https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 05 Sep 2026 01:46:27 -0400https://pop.com.sg/precedence-between-implication-and-bi-implication.htmlhttps://pop.com.sg/precedence-between-implication-and-bi-implication.htmlI came across this question: Let p, q, and r be the propositions p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these ...Sat, 05 Sep 2026 05:33:48 -0400https://pop.com.sg/what-are-the-rules-for-a-tetartoid-pentagon.htmlhttps://pop.com.sg/what-are-the-rules-for-a-tetartoid-pentagon.htmlThe tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry. It has twelve identical ...Sat, 05 Sep 2026 05:28:46 -0400https://pop.com.sg/cartesian-coordinates-of-lemniscate.htmlhttps://pop.com.sg/cartesian-coordinates-of-lemniscate.htmlA problem from Spivak's Calculus: Sketch the graph of the lemniscate $$r^2 = 2a^2\cos 2 \theta.$$ Find an equation for its cartesian coordinates. Show that it is the collection of ......Sat, 05 Sep 2026 05:26:26 -0400https://pop.com.sg/determining-the-area-of-a-right-triangle-perimeter-given-hypotenuse-va.htmlhttps://pop.com.sg/determining-the-area-of-a-right-triangle-perimeter-given-hypotenuse-va.htmlThe problem states: Right Triangle- perimeter of $84$, and the hypotenuse is $2$ greater than the other leg. Find the area of this triangle. I have tried different methods of solving this pro......Sat, 05 Sep 2026 05:26:24 -0400https://pop.com.sg/find-the-partial-fraction-for-frac-1-x-2-1-2-x-1.htmlhttps://pop.com.sg/find-the-partial-fraction-for-frac-1-x-2-1-2-x-1.htmlI want to find the partial fraction of: $$\frac{1}{(x^2+1)^2(x-1)}$$ Although I want this expressed through the format of finding the complex number that goes into $x$ so that I can get this equati......Sat, 05 Sep 2026 05:18:07 -0400https://pop.com.sg/a-box-contains-10-balls-numbered-from-1-to-10.htmlhttps://pop.com.sg/a-box-contains-10-balls-numbered-from-1-to-10.htmlAn urn contains balls numbered $1, \ldots, 10$. Five balls are drawn without replacement. What is the probability that the second largest of the five numbers drawn will be 8? I believe the number of ...Sat, 05 Sep 2026 05:12:31 -0400https://pop.com.sg/why-can-039-t-you-flatten-a-sphere.htmlhttps://pop.com.sg/why-can-039-t-you-flatten-a-sphere.htmlIt's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old? As soon as an explanation starts using terms like "Gauss......Sat, 05 Sep 2026 05:11:40 -0400https://pop.com.sg/extended-euclidean-algorithm-for-modular-inverse.htmlhttps://pop.com.sg/extended-euclidean-algorithm-for-modular-inverse.htmlI'm currently learning how to find the inverse of a modulo with the Extended Euclid Algorithm and I stumbled upon a problem when finding an inverse when the $m>p$ as for $m \equiv 1 \pmod{p}$ For ...Sat, 05 Sep 2026 05:10:37 -0400https://pop.com.sg/converting-augmented-matrix-to-reduced-row-echelon-form.htmlhttps://pop.com.sg/converting-augmented-matrix-to-reduced-row-echelon-form.htmlI am struggling with reducing an augmented matrix to reduced row echelon form. The given linear system is \begin{cases} 3x-2z=-3 \\ -2x+z=-2 \\ -z=2 \\ \end{cases} I can write this as an augmen......Sat, 05 Sep 2026 04:57:52 -0400https://pop.com.sg/find-the-indicated-probability.htmlhttps://pop.com.sg/find-the-indicated-probability.htmlQuestion: On one tropical island, hurricanes occur with a mean of 2.74 per year. Assuming that the number of hurricanes can be modeled by a Poisson distribution, find the probability that during th......Sat, 05 Sep 2026 04:54:31 -0400https://pop.com.sg/find-the-arclength-of-the-curve-defined-by-r-t-i-9t-2j-t-3k-for-0-leq-.htmlhttps://pop.com.sg/find-the-arclength-of-the-curve-defined-by-r-t-i-9t-2j-t-3k-for-0-leq-.htmlFirst I found $r'(t)=\langle 1,18t^2,3t^2\rangle$ and so the magnitude of $r'(t)= \sqrt{1+(18t)^2+(3t^2)^2}$ thus the integral from $0$ to $\sqrt{28}$ of $\sqrt{1+324t^2+9t^4} dt$. When I plugged $......Sat, 05 Sep 2026 04:54:23 -0400https://pop.com.sg/classifying-map.htmlhttps://pop.com.sg/classifying-map.htmlLet $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f......Sat, 05 Sep 2026 04:48:51 -0400https://pop.com.sg/sum-of-multiplication-of-two-binomial-coefficients.htmlhttps://pop.com.sg/sum-of-multiplication-of-two-binomial-coefficients.htmlIs there any formula for calculating $\sum_{k=0}^n {n\choose k} {2n\choose 2k}$ ? One possible way is to use Stirling's approximation, but couldn't reach a reasonable answer....Sat, 05 Sep 2026 04:47:35 -0400https://pop.com.sg/why-do-divisions-like-1-98-and-1-998-give-us-numbers-continuously-bein.htmlhttps://pop.com.sg/why-do-divisions-like-1-98-and-1-998-give-us-numbers-continuously-bein.htmlFor example, when I divided $1$ by $98$, I got an amazing result of $0.0102040816326530612244897...