https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 15 Aug 2026 10:34:56 -0400https://pop.com.sg/prove-that-4-is-the-only-solution-to-2-2-duplicate.htmlhttps://pop.com.sg/prove-that-4-is-the-only-solution-to-2-2-duplicate.htmlThis question was featured on Saturday Morning Breakfast Cereal and I haven't been able to find a proof. Can anyone help?...Sat, 15 Aug 2026 14:21:28 -0400https://pop.com.sg/how-to-find-the-perimeter-of-the-triangle-inscribed-within-the-square.htmlhttps://pop.com.sg/how-to-find-the-perimeter-of-the-triangle-inscribed-within-the-square.htmlIn square $ABCD$, the length of its sides is $5$. $E, F$ are two points on $AB$ and $AD$ in such a way so that $∠ECF = 45^\circ$, find the maximum value of the perimeter of $ΔAEF$....Sat, 15 Aug 2026 14:14:56 -0400https://pop.com.sg/verification-of-induction-proof-for-handshake-lemma.htmlhttps://pop.com.sg/verification-of-induction-proof-for-handshake-lemma.htmlI did the following proof which seems correct to me but does not match the approach of the answer provided by my professor, and seems pretty different from the question here in terms of notation and ...Sat, 15 Aug 2026 14:13:58 -0400https://pop.com.sg/trick-fake-proof-of-2x2-5-algebraic-form-closed.htmlhttps://pop.com.sg/trick-fake-proof-of-2x2-5-algebraic-form-closed.htmlJust ran into "algebraic proof" that 2x2=5 on youtube. They didn't hide that it was a trick. Can't find where is the mistake. Let a=4, b=5, c=1 c=b-a /multiply by (b-a)......Sat, 15 Aug 2026 14:13:23 -0400https://pop.com.sg/finding-connected-components-in-a-graph-using-bfs.htmlhttps://pop.com.sg/finding-connected-components-in-a-graph-using-bfs.htmlI'd like to know how do I change the known BFS algorithm in order to find all of the connected components of a given graph. The original algorithm stops whenever we've colored an entire component in ...Sat, 15 Aug 2026 14:05:16 -0400https://pop.com.sg/how-to-verify-if-a-curve-is-exponential-by-eyeballing.htmlhttps://pop.com.sg/how-to-verify-if-a-curve-is-exponential-by-eyeballing.htmlA plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitat......Sat, 15 Aug 2026 14:03:15 -0400https://pop.com.sg/prove-that-an-upper-triangular-matrix-is-invertible-if-and-only-if-eve.htmlhttps://pop.com.sg/prove-that-an-upper-triangular-matrix-is-invertible-if-and-only-if-eve.htmlProve that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero. I have proved that if every diagonal entry is non-zero, then the matrix is invertible by showing......Sat, 15 Aug 2026 13:47:54 -0400https://pop.com.sg/properties-of-the-number-50.htmlhttps://pop.com.sg/properties-of-the-number-50.htmlI will shortly be engaging with my 50th (!) birthday. 50 = 1+49 = 25+25 can perhaps be described as a "sub-Ramanujan" number. I'm trying to put together a quiz including some mathematical content. ...Sat, 15 Aug 2026 13:46:37 -0400https://pop.com.sg/calculating-the-modular-multiplicative-inverse-without-all-those-stran.htmlhttps://pop.com.sg/calculating-the-modular-multiplicative-inverse-without-all-those-stran.htmlI am sure all those symbols are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me. How could I do this on a basic calculator? or with......Sat, 15 Aug 2026 13:44:35 -0400https://pop.com.sg/littlewood-039-s-principles-and-counterexamples.htmlhttps://pop.com.sg/littlewood-039-s-principles-and-counterexamples.htmlLittlewood’s first principle says that each measurable set E of finite measure is nearly a union of finitely many open intervals. However, if the G for which the symmetric difference with E is les......Sat, 15 Aug 2026 13:29:29 -0400https://pop.com.sg/help-getting-a-zero-out-of-the-denominator-of-a-limit.htmlhttps://pop.com.sg/help-getting-a-zero-out-of-the-denominator-of-a-limit.htmlWhy is the following limit equal to $1/2$. I get undefined. :-( $$\lim_{x\to 0}\frac{(x+1)^{1/2}+1}x$$ this should $= 1/2$. When I multiply the top and bottom by $(x+1)^{1/2} - 1$, I end up with......Sat, 15 Aug 2026 13:28:16 -0400https://pop.com.sg/what-is-the-standard-deviation-of-dice-rolling.htmlhttps://pop.com.sg/what-is-the-standard-deviation-of-dice-rolling.htmlWhen trying to find how to simulate rolling a variable amount of dice with a variable but unique number of sides, I read that the mean is $\dfrac{sides+1}{2}$, and that the standard deviation is $\......Sat, 15 Aug 2026 13:22:30 -0400https://pop.com.sg/growth-of-the-riemann-siegel-theta-function.htmlhttps://pop.com.sg/growth-of-the-riemann-siegel-theta-function.htmlThe Riemann-Siegel theta function is defined by: $ \theta(t) = arg\left(\Gamma(\frac{0.5+it}{2}) \right) - \frac{t}{2}\log{\pi}$ As in the following: https://en.wikipedia.org/wiki/Riemann%E2%80%...Sat, 15 Aug 2026 13:08:07 -0400https://pop.com.sg/calculating-the-variance-using-the-probability-mass-function-pmf.htmlhttps://pop.com.sg/calculating-the-variance-using-the-probability-mass-function-pmf.htmlMy Problem: The random variables X and Y have the simultaneous probability function pX,Y(x, y). pX,Y(x, y) X = 0 X = 1 X = 2 Y = 0 0.1 (Z = 0) 0.2 (Z = 1) 0.2 (Z = 2) Y = 1 0.1 (Z = 1) 0.1 (Z = ......Sat, 15 Aug 2026 13:05:49 -0400https://pop.com.sg/6-team-round-robin.htmlhttps://pop.com.sg/6-team-round-robin.htmlIs it possible to create a schedule where we have 6 teams play 5 different games with each team playing each game and each team only once? Or do we have to increase the amount of games to make this ...Sat, 15 Aug 2026 13:04:50 -0400https://pop.com.sg/what-is-the-difference-between-infinite-and-transfinite.htmlhttps://pop.com.sg/what-is-the-difference-between-infinite-and-transfinite.htmlI have never properly got my head round exactly what the difference is between "transfinite" and "infinite". Wikipedia says: transfinite numbers are numbers that are "infinite" in the sense that t......Sat, 15 Aug 2026 13:01:12 -0400https://pop.com.sg/how-to-construct-a-transcendental-number.htmlhttps://pop.com.sg/how-to-construct-a-transcendental-number.htmlI am doing a project about irrational and transcendental numbers and I was wondering how could I construct a "new" transcendental number. I know that all Liouville numbers are transcendental so this ...Sat, 15 Aug 2026 12:59:17 -0400https://pop.com.sg/a-rectangle-is-twice-as-long-as-it-is-wide.htmlhttps://pop.com.sg/a-rectangle-is-twice-as-long-as-it-is-wide.htmlA Farmer fenced a rectangular area with $75$ metres of chicken wire. The rectangle is twice as long as it is wide. What are its dimensions?...Sat, 15 Aug 2026 12:58:04 -0400https://pop.com.sg/the-well-ordering-principle.htmlhttps://pop.com.sg/the-well-ordering-principle.htmlHere is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered. In the book, it says that the chief advan......Sat, 15 Aug 2026 12:55:07 -0400https://pop.com.sg/what-are-the-rules-of-powers-of-powers-duplicate.htmlhttps://pop.com.sg/what-are-the-rules-of-powers-of-powers-duplicate.htmlWhat would $2^{3^4}$ equate to? I can think of two rules that may apply: $a^{b^c} = a^{(b^c)}$ (Making $2^{3^4} = 2^{81}\approx2.417\cdot10^{24}$) or $a^{b^c} = (a^b)^c = a^{bc}$ (Making $2^{3^4}......Sat, 15 Aug 2026 12:52:37 -0400https://pop.com.sg/what-does-a-colon-mean-as-a-mathematical-symbol-in-the-context-of-mapp.htmlhttps://pop.com.sg/what-does-a-colon-mean-as-a-mathematical-symbol-in-the-context-of-mapp.htmlI am trying to understand the principle of recursion: Let ℕ be the natural numbers. Let T be a set. Let a ∈ T. Let g: T → T be a mapping. $$\forall x \in N: f \left({x}\r......Sat, 15 Aug 2026 12:46:59 -0400https://pop.com.sg/why-does-the-minimum-value-of-x-x-equal-1-e.htmlhttps://pop.com.sg/why-does-the-minimum-value-of-x-x-equal-1-e.htmlThe graph of $y=x^x$ looks like this: As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$. I know that $e$ is the number for expon......Sat, 15 Aug 2026 12:43:31 -0400https://pop.com.sg/why-algebraic-topology-is-also-called-combinatorial-topology.htmlhttps://pop.com.sg/why-algebraic-topology-is-also-called-combinatorial-topology.htmlI remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...Sat, 15 Aug 2026 12:41:55 -0400https://pop.com.sg/largest-inscribed-rectangle-in-sector.htmlhttps://pop.com.sg/largest-inscribed-rectangle-in-sector.htmlWhat is the largest (by area) rectangle that can be inscribed in a circular sector wiith radius $r$ and central angle $\alpha$? I think I have an answer to this question, but would like it confirm......Sat, 15 Aug 2026 12:39:55 -0400https://pop.com.sg/every-subspace-of-mathbb-r-n-has-an-orthonormal-basis.htmlhttps://pop.com.sg/every-subspace-of-mathbb-r-n-has-an-orthonormal-basis.htmlIs there a non-constructive proof of this statement, i.e., one that avoids Gram-Schmidt?...Sat, 15 Aug 2026 12:36:07 -0400https://pop.com.sg/what-is-the-role-of-conjectures-in-modern-mathematics.htmlhttps://pop.com.sg/what-is-the-role-of-conjectures-in-modern-mathematics.htmlToday, I