https://liberty.ioCurated star highlights with fresh energy.en-usThu, 20 Aug 2026 14:45:12 -0400https://liberty.io/prove-that-a-2-b-2-is-rational-if-a-b-and-ab-are-rational-duplicate.htmlhttps://liberty.io/prove-that-a-2-b-2-is-rational-if-a-b-and-ab-are-rational-duplicate.htmlTrying to prove or disprove this statement: For all real numbers A, B, if $AB$ is rational and $A+B$ is rational, then $A^2$+$B^2$ is rational Ive hit a wall as Im assuming A and B must either be ...Thu, 20 Aug 2026 18:37:25 -0400https://liberty.io/frac-1-6-frac-5-6-12-frac-5-8-6-12-18-frac-5-8-11-6-12-18-24-cdots.htmlhttps://liberty.io/frac-1-6-frac-5-6-12-frac-5-8-6-12-18-frac-5-8-11-6-12-18-24-cdots.htmlHow to find the sum $$\frac {1}{6} + \frac {5}{6.12} + \frac {5. 8}{6.12.18} + \frac {5.8.11}{6.12.18.24} + \cdots.$$ I have tried to make this series in the form $1 + n x + \frac {n (n + 1......Thu, 20 Aug 2026 18:34:34 -0400https://liberty.io/how-to-derive-inverse-hyperbolic-trigonometric-functions.htmlhttps://liberty.io/how-to-derive-inverse-hyperbolic-trigonometric-functions.html$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of......Thu, 20 Aug 2026 18:22:00 -0400https://liberty.io/why-rationalize-the-denominator.htmlhttps://liberty.io/why-rationalize-the-denominator.htmlIn grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "ther......Thu, 20 Aug 2026 18:21:10 -0400https://liberty.io/magnitude-of-the-sum-of-vectors-in-3d-space.htmlhttps://liberty.io/magnitude-of-the-sum-of-vectors-in-3d-space.htmlI have three vectors $u_1,u_2,u_3$, $$ ||u_1||=3\\ ||u_2||=1\\ ||u_3||=2\\ \angle u_1,u_2=\frac{\pi}{3}\\ \angle u_3,u_1=\frac{\pi}{4}\\ \angle u_3,u_2=\frac{\pi}{4}. $$ I am asked to calculate the ...Thu, 20 Aug 2026 18:20:35 -0400https://liberty.io/borel-sigma-algebra-definition.htmlhttps://liberty.io/borel-sigma-algebra-definition.htmlDefinition: The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals $\ (a, b]$, where $\ a<b$ in $\mathbb R$ (We ......Thu, 20 Aug 2026 18:17:00 -0400https://liberty.io/what-is-the-definition-of-differentiability.htmlhttps://liberty.io/what-is-the-definition-of-differentiability.htmlSome places define it as: If the Left hand derivative and the Right hand derivative at a point are equal then the function is said to be differentiable at that point. Others define it based on ......Thu, 20 Aug 2026 18:11:48 -0400https://liberty.io/books-for-advanced-algebra-and-advanced-geometry-amc-12-duplicate.htmlhttps://liberty.io/books-for-advanced-algebra-and-advanced-geometry-amc-12-duplicate.htmlI will be taking the American Math Contest. Various topics are covered in this test including advanced geometry and algebra. It would be great if any of you could provide books/references which con......Thu, 20 Aug 2026 18:09:22 -0400https://liberty.io/what-is-gcd-12345-54321.htmlhttps://liberty.io/what-is-gcd-12345-54321.htmlWhat is $\gcd(12345,54321)$? I noticed that after trying $\gcd(12,21),\gcd(123,321),$ and $\gcd(1234,4321)$ that they are all less then or equal to $3$. That leads me to question if there is an ea......Thu, 20 Aug 2026 18:07:53 -0400https://liberty.io/why-are-abelian-groups-of-interest-what-is-their-usefulness.htmlhttps://liberty.io/why-are-abelian-groups-of-interest-what-is-their-usefulness.htmlI am reading about Abelian groups So apparently it is a set, with an associative binary operation, and identity element, an inverse operation and the binary operation must also be symmetric. But it......Thu, 20 Aug 2026 18:07:13 -0400https://liberty.io/jordan-region-problem.htmlhttps://liberty.io/jordan-region-problem.htmlWhat is a Jordan Region? I can't find the definition anywhere. The question asked, if $E\subset \mathbb{R}^n$, bounded and with finitely many accumulation points, then $E$ is Jordan region....Thu, 20 Aug 2026 18:07:02 -0400https://liberty.io/complex-numbers-closed-under-division.htmlhttps://liberty.io/complex-numbers-closed-under-division.htmlMy question is very simple that is: Is Complex number closed under division? Can we consider this 0+0i as complex numbers? or it is not a complex number....Thu, 20 Aug 2026 18:04:42 -0400https://liberty.io/2-dimensional-laplace-039-s-equation-in-polar-coordinates.htmlhttps://liberty.io/2-dimensional-laplace-039-s-equation-in-polar-coordinates.htmlThe problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdo......Thu, 20 Aug 2026 17:55:29 -0400https://liberty.io/triangles-flagpoles-and-heights-oh-my.htmlhttps://liberty.io/triangles-flagpoles-and-heights-oh-my.htmlHere is a math question i got from school: On a horizontal plane, there are two flagpoles. One is 20m, and the other is 10m. There is a wire connected from the top of each flagpole, to the bottom ......Thu, 20 Aug 2026 17:52:08 -0400https://liberty.io/what-is-a-central-projection.htmlhttps://liberty.io/what-is-a-central-projection.htmlWhat is a central projection in the context of a C*-algebra or a W*-algebra? Unfortunately while I see results on this concept, I haven't been able to find a definition. Thanks!...Thu, 20 Aug 2026 17:38:20 -0400https://liberty.io/radius-of-convergence-for-binomial-series.htmlhttps://liberty.io/radius-of-convergence-for-binomial-series.htmlThe question is simple: determine the radius of convergence for the given series. I have done this for other problems using the following rule which our book described: If the given series is $$......Thu, 20 Aug 2026 17:24:01 -0400https://liberty.io/number-of-squares-on-a-rectangular-board-that-are-neither-in-the-4th-r.htmlhttps://liberty.io/number-of-squares-on-a-rectangular-board-that-are-neither-in-the-4th-r.htmlA rectangular game board is composed of identical squares arranged in a rectangular array of $r$ rows and $r+1$ columns. The $r$ rows are numbered from $1$ through $r$, and the $r+1$ columns are ...Thu, 20 Aug 2026 17:23:47 -0400https://liberty.io/proof-by-mathematical-induction-related-with-geometry.htmlhttps://liberty.io/proof-by-mathematical-induction-related-with-geometry.htmlQ:Consider $n+2$ distinct point from the circumference of a circle.If consecutive points along the circle are joined by line segments creating a polygon with $n+2$ sides then the sum of interior an......Thu, 20 Aug 2026 17:17:17 -0400https://liberty.io/difference-between-fourier-transform-and-wavelets.htmlhttps://liberty.io/difference-between-fourier-transform-and-wavelets.htmlWhile understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency wherea......Thu, 20 Aug 2026 17:10:24 -0400https://liberty.io/help-understanding-how-ln-and-e-cancel.htmlhttps://liberty.io/help-understanding-how-ln-and-e-cancel.htmlI realise cancel may be the wrong term and inverse may be more appropriate but these is one situation I really don't get…or rather haven't found a suitable explanation. Most sources I have come acr......Thu, 20 Aug 2026 17:08:11 -0400https://liberty.io/pentagons-and-triangles.htmlhttps://liberty.io/pentagons-and-triangles.htmlIf you have a regular pentagon with all equal sides and you split that pentagon into 5 triangles (inside of that pentagon) would the triangles be equalateral or not?...Thu, 20 Aug 2026 17:05:26 -0400https://liberty.io/growth-of-exponential-functions-vs-polynomial.htmlhttps://liberty.io/growth-of-exponential-functions-vs-polynomial.htmlWill $2^x$ take over $x^{1000}$ ? I thought that exponential functions had the fastest growth rate, however, graphing it on wolfram alpha made it seem as if the initial behaviors of the two functi......Thu, 20 Aug 2026 17:05:17 -0400https://liberty.io/integral-int-0-1-ln-x-dx.htmlhttps://liberty.io/integral-int-0-1-ln-x-dx.htmlI have a question about the integral of $\ln x$. When I try to calculate the integral of $\ln x$ from 0 to 1, I always get the following result. $\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1) ......Thu, 20 Aug 2026 16:59:11 -0400https://liberty.io/using-integral-definition-to-solve-this-integral.htmlhttps://liberty.io/using-integral-definition-to-solve-this-integral.htmlI'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...Thu, 20 Aug 2026 16:58:55 -0400https://liberty.io/best-way-to-find-reduced-row-echelon-form-rref-of-a-matrix.htmlhttps://liberty.io/best-way-to-find-reduced-row-echelon-form-rref-of-a-matrix.htmlI'm sitting here doing rref problems and many of them seem so tedious. Any tricks out there to achieve rref with less effort or am I stuck with rewriting the matrix for every 2/3 operations? I kno......Thu, 20 Aug 2026 16:58:18 -0400https://liberty.io/how-to-find-an-inverse-of-the-following-function.htmlhttps://liberty.io/how-to-find-an-inverse-of-the-following-function.html$$f(x)=x^3+1$$ To find inverse, from what I've learned we change the y to x $$x=y^3+1$$ solve for y $$x-1=y^3$$ Should I cube root the x-1 for this? if i did that I still would not get the answer ...Thu, 20 Aug 2026 16:44:42 -0400https://liberty.io/is-there-a-simple-method-to-do-lu-decomposition-by-hand.htmlhttps://liberty.io/is-there-a-simple-method-to-do-lu-decomposition-by-hand.htmlToday my professor in numerical analysis pointed out that in the exam we will probably have to do LU decomposition by hand. I understand how the decomposition works theoretically, but when it comes ...Thu, 20 Aug 2026 16:43:24 -0400https://liberty.io/is-a-matrix-multiplied-with-its-transpose-something-special.htmlhttps://liberty.io/is-a-matrix-multiplied-with-its-transpose-something-special.htmlIn my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$?...Thu, 20 Aug 2026 16:30:51 -0400https://liberty.io/why-arccos-2-have-value-in-complex-numbers-even-though-its-d-f-1-1.htmlhttps://liberty.io/why-arccos-2-have-value-in-complex-numbers-even-though-its-d-f-1-1.htmlI've seen that arccos has $D_f [-1,1]$ however when putting arccos(2) into wolframalpha, I get the complex number result if I understand it correctly. Why is it so? If true, should not $D_f$ be ...Thu, 20 Aug 2026 16:28:51 -0400https://liberty.io/how-to-find-out-x-in-a-trinomial-closed.htmlhttps://liberty.io/how-to-find-out-x-in-a-trinomial-closed.htmlHow can I find out what X equals in this? $$x^2 - 2x - 3 = 117$$ How would I get started? I'm truly stuck....Thu, 20 Aug 2026 16:28:47 -0400https://liberty.io/subset-vs-subseteq-when-not-referring-to-strict-inclusion.htmlhttps://liberty.io/subset-vs-subseteq-when-not-referring-to-strict-inclusion.htmlInspired by the confusion in the comments on this question: I always thought that the standard was to read $\subset$ as "is a strict subset of", and $\subseteq$ could mean proper or improper inclu......Thu, 20 Aug 2026 16:26:50 -0400https://liberty.io/exponential-distribution-moment-generating-function-to-find-the-mean.htmlhttps://liberty.io/exponential-distribution-moment-generating-function-to-find-the-mean.htmlWith mean = 2 with exponential distribution Calculate $ E(200 + 5Y^2 + 4Y^3) = 432 $ $E(200) = 200 $ $E(5Y^2) = 5E(Y^2) = 5(8) = 40 $ $E(4Y^3) = 4E(Y^3) = 4(48) = 192 $ $E(Y^2) = V(Y) + [E(Y......Thu, 20 Aug 2026 16:21:05 -0400https://liberty.io/what-is-the-formal-definition-of-ordered-tree.htmlhttps://liberty.io/what-is-the-formal-definition-of-ordered-tree.htmlHi I understand that for rooted tree, the definition is as follow: Is a directed graph $D=<V,E>$ $\exists v\in V\rightarrow \deg^-(v)=0$ which indicate the in-degree of $v$ is $0$ $\forall v......Thu, 20 Aug 2026 16:16:46 -0400https://liberty.io/find-an-area-of-a-rectangle-by-its-diagonal-and-quotient-of-its-sides.htmlhttps://liberty.io/find-an-area-of-a-rectangle-by-its-diagonal-and-quotient-of-its-sides.htmlOkay, I've found no formulas for this one online but I'm pretty sure that the area of a rectangle is calculable from its diagonal and ratio (quotient of its sides). Think of it: with a constant ...Thu, 20 Aug 2026 16:09:29 -0400https://liberty.io/why-is-dot-product-multiplication-commutative.htmlhttps://liberty.io/why-is-dot-product-multiplication-commutative.htmlSay I have a unit vector $\hat{u}$ and another vector $\vec{t}$ of magnitude 4 units, then I connect the vectors tail to tail with an acute angle between them. I don't understand how the dot produc......Thu, 20 Aug 2026 15:59:00 -0400https://liberty.io/what-is-an-indeterminate-in-a-polynomial-ring.htmlhttps://liberty.io/what-is-an-indeterminate-in-a-polynomial-ring.htmlI am currently studying polynomial ring. I have a basic doubt. What is $x$ in a polynomial? A polynomial is an expression $\sum_{k = 0}^n a_k x^k$, where $n,k \in \mathbb{N}$ and $a_k \in R$ where......Thu, 20 Aug 2026 15:55:13 -0400https://liberty.io/is-the-center-of-g-the-same-as-the-centralizer-of-g-in-g.htmlhttps://liberty.io/is-the-center-of-g-the-same-as-the-centralizer-of-g-in-g.htmlIs the center, $Z(G)$, of a group $G$ the same as the centralizer, $C(g)$, of an element $g\in G$? I have proven that $C(g)\leq G\forall g\in G$ but my homework, in a later problem, asks me to prov......Thu, 20 Aug 2026 15:52:44 -0400https://liberty.io/triple-integrals-and-cylindrical-coordinates.htmlhttps://liberty.io/triple-integrals-and-cylindrical-coordinates.htmlmy question is can cylindrical coordinates be somehow changed in order? I mean the standard cylindrical coordinates suppose a line of length $r$ that makes an angle $\theta$ with the $x$ axis and l......Thu, 20 Aug 2026 15:47:38 -0400https://liberty.io/find-the-spectral-decomposition-of-a.htmlhttps://liberty.io/find-the-spectral-decomposition-of-a.html$$ A= \begin{pmatrix} -3 & 4\\ 4 & 3 \end{pmatrix} $$ So i am assuming that i must find the evalues and evectors of this matrix first, and that is exactly what i did. The evalues are $5......Thu, 20 Aug 2026 15:43:20 -0400https://liberty.io/converting-piecewise-cdf-to-pdf.htmlhttps://liberty.io/converting-piecewise-cdf-to-pdf.htmlFor instance, I have the following CDF that I am trying to find the mean for, by firstly finding the PDF and then integrating over the interval. $$F(x) = \begin{cases} 0 & x\leq -2 ......Thu, 20 Aug 2026 15:23:57 -0400https://liberty.io/calculate-the-value-of-a-derivative-at-the-origin.htmlhttps://liberty.io/calculate-the-value-of-a-derivative-at-the-origin.htmlI have the following question in a course: An example of the logistic function is defined by $$\varphi(v)=\frac{1}{1+e^{-av}}$$ whose limiting values are $0$ and $1$. Show that the derivative of $\...Thu, 20 Aug 2026 15:21:04 -0400https://liberty.io/memorizing-the-unit-circle.htmlhttps://liberty.io/memorizing-the-unit-circle.htmlI know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. What is the best way to memorize the ...Thu, 20 Aug 2026 15:11:27 -0400https://liberty.io/combinations-of-up-to-n-out-of-m-elements.htmlhttps://liberty.io/combinations-of-up-to-n-out-of-m-elements.htmlGiven a set of $m$ unique elements to choose from, and using at most $n$ elements in a combination, how many combinations can I have? A combination can repeat an element more than once, as long as ......Thu, 20 Aug 2026 14:58:12 -0400https://liberty.io/how-to-diagonalize-this-matrix.htmlhttps://liberty.io/how-to-diagonalize-this-matrix.htmlCan someone show me step-by-step how to diagonalize this matrix? I'm trying to teach myself differential equations + linear algebra, but I'm stumped on how to do this. I'd really appreciate if some......Thu, 20 Aug 2026 14:51:30 -0400https://liberty.io/find-base-of-isosceles-triangle-with-side-length-and-angle.htmlhttps://liberty.io/find-base-of-isosceles-triangle-with-side-length-and-angle.htmlI would like to calculate the length of the side in red on the image. I tried the Law of cosines, but maybe i haven't applied the formula right, because for a side "a" and "b" of size 64 and a angl......Thu, 20 Aug 2026 14:45:09 -0400https://liberty.io/knights-knaves-and-normals-puzzle.htmlhttps://liberty.io/knights-knaves-and-normals-puzzle.htmlProblem Statement You are on an island inhabited by three types of people: knights (always make true statements), knaves (always make false statements) and normals (sometimes make true statements and ...Thu, 20 Aug 2026 14:40:23 -0400https://liberty.io/what-is-the-difference-between-a-scalar-and-a-vector-field.htmlhttps://liberty.io/what-is-the-difference-between-a-scalar-and-a-vector-field.htmlCould someone please indicate precisely the difference between a scalar and a vector field? I find no matter how many times I try to understand, but I always am confused in the end. So what exactly......Thu, 20 Aug 2026 14:17:03 -0400https://liberty.io/in-the-proof-of-irrationality-of-sqrt-2-or-sqrt-7-why-do-the-numerator.htmlhttps://liberty.io/in-the-proof-of-irrationality-of-sqrt-2-or-sqrt-7-why-do-the-numerator.htmlI have been confused by this problem for a very long period of time, and I think I am personally opposed to this concept and refused to agree with it in my introduction to mathematical proof cours......Thu, 20 Aug 2026 14:16:43 -0400https://liberty.io/proving-that-even-numbers-equal-the-sum-of-two-odd-numbers.htmlhttps://liberty.io/proving-that-even-numbers-equal-the-sum-of-two-odd-numbers.htmlDefine E to be the set of even integers; E = {$x$ $\in$ $\mathbb{Z}$ : $x$ = 2$k$, where $k$ $\in$ $\mathbb{Z}$}. Define F to be the set of integers that can be expressed as the sum of two odd n......Thu, 20 Aug 2026 14:09:52 -0400https://liberty.io/what-is-the-difference-between-the-first-derivative-and-the-second-der.htmlhttps://liberty.io/what-is-the-difference-between-the-first-derivative-and-the-second-der.htmlSo a bit confused when I see a question say: "Use the second derivative test to find all relative extrema" When using the second derivative test are we not looking for concavity and points of ...Thu, 20 Aug 2026 14:09:36 -0400https://liberty.io/is-there-negative-base-representation-of-numbers.htmlhttps://liberty.io/is-there-negative-base-representation-of-numbers.htmlEvery real number can be expressed as its base 10 representation, as $$\sum_{i=-\infty}^\infty a_i 10^i\,$$ where $a_i\in\{0,1,\dots,9\}$. We also have base 2 or base 8. Question: Is there a base -2 ...Thu, 20 Aug 2026 14:08:37 -0400https://liberty.io/prime-of-form-n-2-4.htmlhttps://liberty.io/prime-of-form-n-2-4.htmlShow that The only prime of the form $n^2-4$, $n$ being an integer is $3$ We have $n^2-4=(n+2)(n-2)$ Now for $n^2-4$ being prime value of $(n-2)$ must be $1$. Then $n=3$ and putting the value we g......Thu, 20 Aug 2026 14:07:01 -0400https://liberty.io/systems-of-equations-inconsistent.htmlhttps://liberty.io/systems-of-equations-inconsistent.htmlQuestion: Consider the following system of three equations: $$2y+2z=9-2x$$ $$x=12-3y-4z$$ $$Ax+5y+6z=B$$ Find values of A and B which makes the system of equations incons......Thu, 20 Aug 2026 14:05:24 -0400https://liberty.io/is-tautological-bundle-mathcal-o-1-or-mathcal-o-1.htmlhttps://liberty.io/is-tautological-bundle-mathcal-o-1-or-mathcal-o-1.htmlI always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a ......Thu, 20 Aug 2026 14:04:45 -0400https://liberty.io/how-to-determine-particular-months-with-five-weekends.htmlhttps://liberty.io/how-to-determine-particular-months-with-five-weekends.htmlJanuary of 2016 has five Fridays, five Saturdays and five Sundays. What is the next January that will again have five weekends like this?...Thu, 20 Aug 2026 14:03:04 -0400https://liberty.io/find-the-domain-of-a-trig-function.htmlhttps://liberty.io/find-the-domain-of-a-trig-function.htmlFind the Domain of the function $y=\sqrt{1+2 \sin x}$. My solution: $1+2 \sin x \geq 0$ (since square-root of a negative number is not defined) $2 \sin x \geq -1$ $\sin x \geq -1/2$ $x \ge -\frac{......Thu, 20 Aug 2026 13:24:49 -0400https://liberty.io/optimal-strategy-for-dice-10000-dice-game.htmlhttps://liberty.io/optimal-strategy-for-dice-10000-dice-game.htmlThere exists a dice game known as Dice-10000, with alternative names such as Dix Mille or 6-Dice The rules are simple, with each player taking turns rolling dice in order to achieve certain combina......Thu, 20 Aug 2026 13:15:56 -0400https://liberty.io/local-truncation-error-of-euler-method.htmlhttps://liberty.io/local-truncation-error-of-euler-method.htmlWikipedia and this book say the local truncation error of Euler method is $O(h^2)$. But this book and A friendly Introduction to Numerical Analysis say it's $O(h)$. Which is correct? I have a similar ...Thu, 20 Aug 2026 13:13:41 -0400https://liberty.io/can-every-real-number-be-represented-by-a-possibly-infinite-decimal.htmlhttps://liberty.io/can-every-real-number-be-represented-by-a-possibly-infinite-decimal.htmlDoes every real number have a representation within our decimal system? The reason I ask is because, from beginning a mathematics undergraduate degree a lot of 'mathematical facts' I had previously ...Thu, 20 Aug 2026 13:07:14 -0400https://liberty.io/how-does-taylor-series-work-for-sine-and-cosine.htmlhttps://liberty.io/how-does-taylor-series-work-for-sine-and-cosine.htmlI was just wondering if anyone can help me understand taylor series for sine and cosine. I have no background in calculus but I always found it interesting how the ratios of the arc to the radius ......Thu, 20 Aug 2026 13:06:02 -0400https://liberty.io/how-to-calculate-matrix-tr-ab.htmlhttps://liberty.io/how-to-calculate-matrix-tr-ab.htmlI have these Matrix A and B below in the image: And i wonder how do i calculate the tr(AB)? I have done this by taking the diagonal in each Matrix A and B: 3 * 6 + 1 * 5 + 5 * 9 = 67 But after tha......Thu, 20 Aug 2026 13:05:45 -0400https://liberty.io/equation-of-circles-tangent-to-x-axis-with-a-given-radius-and-point.htmlhttps://liberty.io/equation-of-circles-tangent-to-x-axis-with-a-given-radius-and-point.htmlFind the equations of the circles which are tangent to the x-axis, with radius of 5 units and passing through the point $(0,8)$. I know how to formulate an equation with the given radius of 5 and t......Thu, 20 Aug 2026 12:59:52 -0400https://liberty.io/solving-log-6-2x-3-log-6-x-5-log-3x.htmlhttps://liberty.io/solving-log-6-2x-3-log-6-x-5-log-3x.htmlSolve for $x$ I have an equation that I have been working on solving; I know the solution, but I cannot get to it myself. Almost every simplification I do reverts back to a