$ where so many numbers get multiplied by two every time in the right pattern with some carrying. ......Sat, 05 Sep 2026 04:42:17 -0400https://pop.com.sg/positive-and-negative-complex-numbers.htmlhttps://pop.com.sg/positive-and-negative-complex-numbers.htmlCan there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number w......Sat, 05 Sep 2026 04:40:15 -0400https://pop.com.sg/relative-interpretations-alla-kunen.htmlhttps://pop.com.sg/relative-interpretations-alla-kunen.htmlat the moment I try to figure out some details of Kunen's "Relative Interpretation" Definition (within the 2013 Edition of his "Set Theory", p. 99 to 100): Definition If $\Lambda$ is some axio......Sat, 05 Sep 2026 04:38:14 -0400https://pop.com.sg/formula-for-the-sequence-1-1-2-2-3-3-closed.htmlhttps://pop.com.sg/formula-for-the-sequence-1-1-2-2-3-3-closed.htmlI am trying to solve a programming problem where i encounter this pattern 1 1 2 2 3 3 4 4 . What is the formula for computing it for Nth term ?...Sat, 05 Sep 2026 04:38:06 -0400https://pop.com.sg/understanding-the-earth-mover-s-distance-emd-from-the-matlab.htmlhttps://pop.com.sg/understanding-the-earth-mover-s-distance-emd-from-the-matlab.htmlI found difficulty in understanding the Earth Mover’s Distance (EMD) from the MATLAB implementation. The method is actually implemented in this paper. Anyone help me to where EMD is implemented in ......Sat, 05 Sep 2026 04:35:30 -0400https://pop.com.sg/is-there-any-significant-consequence-if-p-does-not-equal-np-complete.htmlhttps://pop.com.sg/is-there-any-significant-consequence-if-p-does-not-equal-np-complete.htmlI hear a lot on these forums about how if P=NP-complete how our lives would be better, is there any significant consequence if we found P to be not equal to NP-Complete?...Sat, 05 Sep 2026 04:30:32 -0400https://pop.com.sg/calculating-inverse-trig-expressions-like-cos-arctan-2.htmlhttps://pop.com.sg/calculating-inverse-trig-expressions-like-cos-arctan-2.htmlI have some problems "connecting dots". All feedback is welcomed and really, really helpful! :) Task 1: calculate $\quad \tan{(\arcsin{(-\frac{3}{4}}))}$ Solution: $\tan{(\arcsin{-\frac{3}{4}})......Sat, 05 Sep 2026 04:25:31 -0400https://pop.com.sg/proof-of-schur-039-s-lemma.htmlhttps://pop.com.sg/proof-of-schur-039-s-lemma.htmlCan someone give me a simplified proof of Schur's lemma in group theory. Sorry if the question looks a standard textbook proof. But I find the proof complicated in books. It would be helpful if som......Sat, 05 Sep 2026 04:24:35 -0400https://pop.com.sg/is-7-5040-the-largest-highly-composite-factorial.htmlhttps://pop.com.sg/is-7-5040-the-largest-highly-composite-factorial.htmlA highly composite number is a natural number $\ n\ge 1\ $ that has more divisors than any smaller natural number $\ m\ge 1$. Checking the $\ 1000\ $ entries in the OEIS-sequence, I noticed that o......Sat, 05 Sep 2026 04:22:49 -0400https://pop.com.sg/extremas-using-lagrange-or-other-methods-with-exponential-functions.htmlhttps://pop.com.sg/extremas-using-lagrange-or-other-methods-with-exponential-functions.html$ \overline{B((0,0),1))}\mapsto \mathbb{R}$ $f(x,y)= e^{(x-1)y}$ How do you find the extremas for this functions? I tried using Lagrange method but I have trouble solving the system of equations ......Sat, 05 Sep 2026 04:18:23 -0400https://pop.com.sg/giving-two-equal-line-segments-ac-and-bd-so-that-they-bisect-each-othe.htmlhttps://pop.com.sg/giving-two-equal-line-segments-ac-and-bd-so-that-they-bisect-each-othe.htmlGiven $AC=BD$ so $\angle AED= \angle CEB$ by Prop I.15 $AE+EC$ and $BE=DE$ by definition of bisect So by $SAS$, $\triangle AED$ is congruent to $\triangle BEC$ (postulate 12) Therefor $AD=BC$ ......Sat, 05 Sep 2026 04:15:46 -0400https://pop.com.sg/complex-space-why-unit-disc.htmlhttps://pop.com.sg/complex-space-why-unit-disc.htmlIn complex space for simplicity the properties of the functions on curves are sometimes considered on the unit circle on complex plane, with the center (0, 0) . So basically I would like to know why ...Sat, 05 Sep 2026 04:11:04 -0400https://pop.com.sg/complete-graph-k-n-can-be-expressed-as-the-union-of-k-bipartite-graphs.htmlhttps://pop.com.sg/complete-graph-k-n-can-be-expressed-as-the-union-of-k-bipartite-graphs.htmlI was going through the text "Introduction to Graph Theory" by Douglas West where I came across the following theorem. Theorem. The complete graph $K_n$ can be expressed as the union of ......Sat, 05 Sep 2026 03:59:08 -0400https://pop.com.sg/simplest-way-to-count-distinct-combinations-2.htmlhttps://pop.com.sg/simplest-way-to-count-distinct-combinations-2.htmlFollowing my previous problem (which was solved) Simplest way to count distinct combinations (Reading this topic is not mandatory). Consider I have these letters: A, B, C, D, E. n is the number......Sat, 05 Sep 2026 03:51:08 -0400https://pop.com.sg/making-sense-of-a-cross-product-of-three-vectors.htmlhttps://pop.com.sg/making-sense-of-a-cross-product-of-three-vectors.htmlBecause of the cross product of two vectors being another vector I can calculate $\vec a\times(\vec b\times\vec c)$ as well as $(\vec a\times\vec b)\times\vec c$. I know that the cross product is not ...Sat, 05 Sep 2026 03:47:49 -0400https://pop.com.sg/why-do-we-disregard-the-remainder-while-finding-an-oblique-asymptote.htmlhttps://pop.com.sg/why-do-we-disregard-the-remainder-while-finding-an-oblique-asymptote.htmlI have to find the oblique asymptote for the following equation: $$y=\frac{2x^{2}-4x-1}{x-3}$$ I applied long division to bring it into the form of $Q + \frac{R}{D}$ leaving me with: $$y=2x+2+\fra......Sat, 05 Sep 2026 03:25:22 -0400https://pop.com.sg/does-the-definition-of-the-angle-between-two-vectors-require-that-they.htmlhttps://pop.com.sg/does-the-definition-of-the-angle-between-two-vectors-require-that-they.htmlI am thinking specifically about $\mathbb{R}^2$ so I can visualize things. By "origin" I mean that they start at the same point. When we graphically represnt vectors we don't care where the start......Sat, 05 Sep 2026 03:22:13 -0400https://pop.com.sg/quotient-groups-of-d10.htmlhttps://pop.com.sg/quotient-groups-of-d10.htmlHow do I find the groups D10/N where N is the normal subgroups to D10? I know that the definition is aN for all a $\in$ D10. But I am unsure what the group, for example, D10/ D10 looks like? Any h......Sat, 05 Sep 2026 03:20:08 -0400https://pop.com.sg/definition-of-measurability.htmlhttps://pop.com.sg/definition-of-measurability.htmlThe alternative definition of measurability is: A subset $E \subset \mathbb{R}^n$ is measurable if for every $\epsilon > 0$ there is a closed set $F \subset E$ such that $m^*(E\setminus F)<\...Sat, 05 Sep 2026 03:10:53 -0400https://pop.com.sg/the-volume-of-a-torus.htmlhttps://pop.com.sg/the-volume-of-a-torus.htmlA torus $\mathscr{T}$ with the equation $$z^2 + \left( \sqrt{x^2 + y^2} - 2 \right)^2 = 1.$$ (a) Give an equation with a close line in the plane $Oxz$ where $\mathscr{T}$ is a surface of ...Sat, 05 Sep 2026 03:07:51 -0400https://pop.com.sg/what-does-ir-mean-in-linear-algebra.htmlhttps://pop.com.sg/what-does-ir-mean-in-linear-algebra.htmlI am new to linear algebra and I was wondering if I could get some help for this question. I understand if it was something like this IR^2 -> IR. I have no idea what IR(=IR^1) means. Could someone ...Sat, 05 Sep 2026 03:02:51 -0400https://pop.com.sg/determine-whether-this-statement-is-true-or-false-forall-x-in-z-exists.htmlhttps://pop.com.sg/determine-whether-this-statement-is-true-or-false-forall-x-in-z-exists.htmlJust beginning in Discrete math stumbled into the symbolic logic section and need some selp on this question, any input is appreciated. $$ \forall x \in {Z}, \exists y , z \in {Z} [x = 5y + 7z] $$ ...Sat, 05 Sep 2026 02:50:06 -0400https://pop.com.sg/why-is-it-important-for-a-matrix-to-be-square.htmlhttps://pop.com.sg/why-is-it-important-for-a-matrix-to-be-square.htmlI am currently trying to self-study linear algebra. I've noticed that a lot of the definitions for terms (like eigenvectors, characteristic polynomials, determinants, and so on) require a square ma......Sat, 05 Sep 2026 02:44:01 -0400https://pop.com.sg/constructing-a-square-from-the-difference-of-diagonal-and-side.htmlhttps://pop.com.sg/constructing-a-square-from-the-difference-of-diagonal-and-side.htmlI have figured it out how to construct a square given a segment which is the difference of diagonal and side: construct an equal sided right triangle where the leg is the given