.com.sg/in-the-proof-of-irrationality-of-sqrt-2-or-sqrt-7-why-do-the-numerator.htmlI have been confused by this problem for a very long period of time, and I think I am personally opposed to this concept and refused to agree with it in my introduction to mathematical proof cours......Thu, 20 Aug 2026 14:16:43 -0400https://pop.com.sg/proving-that-even-numbers-equal-the-sum-of-two-odd-numbers.htmlhttps://pop.com.sg/proving-that-even-numbers-equal-the-sum-of-two-odd-numbers.htmlDefine E to be the set of even integers; E = {$x$ $\in$ $\mathbb{Z}$ : $x$ = 2$k$, where $k$ $\in$ $\mathbb{Z}$}. Define F to be the set of integers that can be expressed as the sum of two odd n......Thu, 20 Aug 2026 14:09:52 -0400https://pop.com.sg/what-is-the-difference-between-the-first-derivative-and-the-second-der.htmlhttps://pop.com.sg/what-is-the-difference-between-the-first-derivative-and-the-second-der.htmlSo a bit confused when I see a question say: "Use the second derivative test to find all relative extrema" When using the second derivative test are we not looking for concavity and points of ...Thu, 20 Aug 2026 14:09:36 -0400https://pop.com.sg/is-there-negative-base-representation-of-numbers.htmlhttps://pop.com.sg/is-there-negative-base-representation-of-numbers.htmlEvery real number can be expressed as its base 10 representation, as $$\sum_{i=-\infty}^\infty a_i 10^i\,$$ where $a_i\in\{0,1,\dots,9\}$. We also have base 2 or base 8. Question: Is there a base -2 ...Thu, 20 Aug 2026 14:08:37 -0400https://pop.com.sg/prime-of-form-n-2-4.htmlhttps://pop.com.sg/prime-of-form-n-2-4.htmlShow that The only prime of the form $n^2-4$, $n$ being an integer is $3$ We have $n^2-4=(n+2)(n-2)$ Now for $n^2-4$ being prime value of $(n-2)$ must be $1$. Then $n=3$ and putting the value we g......Thu, 20 Aug 2026 14:07:01 -0400https://pop.com.sg/systems-of-equations-inconsistent.htmlhttps://pop.com.sg/systems-of-equations-inconsistent.htmlQuestion: Consider the following system of three equations: $$2y+2z=9-2x$$ $$x=12-3y-4z$$ $$Ax+5y+6z=B$$ Find values of A and B which makes the system of equations incons......Thu, 20 Aug 2026 14:05:24 -0400https://pop.com.sg/is-tautological-bundle-mathcal-o-1-or-mathcal-o-1.htmlhttps://pop.com.sg/is-tautological-bundle-mathcal-o-1-or-mathcal-o-1.htmlI always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a ......Thu, 20 Aug 2026 14:04:45 -0400https://pop.com.sg/how-to-determine-particular-months-with-five-weekends.htmlhttps://pop.com.sg/how-to-determine-particular-months-with-five-weekends.htmlJanuary of 2016 has five Fridays, five Saturdays and five Sundays. What is the next January that will again have five weekends like this?...Thu, 20 Aug 2026 14:03:04 -0400https://pop.com.sg/find-the-domain-of-a-trig-function.htmlhttps://pop.com.sg/find-the-domain-of-a-trig-function.htmlFind the Domain of the function $y=\sqrt{1+2 \sin x}$. My solution: $1+2 \sin x \geq 0$ (since square-root of a negative number is not defined) $2 \sin x \geq -1$ $\sin x \geq -1/2$ $x \ge -\frac{......Thu, 20 Aug 2026 13:24:49 -0400https://pop.com.sg/optimal-strategy-for-dice-10000-dice-game.htmlhttps://pop.com.sg/optimal-strategy-for-dice-10000-dice-game.htmlThere exists a dice game known as Dice-10000, with alternative names such as Dix Mille or 6-Dice The rules are simple, with each player taking turns rolling dice in order to achieve certain combina......Thu, 20 Aug 2026 13:15:56 -0400https://pop.com.sg/local-truncation-error-of-euler-method.htmlhttps://pop.com.sg/local-truncation-error-of-euler-method.htmlWikipedia and this book say the local truncation error of Euler method is $O(h^2)$. But this book and A friendly Introduction to Numerical Analysis say it's $O(h)$. Which is correct? I have a similar ...Thu, 20 Aug 2026 13:13:41 -0400https://pop.com.sg/can-every-real-number-be-represented-by-a-possibly-infinite-decimal.htmlhttps://pop.com.sg/can-every-real-number-be-represented-by-a-possibly-infinite-decimal.htmlDoes every real number have a representation within our decimal system? The reason I ask is because, from beginning a mathematics undergraduate degree a lot of 'mathematical facts' I had previously ...Thu, 20 Aug 2026 13:07:14 -0400https://pop.com.sg/how-does-taylor-series-work-for-sine-and-cosine.htmlhttps://pop.com.sg/how-does-taylor-series-work-for-sine-and-cosine.htmlI was just wondering if anyone can help me understand