https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 14 Aug 2026 07:34:14 -0400https://pop.com.sg/how-do-i-find-the-mode-when-my-data-points-have-decimals.htmlhttps://pop.com.sg/how-do-i-find-the-mode-when-my-data-points-have-decimals.htmlMy data has numbers such as 70.35, 70.39, 70.1. None of the numbers are exactly the same. How do I find the mode?...Fri, 14 Aug 2026 11:33:25 -0400https://pop.com.sg/is-it-necessary-for-the-imaginary-axis-to-be-perpendicular-to-the-real.htmlhttps://pop.com.sg/is-it-necessary-for-the-imaginary-axis-to-be-perpendicular-to-the-real.htmlThe Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...Fri, 14 Aug 2026 11:32:36 -0400https://pop.com.sg/what-is-the-definition-of-binary-sequence.htmlhttps://pop.com.sg/what-is-the-definition-of-binary-sequence.htmlCan I write an infinite binary sequence like so: ...0111001001, ...10010 because I saw some people write infinite binary set from left to right like so: 1011000... , 101111... But I was not sure ......Fri, 14 Aug 2026 11:28:24 -0400https://pop.com.sg/find-critical-and-saddle-points-for-a-multivariable-function.htmlhttps://pop.com.sg/find-critical-and-saddle-points-for-a-multivariable-function.htmlThe function is $f(x, y) = x^2 + 2y^2 + 2xy - 4y + 15$ here's what I did, $$f_x = 2x + 2y$$ $$f_y = 4y + 2x - 4$$ $$f_{xx} = 2$$ $$f_{yy} = 4$$ $$f_{xy} = 2$$ Putting $f_x = 0$ and $f_y = 0$, w......Fri, 14 Aug 2026 11:05:06 -0400https://pop.com.sg/when-is-the-equicorrelation-matrix-positive-definite.htmlhttps://pop.com.sg/when-is-the-equicorrelation-matrix-positive-definite.htmlShow that the matrix, $$ A= \begin{bmatrix} 1 & \rho & \dots & \rho \\ \rho & 1 & \dots & \rho \\ \vdots & \rho & \ddots & \vdots \\ \rho & \dot......Fri, 14 Aug 2026 11:04:35 -0400https://pop.com.sg/alternate-definition-of-a-derivative-question.htmlhttps://pop.com.sg/alternate-definition-of-a-derivative-question.htmlThe alternate form of the derivative is $$f'(a)=\lim_\limits{x \to a}\frac{f(x)-f(a)}{x-a}$$ Can this be rewritten to this and still be true? $$f'(a)=\lim_\limits{x \to a}\frac{f(a)-f(x)}{a......Fri, 14 Aug 2026 10:48:50 -0400https://pop.com.sg/formula-for-calculating-total-yearly-payments-with-yearly-escalation-o.htmlhttps://pop.com.sg/formula-for-calculating-total-yearly-payments-with-yearly-escalation-o.htmlMy knowledge on math/terminology is pretty terrible so seeking some help with a formula (which I'll be converting to javascript to be used in a calculator). I'm trying to calculate the total amoun......Fri, 14 Aug 2026 10:18:46 -0400https://pop.com.sg/finding-sum-of-factors-of-a-number-using-prime-factorization.htmlhttps://pop.com.sg/finding-sum-of-factors-of-a-number-using-prime-factorization.htmlGiven a number, there is an algorithm described here to find it's sum and number of factors. For example, let us take the number $1225$ : It's factors are $1, 5, 7, 25, 35, 49, 175, 245, 1225 $ an......Fri, 14 Aug 2026 10:17:21 -0400https://pop.com.sg/how-to-integrate-with-respect-to-x-2.htmlhttps://pop.com.sg/how-to-integrate-with-respect-to-x-2.html$$\int x\,dx^2$$ I'm having trouble comprehending this question. I can perform substitution, but when I do I come out wrong. Here's how I've happened to handle this: $\int x\,dx^2$ $u=x^2$ $du=......Fri, 14 Aug 2026 10:01:32 -0400https://pop.com.sg/which-day-of-the-week-will-be-100-days-from-now.htmlhttps://pop.com.sg/which-day-of-the-week-will-be-100-days-from-now.htmlIf today is Wednesday, what day of the week will it be $100$ days from now? Going forward $1,8,15,\dots,92,99$ days all result in Wednesday. So the $100$th day will be Thursday. But the solution ......Fri, 14 Aug 2026 09:49:16 -0400https://pop.com.sg/notation-for-eigenvalues.htmlhttps://pop.com.sg/notation-for-eigenvalues.htmlIs there a specific notation for eigenvalues? specifically, I'd like to write: $$m\equiv \text{smallest eigenvaue of }H$$ I've seen some sources write this as: $H\succeq mI$, where "$\succeq0$" mea......Fri, 14 Aug 2026 09:48:50 -0400https://pop.com.sg/what-is-the-mean-of-outcomes-of-six-faced-fair-die-closed.htmlhttps://pop.com.sg/what-is-the-mean-of-outcomes-of-six-faced-fair-die-closed.htmlA six faced fair die is tossed a large number of times (tending to infinity). What is the mean of outcomes of the die? My try: let the die be tossed $n$ times then