heard of something so called Goldbach's conjecture from my mathematics teacher in the class. This was one of the most interesting things that I have ever heard in mathematics. This made me ...Sat, 15 Aug 2026 12:32:47 -0400https://pop.com.sg/l-039-hospital-039-s-rule-for-the-indeterminate-form-0-0.htmlhttps://pop.com.sg/l-039-hospital-039-s-rule-for-the-indeterminate-form-0-0.htmlI'm teaching an introductory real analysis course for the first time this year and one of the textbook problems asks to prove the following: Let $f$ be differentiable on $(a,b)$ and let $c \in (a,......Sat, 15 Aug 2026 12:17:24 -0400https://pop.com.sg/limit-of-infinite-sequence-from-partial-sum.htmlhttps://pop.com.sg/limit-of-infinite-sequence-from-partial-sum.htmlI think there was a rule in Calculus that mentions this, but I am not sure. If I need to find $\lim_{n \to \infty} a_n$ and I am only given the nth partial sum: $S_n =\sum_{k=1}^{n} a_k = f(n)$ ......Sat, 15 Aug 2026 12:11:41 -0400https://pop.com.sg/two-lines-that-are-perpendicular-to-a-third-line-do-not-have-to-be-par.htmlhttps://pop.com.sg/two-lines-that-are-perpendicular-to-a-third-line-do-not-have-to-be-par.htmlIn 3-space, two lines that are perpendicular to the same plane must be parallel. However, two lines that are perpendicular to a third line do not have to be parallel. Explain why. I am having a very ...Sat, 15 Aug 2026 12:07:52 -0400https://pop.com.sg/getting-probability-from-z-score-on-ti-83.htmlhttps://pop.com.sg/getting-probability-from-z-score-on-ti-83.htmlI'm able to use a $z$-table, but would like to be able to use my ti-$83+$ to get value. For example, $P(z \leq 2.5) = 0.9938$ , given $2.5$ I would like to get $0.9938$. Any help would be appreci......Sat, 15 Aug 2026 12:07:21 -0400https://pop.com.sg/square-root-inside-a-square-root.htmlhttps://pop.com.sg/square-root-inside-a-square-root.htmlHi guys I just want to ask how to solve this. The given is: $$f(x)=\sqrt{x}$$ then solve for the $$(f \circ f)(x)$$ then it becomes. $$(f \circ f)(x)=f(f(x))$$ $$f(x)=\sqrt{\sqrt{x}}$$ Can the ...Sat, 15 Aug 2026 12:05:51 -0400https://pop.com.sg/what-is-the-difference-between-3-choose-2-and-3-choose-1-2-choose-1.htmlhttps://pop.com.sg/what-is-the-difference-between-3-choose-2-and-3-choose-1-2-choose-1.htmlWhile choosing 2 from 3, we do ${3 \choose 2}$. But what would happen if we do ${3 \choose 1}{2 \choose 1}$? [Choosing 1 from 3 and again 1 from the remaining two)? I know the latter is incorrect b......Sat, 15 Aug 2026 11:57:15 -0400https://pop.com.sg/value-of-summation-of-log-n.htmlhttps://pop.com.sg/value-of-summation-of-log-n.htmlContext: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: For each ...Sat, 15 Aug 2026 11:57:09 -0400https://pop.com.sg/linear-algebra-identity-matrix-and-its-relation-to-eigenvalues-and-eig.htmlhttps://pop.com.sg/linear-algebra-identity-matrix-and-its-relation-to-eigenvalues-and-eig.htmlKindly help me understand this statement made by my prof. "The identity matrix I has the property that any non zero vector V is an eigenvector of eigenvalue 1." My assumption of this statement is......Sat, 15 Aug 2026 11:41:49 -0400https://pop.com.sg/what-precisely-is-the-difference-between-euclidean-geometry-and-non-eu.htmlhttps://pop.com.sg/what-precisely-is-the-difference-between-euclidean-geometry-and-non-eu.htmlI was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that non-Eucl......Sat, 15 Aug 2026 11:41:09 -0400https://pop.com.sg/can-someone-please-help-me-convert-this-triple-integral-to-spherical-c.htmlhttps://pop.com.sg/can-someone-please-help-me-convert-this-triple-integral-to-spherical-c.htmlI'm not sure how to approach the problem, though I know it needs to be broken up into two integrals before it can be evaluated based on the way the answer input is set up. I do not know how to start ...Sat, 15 Aug 2026 11:13:51 -0400https://pop.com.sg/why-are-exact-differential-equations-called-so.htmlhttps://pop.com.sg/why-are-exact-differential-equations-called-so.htmlAs the question says, why are exact differential equations called so? From Wikipedia, I got "The nomenclature of "exact differential equation" refers to the exact differential of a function". That ...Sat, 15 Aug 2026 11:13:51 -0400https://pop.com.sg/closure-of-the-set-of-all-polynomial-with-variable-x-in-0-1.htmlhttps://pop.com.sg/closure-of-the-set-of-all-polynomial-with-variable-x-in-0-1.htmlLet ${P}$ denote the set of all polynomial with variable $x\in [0,1]$, I need to know what is the closure of ${P}$ in $C[0,1]$? Well, Stone-Weierstrass theorem says: If $f\in C[0,1]$ then there exi......Sat, 15 Aug 2026 11:10:29 -0400https://pop.com.sg/what-are-the-mean-and-variance-of-the-log-of-a-random-variable.htmlhttps://pop.com.sg/what-are-the-mean-and-variance-of-the-log-of-a-random-variable.htmlHere's the problem. We have a random variable X that follows a Poisson law. If we take the log of this variable, what are the first two moments (mean and variance) of the law it follows? This looks ...Sat, 15 Aug 2026 11:02:54 -0400https://pop.com.sg/a-double-limit-question.htmlhttps://pop.com.sg/a-double-limit-question.htmlMaybe it's easy, but: Is it true that $$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$ If it is, could you help me prove it? Thanks...Sat, 15 Aug 2026 10:51:50 -0400https://pop.com.sg/recognizing-the-sequence-1-16-1-8-3-16-1-4-5-16.htmlhttps://pop.com.sg/recognizing-the-sequence-1-16-1-8-3-16-1-4-5-16.htmlWhat is the missing number? $$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \ \ \ [?]$$ $$A. \frac{5}{4}\quad B. \frac{3}{4}\quad C. \frac{5}{8}\quad D. \frac{3}{8}$$ Spo......Sat, 15 Aug 2026 10:45:05 -0400https://pop.com.sg/finding-a-spanning-set-for-a-null-space.htmlhttps://pop.com.sg/finding-a-spanning-set-for-a-null-space.htmlI have a matrix $A$ that is like this: \begin{equation} A = \pmatrix{ 1 & 2 & -3 & 1 & 5 \\ 1 & 3 & -1 & 4 & -2 \\ 1 & 1 & -5 & -2 & 12 \\ 1 & 4......Sat, 15 Aug 2026 10:44:41 -0400https://pop.com.sg/is-k-g-1-bg.htmlhttps://pop.com.sg/is-k-g-1-bg.htmlIs an Eilenberg-MacLane space $K(G,1)$ the same as the classifying space $BG$ for a group $G$ ?...Sat, 15 Aug 2026 10:44:01 -0400https://pop.com.sg/entire-function-with-vanishing-derivatives.htmlhttps://pop.com.sg/entire-function-with-vanishing-derivatives.htmlLet $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. And assume that at each point, one of it's derivatives vanishes. What can you say about $f$? A hint suggests that $f$ must be a poly......Sat, 15 Aug 2026 10:43:17 -0400https://pop.com.sg/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthu.htmlhttps://pop.com.sg/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthu.htmlI'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs......Sat, 15 Aug 2026 10:34:12 -0400https://pop.com.sg/question-concerning-linear-combinations-of-vectors-and-linear-independ.htmlhttps://pop.com.sg/question-concerning-linear-combinations-of-vectors-and-linear-independ.htmlMy question concerns the definition of linear combinations and a criteria for linear independence of a Set (either finite or infinite). Here is the following Definition and Criteria given: A vec......Sat, 15 Aug 2026 10:29:03 -0400https://pop.com.sg/is-there-a-term-do-refer-to-a-set-of-sets-in-math-to-distinguish-set-o.htmlhttps://pop.com.sg/is-there-a-term-do-refer-to-a-set-of-sets-in-math-to-distinguish-set-o.htmlIn mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. for example A = {2, 1, 3} B = {blue, white, red} C = {3, 5, 6} D = {A, C} all the ...Sat, 15 Aug 2026 10:28:46 -0400https://pop.com.sg/what-is-the-difference-between-a-point-and-a-vector.htmlhttps://pop.com.sg/what-is-the-difference-between-a-point-and-a-vector.htmlI understand that a vector has direction and magnitude whereas a point doesn't. However, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do c......Sat, 15 Aug 2026 10:24:29 -0400https://pop.com.sg/is-a-simplified-function-the-same-as-the-original.htmlhttps://pop.com.sg/is-a-simplified-function-the-same-as-the-original.htmlIs a simplified function the same as the original? Example: Let $f(x) = \frac{ax}{x}$, and $g(x) = a$ where $a$ and $x$ are real numbers. Does $g$ = $f$?...Sat, 15 Aug 2026 10:20:59 -0400https://pop.com.sg/good-book-for-self-study-of-continued-fractions.htmlhttps://pop.com.sg/good-book-for-self-study-of-continued-fractions.htmlDoes anyone have a recommendation for a rigorous while readable book to use for the self study of continued fractions? PS - As examples of "rigorous while readable book" for self-learning, A. Pr......Sat, 15 Aug 2026 10:20:18 -0400https://pop.com.sg/average-length-of-the-longest-segment.htmlhttps://pop.com.sg/average-length-of-the-longest-segment.htmlThis post is related to a previous SE post If a 1 meter rope …. concerning average length of a smallest