taylor series for sine and cosine. I have no background in calculus but I always found it interesting how the ratios of the arc to the radius ......Thu, 20 Aug 2026 13:06:02 -0400https://pop.com.sg/how-to-calculate-matrix-tr-ab.htmlhttps://pop.com.sg/how-to-calculate-matrix-tr-ab.htmlI have these Matrix A and B below in the image: And i wonder how do i calculate the tr(AB)? I have done this by taking the diagonal in each Matrix A and B: 3 * 6 + 1 * 5 + 5 * 9 = 67 But after tha......Thu, 20 Aug 2026 13:05:45 -0400https://pop.com.sg/equation-of-circles-tangent-to-x-axis-with-a-given-radius-and-point.htmlhttps://pop.com.sg/equation-of-circles-tangent-to-x-axis-with-a-given-radius-and-point.htmlFind the equations of the circles which are tangent to the x-axis, with radius of 5 units and passing through the point $(0,8)$. I know how to formulate an equation with the given radius of 5 and t......Thu, 20 Aug 2026 12:59:52 -0400https://pop.com.sg/solving-log-6-2x-3-log-6-x-5-log-3x.htmlhttps://pop.com.sg/solving-log-6-2x-3-log-6-x-5-log-3x.htmlSolve for $x$ I have an equation that I have been working on solving; I know the solution, but I cannot get to it myself. Almost every simplification I do reverts back to a previous step. Can anyone ...Thu, 20 Aug 2026 12:54:11 -0400https://pop.com.sg/what-039-s-an-intuitive-explanation-of-the-max-flow-min-cut-theorem.htmlhttps://pop.com.sg/what-039-s-an-intuitive-explanation-of-the-max-flow-min-cut-theorem.htmlI'm about to read the proof of the max-flow min-cut theorem that helps solve the maximum network flow problem. Could someone please suggest an intuitive way to understand the theorem?...Thu, 20 Aug 2026 12:45:11 -0400https://pop.com.sg/in-mathematics-is-null-zero.htmlhttps://pop.com.sg/in-mathematics-is-null-zero.htmlKindly consider it soft question as I am a software engineer.and I know only software but I have a doubt in my mind that there would be something like null in mathmatics as well. If Mathematics N......Thu, 20 Aug 2026 12:45:04 -0400https://pop.com.sg/probability-of-a-given-a-union-b.htmlhttps://pop.com.sg/probability-of-a-given-a-union-b.htmlWhat is the probability of A given A union B? We know that p(A) = 0.5 p(B) = 0.3 p(AB) = 0.1 From my understanding of conditional probability i think it should be p(A)/p(A union B) . Is this corre......Thu, 20 Aug 2026 12:35:26 -0400https://pop.com.sg/finding-orthogonal-angles-polar-coordinate-of-an-n-dimensional-vector.htmlhttps://pop.com.sg/finding-orthogonal-angles-polar-coordinate-of-an-n-dimensional-vector.htmlOrthogonal angles are the angles used when converting a vector to polar coordinates. So for vector $(1, 1)$, the orthogonal angle is $45$ degrees. Given a vector $(x_1, x_2, x_3, ..., x_n)$, what ......Thu, 20 Aug 2026 12:35:18 -0400https://pop.com.sg/finding-the-limit-of-1-cos-x-x-as-x-to-0-with-squeeze-theorem.htmlhttps://pop.com.sg/finding-the-limit-of-1-cos-x-x-as-x-to-0-with-squeeze-theorem.htmlHow do I find: $$ \lim_{x\to0}\frac{1-\cos(x)}{x} $$ Using the squeeze theorem. Particularly, how would I arrive at its bounding functions? If possible, please try not to use derivatives....Thu, 20 Aug 2026 12:33:37 -0400https://pop.com.sg/terence-tao-type-books-in-other-fields.htmlhttps://pop.com.sg/terence-tao-type-books-in-other-fields.htmlI have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that ...Thu, 20 Aug 2026 12:30:20 -0400https://pop.com.sg/how-do-you-write-the-summation-of-a-summation.htmlhttps://pop.com.sg/how-do-you-write-the-summation-of-a-summation.htmlLike how should you do it and how do you manipulate it algebraically? $$S = \{1/2, 1/3, \ldots, 1/n\}$$ $$Q = \{S(2), S(3), \ldots, S(n)\}$$...Thu, 20 Aug 2026 12:30:15 -0400https://pop.com.sg/why-teach-linear-algebra-before-abstract-algebra.htmlhttps://pop.com.sg/why-teach-linear-algebra-before-abstract-algebra.htmlIs there a reason why most undergraduate curriculums put linear algebra before abstract algebra? I'm asking this because personally it seems to be much easier to understand the architecture behind ...Thu, 20 Aug 2026 12:26:32 -0400https://pop.com.sg/gre-pobability-question.htmlhttps://pop.com.sg/gre-pobability-question.htmlOne person is to be selected at random from a group of 25 people. The probability that the selected person will be a male is 0.44, and the probability that the selected person will be a male who wa......Thu, 20 Aug 