the mean of outcomes $$=\frac{......Fri, 14 Aug 2026 09:37:45 -0400https://pop.com.sg/can-a-function-be-of-class-c1-even-if-its-partial-derivatives-are-not-.htmlhttps://pop.com.sg/can-a-function-be-of-class-c1-even-if-its-partial-derivatives-are-not-.htmlI know that if all partial derivatives of a function f exist and are continuous then the function is said to be of class C1 (continuously differentiable). However, I was not able to find whether ......Fri, 14 Aug 2026 09:30:59 -0400https://pop.com.sg/visualizing-the-4th-dimension.htmlhttps://pop.com.sg/visualizing-the-4th-dimension.htmlIn a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects? I searched that we can at least see ...Fri, 14 Aug 2026 09:29:19 -0400https://pop.com.sg/deval-function-in-matlab.htmlhttps://pop.com.sg/deval-function-in-matlab.htmlI can't understand how the deval function works in matlab, it's hard for me to understand the code, are there any books or descriptions of numerical methods where I can read about numerical algorit......Fri, 14 Aug 2026 09:29:05 -0400https://pop.com.sg/how-to-find-the-closed-form-to-the-fibonacci-numbers-duplicate.htmlhttps://pop.com.sg/how-to-find-the-closed-form-to-the-fibonacci-numbers-duplicate.htmlPossible Duplicate: Prove this formula for the Fibonacci Sequence How to find the closed form to the fibonacci numbers? I have seen is possible calculate the fibonacci numbers without recursi......Fri, 14 Aug 2026 09:17:00 -0400https://pop.com.sg/help-explaining-structural-induction.htmlhttps://pop.com.sg/help-explaining-structural-induction.htmlI am trying to wrap my head around structural induction. Can someone break it down and explain it around this problem? Let S, a subset of $\mathbb{N}*\mathbb{N}$, be defined recursively by: Base ......Fri, 14 Aug 2026 09:13:55 -0400https://pop.com.sg/method-of-completing-squares-with-3-variables.htmlhttps://pop.com.sg/method-of-completing-squares-with-3-variables.htmlI want to use the method "completing squares" for this term: $x^2-2xy +y^2+z^2*a+2xz-2yz$ The result should be $(x-y+z)^2 +(a-1)*z^3$ Is there a "recipe" behind how to do this? Hope someone coul......Fri, 14 Aug 2026 09:02:32 -0400https://pop.com.sg/number-of-cycles-and-paths-in-a-wheel-graph-closed.htmlhttps://pop.com.sg/number-of-cycles-and-paths-in-a-wheel-graph-closed.htmlSay I have a wheel graph $W_n$ where $n$ represents the number of vertices and $n \geq 3$. Is there a general formula to determine the number of cycles there are in total or the number of cycles of a ...Fri, 14 Aug 2026 08:59:19 -0400https://pop.com.sg/why-is-the-surface-of-a-torus-flat.htmlhttps://pop.com.sg/why-is-the-surface-of-a-torus-flat.htmlWhy is the surface of a torus is said to be flat? If you consider the geometry of the torus, its surface has locally positive (spherical), negative (hyperbolic) and flat curvature....Fri, 14 Aug 2026 08:53:47 -0400https://pop.com.sg/can-something-be-statistically-impossible.htmlhttps://pop.com.sg/can-something-be-statistically-impossible.htmlDoes it make sense when people say "statistically impossible"?...Fri, 14 Aug 2026 08:53:22 -0400https://pop.com.sg/the-tits-cone-geometric-understanding.htmlhttps://pop.com.sg/the-tits-cone-geometric-understanding.htmlI know that the definition of the Tits cone is $Y=\bigcup_{w\in W}{wC}$ with W the Coxeter Groups and C the fundamental chamber. One theorem says that Y is the whole space if W is finite. But how c......Fri, 14 Aug 2026 08:48:23 -0400https://pop.com.sg/why-are-the-nonzero-integers-under-multiplication-not-a-group-closed.htmlhttps://pop.com.sg/why-are-the-nonzero-integers-under-multiplication-not-a-group-closed.htmlI was just checking out a GRE mathematics practise test. I have never had theories about groups, so could someone explain how to test if something is a group and why the 'nonzero integers under ...Fri, 14 Aug 2026 08:45:43 -0400https://pop.com.sg/is-there-a-better-notation-for-integrating-twice.htmlhttps://pop.com.sg/is-there-a-better-notation-for-integrating-twice.htmlI'm studying kinematics with varying acceleration. To calculate the displacement from the acceleration you have to integrate the acceleration with respect to t, then integrate that with respect to t, ...Fri, 14 Aug 2026 08:38:06 -0400https://pop.com.sg/what-is-d-in-equation-of-a-plane.htmlhttps://pop.com.sg/what-is-d-in-equation-of-a-plane.htmlIn