segment. A rope of 1m is divided into three pieces by two random points. Find the average ...Sat, 15 Aug 2026 10:13:46 -0400https://pop.com.sg/splitting-a-circle-into-3-equal-parts-using-2-lines.htmlhttps://pop.com.sg/splitting-a-circle-into-3-equal-parts-using-2-lines.htmlThe picture probably explains my question best. I need to find a way to divide a circle into 3 parts of equal area with only 2 lines that intersect each other on the outline of the circle. Also I n......Sat, 15 Aug 2026 10:10:32 -0400https://pop.com.sg/how-to-solve-equations-of-this-form-x-x-n.htmlhttps://pop.com.sg/how-to-solve-equations-of-this-form-x-x-n.htmlHow would I go about solving equations of this form: $$ x^x = n $$ for values of n that do not have obvious solutions through factoring, such as $27$ ($3^3$) or $256$ ($4^4$). For instance, how ......Sat, 15 Aug 2026 10:06:56 -0400https://pop.com.sg/finding-average-speed-using-the-mean-value-theorem-for-integrals.htmlhttps://pop.com.sg/finding-average-speed-using-the-mean-value-theorem-for-integrals.htmlI am learning how to use the equation $\frac {1}{b-a} \int^b_af(x)dx$. More secifically I dont understand how to apply it to the following problem: A car starting from rest accelerates at a rate ......Sat, 15 Aug 2026 10:01:06 -0400https://pop.com.sg/characterizations-of-local-topology-of-mathbb-r-n.htmlhttps://pop.com.sg/characterizations-of-local-topology-of-mathbb-r-n.htmlWhat topological properties uniquely characterize the local topology of $\mathbb{R}^n$? I know $\mathbb{R}$ is the unique complete ordered field, but that's partly algebraic. Better if the ...Sat, 15 Aug 2026 09:58:40 -0400https://pop.com.sg/verify-using-the-definition-of-convergence-of-a-sequence-that-the-foll.htmlhttps://pop.com.sg/verify-using-the-definition-of-convergence-of-a-sequence-that-the-foll.html(a) lim $\displaystyle\frac{2n^2}{n^3+3} = 0$ (b) lim $\displaystyle\frac{\text{sin}(n^2)}{n^{\frac{1}{3}}} = 0$ For (a) I have: $$\bigg|\displaystyle\frac{2n^2}{n^3+3} - 0\bigg| < \epsilon$$ ......Sat, 15 Aug 2026 09:57:52 -0400https://pop.com.sg/what-is-the-difference-between-a-trace-and-a-contour-in-calculus.htmlhttps://pop.com.sg/what-is-the-difference-between-a-trace-and-a-contour-in-calculus.htmlAs far as I can tell they're exactly the same thing, but the notes here discuss them as if they are separate: The final topic in this section is that of traces.  In some ways these are similar to ...Sat, 15 Aug 2026 09:55:11 -0400https://pop.com.sg/solving-double-derivative-frac-d-2y-dx-2-f-y-by-integration.htmlhttps://pop.com.sg/solving-double-derivative-frac-d-2y-dx-2-f-y-by-integration.htmlI want to ask if a differential equation of second order can be solved by integration? Like equations of the type $\dfrac{d^2y}{dx^2} = f(y)$. I know this can be solved by making equations of the f......Sat, 15 Aug 2026 09:53:01 -0400https://pop.com.sg/plus-or-minus-minus-or-plus-it-is-the-same.htmlhttps://pop.com.sg/plus-or-minus-minus-or-plus-it-is-the-same.htmlAre "Plus or Minus" or "Minus or Plus" the same ? $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$x = \frac{-b\mp \sqrt{b^2-4ac}}{2a}$$ $$x^2=9 \implies x = \pm 3 \quad \text{or}\quad x = \mp3$$ These ...Sat, 15 Aug 2026 09:47:41 -0400https://pop.com.sg/counting-all-6-digit-x-numbers-s-t-x-123456.htmlhttps://pop.com.sg/counting-all-6-digit-x-numbers-s-t-x-123456.htmlWe want to count all the $6$ digits numbers (including ones with $0$'s as digits) that are strictly greater than $123456$ (or even better, some general method of counting for some number $n$ of len......Sat, 15 Aug 2026 09:40:05 -0400https://pop.com.sg/good-examples-of-double-induction.htmlhttps://pop.com.sg/good-examples-of-double-induction.htmlI'm looking for good examples where double induction is necessary. What I mean by double induction is induction on $\omega^2$. These are intended as examples in an "Automatas and Formal Languages" ......Sat, 15 Aug 2026 09:36:34 -0400https://pop.com.sg/surface-area-of-quarter-of-a-sphere.htmlhttps://pop.com.sg/surface-area-of-quarter-of-a-sphere.htmlA quarter sphere with a radius of $10 \text{ units}$. Please help, also remember the sides. I used the normal formula of the total surface area of a sphere and divided it by $4$, then added half ......Sat, 15 Aug 2026 09:31:44 -0400https://pop.com.sg/limit-of-1-x-n-n-when-n-tends-to-infinity-duplicate.htmlhttps://pop.com.sg/limit-of-1-x-n-n-when-n-tends-to-infinity-duplicate.htmlDoes