2026 12:24:47 -0400https://pop.com.sg/what-exactly-is-an-implicitly-defined-function.htmlhttps://pop.com.sg/what-exactly-is-an-implicitly-defined-function.htmlSo I've just started to get into some calculus and I recently came across the topic of implicit differentiation. I am extremely confused on what implicit functions are and there is very little ...Thu, 20 Aug 2026 12:10:21 -0400https://pop.com.sg/difference-between-modulus-and-remainder.htmlhttps://pop.com.sg/difference-between-modulus-and-remainder.html(I've come here from this question.) What's the difference between modulus and remainder? (Bearing in mind until 5 minutes ago I thought they were the same :P )...Thu, 20 Aug 2026 12:00:14 -0400https://pop.com.sg/sum-of-products-expansions-of-these-boolean-functions.htmlhttps://pop.com.sg/sum-of-products-expansions-of-these-boolean-functions.htmlFind the sum-of-products expansions of these Boolean functions. $a)$ $F(x, y) = \text{~}x + y$ $b)$ $F(x, y) = x \text{~}y$ $c)$ $F(x, y) = 1$ $d)$ $F(x, y) = \text{~}y$...Thu, 20 Aug 2026 11:56:22 -0400https://pop.com.sg/why-n-and-what-does-it-mean.htmlhttps://pop.com.sg/why-n-and-what-does-it-mean.htmlMaybe a bit too early to ask, but I give it a shot. I'm trying to study mathematics again during my free time. Now I tried to figure out what exactly is meant with the multiple. Well this formula m......Thu, 20 Aug 2026 11:45:31 -0400https://pop.com.sg/sigmoid-curve-function-calculation.htmlhttps://pop.com.sg/sigmoid-curve-function-calculation.htmlSo I am learning about Sigmoid Curves and in this article it says the formula is: $$\tag{1} \frac{1}{1+e^{-\beta_0-\beta_1x}} $$ Then it gives an example problem: Let’s say you take $ \beta_0 =......Thu, 20 Aug 2026 11:22:35 -0400https://pop.com.sg/mgf-of-a-sum-product-of-mgfs.htmlhttps://pop.com.sg/mgf-of-a-sum-product-of-mgfs.htmlSo my book says the following: The moment generating function for the geometric distribution is: $g(t) = \cfrac{pe^t}{1-e^t(1-p)}$ where $|e^t(1-p)| <1$ of course for the geometric series to con......Thu, 20 Aug 2026 10:55:29 -0400https://pop.com.sg/general-form-of-an-integer.htmlhttps://pop.com.sg/general-form-of-an-integer.htmlFrom the Division Algorithm, We know that if an integer is divided by $3$ , it will leave remainder $0,1$ or $2$. Now, how is one supposed to write a proper proof for the given question? P.S. This......Thu, 20 Aug 2026 10:51:32 -0400https://pop.com.sg/a-scalene-triangle-with-no-right-nor-obtuse-angle.htmlhttps://pop.com.sg/a-scalene-triangle-with-no-right-nor-obtuse-angle.htmlI want to find the area of the perfect triangle, i.e. a triangle with no particularity whatsoever : no side shall be equal to another, no right angle, no obtuse angle. So I gave myself a segment ......Thu, 20 Aug 2026 10:49:16 -0400https://pop.com.sg/how-to-create-a-reflexive-symmetric-and-transitive-closures.htmlhttps://pop.com.sg/how-to-create-a-reflexive-symmetric-and-transitive-closures.htmlI'm working on a task where I need to find out the reflexive, symmetric and transitive closures of R. Statement is given below: Assume that U = {1, 2, 3, a, b} and let the relation R on U which is ...Thu, 20 Aug 2026 10:48:59 -0400https://pop.com.sg/finding-the-p-value-in-a-proportion-statistics-with-binomial-distribut.htmlhttps://pop.com.sg/finding-the-p-value-in-a-proportion-statistics-with-binomial-distribut.htmlGiven that the sample is 40, we observed that 80% of the sample (32 participants) have a successful trial. We want to test the hypothesis $H_{0} = p_{0} = 0.7$ versus $H_{a} \neq 0.7$ at 5% signifi......Thu, 20 Aug 2026 10:44:08 -0400https://pop.com.sg/finding-a-function-039-s-domain-from-the-function-039-s-formula.htmlhttps://pop.com.sg/finding-a-function-039-s-domain-from-the-function-039-s-formula.htmlThere is a question in my calculus 101 textbook that says find the domain of the function. The function formulas are given, eg: $$f(x) = \frac{x+4}{x^2 -9}$$ and $$g(t) = \sqrt[3]{2t - 1}$$ No ot......Thu, 20 Aug 2026 10:38:27 -0400https://pop.com.sg/what-039-s-an-intuitive-way-of-looking-at-quotient-spaces.htmlhttps://pop.com.sg/what-039-s-an-intuitive-way-of-looking-at-quotient-spaces.htmlI understand the concept of $\mathbb{Z}/n\mathbb{Z}$, but I am having a really hard time understanding how this concept of quotients applies to vector spaces. Suppose $V = \mathbb{F}[x]$ is a vector ...Thu, 20 Aug 