previous step. Can anyone ...Thu, 20 Aug 2026 12:54:11 -0400https://liberty.io/what-039-s-an-intuitive-explanation-of-the-max-flow-min-cut-theorem.htmlhttps://liberty.io/what-039-s-an-intuitive-explanation-of-the-max-flow-min-cut-theorem.htmlI'm about to read the proof of the max-flow min-cut theorem that helps solve the maximum network flow problem. Could someone please suggest an intuitive way to understand the theorem?...Thu, 20 Aug 2026 12:45:11 -0400https://liberty.io/in-mathematics-is-null-zero.htmlhttps://liberty.io/in-mathematics-is-null-zero.htmlKindly consider it soft question as I am a software engineer.and I know only software but I have a doubt in my mind that there would be something like null in mathmatics as well. If Mathematics N......Thu, 20 Aug 2026 12:45:04 -0400https://liberty.io/probability-of-a-given-a-union-b.htmlhttps://liberty.io/probability-of-a-given-a-union-b.htmlWhat is the probability of A given A union B? We know that p(A) = 0.5 p(B) = 0.3 p(AB) = 0.1 From my understanding of conditional probability i think it should be p(A)/p(A union B) . Is this corre......Thu, 20 Aug 2026 12:35:26 -0400https://liberty.io/finding-orthogonal-angles-polar-coordinate-of-an-n-dimensional-vector.htmlhttps://liberty.io/finding-orthogonal-angles-polar-coordinate-of-an-n-dimensional-vector.htmlOrthogonal angles are the angles used when converting a vector to polar coordinates. So for vector $(1, 1)$, the orthogonal angle is $45$ degrees. Given a vector $(x_1, x_2, x_3, ..., x_n)$, what ......Thu, 20 Aug 2026 12:35:18 -0400https://liberty.io/finding-the-limit-of-1-cos-x-x-as-x-to-0-with-squeeze-theorem.htmlhttps://liberty.io/finding-the-limit-of-1-cos-x-x-as-x-to-0-with-squeeze-theorem.htmlHow do I find: $$ \lim_{x\to0}\frac{1-\cos(x)}{x} $$ Using the squeeze theorem. Particularly, how would I arrive at its bounding functions? If possible, please try not to use derivatives....Thu, 20 Aug 2026 12:33:37 -0400https://liberty.io/terence-tao-type-books-in-other-fields.htmlhttps://liberty.io/terence-tao-type-books-in-other-fields.htmlI have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that ...Thu, 20 Aug 2026 12:30:20 -0400https://liberty.io/how-do-you-write-the-summation-of-a-summation.htmlhttps://liberty.io/how-do-you-write-the-summation-of-a-summation.htmlLike how should you do it and how do you manipulate it algebraically? $$S = \{1/2, 1/3, \ldots, 1/n\}$$ $$Q = \{S(2), S(3), \ldots, S(n)\}$$...Thu, 20 Aug 2026 12:30:15 -0400https://liberty.io/why-teach-linear-algebra-before-abstract-algebra.htmlhttps://liberty.io/why-teach-linear-algebra-before-abstract-algebra.htmlIs there a reason why most undergraduate curriculums put linear algebra before abstract algebra? I'm asking this because personally it seems to be much easier to understand the architecture behind ...Thu, 20 Aug 2026 12:26:32 -0400https://liberty.io/gre-pobability-question.htmlhttps://liberty.io/gre-pobability-question.htmlOne person is to be selected at random from a group of 25 people. The probability that the selected person will be a male is 0.44, and the probability that the selected person will be a male who wa......Thu, 20 Aug 2026 12:24:47 -0400https://liberty.io/what-exactly-is-an-implicitly-defined-function.htmlhttps://liberty.io/what-exactly-is-an-implicitly-defined-function.htmlSo I've just started to get into some calculus and I recently came across the topic of implicit differentiation. I am extremely confused on what implicit functions are and there is very little ...Thu, 20 Aug 2026 12:10:21 -0400https://liberty.io/difference-between-modulus-and-remainder.htmlhttps://liberty.io/difference-between-modulus-and-remainder.html(I've come here from this question.) What's the difference between modulus and remainder? (Bearing in mind until 5 minutes ago I thought they were the same :P )...Thu, 20 Aug 2026 12:00:14 -0400https://liberty.io/sum-of-products-expansions-of-these-boolean-functions.htmlhttps://liberty.io/sum-of-products-expansions-of-these-boolean-functions.htmlFind the sum-of-products expansions of these Boolean functions. $a)$ $F(x, y) = \text{~}x + y$ $b)$ $F(x, y) = x \text{~}y$ $c)$ $F(x, y) = 1$ $d)$ $F(x, y) = \text{~}y$...Thu, 20 Aug 2026 11:56:22 -0400https://liberty.io/why-n-and-what-does-it-mean.htmlhttps://liberty.io/why-n-and-what-does-it-mean.htmlMaybe a bit too early to ask, but I give it a shot. I'm trying to study mathematics again