https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 02 Sep 2026 11:37:55 -0400https://pop.com.sg/what-is-the-sum-of-n-1-n-2-n-k.htmlhttps://pop.com.sg/what-is-the-sum-of-n-1-n-2-n-k.htmlWhat is the sum of this series ? $(n-1)+(n-2)+(n-3)+...+(n-k)$ $(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$ So how can we find the sum from $n-1$ to $n-k$ ?...Wed, 02 Sep 2026 15:34:29 -0400https://pop.com.sg/the-assumption-in-proof-by-induction.htmlhttps://pop.com.sg/the-assumption-in-proof-by-induction.htmlThe second step in proof by induction is to: Prove that if the statement is true for some integer $n=k$, where $k\ge n_0$ then it is also true for the next larger integer, $n=k+1$ My question......Wed, 02 Sep 2026 15:15:38 -0400https://pop.com.sg/what-is-meant-by-evenly-divisible.htmlhttps://pop.com.sg/what-is-meant-by-evenly-divisible.html"What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?" Is it different from divisible?...Wed, 02 Sep 2026 15:13:59 -0400https://pop.com.sg/prove-all-right-angles-are-congruent.htmlhttps://pop.com.sg/prove-all-right-angles-are-congruent.htmlProve all right angles are congruent. I only have to prove one side to this argument, so I just need to the the other argument. So basically, if two angles are right, then they must be congruent......Wed, 02 Sep 2026 15:10:14 -0400https://pop.com.sg/mod-and-rem-in-maple.htmlhttps://pop.com.sg/mod-and-rem-in-maple.htmlI do not understand this statement "reduce the coefficient modulo 5 and take remainder"? Is the following Maple command to reduce the coefficient modulo 5 , which coefficient to reduce? rem(6*x^2......Wed, 02 Sep 2026 15:08:29 -0400https://pop.com.sg/how-the-recursive-structure-of-apollonian-gaskets-can-be-described-in-.htmlhttps://pop.com.sg/how-the-recursive-structure-of-apollonian-gaskets-can-be-described-in-.htmlThe classical Descartes-Soddy relationship between the signed curvatures $b_k$ ("b" for "bend") of 4 mutually tangent circles (Apollonian configuration): $$\sum_{k=1}^4 b_k^2=\t......Wed, 02 Sep 2026 15:06:19 -0400https://pop.com.sg/finding-h-so-that-a-system-is-consistent-linear-matrix.htmlhttps://pop.com.sg/finding-h-so-that-a-system-is-consistent-linear-matrix.htmlHow can I determine what value of $h$ would make my system consistent. The augmented matrix of my system is: $\begin{bmatrix} 1&4&-2\\ 3&h&6 \end{bmatrix}$ My first step is to add......Wed, 02 Sep 2026 15:05:26 -0400https://pop.com.sg/relationship-between-equicontinuity-and-total-boundedness.htmlhttps://pop.com.sg/relationship-between-equicontinuity-and-total-boundedness.htmlA subspace of a complete metric space is compact iff it is closed and totally bounded. Take a compact space $X$, then its space of continuous functions $C(X)$ is complete. Subspaces of $C(X)$ are ...Wed, 02 Sep 2026 15:04:35 -0400https://pop.com.sg/find-the-singular-points-of-the-differential-equation-x-3-x-1-y-039-03.htmlhttps://pop.com.sg/find-the-singular-points-of-the-differential-equation-x-3-x-1-y-039-03.htmlConsider the second order linear homogeneous equation $$a_0(x)y'' + a_1(x)y'+ a_2(x)y = 0, x \in I \tag{1}$$ Suppose that $a_0$, $a_1$ and $a_2$ are analytic at $x_0 \in I$. If $a_0(x_0) = 0$, then......Wed, 02 Sep 2026 14:58:24 -0400https://pop.com.sg/how-does-the-dot-product-determine-similarity.htmlhttps://pop.com.sg/how-does-the-dot-product-determine-similarity.htmlI want to know how the dot product can determine whether two vectors are similar? I know that the formula $$\cos(\theta) = \frac{u \cdot v }{ ||u||\,||v||}$$ means something, but don't know what....Wed, 02 Sep 2026 14:54:06 -0400https://pop.com.sg/projectile-motion-question-when-launched-at-an-angle.htmlhttps://pop.com.sg/projectile-motion-question-when-launched-at-an-angle.htmlThe question in the calculus class is If the initial velocity of a projectile is $80$ ft/s with an angle of elevation of $55^\circ$, at what height will it strike a building that is $178$ ft away f......Wed, 02 Sep 2026 14:50:15 -0400https://pop.com.sg/inverse-laplace-through-complex-roots.htmlhttps://pop.com.sg/inverse-laplace-through-complex-roots.htmlI have been asked to apply inverse laplace to this: $$ \frac{(4s+5)}{s^2 + 5s +18.5} $$ What I have done is; I found the roots of denominator which are : $$ (-5-7i)/2 $$ and $$ (-5+7i)/2 $$ Then I ...Wed, 02 Sep 2026 14:22:38 -0400https://pop.com.sg/what-does-spans-and-spanned-by-mean-in-the-context-of-linear-algebra.htmlhttps://pop.com.sg/what-does-spans-and-spanned-by-mean-in-the-context-of-linear-algebra.htmlI'm more frequently coming across phrases such as "vector $b$ spanned by $\{b_1, \dots , b_n\}$" and "$A$ spans $B$" while studying linear algebra. What do the terms "spanned by" and "spans" mean i......Wed, 02 Sep 2026 14:12:31 -0400https://pop.com.sg/difference-between-newton-039-s-method-and-gauss-newton-method.htmlhttps://pop.com.sg/difference-between-newton-039-s-method-and-gauss-newton-method.htmlI know that the Gauss-Newton method is essentially Newton's method with the modification that the Gauss-Newton method it uses the approximation $2J^TJ$ (where $J$ is the Jacobian matrix) for the He......Wed, 02 Sep 2026 14:04:04 -0400https://pop.com.sg/is-a-straight-line-the-shortest-distance-between-two-points.htmlhttps://pop.com.sg/is-a-straight-line-the-shortest-distance-between-two-points.htmlQuite simply, I heard a lot of talk about how a straight line isn't necessarily the shortest distance between two points. Is this true, and if it isn't, how would that work?...Wed, 02 Sep 2026 14:03:25 -0400https://pop.com.sg/how-to-find-kolmogorov-forward-equations-given-generator-matrix-q.htmlhttps://pop.com.sg/how-to-find-kolmogorov-forward-equations-given-generator-matrix-q.htmlI am having difficulty in forming Kolmogorov Forward Equations. I understand how the KFE is derived and that $$\frac {d}{ds} p_{ij} (s) = \sum_{k \neq j} p_{ik} (s) \lambda_{k} r_{kj} - p_{ij} (s)\...Wed, 02 Sep 2026 14:00:13 -0400https://pop.com.sg/concrete-fft-polynomial-multiplication-example.htmlhttps://pop.com.sg/concrete-fft-polynomial-multiplication-example.htmlI have read a number of explanations of the steps involved in multiplying two polynomials using fast fourier transform and am not quite getting it in practice. I was wondering if I could get some h......Wed, 02 Sep 2026 13:49:33 -0400https://pop.com.sg/limits-of-a-piecewise-functions.htmlhttps://pop.com.sg/limits-of-a-piecewise-functions.htmlI don't understand how to do this question. I've done a sketch, but wouldn't the limit as x tends to -2 from above be 2? I mean, it's on the actual point that it's tending too... However, I get the ...Wed, 02 Sep 2026 13:48:13 -0400https://pop.com.sg/quickest-way-to-find-coefficient-of-trinomial.htmlhttps://pop.com.sg/quickest-way-to-find-coefficient-of-trinomial.htmlHow can I find the coefficient of $a^3b^2$ in the expansion of: $$ (a+b+3)^5 $$ What is the quickest way to do this without doing the whole expansion?...Wed, 02 Sep 2026 13:44:27 -0400https://pop.com.sg/suppose-ab-ac-where-b-and-c-are-matrices-and-a-is-invertible-show-that.htmlhttps://pop.com.sg/suppose-ab-ac-where-b-and-c-are-matrices-and-a-is-invertible-show-that.htmlSuppose $AB=​AC$, where $B$ and $C$ are matrices, and $A$ is invertible. Show that $B$=enter image description here$C$. Is this​ true, in​ general, when $A$ is not​ invertible? What can be deduced ......Wed, 02 Sep 2026 13:41:38 -0400https://pop.com.sg/motivation-and-examples-for-ramification.htmlhttps://pop.com.sg/motivation-and-examples-for-ramification.htmlI started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...Wed, 02 Sep 2026 13:40:53 -0400https://pop.com.sg/what-is-frac-1-n-called.htmlhttps://pop.com.sg/what-is-frac-1-n-called.htmlI have forgotten the name of this, and Google is not helpful (I've even tried searching "1 divided by n" and the answer has not come up). What is it called when you have a number, $n$, and perform $\...Wed, 02 Sep 2026 13:37:26 -0400https://pop.com.sg/proof-of-the-mean-value-theorem-for-integrals.htmlhttps://pop.com.sg/proof-of-the-mean-value-theorem-for-integrals.htmlMy Single Variable Calc Textbook asked me to prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for Derivatives to the function $F(x)=\displaystyle\int_a^xf(t)dt$. I'm pr......Wed, 02 Sep 2026 13:30:19 -0400https://pop.com.sg/textbooks-for-math-contests-apart-from-aops.htmlhttps://pop.com.sg/textbooks-for-math-contests-apart-from-aops.htmlI am looking for textbooks on math contests that give the theory associated with the topics (such as graph theory,geometry,Trig,combinatorics,etc) before giving a large volley of problems to solve(......Wed, 02 Sep 2026 13:27:45 -0400https://pop.com.sg/stirling-numbers-of-the-first-kind-identity.htmlhttps://pop.com.sg/stirling-numbers-of-the-first-kind-identity.htmlProve the following identity for the signless Stirling numbers of the first kind: $$\sum_{m=k}^n c(n,m){m \choose k}=c(n+1,k+1)$$ I'm trying to rewrite the sum as follows $$c(n,k){k \choose k}+c(n,......Wed, 02 Sep 2026 13:15:30 -0400https://pop.com.sg/boundedness-of-the-solution-of-the-heat-equation.htmlhttps://pop.com.sg/boundedness-of-the-solution-of-the-heat-equation.htmlThe general solution of the heat equation $$\left\{\begin{array}{rcl} \partial_tu-\Delta u &=& 0\\ u(x,0)&=& f \end{array} \right.