2026 10:32:30 -0400https://pop.com.sg/geometric-intuition-of-conjugate-function.htmlhttps://pop.com.sg/geometric-intuition-of-conjugate-function.htmlI am looking for a geometric and intuitive explanation of the conjugate function and how it maps to the below analytical formula. $$ f^*(y)= \sup_{x \in \operatorname{dom} f } (y^Tx-f(x))$$...Thu, 20 Aug 2026 10:26:38 -0400https://pop.com.sg/how-many-tickets-sold-at-a-concert.htmlhttps://pop.com.sg/how-many-tickets-sold-at-a-concert.htmlThe music department of a school sold 750 tickets to the school concert, for a total of \$4755. Students paid \$5 for a ticket and non-students paid \$8 for a ticket. How many non-students attended......Thu, 20 Aug 2026 10:26:38 -0400https://pop.com.sg/x-vs-t-and-y-vs-t-graphs.htmlhttps://pop.com.sg/x-vs-t-and-y-vs-t-graphs.htmlHow would I go about sketching the x vs t and y vs t graphs for the following picture: Our class didn't get the opportunity to solve these types of questions, so I'm on my own for this one. My thi......Thu, 20 Aug 2026 10:23:51 -0400https://pop.com.sg/how-can-e-x-seemingly-approach-0-as-x-approaches-negative-infinity.htmlhttps://pop.com.sg/how-can-e-x-seemingly-approach-0-as-x-approaches-negative-infinity.htmlThe specific problem with regards to this graph is: $$\lim_{x \to -\infty} g(2 + e^x)$$ Now it's true that we technically can't apply "the limit of a sum is the sum of the limits" since the functi......Thu, 20 Aug 2026 10:17:10 -0400https://pop.com.sg/can-anyone-please-explain-the-the-difference-between-a-vector-and-a-ma.htmlhttps://pop.com.sg/can-anyone-please-explain-the-the-difference-between-a-vector-and-a-ma.htmlI just took Calculus 3 last semester at my University and got comfortable with the idea of vectors, vector-valued functions, and basic vector operations like the dot and cross products. This semes......Thu, 20 Aug 2026 10:16:01 -0400https://pop.com.sg/are-natural-numbers-a-field-with-alternate-addition-and-multiplication.htmlhttps://pop.com.sg/are-natural-numbers-a-field-with-alternate-addition-and-multiplication.htmlI've just started P.Halmos' book "Finite Vector Spaces", and at the first chapter, after defining the axioms of a field, there is an exercide that asks: if we consider the non negative integers, ca......Thu, 20 Aug 2026 10:15:47 -0400https://pop.com.sg/can-we-take-images-of-equivalence-relations.htmlhttps://pop.com.sg/can-we-take-images-of-equivalence-relations.htmlGiven a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. Th......Thu, 20 Aug 2026 10:05:32 -0400https://pop.com.sg/how-can-i-find-the-maximum-revenue.htmlhttps://pop.com.sg/how-can-i-find-the-maximum-revenue.htmlAn oil producing country can sell 7 million barrels of oil a day at a price of $90$ per barrel. If each $1$ price increase will result in a sales decrease of $100,000$ barrels per day, what price w......Thu, 20 Aug 2026 09:51:47 -0400https://pop.com.sg/how-to-check-whether-a-relation-is-transitive-from-the-matrix-represen.htmlhttps://pop.com.sg/how-to-check-whether-a-relation-is-transitive-from-the-matrix-represen.html$$\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}$$ This is a matrix representation of a relation on the set $\{1, 2, 3\}$. I have to determine if this relation matrix is ...Thu, 20 Aug 2026 09:50:06 -0400https://pop.com.sg/oriented-graph-vs-directed-graph.htmlhttps://pop.com.sg/oriented-graph-vs-directed-graph.htmlAlright, while the definitions are stated in my lecure notes, textbooks and wiki, I'll be honest, it just explodes my mind with what seems like word sorcery. Definition A directed graph is called an ...Thu, 20 Aug 2026 09:37:45 -0400https://pop.com.sg/confusion-on-the-proofs-of-convolution-theorem-of-fourier-transform.htmlhttps://pop.com.sg/confusion-on-the-proofs-of-convolution-theorem-of-fourier-transform.htmlThe convolution theorem of Fourier transform is stated as follows: Define $h(x):=f(x)*g(x)$, then we have $$\hat{h}(k)=\hat{f}(k)\hat{g}(k).