anyone know the exact proof of this limit result? $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$...Sat, 15 Aug 2026 09:20:18 -0400https://pop.com.sg/inverse-of-sparse-matrix-is-not-generally-sparse.htmlhttps://pop.com.sg/inverse-of-sparse-matrix-is-not-generally-sparse.htmlI have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the in......Sat, 15 Aug 2026 09:18:14 -0400https://pop.com.sg/how-to-find-orthogonal-projection-of-vector-on-a-subspace.htmlhttps://pop.com.sg/how-to-find-orthogonal-projection-of-vector-on-a-subspace.htmlWell, I have this subspace: $V = \operatorname{span}\left\{ \begin{pmatrix}\frac{1}{3} \\\frac{2}{3} \\\frac{2}{3}\end{pmatrix},\begin{pmatrix}1 \\3 \\4\end{pmatrix}\right\}$ and the vector $v = \b......Sat, 15 Aug 2026 09:03:07 -0400https://pop.com.sg/show-that-every-ideal-of-the-ring-mathbb-z-is-principal.htmlhttps://pop.com.sg/show-that-every-ideal-of-the-ring-mathbb-z-is-principal.htmlLet $\mathbb Z$ be the ring of integers. The question asks to show that every ideal of $\mathbb Z$ is principal. I beg someone to help me because it is a new concept to me....Sat, 15 Aug 2026 09:02:58 -0400https://pop.com.sg/memoryless-property-and-geometric-distribution.htmlhttps://pop.com.sg/memoryless-property-and-geometric-distribution.htmlSuppose $X$ is a random variable taking values in $\mathbb N_0$ with the memoryless property,i.e., for each pair of number $s,t \in \mathbb N$, $$P(X\geq s+t\mid X>t)=P(X\geq s)$$ Show that a r......Sat, 15 Aug 2026 08:52:09 -0400https://pop.com.sg/how-to-find-position-of-point-that-is-x-unit-distant-from-ab-line-segm.htmlhttps://pop.com.sg/how-to-find-position-of-point-that-is-x-unit-distant-from-ab-line-segm.htmlI am trying to calculate coordinates of point P, which is x units distant from AB line segment and y units distant from BC line segment. Edit: I am trying to write code for general solution. As ...Sat, 15 Aug 2026 08:46:39 -0400https://pop.com.sg/which-matrices-perform-dilation.htmlhttps://pop.com.sg/which-matrices-perform-dilation.htmlFor each angle $\theta \in \mathbb{R}$, we get two corresponding matrices: $$\mathrm{Rot}^\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos\theta\end{bmatrix}, \qqu......Sat, 15 Aug 2026 08:39:22 -0400https://pop.com.sg/example-of-a-simple-pole.htmlhttps://pop.com.sg/example-of-a-simple-pole.htmlI was told that $\operatorname{sech} x$ has a simple pole. Could someone please explain what that means? I have looked up the definition but it involves too much jargon like holomorphic, etc. Is th......Sat, 15 Aug 2026 08:23:02 -0400https://pop.com.sg/rarefaction-wave-in-nonlinear-pde-u-t-u-3-2-x-0.htmlhttps://pop.com.sg/rarefaction-wave-in-nonlinear-pde-u-t-u-3-2-x-0.htmlI am dealing with the following PDE: $$u_t + \big(u^{\frac{3}{2}}\big)_{x} = 0$$ subject to: $$u(x,0) = \begin{cases} 1 & x\leq 0 \\ 4 & 0\leq x\leq 10 \\ 1 & x > ......Sat, 15 Aug 2026 08:22:27 -0400https://pop.com.sg/in-a-family-with-3-children-what-is-the-probability-that-they-have-2-b.htmlhttps://pop.com.sg/in-a-family-with-3-children-what-is-the-probability-that-they-have-2-b.htmlI'm doing some regents practice questions, and one of them asked In a family with 3 children, what is the probability that they have 2 boys and 1 girl? And the answer choices are 3/8 1/4 1......Sat, 15 Aug 2026 08:20:34 -0400https://pop.com.sg/about-plus-minus-sign.htmlhttps://pop.com.sg/about-plus-minus-sign.htmlIf, say, we know that $x=1$, is then the expression $x=\pm 1$ mathematically incorrect? I ask this because when we use $\pm$ sign in the discriminant formula we imply that any of the plus or minus ......Sat, 15 Aug 2026 08:17:39 -0400https://pop.com.sg/organic-chemistry-for-mathematicians.htmlhttps://pop.com.sg/organic-chemistry-for-mathematicians.htmlRecently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do ...Sat, 15 Aug 2026 08:11:17 -0400https://pop.com.sg/find-t-v-using-the-standard-matrix-and-the-matrix-relative-to-b-and-b-.htmlhttps://pop.com.sg/find-t-v-using-the-standard-matrix-and-the-matrix-relative-to-b-and-b-.htmlFind T(V) by using the standard matrix and the matrix relative to B and B' $T: R^3 -> R^2, T(x,y,z) = (x-y + 0, 0 + y-z), v = (2,4,6)$ $B = {(1,1,1), (1,1,0), (0,1,1)}$ $B' = {(1,1),(2,1)}$ ...Sat, 15 Aug 2026 08:07:26 -0400https://pop.com.sg/finding-limit-of-a-sequence-defined-by-another-sequence.htmlhttps://pop.com.sg/finding-limit-of-a-sequence-defined-by-another-sequence.htmlLet