segment and then add......Sat, 05 Sep 2026 02:41:29 -0400https://pop.com.sg/what-are-the-differences-between-rings-groups-and-fields.htmlhttps://pop.com.sg/what-are-the-differences-between-rings-groups-and-fields.htmlRings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?...Sat, 05 Sep 2026 02:37:39 -0400https://pop.com.sg/turn-partial-differential-equation-into-ordinary-differential-equation.htmlhttps://pop.com.sg/turn-partial-differential-equation-into-ordinary-differential-equation.html(I looked for a post similar but couldn't find one. I apologize in advance if there's a post similar to this. Apologizes to the moderators for a weak post.) I'm taking an ordinary differential equ......Sat, 05 Sep 2026 02:37:00 -0400https://pop.com.sg/square-root-of-sqrt-1-4-sqrt-3-i.htmlhttps://pop.com.sg/square-root-of-sqrt-1-4-sqrt-3-i.htmlHow can we find square root of the complex number $$\sqrt{1-4\sqrt{3}i}?$$ Now here if I assume square root to be $a+ib$ i.e. $a+ib=\sqrt{\sqrt{1-4\sqrt{3}i}}$, then after squaring both sides, ho......Sat, 05 Sep 2026 02:30:41 -0400https://pop.com.sg/how-to-use-double-angle-identities-to-find-length-039-d-039.htmlhttps://pop.com.sg/how-to-use-double-angle-identities-to-find-length-039-d-039.htmlEdit: Ok the question has now been amended by my tutor as I highlighted it was not possible to solve. A pole is tensioned at point A as shown below, if the angles are to be kept the same, then usi......Sat, 05 Sep 2026 02:17:25 -0400https://pop.com.sg/how-to-prove-distance-from-foci-on-an-ellipse-is-equal-to-twice-the-se.htmlhttps://pop.com.sg/how-to-prove-distance-from-foci-on-an-ellipse-is-equal-to-twice-the-se.htmlProve that for any point (x,y) on the conic, the sum of the distances to the two foci is always twice the semi-major axis. I know that this can be proven in general for all ellipses but the practice ...Sat, 05 Sep 2026 02:17:20 -0400https://pop.com.sg/linear-approximation-with-two-variables.htmlhttps://pop.com.sg/linear-approximation-with-two-variables.htmlThe problem I have is this: Use suitable linear approximation to find the approximate values for given functions at the points indicated: $f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$ I know how to do ...Sat, 05 Sep 2026 02:09:52 -0400https://pop.com.sg/definition-of-an-angle-vs-measurement-of-an-angle.htmlhttps://pop.com.sg/definition-of-an-angle-vs-measurement-of-an-angle.htmlThe definition of an angle in Euclidean geometry is given as follows: Angle. Definition: A shape, formed by two lines or rays diverging from a common point (the vertex). However, the definition ...Sat, 05 Sep 2026 02:01:11 -0400https://pop.com.sg/how-could-a-square-root-of-fraction-have-a-negative-root.htmlhttps://pop.com.sg/how-could-a-square-root-of-fraction-have-a-negative-root.htmlWhat I know If I'm not mistaken: $\pm \sqrt{\frac{x}{y}}=\frac{\pm \sqrt{x}}{\pm \sqrt{y}}$ a repeated $\pm$ sign in an equation means make every "$\pm$" sign a plus or make every one of them a m......Sat, 05 Sep 2026 02:00:56 -0400https://pop.com.sg/piecewise-function-plot-in-matlab.htmlhttps://pop.com.sg/piecewise-function-plot-in-matlab.htmlIs it possible in Matlab to plot an even piecewise function like: $ f(x) = \begin{cases} 3t , 0 < t < \pi \\ -3t , -\pi \le t \le 0 \end{cases}$ which has a period of $2\pi$. I ......Sat, 05 Sep 2026 01:59:36 -0400https://pop.com.sg/how-to-solve-this-limit-lim-limits-x-to0-frac-1-x-1-x-e-x-duplicate.htmlhttps://pop.com.sg/how-to-solve-this-limit-lim-limits-x-to0-frac-1-x-1-x-e-x-duplicate.htmlI tried to evaluate $$\lim_{x\to0}\frac{(1+x)^{1/x}-e}x$$but all my efforts were in vain . L'Hospital rule becomes very complicated....Sat, 05 Sep 2026 01:50:26 -0400https://pop.com.sg/even-and-odd-cubic-polynomials.htmlhttps://pop.com.sg/even-and-odd-cubic-polynomials.htmlBook says approximating even function $P_3$ must be of form $ax^2+b\ $so they neglected $x,x^3\ $. This function approximate even function $|x|^3$ very nicely in the book after some tweaking. I am ...Sat, 05 Sep 2026 01:32:58 -0400https://pop.com.sg/volume-of-a-parabola-rotated-around-a-line.htmlhttps://pop.com.sg/volume-of-a-parabola-rotated-around-a-line.htmlThe question is as follows: Find the volume of the solid generated by revolving the region bounded by the parabola $y = x^2$ and the line $y = 1$ about the line $y = -1$. My attempt: Using the was......Sat, 05 Sep 2026 01:32:13 -0400https://pop.com.sg/evaluate-the-indefinite-integral-int-1-x-2-3-2-dx.htmlhttps://pop.com.sg/evaluate-the-indefinite-integral-int-1-x-2-3-2-dx.htmlEvaluate the indefinite integral $$\int \frac {dx}{(1-x^2)^{3/2}} .