$$ I have a confusion of the proofs of this theorem. Most ...Thu, 20 Aug 2026 09:35:36 -0400https://pop.com.sg/sum-of-a-matrix-with-its-transpose-duplicate.htmlhttps://pop.com.sg/sum-of-a-matrix-with-its-transpose-duplicate.htmlI've a question that many of yours could consider stupid: if i sum a matrix with its transpose, I obtain a particular result? E.g. $A + A^T = B$, $B$ has some particular properties?...Thu, 20 Aug 2026 09:32:54 -0400https://pop.com.sg/what-does-f-max-f-0-and-f-max-f-0-mean.htmlhttps://pop.com.sg/what-does-f-max-f-0-and-f-max-f-0-mean.htmlLet $f$ be a continious function over an intervall with $f^+:=\max(f,0)$ and $f^-:=\max(-f,0)$ What does that mean graphically?...Thu, 20 Aug 2026 09:29:12 -0400https://pop.com.sg/limit-of-a-multivariable-function-with-log.htmlhttps://pop.com.sg/limit-of-a-multivariable-function-with-log.htmlWhat is the limit of $$f(x,y) = (x^2+y^2)\log|x+y|$$ as $(x,y) \rightarrow (0,0)$ I tried using Squeeze Theorem but couldn't think of an upper bound for it. I know the limit of $t log(t)$ is 0......Thu, 20 Aug 2026 09:25:14 -0400https://pop.com.sg/fundamental-group-of-mathbb-r-3-minus-trefoil-knot.htmlhttps://pop.com.sg/fundamental-group-of-mathbb-r-3-minus-trefoil-knot.htmlLet $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van K......Thu, 20 Aug 2026 09:14:55 -0400https://pop.com.sg/proof-of-the-central-limit-theorem-using-moment-generating-functions.htmlhttps://pop.com.sg/proof-of-the-central-limit-theorem-using-moment-generating-functions.htmlBelow is a method of proving the Central Limit Theorem using moment generating functions. Let $$X_{1},X_{2},...,X_{n}$$ be a sequence of i.i.d. random variables with expected value and variance ......Thu, 20 Aug 2026 09:09:20 -0400https://pop.com.sg/matrix-determinant-lemma-derivation.htmlhttps://pop.com.sg/matrix-determinant-lemma-derivation.htmlWhile reading this wikipedia article on the determinant lemma, I stumbled upon this expression (in a proof section): \begin{equation} \begin{pmatrix} \mathbf{I} & 0 \\ \mathbf{v}^\mathrm{T} &am......Thu, 20 Aug 2026 08:49:58 -0400https://pop.com.sg/modular-arithmetic-find-the-square-root.htmlhttps://pop.com.sg/modular-arithmetic-find-the-square-root.htmlOkay so I am in college and in the notes it shows this example: // Example: Compute the square root of 3 modulo 143 3 modulo 143 = 3 mod (11*13) Then he jumps to this: √3 (mod 11) = ± ...Thu, 20 Aug 2026 08:20:10 -0400https://pop.com.sg/what-is-the-difference-between-sqrt-x-2-and-sqrt-x-2-closed.htmlhttps://pop.com.sg/what-is-the-difference-between-sqrt-x-2-and-sqrt-x-2-closed.html$\sqrt{x^2}\;$ and $(\sqrt{x})^2$; I mean, are they equal? because what about the following facts: $\sqrt{x^2} = (x^2)^{1/2} = x^{(2)(\frac{1}{2})} = x^{\frac{2}{2}} = x \\ (\sqrt{x})^2 = x^{\frac......Thu, 20 Aug 2026 08:08:57 -0400https://pop.com.sg/outer-automorphisms-of-semi-simple-lie-algebras.htmlhttps://pop.com.sg/outer-automorphisms-of-semi-simple-lie-algebras.htmlIt is known that outer automorphisms of semi-simple Lie algebras are automorphisms of their corresponding Dynkin diagrams. But would it be correct to say that for a semi-simple Lie algebra all outer ...Thu, 20 Aug 2026 08:01:39 -0400https://pop.com.sg/what-is-the-precise-definition-of-a-rigid-shape.htmlhttps://pop.com.sg/what-is-the-precise-definition-of-a-rigid-shape.htmlWikipedia's section on rigid shapes does not appear to actually define what a rigid shape is. Rather it defines 'same shape' and 'rigid transformations' without giving any definitions of what is ...Thu, 20 Aug 2026 08:00:26 -0400https://pop.com.sg/euler-039-s-identity-why-is-the-e-in-e-ix-what-if-it-were-some-other-c.htmlhttps://pop.com.sg/euler-039-s-identity-why-is-the-e-in-e-ix-what-if-it-were-some-other-c.html$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that $\frac{d}......Thu, 20 Aug 2026 07:59:50 -0400https://pop.com.sg/finding-irrational-and-complex-roots-of-a-cubic-polynomial.htmlhttps://pop.com.sg/finding-irrational-and-complex-roots-of-a-cubic-polynomial.htmlI've got a question which shows short answers and no method so I'm trying to find a hand performed method of solving the cubic polynomial for the roots: $$f(x) = 2x^3 + 8x^2 + 10x - 6$$ From the ...Thu, 20 Aug 2026 07:54:33 -0400https://pop.com.sg/calculate-the-length-of-ac.htmlhttps://pop.com.sg/calculate-the-length-of-ac.htmlThe diameter $AB$ of the circle is $10\,\text{cm}$. The length of $BC$ is $6\,\text{cm}$. Calculate the length of $AC$. I'm doing a mock exam and I'm not sure how to work out the length of $AC$. Any ...Thu, 20 Aug 2026 07:53:28 -0400https://pop.com.sg/does-the-gamma-function-have-an-inverse.htmlhttps://pop.com.sg/does-the-gamma-function-have-an-inverse.htmlDoes the Gamma Function have an Inverse? (Is there an "arc-gamma" function?) Where $\Gamma(x) = y... \Gamma^{-1}(y) = x\ (arc\Gamma(y)=x)$. I've searched and found something called DiGamma Functi......Thu, 20 Aug 2026 07:50:45 -0400https://pop.com.sg/proof-for-if-a-is-invertible-then-ab-is-invertible.htmlhttps://pop.com.sg/proof-for-if-a-is-invertible-then-ab-is-invertible.htmlFirst, $A$ and $B$ are square matrices So to prove if $AB$ is invertible, then $A$ is invertible: I let $C=(AB^{-1})B$ Then $CA=(AB^{-1})AB=I$ And $C=A^{-1}$ , so A is invertible. But how do I......Thu, 20 Aug 2026 07:49:19 -0400https://pop.com.sg/what-can-be-a-real-world-example-of-permutation-without-repetition-clo.htmlhttps://pop.com.sg/what-can-be-a-real-world-example-of-permutation-without-repetition-clo.htmlI know that a Permutation lock is a concrete real-world example of "permutation with repetition". What can be a concrete real-world example of Permutation without repetition?...Thu, 20 Aug 2026 07:46:03 -0400https://pop.com.sg/prove-or-disprove-there-exists-a-ring-homomorphism-phi-mathbb-c-righta.htmlhttps://pop.com.sg/prove-or-disprove-there-exists-a-ring-homomorphism-phi-mathbb-c-righta.htmlProve or disprove: There exists a ring homomorphism $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$. I think it is intuitive to try $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mat......Thu, 20 Aug 2026 07:42:23 -0400https://pop.com.sg/definition-of-definition.htmlhttps://pop.com.sg/definition-of-definition.htmlI was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topi......Thu, 20 Aug 2026 07:38:54 -0400https://pop.com.sg/numbers-that-can-be-expressed-as-the-sum-of-two-cubes-in-exactly-two-d.htmlhttps://pop.com.sg/numbers-that-can-be-expressed-as-the-sum-of-two-cubes-in-exactly-two-d.htmlIt seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: Number Theory Taxicab Number)......Thu, 20 Aug 2026 07:38:28 -0400https://pop.com.sg/simultaneously-solenoidal-and-irrotational-vector.htmlhttps://pop.com.sg/simultaneously-solenoidal-and-irrotational-vector.htmlCan we have a vector field $\vec{V}(\vec{r})$ such that $\nabla\times\vec{V}=\vec{0}$ and $\nabla\cdot\vec{V}=0$...Thu, 20 Aug 2026 07:37:37 -0400https://pop.com.sg/number-of-solutions-of-the-equation-3-x-4-x-5-x-in-the-set-of-positive.htmlhttps://pop.com.sg/number-of-solutions-of-the-equation-3-x-4-x-5-x-in-the-set-of-positive.htmlI came across the following question: Find the number of solutions of the equation $3^x+4^x=5^x$ in the set of positive real numbers. I tried the above question by taking log on both sides and then ...Thu, 20 Aug 2026 07:34:52 -0400https://pop.com.sg/finding-the-moment-generating-function-of-a-transformed-random-variabl.htmlhttps://pop.com.sg/finding-the-moment-generating-function-of-a-transformed-random-variabl.htmlSuppose I have a random variable $X$ governed by the Gamma distribution. If I apply a transformation like $Y = \ln\bigg( \frac{X}{1+X}\bigg)$, could I use properties of moment generating functions to ...Thu, 20 Aug 2026 07:30:45 -0400https://pop.com.sg/height-and-velocity-of-ball-thrown-vertically.htmlhttps://pop.com.sg/height-and-velocity-of-ball-thrown-vertically.htmlA ball is thrown upward from roof of 32 foot building with velocity of $112$ ft/sec. The height after $t$ seconds is: $s(t)=32+112t-16t^2$. (a) Find the maximum height that the ball reaches. (ans......Thu, 20 Aug 2026 07:30:21 -0400https://pop.com.sg/is-there-an-existing-formula-in-finding-the-area-of-a-rhombus-wherein-.htmlhttps://pop.com.sg/is-there-an-existing-formula-in-finding-the-area-of-a-rhombus-wherein-.htmlis there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given....Thu, 20 Aug 2026 07:23:58 -0400https://pop.com.sg/expected-value-of-a-uniform-distribution.htmlhttps://pop.com.sg/expected-value-of-a-uniform-distribution.htmlLet $f(x) = 0.025x + 0.15$ for $2 < x < 6$. The expected value formula is $1/2 \cdot (b-a)$. So is the expected value just $1/2 \cdot (6-2) = 4$ or do I have to integrate $f(x)$ first?...Thu, 20 Aug 2026 07:23:48 -0400https://pop.com.sg/how-to-derive-poisson-distribution-from-binomial-distribution.htmlhttps://pop.com.sg/how-to-derive-poisson-distribution-from-binomial-distribution.htmlI am familiar with another answer about this topic: How to prove Poisson Distribution is the approximation of Binomial Distribution? , but i don't understand one part of it. Could anyone explain t......Thu, 20 Aug 2026 07:23:04 -0400https://pop.com.sg/is-an-anti-symmetric-and-asymmetric-relation-the-same-are-irreflexive-.htmlhttps://pop.com.sg/is-an-anti-symmetric-and-asymmetric-relation-the-same-are-irreflexive-.htmlI don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x). But what if you have......Thu, 20 Aug 2026 07:18:28 -0400https://pop.com.sg/r-and-r-radr-have-the-same-simple-left-modules.htmlhttps://pop.com.sg/r-and-r-radr-have-the-same-simple-left-modules.htmlLet $R$ be a ring (not necessarily commutative). $\operatorname{rad}R:=\bigcap \operatorname{ann}M$, where $M$ ranges over all the simple left $R$-modules. My question is as follow: $R$ and $R/\...Thu, 20 Aug 2026 07:17:01 -0400https://pop.com.sg/what-are-some-examples-of-proof-by-contrapositive.htmlhttps://pop.com.sg/what-are-some-examples-of-proof-by-contrapositive.htmlApplying the Modus Tollens argument to Fermat's Little Theorem really helped me to understand logical implication. I never knew that FLT was actually a compositality test. Theorem (FLT): given int......Thu, 20 Aug 2026 07:13:50 -0400https://pop.com.sg/definition-of-the-j-invariant-of-an-elliptic-curve.htmlhttps://pop.com.sg/definition-of-the-j-invariant-of-an-elliptic-curve.htmlIt seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of ......Thu, 20 Aug 2026 07:11:58 -0400https://pop.com.sg/what-is-and-how-can-i-find-an-orthogonal-component.htmlhttps://pop.com.sg/what-is-and-how-can-i-find-an-orthogonal-component.htmlI have the following task : Find the orthogonal projection and orthogonal component of vector $ \overline x $ relatively to linear subspace generated by vectors $ \overline a_{1}, \overl......Thu, 20 Aug 2026 07:08:58 -0400https://pop.com.sg/confused-about-effective-rate-of-discount-theory-of-interest.htmlhttps://pop.com.sg/confused-about-effective-rate-of-discount-theory-of-interest.htmlI'm currently reading Kellison's book, The Theory of Interest. I've reached the chapter on Effective Rate of Discount and it's somewhat confusing. The book explains it as a loan where interest is p......Thu, 20 Aug 2026 07:08:24 -0400https://pop.com.sg/why-is-the-square-root-of-a-number-not-plus-or-minus-duplicate.htmlhttps://pop.com.sg/why-is-the-square-root-of-a-number-not-plus-or-minus-duplicate.htmlFor example, $\sqrt{4}$. I've asked a bunch of people and I get mixed answers all the time, as to whether it is $-2$ and $+2$ or just $+2$. How about if there's a negative in front of the square r......Thu, 20 Aug 2026 07:04:38 -0400https://pop.com.sg/quadrilateral-inscribed-angles-calculation-with-one-arc-angle.htmlhttps://pop.com.sg/quadrilateral-inscribed-angles-calculation-with-one-arc-angle.htmlI am trying desperately to solve following problem. How can I solve it, the image and question is included in image...Thu, 20 Aug 2026 07:03:13 -0400https://pop.com.sg/x-3-3x-3-0-prove-that-10-x-127.htmlhttps://pop.com.sg/x-3-3x-3-0-prove-that-10-x-127.html$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual valu......Thu, 20 Aug 2026 06:54:45 -0400https://pop.com.sg/find-an-exponential-equation-that-passes-through-the-points-2-2-25-and.htmlhttps://pop.com.sg/find-an-exponential-equation-that-passes-through-the-points-2-2-25-and.htmlI am asked to find an exponential equation that passes through $(2, 2.25)$ and $(5, 60.75)$. My textbook says the solution it's $y=0.25(3)^x$ whereas I got $y=0.028(9)^x$. Here is my working: Express ...Thu, 20 Aug 2026 06:51:18 -0400https://pop.com.sg/what-039-s-the-difference-between-f-cdot-g-and-f-g-x.htmlhttps://pop.com.sg/what-039-s-the-difference-between-f-cdot-g-and-f-g-x.htmlFor example if $f(x) = x + 2$ and $g(x) = 4x - 1$ Then what would be the difference in $f \cdot g$ and $f(g(x))$?...Thu, 20 Aug 2026 06:37:33 -0400https://pop.com.sg/how-to-check-if-these-vectors-are-normal-or-orthogonal.htmlhttps://pop.com.sg/how-to-check-if-these-vectors-are-normal-or-orthogonal.htmlIf we have number of vectors. $a = (1, 0, 2, 0)$ $b = (-2, 1, 0, -1)$ $c = (-1, 1, 2, -1)$ How to check if they are normal, or orthogonal vectors?...Thu, 20 Aug 2026 06:33:50 -0400https://pop.com.sg/quaternions-why-does-ijk-1-and-ij-k-and-ji-k.htmlhttps://pop.com.sg/quaternions-why-does-ijk-1-and-ij-k-and-ji-k.htmlCurrently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: $$\begin{matrix}ij=k,&ji=-k,\\jk=i,&amp......Thu, 20 Aug 2026 06:32:41 -0400https://pop.com.sg/notation-for-rounding-function.htmlhttps://pop.com.sg/notation-for-rounding-function.htmlThe