https://liberty.ioCurated star highlights with fresh energy.en-usFri, 14 Aug 2026 22:46:58 -0400https://liberty.io/what-does-the-exclamation-mark-do.htmlhttps://liberty.io/what-does-the-exclamation-mark-do.htmlI've seen this but never knew what it does. Can any one let me in on the details? Thanks....Sat, 15 Aug 2026 02:39:18 -0400https://liberty.io/cyclic-subgroup-of-s-n.htmlhttps://liberty.io/cyclic-subgroup-of-s-n.htmlFind the smallest natural number n ∈ ℕ such that $S_n$ contains a cyclic subgroup of order 101. Proof: We seek the smallest n such that Sn contains a permutation of order 101. Permutation can be w......Sat, 15 Aug 2026 02:38:41 -0400https://liberty.io/questions-tagged-number-theory.htmlhttps://liberty.io/questions-tagged-number-theory.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 15 Aug 2026 02:38:20 -0400https://liberty.io/are-one-by-one-matrices-equivalent-to-scalars.htmlhttps://liberty.io/are-one-by-one-matrices-equivalent-to-scalars.htmlI am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are of......Sat, 15 Aug 2026 02:29:50 -0400https://liberty.io/how-do-i-represent-an-equilateral-triangle-in-cartesian-coordinates-ce.htmlhttps://liberty.io/how-do-i-represent-an-equilateral-triangle-in-cartesian-coordinates-ce.htmlI am trying to figure out how to find the Cartesian coordinates of an equilateral triangle centered around (0,0)....Sat, 15 Aug 2026 02:14:23 -0400https://liberty.io/cartesian-products-and-sets.htmlhttps://liberty.io/cartesian-products-and-sets.htmlI'm really confused on how to approach this question: Recall that the Cartesian product $A \times A$ is defined as the set $\{(x, y) : x \in A \land y \in A \}$. Thus if for example $A = \{1, 2,......Sat, 15 Aug 2026 02:12:59 -0400https://liberty.io/how-to-find-complex-eigenvectors-from-complex-eigenvalues.htmlhttps://liberty.io/how-to-find-complex-eigenvectors-from-complex-eigenvalues.htmlI have this matrix that represents a $2 \times 2$ linear system and I am supposed to solve to find what $x(t)$ and $y(t)$ are. $\left[ {\begin{array}{cc} 1 & 5 \\ -1 & -3 \\ \end{......Sat, 15 Aug 2026 02:08:02 -0400https://liberty.io/how-to-express-an-angle-of-90-degrees-between-two-lines.htmlhttps://liberty.io/how-to-express-an-angle-of-90-degrees-between-two-lines.htmlIf I would extend two lines $l_1$ and $l_2$ they would intersect with an angle of 90 degrees. How should I write with math terms that there would be a 90 degree angle. I assume $l_1 \perp l_2$ is w......Sat, 15 Aug 2026 02:06:06 -0400https://liberty.io/deriving-mean-and-variance-of-laplace-distribution.htmlhttps://liberty.io/deriving-mean-and-variance-of-laplace-distribution.htmlIt has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 \sigma}\,e^{-\Large\......Sat, 15 Aug 2026 02:06:03 -0400https://liberty.io/trivial-and-non-trivial-closed.htmlhttps://liberty.io/trivial-and-non-trivial-closed.htmlHow do we know what to do when asked to find the solutions of given equations saying find the solution when it has non trivial solution??...Sat, 15 Aug 2026 01:58:58 -0400https://liberty.io/function-for-second-quadrant-in-unit-circle.htmlhttps://liberty.io/function-for-second-quadrant-in-unit-circle.htmlI am no "math-guy" and would really appreciate some help in writing a function for the second quadrant of the unit circle. Conditions I want to be met: Centre of the circle is moved so s......Sat, 15 Aug 2026 01:57:44 -0400https://liberty.io/why-does-the-imaginary-number-i-satisfy-i-times-0-0.htmlhttps://liberty.io/why-does-the-imaginary-number-i-satisfy-i-times-0-0.htmlWhy does the imaginary number $i$ satisfy $i\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that....Sat, 15 Aug 2026 01:54:05 -0400https://liberty.io/what-is-the-name-for-the-shape-formed-by-removing-a-square-from-the-co.htmlhttps://liberty.io/what-is-the-name-for-the-shape-formed-by-removing-a-square-from-the-co.htmlSquares of consecutive numbers differ by the sum of those numbers, so $6^2 = 5^2 + 5 + 6$. Geometrically, this is because the difference between