$$ My answer: I have taken $u = 1-x^2$ but I arrived at the integral of $\frac{-1}{(u^3 - u^4)^{1/2}}$. What should I do next?...Sat, 05 Sep 2026 01:26:57 -0400https://pop.com.sg/riesz-functional-calculus-vs-holomorphic-functional-calculus.htmlhttps://pop.com.sg/riesz-functional-calculus-vs-holomorphic-functional-calculus.html"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...Sat, 05 Sep 2026 01:26:56 -0400https://pop.com.sg/find-the-matrix-s-of-stretch-by-a-factor-of-3.htmlhttps://pop.com.sg/find-the-matrix-s-of-stretch-by-a-factor-of-3.htmlAll mappings are from $\mathbb{R}^2$ to $\mathbb{R}^2$. Find the matrix $S$ of stretch by a factor of $3$ in the $y$-direction and the matrix $S^{-1}$. So the matrix $S$ is a $2\times2$ matrix. ......Sat, 05 Sep 2026 01:21:04 -0400https://pop.com.sg/deriving-the-midpoint-formula-in-3d-using-the-distance-formula.htmlhttps://pop.com.sg/deriving-the-midpoint-formula-in-3d-using-the-distance-formula.htmlI want to derive the midpoint of the line segment connecting the points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ using only the distance formula (since this is all that has been taught to students......Sat, 05 Sep 2026 01:17:16 -0400https://pop.com.sg/cosine-and-sine-of-the-angle-multiplied-by-a-scalar.htmlhttps://pop.com.sg/cosine-and-sine-of-the-angle-multiplied-by-a-scalar.htmlCan you write $\cos \alpha x$ in terms of $\cos x$ ? similarly for $\sin \alpha x$. where $\alpha$ is a scalar in $\mathbb{R}$. (not necessarily integer)....Sat, 05 Sep 2026 01:02:29 -0400https://pop.com.sg/what-039-s-the-formula-to-solve-summation-of-logarithms.htmlhttps://pop.com.sg/what-039-s-the-formula-to-solve-summation-of-logarithms.htmlI'm studying summation. Everything I know so far is that: $\sum_{i=1}^n\ k = \frac{n(n+1)}{2}\ $ $\sum_{i=1}^{n}\ k^2 = \frac{n(n+1)(2n+1)}{6}\ $ $\sum_{i=1}^{n}\ k^3 = \frac{n^2(n+1)^2}{4}\ $ ...Sat, 05 Sep 2026 00:59:25 -0400https://pop.com.sg/how-to-get-the-exact-value-of-sin-x-if-sin-2x-frac-24-25.htmlhttps://pop.com.sg/how-to-get-the-exact-value-of-sin-x-if-sin-2x-frac-24-25.htmlHow to get the exact value of $\sin(x)$ if $\sin(2x) = \frac{24}{25}$ ? I checked various trigonometric identities, but I am unable to derive $\sin(x)$ based on the given information. For instanc......Sat, 05 Sep 2026 00:46:46 -0400https://pop.com.sg/matrix-for-reflection-about-the-line-y-tan-theta-x.htmlhttps://pop.com.sg/matrix-for-reflection-about-the-line-y-tan-theta-x.htmlHow would I show that a reflection about the line $y = \tan (\theta) \, x$ is the following? \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}...Sat, 05 Sep 2026 00:42:14 -0400https://pop.com.sg/some-integral-representations-of-the-euler-mascheroni-constant.htmlhttps://pop.com.sg/some-integral-representations-of-the-euler-mascheroni-constant.htmlWhat kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\inf......Sat, 05 Sep 2026 00:33:42 -0400https://pop.com.sg/proving-a-cup-b-cap-c-a-cup-b-cap-a-cup-c.htmlhttps://pop.com.sg/proving-a-cup-b-cap-c-a-cup-b-cap-a-cup-c.htmlProve the distributive property for sets: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ I'm not good with proofs but my understanding is that I have to prove 2 things: (1) $A \cup (B \cap C) \...Sat, 05 Sep 2026 00:25:52 -0400https://pop.com.sg/90-degree-counter-clockwise-rotation-around-a-point.htmlhttps://pop.com.sg/90-degree-counter-clockwise-rotation-around-a-point.htmlHow do you do a 90 degree counter-clockwise rotation around a point? I know around the origin it's $(-y,x)$, but what would it be around a point? $$(-y - a,x - b)$$ Where $(a,b)$ is the rotation p......Sat, 05 Sep 2026 00:23:49 -0400https://pop.com.sg/the-thinking-behind-4-times-5-is-12-and-4-times-6-is-13-and-4-times-7-.htmlhttps://pop.com.sg/the-thinking-behind-4-times-5-is-12-and-4-times-6-is-13-and-4-times-7-.htmlExcerpt from Lewis Carroll's Alice in Wonderland: "Let me see: four times five is twelve, and four times six is thirteen, and four times seven is-oh dear! I shall never get to twenty at that ......Sat, 05 Sep 2026 00:20:25 -0400https://pop.com.sg/solve-the-following-differential-equation-y-frac-dx-dy-x-2y-2-with-the.htmlhttps://pop.com.sg/solve-the-following-differential-equation-y-frac-dx-dy-x-2y-2-with-the.htmlSolve