$$ is given by $$u(x,t)=\int\limits_{\mathbb R^n}\P......Wed, 02 Sep 2026 13:13:03 -0400https://pop.com.sg/find-the-axes-of-the-ellipse.htmlhttps://pop.com.sg/find-the-axes-of-the-ellipse.htmlWhat are the equations of the major and minor axes of the ellipse $x^2+2y^2-2xy-1=0$. The centre of the ellipse is $(0,0)$ but the axes are tilted (with respect to $x-y$ axes). I don't know how to ......Wed, 02 Sep 2026 13:12:20 -0400https://pop.com.sg/techniques-to-prove-a-function-is-uniformly-continuous.htmlhttps://pop.com.sg/techniques-to-prove-a-function-is-uniformly-continuous.htmlSo I understand the definition of uniform continuity, but wanted some suggestions to prove that a function is or isn't uniformly continuous. I have looked ahead and have seen that if a function is ...Wed, 02 Sep 2026 13:11:13 -0400https://pop.com.sg/can-an-alternating-series-ever-be-absolutely-convergent.htmlhttps://pop.com.sg/can-an-alternating-series-ever-be-absolutely-convergent.htmlCan an alternating series EVER be absolutely convergent? I am examining practice problems in my calculus book and I haven't yet come across a case where this is so. It might be because they are s......Wed, 02 Sep 2026 13:05:05 -0400https://pop.com.sg/if-integral-is-zero-and-function-is-continuous-and-non-negative-then-w.htmlhttps://pop.com.sg/if-integral-is-zero-and-function-is-continuous-and-non-negative-then-w.htmlIf $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$ then prove that $f(x)=0$ for all $x \in [a,b]$. I tried with Riemann's definite integral definition but couldn't ...Wed, 02 Sep 2026 12:49:37 -0400https://pop.com.sg/linearize-an-exponential-looking-graph-with-log-function.htmlhttps://pop.com.sg/linearize-an-exponential-looking-graph-with-log-function.htmlThis may be a beginner question, but I can't quite wrap my head around logs... I have a set of data (from an experiment) which gives me an exponential-looking graph (Fig 1). I'd like to "linearize"......Wed, 02 Sep 2026 12:49:06 -0400https://pop.com.sg/how-to-put-9-pigs-into-4-pens-so-that-there-are-an-even-number-of-pigs.htmlhttps://pop.com.sg/how-to-put-9-pigs-into-4-pens-so-that-there-are-an-even-number-of-pigs.htmlPut 9 pigs into 4 pens so that there are an even number of pigs in each pen. Read this for inspiration....Wed, 02 Sep 2026 12:47:12 -0400https://pop.com.sg/pseudometric-and-metric.htmlhttps://pop.com.sg/pseudometric-and-metric.htmlPSEUDOMETRIC Any explanation i will be so thankful . a finite pseudometric on a set X is a function satisfying (M1) ,(M2),(M3) and (M2*) d(x,x)=0 What is the difference between a metric and ...Wed, 02 Sep 2026 12:46:58 -0400https://pop.com.sg/is-there-an-intuitive-way-of-visualising-complex-roots.htmlhttps://pop.com.sg/is-there-an-intuitive-way-of-visualising-complex-roots.htmlConsider the function $f(x)$ such that $f(x) = x^2-4x+13$. By considering the discriminant, it can immediately be seen that the function has no real roots, since $b^2-4ac = (-4)^2-4(13) = -36$ and ......Wed, 02 Sep 2026 12:44:19 -0400https://pop.com.sg/does-the-alphabet-of-the-language-of-propositional-logic-have-no-funct.htmlhttps://pop.com.sg/does-the-alphabet-of-the-language-of-propositional-logic-have-no-funct.htmlIn Ebbinghaus' Mathematical Logic, II.2.1 on p14 says that the alphabet of a first order language contains function symbols, relation symbols, and constants, whereas I don't see these symbols in &q......Wed, 02 Sep 2026 12:41:57 -0400https://pop.com.sg/what-is-the-definition-of-a-mathematical-point.htmlhttps://pop.com.sg/what-is-the-definition-of-a-mathematical-point.htmlA point is defined in the elements as that which have no parts, but what does this mean? There is nothing in the physical world that doesn't have an extension. You can't say a point is primitive no......Wed, 02 Sep 2026 12:35:14 -0400https://pop.com.sg/complete-the-squares-to-find-the-center-and-radius-of-the-circle.htmlhttps://pop.com.sg/complete-the-squares-to-find-the-center-and-radius-of-the-circle.htmlI've had trouble on this one problem for a couple days. Complete the square on the X and Y terms to find the center and radius of the circle. $x^2+2x+y^2-4y=-4\:\:$...Wed, 02 Sep 2026 12:31:21 -0400https://pop.com.sg/how-to-prove-that-the-gram-matrix-a-ta-is-symmetric-with-nonnegative-d.htmlhttps://pop.com.sg/how-to-prove-that-the-gram-matrix-a-ta-is-symmetric-with-nonnegative-d.htmlIf A is any real $m \times n$ matrix, then $A^TA$ is called the Gram matrix of $A$. Show that the Gram matrix of $A$ is always symmetric with nonnegative diagonal elements. I have tried several ...Wed, 02 Sep 2026 12:28:43 -0400https://pop.com.sg/what-is-the-relationship-between-the-leibniz-integral-rule-and-the-dom.htmlhttps://pop.com.sg/what-is-the-relationship-between-the-leibniz-integral-rule-and-the-dom.htmlThe Leibniz integral rule states $$ {\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt} $$ when the integral bounds are not a ...Wed, 02 Sep 2026 12:26:17 -0400https://pop.com.sg/sum-of-consecutive-numbers.htmlhttps://pop.com.sg/sum-of-consecutive-numbers.htmlI was wondering if there a way to figure out the number of ways to express an integer by adding up consecutive natural numbers. For example for $N=3$ there is one way to express it $1+2 = 3$ I ha......Wed, 02 Sep 2026 12:23:55 -0400https://pop.com.sg/prove-this-identity-tan-2x-sec-2x-tan-x-pi-4.htmlhttps://pop.com.sg/prove-this-identity-tan-2x-sec-2x-tan-x-pi-4.htmlI've been having a time with this problem. I tried to start with the left side but I hit a dead end quick... I then tried the right side and had a little more luck but I've hit a block. I first use......Wed, 02 Sep 2026 12:16:21 -0400https://pop.com.sg/how-to-039-easily-039-obtain-the-quadratic-form-of-an-inverse.htmlhttps://pop.com.sg/how-to-039-easily-039-obtain-the-quadratic-form-of-an-inverse.htmlHow to 'easily' obtain the quadratic form of an inverse? x = $v^TA^{-1}\,v$ $A$ is a symmetric invertible matrix of real numbers, it is also positive-definite and sparse. $v$ is a vector with few......Wed, 02 Sep 2026 12:10:52 -0400https://pop.com.sg/gittins-index-for-a-simple-example.htmlhttps://pop.com.sg/gittins-index-for-a-simple-example.htmlEverything I can find on the Gittins Index is extremely in depth and abstract with almost no examples. I have spent hours digging through academic papers, lecture notes, and Wikipedia sources. I ...Wed, 02 Sep 2026 11:58:58 -0400https://pop.com.sg/counter-examples-related-to-slutsky-039-s-theorem.htmlhttps://pop.com.sg/counter-examples-related-to-slutsky-039-s-theorem.htmlConsider some real-valued random variables $(X_n)$, $(Y_n)$, $X$ and $Y$, defined on the same probability space. What is a counterexample to the claim that $X_n\to X$ in distribution and $Y_n \......Wed, 02 Sep 2026 11:57:09 -0400https://pop.com.sg/dividing-both-sides-of-an-inequality-by-a-vector.htmlhttps://pop.com.sg/dividing-both-sides-of-an-inequality-by-a-vector.htmlAs ridiculous as the question sounds, I'm just referring to a very specific case. Suppose that I'm given two $n \times 1$ vectors $x$ and $c$, an $m \times n$ matrix $A$ and a $m \times 1$ vector $......Wed, 02 Sep 2026 11:55:12 -0400https://pop.com.sg/jacobian-transformation-to-find-joint-pdf-of-y-1-and-y-2.htmlhttps://pop.com.sg/jacobian-transformation-to-find-joint-pdf-of-y-1-and-y-2.htmlLet $X_1$ and $X_2$ have the following joint density: $$f_{X_{1},X_{2}}(x_1,x_2) = \begin{cases} {\frac{4}{\pi}e^{-(x_1^2+x_2^2)}} & \text{$x_1 \gt 0$, $x_2 \gt 0$} \\{0} & \text{...Wed, 02 Sep 2026 11:38:37 -0400https://pop.com.sg/how-to-know-limit-of-ln-x-approaches-negative-infinity-as-x-to-0.htmlhttps://pop.com.sg/how-to-know-limit-of-ln-x-approaches-negative-infinity-as-x-to-0.htmlSo $\lim_{x \to 0^+} \ln x = -\infty$. If you draw the graph then it's very obvious. But without drawing a graph, how do we know it's $-\infty$ as opposed to $+\infty$?...Wed, 02 Sep 2026 11:34:19 -0400https://pop.com.sg/calculating-the-perimeter-of-an-ellipse.htmlhttps://pop.com.sg/calculating-the-perimeter-of-an-ellipse.htmlThe hydraulic diameter is defined as $$D_h=\dfrac{4A}{P}$$ where $A$ is the area and $P$ is the perimeter. The area of an ellipse is simply $\pi ab$. The perimeter has many many different ...Wed, 02 Sep 2026 11:29:51 -0400https://pop.com.sg/int-0-infty-1-x-2-converge.htmlhttps://pop.com.sg/int-0-infty-1-x-2-converge.htmlI need to tell if $\int_{0}^{\infty}(-1)^{[x^2]}dx$ converge, diverge or absolutely converges. I managed to say it does not absolutely converges. for it's converges, diverges i tried substituting $......Wed, 02 Sep 2026 11:28:48 -0400https://pop.com.sg/finding-the-exact-value-of-sin-1.htmlhttps://pop.com.sg/finding-the-exact-value-of-sin-1.htmlI have a question that says to find the exact value of $\sin^{-1} (1/\sqrt{2})$ For the answer it shows $\frac{\pi}{4} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ Where does that last part i......Wed, 02 Sep 2026 11:25:47 -0400https://pop.com.sg/does-the-coend-have-a-right-adjoint.htmlhttps://pop.com.sg/does-the-coend-have-a-right-adjoint.htmlLet $ C $ be a small category and $ \mathfrak{X} $ be complete and cocomplete category. The coend may be seen as a functor $ \int ^C : Fun (C\times C^{op}, \mathfrak{X} )\to \mathfrak{X} $ Indeed......Wed, 02 Sep 2026 11:23:51 -0400https://pop.com.sg/integration-by-substitution-fail-for-int-frac-1-1-sin-x-mathrm-dx.htmlhttps://pop.com.sg/integration-by-substitution-fail-for-int-frac-1-1-sin-x-mathrm-dx.htmlWe tried to do an integral with the universal trigonometric substitution $$\int \frac{1}{(1+\sin x)}\, \mathrm dx$$ Meaning, we substituted: $ t = \tan \frac{x}{2} \Rightarrow$ $$\int \frac{1}{......Wed, 02 Sep 2026 11:21:02 -0400https://pop.com.sg/how-to-construct-magic-squares-of-even-order.htmlhttps://pop.com.sg/how-to-construct-magic-squares-of-even-order.htmlCould someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?...Wed, 02 Sep 2026 11:14:07 -0400https://pop.com.sg/prove-that-a-piecewise-function-f-is-differentiable-at-0.htmlhttps://pop.com.sg/prove-that-a-piecewise-function-f-is-differentiable-at-0.html$f(x)= e^{-\frac{1}{x}}$, $x>0$ $f(x)=-x^2$, $x\leq0$ This is a function that is defined for $x$ here above . How can I prove that the derivative of $f$ exists at $x=0$....Wed, 02 Sep 2026 11:13:52 -0400https://pop.com.sg/prove-variance-in-uniform-distribution-continuous.htmlhttps://pop.com.sg/prove-variance-in-uniform-distribution-continuous.htmlI read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?...Wed, 02 Sep 2026 11:10:50 -0400https://pop.com.sg/is-a-negative-number-squared-negative-duplicate.htmlhttps://pop.com.sg/is-a-negative-number-squared-negative-duplicate.html$-3^2 = -9$ I found this problem while doing an algebra refresher in the book The Complete Idiot's Guide to Algebra. I asked my engineer brother this problem and he got it wrong. A search on Goo......Wed, 02 Sep 2026 11:07:12 -0400https://pop.com.sg/zero-probability-and-impossibility.htmlhttps://pop.com.sg/zero-probability-and-impossibility.htmlI read a comment under this question: There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never been......Wed, 02 Sep 2026 10:56:59 -0400https://pop.com.sg/given-coordinates-of-vertices-a-convex-quadrilateral-find-which-pairs-.htmlhttps://pop.com.sg/given-coordinates-of-vertices-a-convex-quadrilateral-find-which-pairs-.htmlI'm writing a program that will take in four $(x, y, z)$ points that form a quadrilateral, and produce two sets of $(x, y, z)$ points that form the two largest triangles that comprise the quadrilat......Wed, 02 Sep 2026 10:55:03 -0400https://pop.com.sg/finding-the-leading-digit-s-in-any-number.htmlhttps://pop.com.sg/finding-the-leading-digit-s-in-any-number.htmlI am somewhat perplexed by this (presumably very simple) issue. Simply, I am trying to find the leading digit(s) in any number (related question). Here is some nice python code (from this question......Wed, 02 Sep 2026 10:54:22 -0400https://pop.com.sg/if-5-sqrt-x-125-sqrt-x-5-1-then-x-equals-4.htmlhttps://pop.com.sg/if-5-sqrt-x-125-sqrt-x-5-1-then-x-equals-4.htmlFor the equation $$5\sqrt [ x ]{ 125 } =\sqrt [ x ]{ { 5 }^{ -1 } } $$ $x$ is equal to $-4$, but I'm not sure why. I've taken the right side of the equation ${ \left( \frac { 1 }{ 5 } \righ......Wed, 02 Sep 2026 10:42:23 -0400https://pop.com.sg/how-to-find-the-equation-of-a-parabola-with-only-one-solution-and-a-po.htmlhttps://pop.com.sg/how-to-find-the-equation-of-a-parabola-with-only-one-solution-and-a-po.htmlI have been given the points $(5, 0)$ and $(10, 5)$. I'm also given that the $a$ value in $ax^2 \pm bx \pm c$ is $-1$. I have been trying to work out to how to do