the two squares is a pair of strips, $5$ and $6$ un......Sat, 15 Aug 2026 01:51:02 -0400https://liberty.io/equation-of-a-circle-tangent-to-the-y-axis.htmlhttps://liberty.io/equation-of-a-circle-tangent-to-the-y-axis.htmlThe problem is: Find the general equation of a circumference with center at $(-4,3)$ and tanget to the $y$ axis I know that calculating the distance between the center and any point on the ...Sat, 15 Aug 2026 01:34:12 -0400https://liberty.io/prove-that-every-convex-function-is-continuous.htmlhttps://liberty.io/prove-that-every-convex-function-is-continuous.htmlA function $f : (a,b) \to \Bbb R$ is said to be convex if $$f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$$ whenever $a < x, y < b$ and $0 < \lambda <1$. Prove that every co......Sat, 15 Aug 2026 01:31:13 -0400https://liberty.io/finding-integrating-factor-for-inexact-differential-equation.htmlhttps://liberty.io/finding-integrating-factor-for-inexact-differential-equation.htmlSuppose we have the following differential equation that is NOT exact, i.e. $M_y \ne N_x$: $2xy^3+y^4+(xy^3-2y)y'=0$ How would I find an integrating factor $μ(x,y)$ so that when I multiply this ...Sat, 15 Aug 2026 01:31:11 -0400https://liberty.io/finding-the-limit-of-a-piecewise-function.htmlhttps://liberty.io/finding-the-limit-of-a-piecewise-function.htmlI had a question about finding limits of piecewise functions through graphs. I believe, I am missing something in my fundamentals about finding limits for these functions. Firstly, I would like to ...Sat, 15 Aug 2026 01:27:49 -0400https://liberty.io/invertability-of-singular-2x2-matrix-with-all-same-real-values.htmlhttps://liberty.io/invertability-of-singular-2x2-matrix-with-all-same-real-values.htmlQuestion: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...Sat, 15 Aug 2026 01:20:20 -0400https://liberty.io/what-039-s-the-degree-of-a-multivariate-polynomial-in-artin-algebra.htmlhttps://liberty.io/what-039-s-the-degree-of-a-multivariate-polynomial-in-artin-algebra.htmlAccording to Degree of a polynomial, it is highest degree among the monomials. Where is this in Artin Algebra? In Chapter 11.9, an exercise gives a degree to an irreducible complex polynomial of two ...Sat, 15 Aug 2026 01:16:33 -0400https://liberty.io/how-do-i-evaluate-this-integral-given-a-table-of-values-for-f-x-and-f-.htmlhttps://liberty.io/how-do-i-evaluate-this-integral-given-a-table-of-values-for-f-x-and-f-.htmlThe problem gives me a list of values for $f(x)$ and $f'(x)$. I tried solving the problem using integration by parts, allowing $u=\sin(2x), u'=2\cos(2x), v'=f'(\cos(2x))$, and $v=f(\cos(2x))$. I use ...Sat, 15 Aug 2026 01:13:22 -0400https://liberty.io/can-different-quaternions-represent-the-same-orientation.htmlhttps://liberty.io/can-different-quaternions-represent-the-same-orientation.htmlFor a current project I'm working on I have to use quaternions to represent the orientation of an object. The piece of code I'm working on now integrates rotation rates to the quaternion represent......Sat, 15 Aug 2026 01:09:26 -0400https://liberty.io/why-bbb-z-integers-is-an-integral-domain.htmlhttps://liberty.io/why-bbb-z-integers-is-an-integral-domain.htmlIs there a proof for $\Bbb Z$ being an integral domain? Or is it an axiom that if $a*b=0$ then $a=0$ or $b=0$ where $a$, $b$ belong to $\Bbb Z$. Is it so obvious?...Sat, 15 Aug 2026 01:01:08 -0400https://liberty.io/proving-holder-039-s-inequality-for-sums.htmlhttps://liberty.io/proving-holder-039-s-inequality-for-sums.htmlI want to prove the Holder's inequality for sums: Let $p\ge1$ be a real number. Let $(x_{k})\in l_{p}$ and $(y_{k})\in l_{q}$ . Then, $$\overset{\infty}{\underset{k=1}{\sum}}\vert x......Sat, 15 Aug 2026 00:58:52 -0400https://liberty.io/how-to-graph-gradient-vector.htmlhttps://liberty.io/how-to-graph-gradient-vector.htmlI'm working on a practice problem for my Calculus 3 course. It gives the function: $z=x^2+y^2$, and asks to graph the contours for $c=1,2,3$. Than asks to calculate the gradient at point $(2,1)$ ......Sat, 15 