a previous Quora post on this topic (https://www.quora.com/How-does-the-equation-ax-+-by-+-cz-d-define-a-plane), a respondent states that $D$ represents the distance from the origin to a plane. ......Fri, 14 Aug 2026 08:36:17 -0400https://pop.com.sg/probability-of-deck-of-cards-such-that-each-person-receives-one-ace.htmlhttps://pop.com.sg/probability-of-deck-of-cards-such-that-each-person-receives-one-ace.htmlSuppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so that each player receives 13 cards. Determine the probability that ...Fri, 14 Aug 2026 08:33:09 -0400https://pop.com.sg/strong-induction-doesn-039-t-require-a-base-case.htmlhttps://pop.com.sg/strong-induction-doesn-039-t-require-a-base-case.htmlI'm considering the natural numbers to be the nonnegative integers. The principle of strong induction can be stated as follows, "If $P$ is a property such that for any $x$, if $P$ holds for all n......Fri, 14 Aug 2026 08:28:09 -0400https://pop.com.sg/derivative-of-frac-d-dx-ln-x-sqrt-x-2-1.htmlhttps://pop.com.sg/derivative-of-frac-d-dx-ln-x-sqrt-x-2-1.htmlShow that $\frac{d}{dx}\ln (x+\sqrt{x^2+1}) = \frac{1}{\sqrt{x^2+1}}$ So I've done this so far: $$\frac{d}{dx}\ln (x+\sqrt{x^2+1}) = \frac{1+x(x^2+1)^{-0.5}}{x+(x^2+1)^{0.5}}$$ I have tried a few ...Fri, 14 Aug 2026 08:19:26 -0400https://pop.com.sg/prove-that-f-x0-g-x0.htmlhttps://pop.com.sg/prove-that-f-x0-g-x0.htmlLet f and g be differentiable functions on an interval $(a,b)$ such that $a < b$. If there is $x0 ∈ (a,b)$ for which $f(x0) =g(x0)$ and $f(x) ≤ g(x)$ for all $x ∈ (a,b)$, prove that $f’(x0) =g’(......Fri, 14 Aug 2026 08:14:32 -0400https://pop.com.sg/what-is-the-purpose-of-finding-roots.htmlhttps://pop.com.sg/what-is-the-purpose-of-finding-roots.htmlI think I know what a root is. It's the value of elements in the domain of a function that map to 0 in the range of the function. At least that's how I see it. Do correct me if I'm wrong. However......Fri, 14 Aug 2026 08:13:17 -0400https://pop.com.sg/tricks-for-constructing-hilbert-style-proofs.htmlhttps://pop.com.sg/tricks-for-constructing-hilbert-style-proofs.htmlSeveral times in my studies, I've come across Hilbert-style proof systems for various systems of logic, and when an author says, "Theorem: $\varphi$ is provable in system $\cal H$," or "Theorem: the ...Fri, 14 Aug 2026 07:53:32 -0400https://pop.com.sg/is-mathbb-z-a-solvable-group.htmlhttps://pop.com.sg/is-mathbb-z-a-solvable-group.htmlOk, so abelian groups are solvable. And Thm II.8.5 of Hungerford says A group is solvable iff it has a solvable series. (The group may be finite or infinite.) However, I can't seem to find a solv......Fri, 14 Aug 2026 07:53:21 -0400https://pop.com.sg/what-is-the-solution-to-this-inequality.htmlhttps://pop.com.sg/what-is-the-solution-to-this-inequality.htmlThis is the given inequality I've been trying to solve $$1/6\leq \frac{1}{\mid x \mid} \leq 1/2$$ However the answer I get is $(0,6] \cup [2,6]$ which is not the answer given in my book. Could you ...Fri, 14 Aug 2026 07:53:13 -0400https://pop.com.sg/proofs-that-every-mathematician-should-know-closed.htmlhttps://pop.com.sg/proofs-that-every-mathematician-should-know-closed.htmlThere are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc. So I'd like to know what ...Fri, 14 Aug 2026 07:51:22 -0400https://pop.com.sg/pythagorean-triple-where-a-b.htmlhttps://pop.com.sg/pythagorean-triple-where-a-b.htmlI cannot find even a single webpage mentioning this topic. I'm a programmer and I'm looking for a 45-45-90 triangle where all of the sides are whole numbers. In the video I am watching, they say to......Fri, 14 Aug 2026 07:50:35 -0400https://pop.com.sg/joint-entropy-of-2-independent-random-variables.htmlhttps://pop.com.sg/joint-entropy-of-2-independent-random-variables.htmlSay we have two independent random variables $X$ and $Y$. What is their joint entropy $H(X,Y)$? I worked this out, but I am not sure if the result I reached is correct. The definitions of entrop......Fri, 14 Aug 2026 07:39:42 -0400https://pop.com.sg/how-do-i-eliminate-the-parameter-to-find-a-cartesian-equation-closed.htmlhttps://pop.com.sg/how-do-i-eliminate-the-parameter-to-find-a-cartesian-equation-closed.htmlGiven the two parametric equations $$x=t^2+1$$ $$y=t+1 \iff t= y-1$$ How