this, but, when I graph the parabo......Wed, 02 Sep 2026 10:40:38 -0400https://pop.com.sg/what-is-a-direct-correlation.htmlhttps://pop.com.sg/what-is-a-direct-correlation.htmlI have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the var......Wed, 02 Sep 2026 10:35:44 -0400https://pop.com.sg/ideals-in-polynomial-rings-generated-by-linear-polynomials.htmlhttps://pop.com.sg/ideals-in-polynomial-rings-generated-by-linear-polynomials.htmlLet $K$ be any field. Is it true that: If $f_1,\dots, f_k \in K[x_1,\dots,x_n]$ are polynomials of degree $1$, then $I=(f_1,\dots, f_k)$ is all $K[x_1,\dots,x_n]$, or, if it is a proper ideal, the......Wed, 02 Sep 2026 10:16:58 -0400https://pop.com.sg/minkowski-functional-being-equivalent-to-original-norm.htmlhttps://pop.com.sg/minkowski-functional-being-equivalent-to-original-norm.htmlDefinition1: A subset $ A$ of a normed space $E$ is absolutely convex if $ \forall x,y \in A, \quad \forall \alpha ,\beta $ s.t $|\alpha |+ |\beta| \leq 1, \quad $ $ \alpha x +\beta y \in A.$ ...Wed, 02 Sep 2026 10:16:00 -0400https://pop.com.sg/what-is-the-name-of-a-fully-twisted-strip.htmlhttps://pop.com.sg/what-is-the-name-of-a-fully-twisted-strip.htmlI know that an odd number of twists generate a Möbius strip, and that an even number of twists generate a shape with two sides. Now, as far as I could find, for the strip with an even number of twi......Wed, 02 Sep 2026 10:12:38 -0400https://pop.com.sg/simple-dice-game-optimal-strategy.htmlhttps://pop.com.sg/simple-dice-game-optimal-strategy.htmlHere's the description of a dice game which puzzles me since quite some time (the game comes from a book which offered a quite unsatisfactory solution — but then, its focus was on programming, so t......Wed, 02 Sep 2026 10:10:49 -0400https://pop.com.sg/the-need-for-the-formal-definition-of-a-limit-duplicate.htmlhttps://pop.com.sg/the-need-for-the-formal-definition-of-a-limit-duplicate.htmlThe intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$. the formal ...Wed, 02 Sep 2026 10:01:13 -0400https://pop.com.sg/how-to-find-4th-degree-polynomial-equation-from-given-points.htmlhttps://pop.com.sg/how-to-find-4th-degree-polynomial-equation-from-given-points.htmlI am trying to find 4th degree polynomial equation from given points. I do not own a graphing calculator so this task is very difficult for me to solve. So far I would out what points I need. The p......Wed, 02 Sep 2026 09:54:37 -0400https://pop.com.sg/lebesgue-integral-basics.htmlhttps://pop.com.sg/lebesgue-integral-basics.htmlI'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's tak......Wed, 02 Sep 2026 09:46:12 -0400https://pop.com.sg/why-is-pi-considered-irrational-if-it-can-be-expressed-as-ratio-of-cir.htmlhttps://pop.com.sg/why-is-pi-considered-irrational-if-it-can-be-expressed-as-ratio-of-cir.htmlPi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where b......Wed, 02 Sep 2026 09:41:52 -0400https://pop.com.sg/can-i-optimize-area-of-cylinder-with-no-givens.htmlhttps://pop.com.sg/can-i-optimize-area-of-cylinder-with-no-givens.htmlI have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads: A metal can is in the form of a cylinder. It has a bottom ......Wed, 02 Sep 2026 09:38:54 -0400https://pop.com.sg/example-of-an-inverse-semigroup.htmlhttps://pop.com.sg/example-of-an-inverse-semigroup.htmlAn inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...Wed, 02 Sep 2026 09:33:49 -0400https://pop.com.sg/what-039-s-the-intuition-behind-the-co-area-formula.htmlhttps://pop.com.sg/what-039-s-the-intuition-behind-the-co-area-formula.htmlI mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer. If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \ge......Wed, 02 Sep 2026 09:30:56 -0400https://pop.com.sg/how-to-rotate-a-vector-along-another-vector.htmlhttps://pop.com.sg/how-to-rotate-a-vector-along-another-vector.htmlI want to rotate a vector e.g $V = Ai + Bj + Ck$ along an another vector $U = xi + yj + zk$. The angle between $U$ and $V$ is $\theta$. How can I do that and how do I know the new value of $A$, $......Wed, 02 Sep 2026 09:20:21 -0400https://pop.com.sg/mean-and-variance-from-a-cumulative-distribution-function.htmlhttps://pop.com.sg/mean-and-variance-from-a-cumulative-distribution-function.htmlI'm trying to find the mean (expected value) and variance for the following distribution function: $F(x)=\begin{cases} 0 & \text{for } x \lt 0\\ x/4 & \text{for } 0 \le x \lt 1\\ ......Wed, 02 Sep 2026 09:14:39 -0400https://pop.com.sg/what-is-the-negation-of-the-implication-statement.htmlhttps://pop.com.sg/what-is-the-negation-of-the-implication-statement.htmlIn a course on logic and proofs the professor presented on the following lines to show an example of negation: $$ \neg (P \Rightarrow Q) \ \ \ \ \Longleftrightarrow \ \ \ \ P \wedge \neg Q $$ I ca......Wed, 02 Sep 2026 09:10:28 -0400https://pop.com.sg/area-between-two-curves-with-respect-to-y.htmlhttps://pop.com.sg/area-between-two-curves-with-respect-to-y.htmlLet $x_1$ be the red curve and $x_2$ be the blue curve. I've tried integrating $x_1-x_2$ as well as $x_2-x_1$ from $y=0$ to $y=9/2$ and both answers were marked wrong. What do I do here? I'm thinki......Wed, 02 Sep 2026 09:08:55 -0400https://pop.com.sg/hanson-wright-inequality-any-relation-between-the-original-quadratic-f.htmlhttps://pop.com.sg/hanson-wright-inequality-any-relation-between-the-original-quadratic-f.htmlHanson-Wright's inequality such as this provides a bound on the second-order chaos of the form $$P(x^\top A x -\mathbb{E}[x^\top A x] > t) $$ For example, $x\sim N(0, I_n)$ and $A$ is some $n\ti......Wed, 02 Sep 2026 09:03:52 -0400https://pop.com.sg/surface-integral-of-a-right-circular-cone.htmlhttps://pop.com.sg/surface-integral-of-a-right-circular-cone.htmlUse a surface integral to show that the surface area of a right circular cone of radius $R$ and height $h$ is $\pi R \sqrt{h^2+R^2}$. Hint -- Use the parametrization $x=r\cos\theta$, $y=r\sin\theta......Wed, 02 Sep 2026 08:58:46 -0400https://pop.com.sg/problems-that-are-easy-with-trigonometry-and-too-difficult-without-it.htmlhttps://pop.com.sg/problems-that-are-easy-with-trigonometry-and-too-difficult-without-it.htmlI was looking for examples that shows importance of trigonometry in an elementary level. Like if you wanted to surprise your high school students that how problems are made easy using trigonometry ......Wed, 02 Sep 2026 08:56:50 -0400https://pop.com.sg/does-euler-039-s-formula-give-e-ix-cos-x-i-sin-x.htmlhttps://pop.com.sg/does-euler-039-s-formula-give-e-ix-cos-x-i-sin-x.htmlDoes Eulers formula give $$e^{-ix}=\cos(x) -i\sin(x)$$ I know that $$e^{ix}=\cos(x)+i\sin(x)$$ But how does it work when we have a $-$ in front...Wed, 02 Sep 2026 08:54:40 -0400https://pop.com.sg/what-is-the-definition-of-a-circular-wedge.htmlhttps://pop.com.sg/what-is-the-definition-of-a-circular-wedge.htmlIn Ahlfors' Complex Analysis, chapter 3, section 4, the author claims that a region whose boundary consists of two circular arcs with common end points is either a "circular wedge" or its complemen......Wed, 02 Sep 2026 08:30:35 -0400https://pop.com.sg/how-do-i-properly-read-a-clinometer.htmlhttps://pop.com.sg/how-do-i-properly-read-a-clinometer.htmlIf the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?...Wed, 02 Sep 2026 08:27:16 -0400https://pop.com.sg/understanding-quantifier-elimination.htmlhttps://pop.com.sg/understanding-quantifier-elimination.htmlI am struggling to understand Quantifier Elimination as it is treated in Hodges' "A Shorter Model Theory". The relevant definitions are : Definition: Take $K$ to be a class of $L$-struct......Wed, 02 Sep 2026 08:26:00 -0400https://pop.com.sg/why-is-the-method-of-prime-factorization-help-in-finding-the-lcm-dupli.htmlhttps://pop.com.sg/why-is-the-method-of-prime-factorization-help-in-finding-the-lcm-dupli.htmlwhy does finding the prime factors help in finding the lcm of two or more numbers. I know how to do it but why is this the case? Edit I found this answer from just thinking on my own.I don't kno......Wed, 02 Sep 2026 08:24:04 -0400https://pop.com.sg/definition-laplace-operator-on-h-1-0-omega.htmlhttps://pop.com.sg/definition-laplace-operator-on-h-1-0-omega.htmlLet $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Is the Laplace operador defined in the space $H^2(\Omega)$ as a sum of second weak derivatives? How is the Laplace operador defined in the......Wed, 02 Sep 2026 08:22:09 -0400https://pop.com.sg/proof-of-congruence-with-mod-3.htmlhttps://pop.com.sg/proof-of-congruence-with-mod-3.htmlOk so I've tried making this one work for a while and keep getting stuck. I have the following: Prove: ($x^2$ is congruent to $x\mod3$) $\iff$ ($x$ is not congruent to $2\mod3$). So I know it is bi-...Wed, 02 Sep 2026 08:19:49 -0400https://pop.com.sg/pigeonhole-principle-formula-using-propositonal-logic.htmlhttps://pop.com.sg/pigeonhole-principle-formula-using-propositonal-logic.htmlAccording to the Pigeonhole Principle, if we try to place $n+1$ pigeons in $n$ pigeonholes, then at least one pigeonhole must have two or more pigeons. For $i \in \{1, 2, \dots, n+1\}$ and $j \in \......Wed, 02 Sep 2026 08:12:48 -0400https://pop.com.sg/what-is-the-mathematical-term-and-symbol-for-division-without-remainde.htmlhttps://pop.com.sg/what-is-the-mathematical-term-and-symbol-for-division-without-remainde.htmlWhat is the mathematical term and symbol for division without remainders? Take for example 322 / 100. I know 322 / 100 is division and the result is 3.22. I know 322 % 100 is a modulo and the resul......Wed, 02 Sep 2026 08:02:00 -0400https://pop.com.sg/express-99-2-3-as-a-fraction-no-calculator.htmlhttps://pop.com.sg/express-99-2-3-as-a-fraction-no-calculator.htmlMy 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is ...Wed, 02 Sep 2026 07:58:24 -0400https://pop.com.sg/please-verify-my-proof-sqrt-18-is-irrational.htmlhttps://pop.com.sg/please-verify-my-proof-sqrt-18-is-irrational.htmlThe following is one of my exam questions and I would like to know whether my proof is correct or not. Prove $\sqrt{18}$ is irrational using proof by contradiction. Following is my proof......Wed, 02 Sep 2026 07:52:34 -0400https://pop.com.sg/vertical-pipe-through-a-45-roof-how-to-determine-the-roof-hole-039-s-s.htmlhttps://pop.com.sg/vertical-pipe-through-a-45-roof-how-to-determine-the-roof-hole-039-s-s.htmlI have a tent roof that is at a 45° angle. In the tent, I have a woodstove that has a vertical stove pipe. I want to cut a hole in the roof of the tent that will be the exact shape of the protru......Wed, 02 Sep 2026 07:45:36 -0400https://pop.com.sg/mixtures-and-alligations-acid-concentration.htmlhttps://pop.com.sg/mixtures-and-alligations-acid-concentration.htmlFrom a vessel containing $1$ litre of pure acid, $100$ ml pure acid was drawn out in each of the beakers A and B. The acid in both the beakers was diluted by adding water in different proportions. ......Wed, 02 Sep 2026 07:43:55 -0400https://pop.com.sg/keeping-the-arc-length-constant-between-points-in-a-spiral.htmlhttps://pop.com.sg/keeping-the-arc-length-constant-between-points-in-a-spiral.htmlI'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is s = r * th......Wed, 02 Sep 2026 07:31:09 -0400https://pop.com.sg/find-max-min-volume-of-rectangular-box-using-lagrange-multipliers.htmlhttps://pop.com.sg/find-max-min-volume-of-rectangular-box-using-lagrange-multipliers.htmlCan anyone help me solve the problem below? This is question number 14.8.42 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the maximum and minimum volumes of a ...Wed, 02 Sep 2026 07:25:58 -0400https://pop.com.sg/determining-all-the-elements-of-s4.htmlhttps://pop.com.sg/determining-all-the-elements-of-s4.htmlWhat is an easy way to determine the elements of $S_{4}$? While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this. I trie......Wed, 02 Sep 2026 07:25:34 -0400https://pop.com.sg/union-of-intersections-and-intersection-of-unions.htmlhttps://pop.com.sg/union-of-intersections-and-intersection-of-unions.htmlWould someone be able to help me understand the (concept of) "union of intersections" and the "intersection of unions". More specifically, how to approach (prove) the following equality of two sets......Wed, 02 Sep 2026 07:20:38 -0400https://pop.com.sg/a-letter-is-picked-at-random-from-the-alphabet.htmlhttps://pop.com.sg/a-letter-is-picked-at-random-from-the-alphabet.htmlJust a book problem I need help on: A letter is picked at random from the alphabet. Find the probability that the letter is contained in the word "house" or in the word "phone". I know this probl......Wed, 02 Sep 2026 07:17:43 -0400https://pop.com.sg/cos-cos-x-sin-sin-x.htmlhttps://pop.com.sg/cos-cos-x-sin-sin-x.htmlShow that $\forall x \in \mathbb{R}\ \cos (\cos x) > \sin (\sin x)$ I've tried the obvious thing to do: $\varphi: x \mapsto \cos(\cos x) - \sin (\sin x)$ $\varphi ' (x) = \sin (\cos x) \sin ......Wed, 02 Sep 2026 07:14:18 -0400https://pop.com.sg/incremental-averaging.htmlhttps://pop.com.sg/incremental-averaging.htmlIs there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance? For example you have a = [4 6 3 9 4 12 4 18] and you wa......Wed, 02 Sep 2026 07:06:19 -0400