there be $\{a_n\}$ such that $a_{n+2}=2a_{n+1}+2a_n$ and $\{b_n\}$ such that $b_n=\frac{a_{n+1}}{a_n}$ if $\lim_{n \to \infty}b_n=L$ what is the limit $L$? I thought about if $L$ exist so I ca......Sat, 15 Aug 2026 08:07:04 -0400https://pop.com.sg/how-does-a-floor-function-work.htmlhttps://pop.com.sg/how-does-a-floor-function-work.htmlI understand what a floor function does, and got a few explanations here, but none of them had a explanation, which is what i'm after. Can someone explain to me what is going on behind the scenes ......Sat, 15 Aug 2026 08:01:40 -0400https://pop.com.sg/can-someone-prove-why-sqrt-ab-sqrt-a-sqrt-b-is-only-valid-when-a-and-b.htmlhttps://pop.com.sg/can-someone-prove-why-sqrt-ab-sqrt-a-sqrt-b-is-only-valid-when-a-and-b.htmlI have seen many people say that a and b can't be positive for example in this false proof : $$1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1} \sqrt{-1} = i^2 = -1$$ Trust me, I understand that $1\neq -1$ ......Sat, 15 Aug 2026 07:48:56 -0400https://pop.com.sg/inversion-formula-for-the-abel-transform.htmlhttps://pop.com.sg/inversion-formula-for-the-abel-transform.htmlI need an inversion formula for the Abel transform $$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$ Hint: The inversion formula found on Wikipedia appears to be incorrect. The "Verific......Sat, 15 Aug 2026 07:36:32 -0400https://pop.com.sg/is-there-a-definition-of-a-pseudo-period-for-f-x-sin-3x-sin-pi-x.htmlhttps://pop.com.sg/is-there-a-definition-of-a-pseudo-period-for-f-x-sin-3x-sin-pi-x.htmlSums of trigonometric functions may or may not be periodic functions; in particular, $\sin(ax)+\sin(bx)$ is periodic if $a/b$ is rational. If we consider the function \begin{equation} f(x) = \sin(......Sat, 15 Aug 2026 07:34:18 -0400https://pop.com.sg/derivative-of-x-mapsto-x-sin-y-x.htmlhttps://pop.com.sg/derivative-of-x-mapsto-x-sin-y-x.htmlIf I differentiate $x\sin(y)$ with respect to $x$ normally then what happens to the $\sin(y)$? In partial differentiation it's taken as constant thus, partially differentiating with respect to $x$ ......Sat, 15 Aug 2026 07:25:09 -0400https://pop.com.sg/a-in-b-vs-a-subset-b-for-proofs.htmlhttps://pop.com.sg/a-in-b-vs-a-subset-b-for-proofs.htmlI have to prove a few different statements. The first is if $A \subset B$ and $B \subset C$ then prove $A \subset C$. This one is fairly straight forward, but I'm stuck on how the next one differ......Sat, 15 Aug 2026 07:09:43 -0400https://pop.com.sg/int-0-infty-sin-ax-sin-bx-dx.htmlhttps://pop.com.sg/int-0-infty-sin-ax-sin-bx-dx.htmlHow can I calculate the integral: $$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx$$ ?? I got stuck.. :/ Could you give me some hint?? Do I have to use the following formula?? $\displaystyle{\sin{(......Sat, 15 Aug 2026 07:03:24 -0400https://pop.com.sg/proof-of-descartes-039-theorem.htmlhttps://pop.com.sg/proof-of-descartes-039-theorem.htmlI came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any proof.Even Wikipedia also, just states the theorem!!I want to know ......Sat, 15 Aug 2026 06:51:58 -0400https://pop.com.sg/anyone-familiar-with-a-mathematical-symbol-for-that-is-to-clarify-wher.htmlhttps://pop.com.sg/anyone-familiar-with-a-mathematical-symbol-for-that-is-to-clarify-wher.htmlMany years ago, a math teacher of mine thought us that there was a mathematical symbol for "that is" / "to clarify" / "whereby" (I'm not from an English speaking country, so I'm writing several ...Sat, 15 Aug 2026 06:49:23 -0400https://pop.com.sg/using-lagrange-multipliers-to-determine-the-point-on-a-surface-nearest.htmlhttps://pop.com.sg/using-lagrange-multipliers-to-determine-the-point-on-a-surface-nearest.htmlI'm attempting to figure this problem out. I would appreciate some guidance on how to get the answer. Thanks. Consider the surface defined as $S: x^2+y^2+z^2 = 8$. If we have a $P = (0,1,1)$, use ...Sat, 15 Aug 2026 06:48:57 -0400https://pop.com.sg/e-2x-e-x-2-can-some-one-explain-how-these-two-are-the-same-i-know-it-i.htmlhttps://pop.com.sg/e-2x-e-x-2-can-some-one-explain-how-these-two-are-the-same-i-know-it-i.html$ (e^{2x})=(e^x)^2$ Can Some one explain how these two are the same? I know it is trick, but I cannot see how it fits into the exponent rules....Sat, 15 Aug 2026 06:27:47 -0400https://pop.com.sg/derivative-of-exponent.htmlhttps://pop.com.sg/derivative-of-exponent.htmlLooking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!...Sat, 