the following differential equation: $y\dfrac{dx}{dy}-x=2y^2$, with the initial condition $y(1)=5$. The thing that is throwing me off (I think) is the $\dfrac{dx}{dy}$ instead of what I am m......Sat, 05 Sep 2026 00:20:18 -0400https://pop.com.sg/methods-of-enumeration-stars-and-bars.htmlhttps://pop.com.sg/methods-of-enumeration-stars-and-bars.htmlI'm having trouble on deciding when it's appropriate to use the stars and bars method for enumeration. For example, the following question: Suppose each student in a 50-student class selects a num......Sat, 05 Sep 2026 00:20:00 -0400https://pop.com.sg/syllogism-in-mathematics.htmlhttps://pop.com.sg/syllogism-in-mathematics.htmlSyllogism is defined as: "All A are B", and "All C are A", thus "All C are B". For example: "All birds have features", "Penguins are birds", thus "Penguins have feather" How do we represent this ...Sat, 05 Sep 2026 00:11:40 -0400https://pop.com.sg/finding-anchor-and-reference-point-of-a-cube-root-function.htmlhttps://pop.com.sg/finding-anchor-and-reference-point-of-a-cube-root-function.htmlMy brother sent me a question from his Pre-Calculus Study Guide. I have been studying math for $4$ years and don't ever recall seeing a question like this. The question: Graph $f(x) = (-1/2)(x+......Sat, 05 Sep 2026 00:02:53 -0400https://pop.com.sg/evaluate-the-following-definite-integrals-using-the-fundamental-theore.htmlhttps://pop.com.sg/evaluate-the-following-definite-integrals-using-the-fundamental-theore.htmlEvaluate the following definite integrals using the Fundamental Theorem of Calculus $$ \int_{-10}^1 s | 25 - s^2 | \; \mathrm d s. $$ my work: $$ s=\pm 5 $$ $$ \int^{-5}_{-10} f(s) + \int^5_{-5}......Fri, 04 Sep 2026 23:53:44 -0400https://pop.com.sg/find-the-solution-of-the-given-initial-value-problem-differential-equa.htmlhttps://pop.com.sg/find-the-solution-of-the-given-initial-value-problem-differential-equa.htmlQuestion: $$\frac{dy}{dt}+ty=1+t$$ where $y(\frac{3}{2})=0$. So I know this is a non-homogeneous equation and as I started working on it, I ended up with: $$e^{\frac{t^2}{2}}y=\int_{\frac{3}{2}}^......Fri, 04 Sep 2026 23:51:26 -0400https://pop.com.sg/what-is-the-difference-between-singular-matrices-and-degenerate-matric.htmlhttps://pop.com.sg/what-is-the-difference-between-singular-matrices-and-degenerate-matric.htmlIn my opinion, both are the matrices that the number of the directions of eigenvector is smaller than their size. So, I guess singular matrices and degenerate matrices refer to the same thing. Is t......Fri, 04 Sep 2026 23:49:57 -0400https://pop.com.sg/notation-for-a-vector-with-constant-equal-components-of-arbitrary-dime.htmlhttps://pop.com.sg/notation-for-a-vector-with-constant-equal-components-of-arbitrary-dime.htmlIs there a standard notation to specify vectors of the form $(c,c,\ldots,c)$, where $c$ is some constant?, e.g. suppose I have the vector $(c,c,c,c,c)$ which has 5 components. Is there a standard (......Fri, 04 Sep 2026 23:47:52 -0400https://pop.com.sg/what-is-the-difference-between-projected-gradient-descent-and-ordinary.htmlhttps://pop.com.sg/what-is-the-difference-between-projected-gradient-descent-and-ordinary.htmlI just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of proje......Fri, 04 Sep 2026 23:36:59 -0400https://pop.com.sg/total-graph-definition.htmlhttps://pop.com.sg/total-graph-definition.htmlI am studying a paper, and there is a definition for a total graph. It states The total graph $T(G)$ of a graph $G$ is the graph whose vertices correspond to the vertices and edges of $G$, and ......Fri, 04 Sep 2026 23:33:32 -0400https://pop.com.sg/proof-of-the-fundamental-theorem-of-line-integrals.htmlhttps://pop.com.sg/proof-of-the-fundamental-theorem-of-line-integrals.htmlSuppose $C$ is a smooth curve given by $\vec{r}(t)$, $a \leq t \leq b$. Also suppose that $\Phi$ is a function whose gradient vector, $\nabla \Phi=f$, is continuous on $C$. Then $$ \int_C f \......Fri, 04 Sep 2026 23:27:30 -0400https://pop.com.sg/what-does-it-mean-that-multiplication-and-division-have-the-same-prece.htmlhttps://pop.com.sg/what-does-it-mean-that-multiplication-and-division-have-the-same-prece.htmlI have read and heard in many places that multiplication and division are of equal precedence. So what is the value of: $$ 80 \div 10 \times 2 $$ Is it $$(80 \div 10) \times 2 = 16$$ or $$ 80......Fri, 04 Sep 2026 23:25:08 -0400https://pop.com.sg/plane-parallel-to-the-base-of-a-triangular-pyramid.htmlhttps://pop.com.sg/plane-parallel-to-the-base-of-a-triangular-pyramid.htmlA triangular pyramid $ABCD$ is given. A plane parallel to the plane $(ABC)$ intersects $DA,DB$ and $DC$ at $A_1,B_1$ and $C_1,$ respectively. Show that $\dfrac{DA_1}{DA}=\dfrac{DB_1}{DB}=\dfrac{DC_......Fri, 04 Sep 2026 23:20:49 -0400https://pop.com.sg/how-to-solve-an-exponential-differential-equation.htmlhttps://pop.com.sg/how-to-solve-an-exponential-differential-equation.htmlMy electrical engineering class just moved into differential equations from linear algebra, which is a topic I've never touched on before. The professor doesn't work hardly any examples on the boar......Fri, 04 Sep 2026 23:14:19 -0400https://pop.com.sg/how-do-i-explain-2-to-the-power-of-zero-equals-1-to-a-child.htmlhttps://pop.com.sg/how-do-i-explain-2-to-the-power-of-zero-equals-1-to-a-child.htmlMy daughter is stuck on the concept that $$2^0 = 1,$$ having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough. How would you explain the ...Fri, 04 Sep 2026 23:11:49 -0400https://pop.com.sg/exp-i-infty-is-not-a-number-according-to-matlab-why-closed.htmlhttps://pop.com.sg/exp-i-infty-is-not-a-number-according-to-matlab-why-closed.htmlI ran into this problem while trying to understand the Laplace transform via MATLAB. exp(-i*Inf) NaN + NaNi But, exp(-Inf) 0 Furthermore, syms t fun= exp(-(0+i)*t) answer=int(fun,0......Fri, 04 Sep 2026 23:06:35 -0400https://pop.com.sg/what-039-s-the-definition-of-omega.htmlhttps://pop.com.sg/what-039-s-the-definition-of-omega.htmlThis is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of $\alep......Fri, 04 Sep 2026 23:00:47 -0400https://pop.com.sg/the-shortest-distance-between-any-two-distinct-points-is-the-line-segm.htmlhttps://pop.com.sg/the-shortest-distance-between-any-two-distinct-points-is-the-line-segm.htmlOn a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?...Fri, 04 Sep 2026 22:33:16 -0400https://pop.com.sg/intrinsic-and-extrinsic-curvature.htmlhttps://pop.com.sg/intrinsic-and-extrinsic-curvature.htmlI want to understand the basic conceptual idea about intrinsic and extrinsic curvature. If we consider a plane sheet of paper (whose intrinsic curvature is zero) rolled into a cylindrical shape, th......Fri, 04 Sep 2026 22:32:36 -0400https://pop.com.sg/time-taken-for-ball-to-reach-its-maximum-height.htmlhttps://pop.com.sg/time-taken-for-ball-to-reach-its-maximum-height.htmlI have the question ``A football is kicked at a speed of 20 $m/s$ at and angle of 45 degrees. (Assume negligible air resistance). Calculate the time taken for the ball to reach its maximum height'......Fri, 04 Sep 2026 22:27:36 -0400https://pop.com.sg/quadratic-formula-and-factoring-are-leading-to-different-answers.htmlhttps://pop.com.sg/quadratic-formula-and-factoring-are-leading-to-different-answers.html$$x^{ 2 }-2x-15=0$$ By factoring, I get: $$(x-5)(x+3)$$ Which has the solutions: $$x=5, x=-3$$ However when I use the quadratic formula (which is what the book saids to use), I get $$\frac ......Fri, 04 Sep 2026 22:08:20 -0400https://pop.com.sg/why-can-a-venn-diagram-for-4-sets-not-be-constructed-using-circles.htmlhttps://pop.com.sg/why-can-a-venn-diagram-for-4-sets-not-be-constructed-using-circles.htmlThis page gives a few examples of Venn diagrams for $4$ sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complet......Fri, 04 Sep 2026 22:07:07 -0400https://pop.com.sg/why-does-soh-cah-toa-hold.htmlhttps://pop.com.sg/why-does-soh-cah-toa-hold.htmlEven looking on the Wiki pages I have a hard time figuring this one out. But why does SOH CAH TOA hold? As in, why is $\sin(x)$ the same as the opposite over hypotenuse of a right triangle? Why is......Fri, 04 Sep 2026 21:54:41 -0400https://pop.com.sg/invertibility-and-rank.htmlhttps://pop.com.sg/invertibility-and-rank.htmlHow do you formally prove that a matrix A is invertible if and only if it has full rank, without using determinants?...Fri, 04 Sep 2026 21:52:32 -0400https://pop.com.sg/baker-hausdorff-lemma-from-sakurai-039-s-book.htmlhttps://pop.com.sg/baker-hausdorff-lemma-from-sakurai-039-s-book.htmlI'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\...Fri, 04 Sep 2026 21:50:49 -0400https://pop.com.sg/is-a-proof-required-for-the-division-algorithm-for-polynomials.htmlhttps://pop.com.sg/is-a-proof-required-for-the-division-algorithm-for-polynomials.htmlI