do I eliminate parameter $t$ to find a Cartesian equation? Do I substitute? This is confusing me, so I would appreciate......Fri, 14 Aug 2026 07:36:31 -0400https://pop.com.sg/exercise-3-4-in-rotman-039-s-an-introduction-to-algebraic-topology.htmlhttps://pop.com.sg/exercise-3-4-in-rotman-039-s-an-introduction-to-algebraic-topology.htmlI am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution. Exercise 3.4: Le......Fri, 14 Aug 2026 07:28:54 -0400https://pop.com.sg/solve-sin-2x-sin-x-frac-pi-6.htmlhttps://pop.com.sg/solve-sin-2x-sin-x-frac-pi-6.htmlI got a this equation to solve and $\sin(2x) = \sin(x-\frac{\pi}{6})$ They was nice to give me for possible answers it could be. x = $\frac{\pi}{6}$ x = $\frac{\pi}{18}$ x = $\frac{7\pi}{18}$ ......Fri, 14 Aug 2026 07:27:31 -0400https://pop.com.sg/what-is-the-difference-between-an-array-and-a-vector.htmlhttps://pop.com.sg/what-is-the-difference-between-an-array-and-a-vector.htmlOkay so I'm doing a little bit of vector calculus at university (mainly with neural networks and the k-means clustering for cluster analysis in a 3 dimensional field or hyperplane) And from what I ...Fri, 14 Aug 2026 07:26:23 -0400https://pop.com.sg/prove-a-countable-intersection-of-open-and-dense-sets-in-a-compact-hau.htmlhttps://pop.com.sg/prove-a-countable-intersection-of-open-and-dense-sets-in-a-compact-hau.htmlLet $X$ be a compact Hausdorff space and let $\{U_n\}$ be a countable collection of subsets that are open and dense in $X$. Show that the intersection $$\bigcap\limits_{n=1}^\infty U_n$$ is dense. I ...Fri, 14 Aug 2026 07:24:49 -0400https://pop.com.sg/difference-between-postulates-axioms-and-theorems.htmlhttps://pop.com.sg/difference-between-postulates-axioms-and-theorems.htmlI'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the ...Fri, 14 Aug 2026 07:10:23 -0400https://pop.com.sg/convolution-of-a-function-with-itself.htmlhttps://pop.com.sg/convolution-of-a-function-with-itself.htmlFunction $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I t......Fri, 14 Aug 2026 07:01:28 -0400https://pop.com.sg/graphing-the-surface-z-xy.htmlhttps://pop.com.sg/graphing-the-surface-z-xy.htmlLet the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the boun......Fri, 14 Aug 2026 06:57:36 -0400https://pop.com.sg/probability-of-2-digit-appearing-in-random-3-digit-number.htmlhttps://pop.com.sg/probability-of-2-digit-appearing-in-random-3-digit-number.htmlLet's say I have {1, 2} what is the probability of them both appearing in three digit number? [1x2, 2x1, 12x, 21x, x12, x21] I tried the following but is not getting the correct answer 1 - P(Bo......Fri, 14 Aug 2026 06:45:57 -0400https://pop.com.sg/difference-between-a-reduced-residue-class-and-a-reduced-residue-syste.htmlhttps://pop.com.sg/difference-between-a-reduced-residue-class-and-a-reduced-residue-syste.htmlCurrently studying Dirichlets proof of Primes in arithmetic progressions. Is there a difference between a Reduced Residue Class and a Reduced Residue System? Any subset R of the integers is called a ...Fri, 14 Aug 2026 06:43:50 -0400https://pop.com.sg/finding-the-moment-generating-function-of-standard-normal-random-varia.htmlhttps://pop.com.sg/finding-the-moment-generating-function-of-standard-normal-random-varia.htmlGiven $Y\sim N(\mu, \sigma^2)$. I'm trying to find the moment generating function of $Z=\frac{Y-\mu}{\sigma}$ using the MGF transform method. Here's what I've tried: $$M_Y(t)=e^{\mu t+\frac{\sigma^......Fri, 14 Aug 2026 06:40:19 -0400https://pop.com.sg/what-technique-should-i-apply-to-find-the-derivative-of-a-ceiling-or-f.htmlhttps://pop.com.sg/what-technique-should-i-apply-to-find-the-derivative-of-a-ceiling-or-f.htmlMay I know the technique to apply to find the derivative, whenever I see a ceiling function of floor function. Thank You! e.g $$ \frac{d}{dx}(x*\lceil x \rceil )$$ and $$ \frac{d}{dx}(x*\lfloor......Fri, 14 Aug 2026 06:39:55 -0400https://pop.com.sg/subtracting-two-points-the-math.htmlhttps://pop.com.sg/subtracting-two-points-the-math.htmlI would like to confirm the math of subtracting two points. I've looked online but all of the sources I have seen want to end up with a vector, I need to end with a point. I am looking at this fu......Fri, 14 Aug 2026 06:38:55 -0400https://pop.com.sg/i-have-100-hundred-different-items-how-to-i-calculate-all-possible-com.htmlhttps://pop.com.sg/i-have-100-hundred-different-items-how-to-i-calculate-all-possible-com.htmlELI5 the formula to figure out all possible combinations without repeating any items but no upper or lower limit of the number to include. Order does not matter....Fri, 14 Aug 2026 06:38:54 -0400https://pop.com.sg/how-would-i-set-up-the-taylor-039-s-inequality-to-prove-that-the-funct.htmlhttps://pop.com.sg/how-would-i-set-up-the-taylor-039-s-inequality-to-prove-that-the-funct.htmlHow would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but I'm......Fri, 14 Aug 2026 06:36:36 -0400https://pop.com.sg/find-area-of-the-shaded-portion.htmlhttps://pop.com.sg/find-area-of-the-shaded-portion.htmlThe distances between the dots in the figure are equal to 1, both horizontally and vertically. What is the measure of the common area of the triangle and the square which you can see shaded in the ......Fri, 14 Aug 2026 06:35:55 -0400https://pop.com.sg/infinite-row-of-squares-some-of-them-white-with-probability-p-what-is-.htmlhttps://pop.com.sg/infinite-row-of-squares-some-of-them-white-with-probability-p-what-is-.htmlSuppose that you have an infinite horizontal row of squares, each square of size $1\times1$ (like a row of squares in a grid of infinite size). The probability of square being white is $p$. Some wh......Fri, 14 Aug 2026 06:25:51 -0400https://pop.com.sg/difference-between-mathrm-rot-and-mathrm-rot-curl.htmlhttps://pop.com.sg/difference-between-mathrm-rot-and-mathrm-rot-curl.htmlI'm not sure whether you've heard of that difference at all and if it's of importance in English notations. In German at least you define: $$\mathrm{rot\,\textbf{V}} = \nabla \times \textbf{V}, \quad\...Fri, 14 Aug 2026 06:25:36 -0400https://pop.com.sg/does-sum-frac-4-n-n-2-2n-converge.htmlhttps://pop.com.sg/does-sum-frac-4-n-n-2-2n-converge.htmlDoes $\sum\frac{4^n(n!)^2}{(2n)!}$ converge? The ratio test is inconclusive. I know that $\sum\frac{2^n(n!)^2}{(2n)!}$ converges, if that's of any help....Fri, 14 Aug 2026 06:20:50 -0400https://pop.com.sg/number-of-directed-graph-with-n-vertices-closed.htmlhttps://pop.com.sg/number-of-directed-graph-with-n-vertices-closed.htmlIn terms of $n$, how many distinct digraphs are there whose vertex set is $X$, where $X$ is a nonempty finite set with $|X| = n$? How does the answer change if we allow loops, but not parallel ar......Fri, 14 Aug 2026 06:13:52 -0400https://pop.com.sg/if-x-in-mathbb-r-solve-4x-2-40-lfloor-x-rfloor-51-0.htmlhttps://pop.com.sg/if-x-in-mathbb-r-solve-4x-2-40-lfloor-x-rfloor-51-0.htmlIf $x\in\mathbb R$, solve $$4x^2-40\lfloor x\rfloor+51=0$$ where $\lfloor x\rfloor$ denotes the integer part of the number. $\lfloor x\rfloor\le x$ and $\lfloor x\rfloor=x-\{x\}$, where $\{x\}$ m......Fri, 14 Aug 2026 06:07:34 -0400https://pop.com.sg/using-the-inverse-matrix-to-solve-a-linear-system.htmlhttps://pop.com.sg/using-the-inverse-matrix-to-solve-a-linear-system.htmlI have the following question: $$x - 2y = 1$$ $$2x + 3y = 4$$ So let's say: $$A = \left[ \begin{matrix} 1 & -2 \\ 2 & 3 \end{matrix} \right]$$ So according to some formula to $2 \ti......Fri, 14 Aug 2026 06:01:23 -0400https://pop.com.sg/calculate-the-distance-d-from-vec-y-to-the-straight-line-through-vec-u.htmlhttps://pop.com.sg/calculate-the-distance-d-from-vec-y-to-the-straight-line-through-vec-u.htmlBackground Let $$\vec{y}= \left[ {\begin{array}{cc} 5 \\ 1 \\ 1......Fri, 14 Aug 2026 05:59:57 -0400https://pop.com.sg/evaluate-int-sqrt-sec-x-tan-x-mathrm-dx.htmlhttps://pop.com.sg/evaluate-int-sqrt-sec-x-tan-x-mathrm-dx.htmlI tried to use integration by parts but it led to results which I could not integrate further. Also substituting $\sec x$ in terms of $\tan x$ or vice-versa led me to nothing proper. Update: Takin......Fri, 14 Aug 2026 05:54:20 -0400https://pop.com.sg/if-b-is-nilpotent-and-ab-ba-then-det-a-b-det-a-duplicate.htmlhttps://pop.com.sg/if-b-is-nilpotent-and-ab-ba-then-det-a-b-det-a-duplicate.htmlThe following stumps me: Let $\mathbb K$ be a field. Let $A, B \in \mathbb K^{n \times n}$ where $B$ is nilpotent and commutes with $A$, i.e., $A B = B A$. Show that $$ \det(A+B)=\det(A) $$ I h......Fri, 14 Aug 2026 05:53:04 -0400https://pop.com.sg/what-is-the-difference-between-a-random-variable-and-an-event.htmlhttps://pop.com.sg/what-is-the-difference-between-a-random-variable-and-an-event.htmlIn