15 Aug 2026 06:18:49 -0400https://pop.com.sg/solving-sin-3-theta-1-2-on-the-interval-0-2-pi-i-don-039-t-understand-.htmlhttps://pop.com.sg/solving-sin-3-theta-1-2-on-the-interval-0-2-pi-i-don-039-t-understand-.htmlI am learning precalculus, and I understand how to obtain first two solutions, but I don't understand where did last four solutions came from: All values of $\theta$ in the interval $[0,2\pi]$ t......Sat, 15 Aug 2026 06:18:06 -0400https://pop.com.sg/problem-using-casio-fx-991ce-x-calculator-to-solve-e-i-pi-closed.htmlhttps://pop.com.sg/problem-using-casio-fx-991ce-x-calculator-to-solve-e-i-pi-closed.htmlI'm trying to solve e^(i*pi) on a Casio fx-991CE X calculator. The calculator is in Complex mode (MENU>2:Complex). I Input e^(i*pi) and press "=" and get an error message instead of -1......Sat, 15 Aug 2026 06:16:41 -0400https://pop.com.sg/finding-a-basis-for-w-perp-and-understanding-it.htmlhttps://pop.com.sg/finding-a-basis-for-w-perp-and-understanding-it.htmlGiven $$ w_1 = \begin{bmatrix} 1 \\ -1 \\ 1 \\ 1 \end{bmatrix},w_2= \begin{bmatrix} 0 \\ 1\\2\\3 \end{bmatrix} $$ let $W$ be the subspace spanned by the given vectors. Find a basis for $W^\pe......Sat, 15 Aug 2026 06:15:49 -0400https://pop.com.sg/what-is-x-tx-called-where-x-is-a-matrix.htmlhttps://pop.com.sg/what-is-x-tx-called-where-x-is-a-matrix.htmlWith $X$ being a matrix, is there a general name for the expression $X^TX$?...Sat, 15 Aug 2026 06:09:50 -0400https://pop.com.sg/limit-with-fraction-in-numerator.htmlhttps://pop.com.sg/limit-with-fraction-in-numerator.htmlI need to find the limit for $$ \lim_{x\,\rightarrow\,0} \frac {\frac{1}{x+3} - \frac{1}{3}}{x} $$ It's supposed to be $-1/9$. I've tried changing it around multiple ways, and get the $9$ but nev......Sat, 15 Aug 2026 06:06:29 -0400https://pop.com.sg/calculate-the-dimension-of-the-vector-subspace-u-text-span-left-v-1-v-.htmlhttps://pop.com.sg/calculate-the-dimension-of-the-vector-subspace-u-text-span-left-v-1-v-.htmlGiven are the vectors $v_{1}=\begin{pmatrix} 1\\ 3\\ 5 \end{pmatrix}, v_{2}=\begin{pmatrix} 4\\ 5\\ 6 \end{pmatrix}, v_{3}=\begin{pmatrix} 6\\ 4\\ 2 \end{pmatrix}$ from $\mathbb{R}^{3}$. ...Sat, 15 Aug 2026 05:52:30 -0400https://pop.com.sg/show-that-the-area-of-a-triangle-is-given-by-this-determinant.htmlhttps://pop.com.sg/show-that-the-area-of-a-triangle-is-given-by-this-determinant.htmlI'm not sure how to solve this problem. Can you guys provide some input/hints? Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$. Show that the area of $\...Sat, 15 Aug 2026 05:48:33 -0400https://pop.com.sg/continuity-correction-factor-adding-or-subtracting-0-5.htmlhttps://pop.com.sg/continuity-correction-factor-adding-or-subtracting-0-5.htmlIn this file :http://mkarefin.weebly.com/uploads/2/0/7/9/20793168/assignment_6-solution.pdf At the solution of problem number 11 on top of page 7. The author writes that $$p(x >200) =p(z>......Sat, 15 Aug 2026 05:38:11 -0400https://pop.com.sg/why-is-this-way-used-to-solve-for-base-10-to-base-16.htmlhttps://pop.com.sg/why-is-this-way-used-to-solve-for-base-10-to-base-16.htmlIn one of my courses there is a problem to convert $142$ base $10$ to base $16$, I know the answer is $8E$ base $16$ just through dividing $142$ by $16$ but the solution is show to use base $2$ lik......Sat, 15 Aug 2026 05:37:40 -0400https://pop.com.sg/calculate-the-volume-a-trough-of-length-250cm-and-cross-sectional-area.htmlhttps://pop.com.sg/calculate-the-volume-a-trough-of-length-250cm-and-cross-sectional-area.htmlI am working on volumes of prisms, using $\pi$, diameter, and radius; however, I haven't done any problems with cross-sectional areas before... Also, I'm only 14 and am finding this really hard! ......Sat, 15 Aug 2026 05:30:09 -0400https://pop.com.sg/i-i-d-what-does-this-stand-for.htmlhttps://pop.com.sg/i-i-d-what-does-this-stand-for.htmlSo almost everywhere in the book it's written "random variables are IID", what does this mean? I think it means independent and identically distributed but not sure. So by definition A and B R.V ......Sat, 15 Aug 2026 05:11:28 -0400https://pop.com.sg/can-the-word-derive-be-used-to-mean-take-the-derivative-of.htmlhttps://pop.com.sg/can-the-word-derive-be-used-to-mean-take-the-derivative-of.htmlBack when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be "different......Sat, 15 Aug 2026 05:07:00 -0400