during my free time. Now I tried to figure out what exactly is meant with the multiple. Well this formula m......Thu, 20 Aug 2026 11:45:31 -0400https://liberty.io/sigmoid-curve-function-calculation.htmlhttps://liberty.io/sigmoid-curve-function-calculation.htmlSo I am learning about Sigmoid Curves and in this article it says the formula is: $$\tag{1} \frac{1}{1+e^{-\beta_0-\beta_1x}} $$ Then it gives an example problem: Let’s say you take $ \beta_0 =......Thu, 20 Aug 2026 11:22:35 -0400https://liberty.io/mgf-of-a-sum-product-of-mgfs.htmlhttps://liberty.io/mgf-of-a-sum-product-of-mgfs.htmlSo my book says the following: The moment generating function for the geometric distribution is: $g(t) = \cfrac{pe^t}{1-e^t(1-p)}$ where $|e^t(1-p)| <1$ of course for the geometric series to con......Thu, 20 Aug 2026 10:55:29 -0400https://liberty.io/general-form-of-an-integer.htmlhttps://liberty.io/general-form-of-an-integer.htmlFrom the Division Algorithm, We know that if an integer is divided by $3$ , it will leave remainder $0,1$ or $2$. Now, how is one supposed to write a proper proof for the given question? P.S. This......Thu, 20 Aug 2026 10:51:32 -0400https://liberty.io/a-scalene-triangle-with-no-right-nor-obtuse-angle.htmlhttps://liberty.io/a-scalene-triangle-with-no-right-nor-obtuse-angle.htmlI want to find the area of the perfect triangle, i.e. a triangle with no particularity whatsoever : no side shall be equal to another, no right angle, no obtuse angle. So I gave myself a segment ......Thu, 20 Aug 2026 10:49:16 -0400https://liberty.io/how-to-create-a-reflexive-symmetric-and-transitive-closures.htmlhttps://liberty.io/how-to-create-a-reflexive-symmetric-and-transitive-closures.htmlI'm working on a task where I need to find out the reflexive, symmetric and transitive closures of R. Statement is given below: Assume that U = {1, 2, 3, a, b} and let the relation R on U which is ...Thu, 20 Aug 2026 10:48:59 -0400https://liberty.io/finding-the-p-value-in-a-proportion-statistics-with-binomial-distribut.htmlhttps://liberty.io/finding-the-p-value-in-a-proportion-statistics-with-binomial-distribut.htmlGiven that the sample is 40, we observed that 80% of the sample (32 participants) have a successful trial. We want to test the hypothesis $H_{0} = p_{0} = 0.7$ versus $H_{a} \neq 0.7$ at 5% signifi......Thu, 20 Aug 2026 10:44:08 -0400https://liberty.io/finding-a-function-039-s-domain-from-the-function-039-s-formula.htmlhttps://liberty.io/finding-a-function-039-s-domain-from-the-function-039-s-formula.htmlThere is a question in my calculus 101 textbook that says find the domain of the function. The function formulas are given, eg: $$f(x) = \frac{x+4}{x^2 -9}$$ and $$g(t) = \sqrt[3]{2t - 1}$$ No ot......Thu, 20 Aug 2026 10:38:27 -0400https://liberty.io/what-039-s-an-intuitive-way-of-looking-at-quotient-spaces.htmlhttps://liberty.io/what-039-s-an-intuitive-way-of-looking-at-quotient-spaces.htmlI understand the concept of $\mathbb{Z}/n\mathbb{Z}$, but I am having a really hard time understanding how this concept of quotients applies to vector spaces. Suppose $V = \mathbb{F}[x]$ is a vector ...Thu, 20 Aug 2026 10:32:30 -0400https://liberty.io/geometric-intuition-of-conjugate-function.htmlhttps://liberty.io/geometric-intuition-of-conjugate-function.htmlI am looking for a geometric and intuitive explanation of the conjugate function and how it maps to the below analytical formula. $$ f^*(y)= \sup_{x \in \operatorname{dom} f } (y^Tx-f(x))$$...Thu, 20 Aug 2026 10:26:38 -0400https://liberty.io/how-many-tickets-sold-at-a-concert.htmlhttps://liberty.io/how-many-tickets-sold-at-a-concert.htmlThe music department of a school sold 750 tickets to the school concert, for a total of \$4755. Students paid \$5 for a ticket and non-students paid \$8 for a ticket. How many non-students attended......Thu, 20 Aug 2026 10:26:38 -0400https://liberty.io/x-vs-t-and-y-vs-t-graphs.htmlhttps://liberty.io/x-vs-t-and-y-vs-t-graphs.htmlHow would I go about sketching the x vs t and y vs t graphs for the following picture: Our class didn't get the opportunity to solve these types of questions, so I'm on my own for this one. My thi......Thu, 20 Aug 2026 10:23:51 -0400https://liberty.io/how-can-e-x-seemingly-approach-0-as-x-approaches-negative-infinity.htmlhttps://liberty.io/how-can-e-x-seemingly-approach-0-as-x-approaches-negative-infinity.htmlThe specific problem with regards to this graph is: $$\lim_{x \to -\infty} g(2 + e^x)$$ Now it's true that we technically can't apply "the limit of a sum is the sum of the limits" since the functi......Thu, 20 Aug 2026 10:17:10 -0400https://liberty.io/can-anyone-please-explain-the-the-difference-between-a-vector-and-a-ma.htmlhttps://liberty.io/can-anyone-please-explain-the-the-difference-between-a-vector-and-a-ma.htmlI just took Calculus 3 last semester at my University and got comfortable with the idea of vectors, vector-valued functions, and basic vector operations like the dot and cross products. This semes......Thu, 20 Aug 2026 10:16:01 -0400https://liberty.io/are-natural-numbers-a-field-with-alternate-addition-and-multiplication.htmlhttps://liberty.io/are-natural-numbers-a-field-with-alternate-addition-and-multiplication.htmlI've just started P.Halmos' book "Finite Vector Spaces", and at the first chapter, after defining the axioms of a field, there is an exercide that asks: if we consider the non negative integers, ca......Thu, 20 Aug 2026 10:15:47 -0400https://liberty.io/can-we-take-images-of-equivalence-relations.htmlhttps://liberty.io/can-we-take-images-of-equivalence-relations.htmlGiven a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. Th......Thu, 20 Aug 2026 10:05:32 -0400https://liberty.io/how-can-i-find-the-maximum-revenue.htmlhttps://liberty.io/how-can-i-find-the-maximum-revenue.htmlAn oil producing country can sell 7 million barrels of oil a day at a price of $90$ per barrel. If each $1$ price increase will result in a sales decrease of $100,000$ barrels per day, what price w......Thu, 20 Aug 2026 09:51:47 -0400https://liberty.io/how-to-check-whether-a-relation-is-transitive-from-the-matrix-represen.htmlhttps://liberty.io/how-to-check-whether-a-relation-is-transitive-from-the-matrix-represen.html$$\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}$$ This is a matrix representation of a relation on the set $\{1, 2, 3\}$. I have to determine if this relation matrix is ...Thu, 20 Aug 2026 09:50:06 -0400https://liberty.io/oriented-graph-vs-directed-graph.htmlhttps://liberty.io/oriented-graph-vs-directed-graph.htmlAlright, while the definitions are stated in my lecure notes, textbooks and wiki, I'll be honest, it just explodes my mind with what seems like word sorcery. Definition A directed graph is called an ...Thu, 20 Aug 2026 09:37:45 -0400https://liberty.io/confusion-on-the-proofs-of-convolution-theorem-of-fourier-transform.htmlhttps://liberty.io/confusion-on-the-proofs-of-convolution-theorem-of-fourier-transform.htmlThe convolution theorem of Fourier transform is stated as follows: Define $h(x):=f(x)*g(x)$, then we have $$\hat{h}(k)=\hat{f}(k)\hat{g}(k).$$ I have a confusion of the proofs of this theorem. Most ...Thu, 20 Aug 2026 09:35:36 -0400https://liberty.io/sum-of-a-matrix-with-its-transpose-duplicate.htmlhttps://liberty.io/sum-of-a-matrix-with-its-transpose-duplicate.htmlI've a question that many of yours could consider stupid: if i sum a matrix with its transpose, I obtain a particular result? E.g. $A + A^T = B$, $B$ has some particular properties?...Thu, 20 Aug 2026 09:32:54 -0400https://liberty.io/what-does-f-max-f-0-and-f-max-f-0-mean.htmlhttps://liberty.io/what-does-f-max-f-0-and-f-max-f-0-mean.htmlLet $f$ be a continious function over an intervall with $f^+:=\max(f,0)$ and $f^-:=\max(-f,0)$ What does that mean graphically?...Thu, 20 Aug 2026 09:29:12 -0400https://liberty.io/limit-of-a-multivariable-function-with-log.htmlhttps://liberty.io/limit-of-a-multivariable-function-with-log.htmlWhat is the limit of $$f(x,y) = (x^2+y^2)\log|x+y|$$ as $(x,y) \rightarrow (0,0)$ I tried using Squeeze Theorem but couldn't think of an upper bound for it. I know the limit of $t log(t)$ is 0......Thu, 20 Aug 2026 09:25:14 -0400https://liberty.io/fundamental-group-of-mathbb-r-3-minus-trefoil-knot.htmlhttps://liberty.io/fundamental-group-of-mathbb-r-3-minus-trefoil-knot.htmlLet $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van K......Thu, 20 Aug 2026 09:14:55 -0400https://liberty.io/proof-of-the-central-limit-theorem-using-moment-generating-functions.htmlhttps://liberty.io/proof-of-the-central-limit-theorem-using-moment-generating-functions.htmlBelow is a method of proving the Central Limit Theorem using moment generating functions. Let $$X_{1},X_{2},...,X_{n}$$ be a sequence of i.i.d. random variables with expected value and variance ......Thu, 20 Aug 2026 09:09:20 -0400