layman terms, what is the difference between a random variable and an event? To my understanding a random variable is a function outputting a real number. And an event is an outcome or a set of ...Fri, 14 Aug 2026 05:33:29 -0400https://pop.com.sg/explanation-for-the-wilson-score-interval.htmlhttps://pop.com.sg/explanation-for-the-wilson-score-interval.htmlI'm looking at this blog to try to understand the Wilson Score interval. I understand it somewhat, but I'm confused by the part under the title "Excerpt". In particular, I don't understand what he's ...Fri, 14 Aug 2026 05:32:55 -0400https://pop.com.sg/dot-product-for-3-vectors.htmlhttps://pop.com.sg/dot-product-for-3-vectors.htmlThe dot product can be used to write the sum: $$\sum_{i=1}^n a_i b_i$$ as $$a^T b$$ Is there an equivalent notation for the following sum: $$\sum_{i=1}^n a_i b_i c_i$$...Fri, 14 Aug 2026 05:31:32 -0400https://pop.com.sg/difference-between-algebra-and-geometry.htmlhttps://pop.com.sg/difference-between-algebra-and-geometry.htmlLooking up dictionary definitions of algebra geometry is pretty unsatisfying as they are usually along the lines of "the branch of mathematics in which letters and other general symbols are us......Fri, 14 Aug 2026 05:31:08 -0400https://pop.com.sg/is-there-a-sense-in-which-4-2-16-closed.htmlhttps://pop.com.sg/is-there-a-sense-in-which-4-2-16-closed.htmlHere's my train of thought: I. $4^2=4^{(4/2)}$ II. $4^{(4/2)}=(4^4)^{(1/2)}$ III. $(4^4)^{(1/2)}=256^{(1/2)}$ IV. $256^{(1/2)}=\sqrt{256}$ square root of 256 has 2 solutions: 16 and -16 Can you exp......Fri, 14 Aug 2026 05:26:34 -0400https://pop.com.sg/questions-tagged-pochhammer-symbol.htmlhttps://pop.com.sg/questions-tagged-pochhammer-symbol.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 14 Aug 2026 05:25:25 -0400https://pop.com.sg/dual-quaternion-inverse.htmlhttps://pop.com.sg/dual-quaternion-inverse.htmlIs it true that for every dual quaternion $Q$ I can find it's inverse such that $QQ^{-1} = 1?$ Using the usual definition $Q^{-1}=\frac{Q^{*}}{||Q||^2}$ doesn't work for me, since the dual part doe......Fri, 14 Aug 2026 05:20:49 -0400https://pop.com.sg/optimization-problem-solution-water-filling.htmlhttps://pop.com.sg/optimization-problem-solution-water-filling.htmlLet $x\in\mathbb{R}^n$ be an optimization variable and $\alpha\in\mathbb{R}^n$ be an n-dimensional vector. The standard water-filling problem is formulated as \begin{equation} \begin{array}{ll} \...Fri, 14 Aug 2026 05:16:11 -0400https://pop.com.sg/how-do-you-explain-to-a-5th-grader-why-division-by-zero-is-meaningless.htmlhttps://pop.com.sg/how-do-you-explain-to-a-5th-grader-why-division-by-zero-is-meaningless.htmlI wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing di......Fri, 14 Aug 2026 05:15:24 -0400https://pop.com.sg/proving-if-and-only-if-when-x-and-y-are-independent.htmlhttps://pop.com.sg/proving-if-and-only-if-when-x-and-y-are-independent.htmlLet $X, Y$ be two random variables, defined on some probability space $(\Omega , A , P)$., each only has two district values: $X \to \{x_1,x_2\}$ , $Y \to \{y_1,y_2\}$. Recall that, in this ca......Fri, 14 Aug 2026 05:13:13 -0400https://pop.com.sg/two-doors-2-guards-logic-problem-is-xor-a-hypothetical-statement.htmlhttps://pop.com.sg/two-doors-2-guards-logic-problem-is-xor-a-hypothetical-statement.htmlYou're trapped in a room with 2 doors, one on your left and one on your right. One door leads to heaven, one door leads to hell. There are two birds in the room. One is a bad bird, which always lie......Fri, 14 Aug 2026 05:13:03 -0400https://pop.com.sg/how-can-i-generate-a-random-number-on-my-ti-84-without-the-rand-functi.htmlhttps://pop.com.sg/how-can-i-generate-a-random-number-on-my-ti-84-without-the-rand-functi.htmlIs there an equation that is possible where I enter a seed, and it will give me the number from the seed. I want it to give me the same results as the rand function would give with the seed as 0. I......Fri, 14 Aug 2026 04:57:35 -0400https://pop.com.sg/how-to-multiply-radicals.htmlhttps://pop.com.sg/how-to-multiply-radicals.htmlThe problem is $$0 = -(\sqrt{7 -2 b} - 2) * -\sqrt{7-2 b} - 2) $$ the original question was this $$-2 = \sqrt{7 -2 b} - \sqrt{2 b + 3}$$ and than i found out the you had to isolate one radical v......Fri, 