Wikipedia article http://en.wikipedia.org/wiki/Nearest_integer_function mentions the following notation for the rounding or "nearest integer" function. (That is, the function that corresponds t......Thu, 20 Aug 2026 06:15:06 -0400https://pop.com.sg/what-is-the-difference-between-a-polynomial-and-a-function-or-can-they.htmlhttps://pop.com.sg/what-is-the-difference-between-a-polynomial-and-a-function-or-can-they.htmlI have been wondering over this basic question (seems rather trivial at first sight) for a long time: What is the difference between a polynomial and function? My confusion arises form the follow......Thu, 20 Aug 2026 06:06:52 -0400https://pop.com.sg/christoffel-symbols-for-spherical-polar-coordinates.htmlhttps://pop.com.sg/christoffel-symbols-for-spherical-polar-coordinates.htmlIf we are given a line element; $$ds^2=dr^2+r^2d\theta^2+r^2sin^2\theta d\varphi^2$$ We can easily then see that the metric and the inverse metric are; $$g=\begin{pmatrix}1&0&0\\0&r^2&a......Thu, 20 Aug 2026 05:46:54 -0400https://pop.com.sg/pythagoras-theorem-proof.htmlhttps://pop.com.sg/pythagoras-theorem-proof.htmlIs my logic correct below to prove the Pythagoras Theorem? Thanks. Area Rectangle R \begin{align*} R &= WL\\ &=(2a+b)(2b+a)\\ &=4ab+2a^2+2b^2+ab\\ &=5ab+2a^2+2b^2 \end{align*}......Thu, 20 Aug 2026 05:44:09 -0400https://pop.com.sg/how-to-compute-homography-matrix-h-from-corresponding-points-2d-2d-pla.htmlhttps://pop.com.sg/how-to-compute-homography-matrix-h-from-corresponding-points-2d-2d-pla.htmlI went through this thread Mapping Irregular Quadrilateral to a Rectangle If i know the 4 corresponding points in image say p1->p1' p2->p2' p3->p3' p4->p4' then how to compute pi(x,y) from pi'(x......Thu, 20 Aug 2026 05:41:59 -0400https://pop.com.sg/how-is-prisoner-039-s-dilemma-different-from-chicken.htmlhttps://pop.com.sg/how-is-prisoner-039-s-dilemma-different-from-chicken.htmlChicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. ......Thu, 20 Aug 2026 05:38:03 -0400https://pop.com.sg/homogeneous-polynomial-in-k-x-y-z-can-factor-into-linear-polynomials.htmlhttps://pop.com.sg/homogeneous-polynomial-in-k-x-y-z-can-factor-into-linear-polynomials.htmlMy question is quite simple. Let $k$ be an algebraically closed field and $f\in k[X,Y]$. We know that $f$ can factor into linear polynomials. I would like to know if there is some generalization o......Thu, 20 Aug 2026 05:36:59 -0400https://pop.com.sg/showing-a-sequence-is-convergent-using-the-epsilon-n-definition.htmlhttps://pop.com.sg/showing-a-sequence-is-convergent-using-the-epsilon-n-definition.htmlI need to prove that the following sequence converges: $\lim_{n\rightarrow \infty} \frac{2n^2+3n+1}{n^2+n+1}=2$ So for the proof/solution I have the following: Let $\epsilon >0$. Then let $N=......Thu, 20 Aug 2026 05:35:15 -0400https://pop.com.sg/two-rectangles-the-1-st-has-twice-the-perimeter-of-the-2-nd-and-the-2-.htmlhttps://pop.com.sg/two-rectangles-the-1-st-has-twice-the-perimeter-of-the-2-nd-and-the-2-.htmlHow can this be solved using just algebra, where the first rectangle has sides $a$ and $b$, and the second rectangle has sides $c$ and $d$? These are the two equations that follow: $$a + b = 2(c +......Thu, 20 Aug 2026 05:33:05 -0400https://pop.com.sg/how-many-barcodes-are-there.htmlhttps://pop.com.sg/how-many-barcodes-are-there.htmlA barcode is made of white and black lines. A barcode always begins and ends width a black line. Each line has is of thickness 1 or 2, and the whole barcode is of thickness 12. How many different ...Thu, 20 Aug 2026 05:28:15 -0400https://pop.com.sg/what-039-s-the-formula-to-work-out-number-of-fixtures-for-any-number-o.htmlhttps://pop.com.sg/what-039-s-the-formula-to-work-out-number-of-fixtures-for-any-number-o.htmlIf there are $20$ teams in a league, the total number of fixtures are $380$ if every team plays each other twice. How would you come to that number where there are any number of teams?...Thu, 20 Aug 2026 05:24:18 -0400https://pop.com.sg/number-of-line-segments-to-approximate-a-circle.htmlhttps://pop.com.sg/number-of-line-segments-to-approximate-a-circle.htmlThis is part of a homework question, so naturally, not looking for a solution. But I have no idea how to approach the problem. The question is: how many line segments are necessary to approximate a ...Thu, 20 Aug 2026 05:12:42 -0400