understand that the Division Algorithm can be applied to polynomials. Namely, for polynomials, for any polynomials $f,g$, there exist polynomials $q,r$ such that $f = gq + r$, with $\deg(f) > ......Fri, 04 Sep 2026 21:49:37 -0400https://pop.com.sg/deriving-probability-from-logit-or-log-odds.htmlhttps://pop.com.sg/deriving-probability-from-logit-or-log-odds.htmlHow can I motivate the derivation of the $p$ below? From: https://en.wikipedia.org/wiki/Logistic_regression#Logistic_model $l = log_b\frac{p}{1-p}=\beta_0 + \beta_1x_1+\beta_2x_2 $ $p = \frac{b^{\......Fri, 04 Sep 2026 21:31:11 -0400https://pop.com.sg/user-cortizol-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-cortizol-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 04 Sep 2026 21:14:57 -0400https://pop.com.sg/how-to-use-the-aops-books.htmlhttps://pop.com.sg/how-to-use-the-aops-books.htmlI just acquired the art of problem solving prealgebra book. I would like to ask, how does one use the AOPS books, are they meant to be supplementary to a full textbook? Or are they used for introd......Fri, 04 Sep 2026 21:09:10 -0400https://pop.com.sg/parametrization-of-the-hyperboloid-of-two-sheets.htmlhttps://pop.com.sg/parametrization-of-the-hyperboloid-of-two-sheets.htmlFind the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$. Ok so I saw two answers for this question: $x(u,v)=(\sinh u \cos v, \sinh u \sin v, \cosh u)......Fri, 04 Sep 2026 21:08:56 -0400https://pop.com.sg/integrate-sin-x-cos-x-using-trig-identity.htmlhttps://pop.com.sg/integrate-sin-x-cos-x-using-trig-identity.htmlBook tells me the answer is: $$ \int \sin(x)\cos(x) dx = \frac{1}{2} \sin^{2}(x) + C $$ however, I get the result: $$ \sin(A)\cos(B) = \frac{1}{2} \sin(A-B)+\frac{1}{2}\sin(A+B) $$ $$ \begin{s......Fri, 04 Sep 2026 21:07:15 -0400https://pop.com.sg/what-is-the-uv-pair-or-uv-plane-exactly.htmlhttps://pop.com.sg/what-is-the-uv-pair-or-uv-plane-exactly.htmlMaybe the answer to this question is easier than computing $1+1$, but I often find this $uv$ pair on pretty much all the parametric equations that have something to do with 3D geometry and all the ...Fri, 04 Sep 2026 21:04:20 -0400https://pop.com.sg/grothendieck-topology-and-relation-with-usual-topologies.htmlhttps://pop.com.sg/grothendieck-topology-and-relation-with-usual-topologies.htmlRecently I stumbled upon the definition of $\textbf{Grothendieck}$ $\textbf{topologies}$ of a category $\mathcal{C}$. I do know that is one of the most interesting parts of the contemporary algebraic ...Fri, 04 Sep 2026 20:53:47 -0400https://pop.com.sg/where-does-the-ln-c-came-from.htmlhttps://pop.com.sg/where-does-the-ln-c-came-from.htmlI'm trying to solve problems which is about the application of indefinite integral. I do quite understand how to solve it but at some point I'm curious where does the $ln(C)$ came from. On the first ...Fri, 04 Sep 2026 20:49:25 -0400https://pop.com.sg/books-on-logic-proof-theory-and-set-theory.htmlhttps://pop.com.sg/books-on-logic-proof-theory-and-set-theory.htmlI graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have o......Fri, 04 Sep 2026 20:44:15 -0400https://pop.com.sg/integral-cohomology-ring-of-real-projective-space.htmlhttps://pop.com.sg/integral-cohomology-ring-of-real-projective-space.htmlWhat is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology rin......Fri, 04 Sep 2026 20:44:00 -0400https://pop.com.sg/in-the-principle-of-mathematical-induction-why-do-we-take-the-base-cas.htmlhttps://pop.com.sg/in-the-principle-of-mathematical-induction-why-do-we-take-the-base-cas.htmlI kinda understand the logic and motivation behind the proof, but what bothers me is the fact, why is the base case (the first statement that we write) is always $P(1)$ when we are proving a propos......Fri, 04 Sep 2026 20:30:12 -0400https://pop.com.sg/solve-the-system-x1-x2-x3.htmlhttps://pop.com.sg/solve-the-system-x1-x2-x3.html$$\left\{\begin{align}x_{1} + x_{2} +4x_{3} &= 1 \\ 3x_{1} +2x_{2} -5x_{3} &= -8 \end{align}\right.$$ $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} \quad \\ \quad \\ \...Fri, 04 Sep 2026 20:28:55 -0400https://pop.com.sg/fooling-set-method.htmlhttps://pop.com.sg/fooling-set-method.htmlIn the book Computational Complexity (By Sanjeev Arora and Boaz Barak) a method called "The fooling set" is mentioned, when trying to bound the communication complexity of a function from below. A ......Fri, 04 Sep 2026 20:27:17 -0400