14 Aug 2026 04:56:19 -0400https://pop.com.sg/what-is-a-fourier-symbol.htmlhttps://pop.com.sg/what-is-a-fourier-symbol.htmlI was wondering what exactly the term Fourier symbol, or sometimes said the symbol of the Fourier transform means. Also, are any books that cover this topic in detail? Edit: After the comments, I h......Fri, 14 Aug 2026 04:46:32 -0400https://pop.com.sg/what-does-sign-mean-on-a-vector.htmlhttps://pop.com.sg/what-does-sign-mean-on-a-vector.htmlI have a maths homework question Given the vector b find bˆ The symbol is actually above the b, but I am not sure what it means. Guessing it means to find the unit vector of b?...Fri, 14 Aug 2026 04:44:24 -0400https://pop.com.sg/circle-revolutions-rolling-around-another-circle.htmlhttps://pop.com.sg/circle-revolutions-rolling-around-another-circle.htmlI just watched this video, and I'm a bit perplexed. Problem: The radius of Circle A is 1/3 the radius of Circle B. Circle A rolls around Circle B one trip back to its starting point. How many t......Fri, 14 Aug 2026 04:36:47 -0400https://pop.com.sg/how-do-i-integrate-sec-x.htmlhttps://pop.com.sg/how-do-i-integrate-sec-x.htmlMy HW asks me to integrate $\sin(x)$, $\cos(x)$, $\tan(x)$, but when I get to $\sec(x)$, I'm stuck....Fri, 14 Aug 2026 04:26:49 -0400https://pop.com.sg/let-b-be-the-solid-tetrahedron-bounded-by-x-0-y-0-z-0-and-x-y-z-1.htmlhttps://pop.com.sg/let-b-be-the-solid-tetrahedron-bounded-by-x-0-y-0-z-0-and-x-y-z-1.htmlLet $B$ be the solid tetrahedron bounded by $x = 0, y = 0, z = 0$, and $x+y+z = 1$. Assume that the density of $B$ is given by the function $y$. Compute the mass of $B$. I am having trouble solvin......Fri, 14 Aug 2026 04:22:12 -0400https://pop.com.sg/proof-by-induction-matrices.htmlhttps://pop.com.sg/proof-by-induction-matrices.htmlGiven the matrix $A=\begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}$, I want to prove that $A^k = \begin{pmatrix}1 & 2k \\ 0 & 1 \end{pmatrix}$ (=induction hypothesis). Since I struggle......Fri, 14 Aug 2026 04:21:27 -0400https://pop.com.sg/math-error-evaluating-int-cot-3-ax-csc-2-ax-dx-closed.htmlhttps://pop.com.sg/math-error-evaluating-int-cot-3-ax-csc-2-ax-dx-closed.htmlFor a National Board Exam Review: Evaluate: $${ \int \cot^3(ax) csc^2(ax)dx }$$ Answer is $${ -\frac{1}{4a}cot^4ax +C }$$ Ok. I certainly know how to integrate it from scratch. What I'm ......Fri, 14 Aug 2026 04:12:24 -0400https://pop.com.sg/multivariable-calculus-find-derivative-using-implicit-differentiation.htmlhttps://pop.com.sg/multivariable-calculus-find-derivative-using-implicit-differentiation.htmlShort simple question which i managed to solve partially. we are given the equation $x^2+y^2-z^2+xz-yz-1=0$. Show using the implicit function theorem that this equation sets in the neighborhood o......Fri, 14 Aug 2026 04:06:38 -0400https://pop.com.sg/weighted-jacobi-method.htmlhttps://pop.com.sg/weighted-jacobi-method.htmlConsider solving a linear system $A \boldsymbol{x}=\boldsymbol{b}$. Let $D$ be the diagonal matrix whose diagonal entries are those of $A,-L$ be the strictly lower triangular part of $A$, and $-U$ ......Fri, 14 Aug 2026 03:53:53 -0400https://pop.com.sg/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-.htmlhttps://pop.com.sg/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-.htmlI have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?...Fri, 14 Aug 2026 03:43:59 -0400https://pop.com.sg/convert-a-n-by-n-matrix-to-upper-triangular.htmlhttps://pop.com.sg/convert-a-n-by-n-matrix-to-upper-triangular.htmlHow can we mathematically and algorithmic-ally convert a $n\times n$ matrix to a upper triangular matrix...Fri, 14 Aug 2026 03:38:07 -0400https://pop.com.sg/solve-the-inequality-x-2-3-0.htmlhttps://pop.com.sg/solve-the-inequality-x-2-3-0.htmlFor the inequality $x^2 - 3 > 0$, we have \begin{align} x^2 - 3 & = (x+\sqrt 3)(x- \sqrt 3) > 0 \end{align} Therefore, \begin{align} x > -\sqrt 3 \end{align} and \begin{align} x ......Fri, 14 Aug 2026 03:35:29 -0400https://pop.com.sg/can-decimal-numbers-be-considered-even-or-odd.htmlhttps://pop.com.sg/can-decimal-numbers-be-considered-even-or-odd.htmlIs the concept of even/odd numbers applicable to decimal numbers? For e.g. - 4.222 is a even number?...Fri, 14 Aug 2026 03:33:45 -0400https://pop.com.sg/the-number-of-integers-039-a-039-1-leq-a-leq-200-such-that-a-a-is-a-pe.htmlhttps://pop.com.sg/the-number-of-integers-039-a-039-1-leq-a-leq-200-such-that-a-a-is-a-pe.htmlThe number of integers $a$ ($1\leq a\leq 200$) such that $a^a$ is a perfect square are is: $A)\;105$ $B)\;103$ $C)\;107$ $D)\;109$ My work $a^a$ is a perfect square when $a$ is even. $2^2 = 4. 4^4......Fri, 14 Aug 2026 03:32:05 -0400https://pop.com.sg/simplifying-finding-the-natural-log-of-two-terms-without-logarithm-law.htmlhttps://pop.com.sg/simplifying-finding-the-natural-log-of-two-terms-without-logarithm-law.htmlIf there's a natural log of two terms, which I cannot simplify with the laws of logarithms, how should I simplify it? e.g. $\ln(e^{6x} + 17)$ The full equation could be something like this: $$\......Fri, 14 Aug 2026 03:22:44 -0400https://pop.com.sg/cayley-table-for-d-4-langle-r-2-rangle.htmlhttps://pop.com.sg/cayley-table-for-d-4-langle-r-2-rangle.htmlWrite out the Cayley table for $D_4/\langle r^2\rangle$. I know that $D_4$ is the dihedral group with 4 sides (square) and $\langle r^2\rangle$ is the algebra generated by $r^2$ (a.k.a. taking the ...Fri, 14 Aug 2026 03:11:08 -0400https://pop.com.sg/interior-extremum-theorem.htmlhttps://pop.com.sg/interior-extremum-theorem.htmlInterior Extremum Theorem. Let $f$ be differentiable on an open interval $(a,b)$. If $f$ attains a maximum value at some point $c \in (a,b)$ ($f(c) \ge f(x)$ for all $x \in(a,b)$), then $f'(c) = 0$......Fri, 14 Aug 2026 02:59:29 -0400https://pop.com.sg/in-simple-english-what-does-it-mean-to-be-transcendental.htmlhttps://pop.com.sg/in-simple-english-what-does-it-mean-to-be-transcendental.htmlFrom Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these ...Fri, 14 Aug 2026 02:58:42 -0400https://pop.com.sg/partial-orders-and-hasse-diagrams-totally-confused.htmlhttps://pop.com.sg/partial-orders-and-hasse-diagrams-totally-confused.htmlSeems I've stumbled upon one of the most confusing topics in discrete mathematics...My book does a terrible job of explaining how you actually create Hasse diagrams and only shows premade diagrams ...Fri, 14 Aug 2026 02:38:20 -0400https://pop.com.sg/how-can-i-find-the-value-of-ln-x-without-using-the-calculator.htmlhttps://pop.com.sg/how-can-i-find-the-value-of-ln-x-without-using-the-calculator.htmlI want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks...Fri, 14 Aug 2026 02:34:52 -0400https://pop.com.sg/upside-down-triangle-symbol-on-function.htmlhttps://pop.com.sg/upside-down-triangle-symbol-on-function.htmlI came across the symbol that looks like an upside-down triangle, and coming in front of a function $f(x,y)$. What does that mean?...Fri, 14 Aug 2026 02:32:17 -0400https://pop.com.sg/principal-branch-of-logarithm.htmlhttps://pop.com.sg/principal-branch-of-logarithm.htmlI need to compute $\log(1+i)$ and $\text{Log}(1+i)$, $\text{Log}$ meaning the principal branch. What I do is to express $(1+i)$ in polar coordinates, then equate it with $$\sqrt {2}(\cos \pi/4+ i\......Fri, 14 Aug 2026 02:29:11 -0400https://pop.com.sg/negation-of-if-and-only-if.htmlhttps://pop.com.sg/negation-of-if-and-only-if.htmlLet a statement P is "X is true if and only if Y is true". What is the negation of P? I am little confused. It seems that digital equivalent of this statement is P = X and Y. Hence negation of P is......Fri, 14 Aug 2026 02:25:38 -0400https://pop.com.sg/equidistant-points-on-a-circle.htmlhttps://pop.com.sg/equidistant-points-on-a-circle.htmlI would like to obtain/generate points on a circle in Cartesian coordinates such that the distance between two consecutive points will be always equal. For example, plotting a circle with radius 10......Fri, 14 Aug 2026 02:25:19 -0400https://pop.com.sg/relative-version-of-kunneth-theorem-for-cw-pairs.htmlhttps://pop.com.sg/relative-version-of-kunneth-theorem-for-cw-pairs.htmlI am working on problem 3.2.12 in Hatcher's algebraic topology which asks to show that the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times \{x_0\})$ and $S^3 \times \mathbb{C}P^{\infty}$ have ...Fri, 14 Aug 2026 02:25:05 -0400https://pop.com.sg/what-is-the-relationship-between-cubic-b-splines-and-cubic-splines.htmlhttps://pop.com.sg/what-is-the-relationship-between-cubic-b-splines-and-cubic-splines.htmlWhat is the relationship between cubic B-splines and cubic splines?...Fri, 14 Aug 2026 02:21:34 -0400