Aug 2026 00:57:18 -0400https://liberty.io/decomposition-group-and-inertia-group.htmlhttps://liberty.io/decomposition-group-and-inertia-group.htmlLet $L/K$ be a Galois extension with Galois group $G$. Let $O_K$ and $O_L$ be the ring of algebraic integers of $K$ and $L$ respectively. Let $P\subseteq O_K$ be a prime. Let $Q\subseteq O_L$ be a ......Sat, 15 Aug 2026 00:53:33 -0400https://liberty.io/euler-path-for-directed-graph.htmlhttps://liberty.io/euler-path-for-directed-graph.htmlHow do we find Euler path for directed graphs? I don't seem to get the algorithm below! Algorithm To find the Euclidean cycle in a digraph (enumerate the edges in the cycle), using a greedy proc......Sat, 15 Aug 2026 00:48:58 -0400https://liberty.io/depressed-cubic-roots.htmlhttps://liberty.io/depressed-cubic-roots.htmlAccording to Wikipedia, a depressed cubic has only one root and 2 imaginary roots. Is this true? Can a depressed cubic of the form $x^3+px+q=0$ have 2 or 3 real roots? Edit: Here is a screenshot o......Sat, 15 Aug 2026 00:48:11 -0400https://liberty.io/how-to-self-study-topology.htmlhttps://liberty.io/how-to-self-study-topology.htmlI'm a first year undergraduate student and I'm a math major. Currently, I'm taking an intro to analysis class and a linear algebra class. However, I often feel constrained by what I do in class and......Sat, 15 Aug 2026 00:43:29 -0400https://liberty.io/proof-of-the-dirichlet-dini-criterion-for-pointwise-convergence-of-fou.htmlhttps://liberty.io/proof-of-the-dirichlet-dini-criterion-for-pointwise-convergence-of-fou.htmlI have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (source: Wikipedia) if $f$ is $2\pi$–periodic, locally integrable and ...Sat, 15 Aug 2026 00:42:49 -0400https://liberty.io/do-other-trigonometric-functions-than-tan-sin-cos-and-their-derivative.htmlhttps://liberty.io/do-other-trigonometric-functions-than-tan-sin-cos-and-their-derivative.htmlI remember my physics teacher mentioning that other trigonometric functions exist apart from the Sin Cos and Tan, he mentioned a few and they did not sound familiar, nothing like Sec Csc and Cot. I......Sat, 15 Aug 2026 00:42:26 -0400https://liberty.io/definition-of-sheaf-using-equalizer.htmlhttps://liberty.io/definition-of-sheaf-using-equalizer.htmlWikipedia give sheaf property using equalizer diagram by saying sheaf property means for any open cover $\{U_i\}$ of $U$ $$F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\...Sat, 15 Aug 2026 00:36:42 -0400https://liberty.io/questions-tagged-sequences-and-series.htmlhttps://liberty.io/questions-tagged-sequences-and-series.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 15 Aug 2026 00:25:17 -0400https://liberty.io/solving-the-congruence-7x-3-1-mod-31.htmlhttps://liberty.io/solving-the-congruence-7x-3-1-mod-31.htmlI am having a problem when the LHS has an addition function; if the question is just a multiple of $x$ it's fine. But when I have questions like $3x+3$ or $4x+7$, I don't seem to get the right ans......Sat, 15 Aug 2026 00:21:55 -0400https://liberty.io/finding-an-orthogonal-vector-by-inspection.htmlhttps://liberty.io/finding-an-orthogonal-vector-by-inspection.htmlIf I have a vector say: $$ \begin{bmatrix} -1 \\ -\frac{1}{2} \\ 1 \end{bmatrix} $$ How can I identify a vector orthogonal to this by inspection? For example if I have this vector: $ \begin{bmatr......Sat, 15 Aug 2026 00:16:44 -0400https://liberty.io/will-this-be-equivalent-to-d-f-x-dx.htmlhttps://liberty.io/will-this-be-equivalent-to-d-f-x-dx.htmlJust wondering if $$d(f(e^x))/d(e^x)$$ would be equivalent to $$d(f(x))/d(x)$$ and if not what could you do to it to find $$d(f(x))/d(x)$$...Sat, 15 Aug 2026 00:12:32 -0400https://liberty.io/in-the-definition-of-k-partite-graph-is-it-necessary-that-k-should-be-.htmlhttps://liberty.io/in-the-definition-of-k-partite-graph-is-it-necessary-that-k-should-be-.htmlIt is well known that a graph is said to be $k$-partite if its vertex set can be partitioned in to k non empty sets such that no two vertices in the same set are adjacent. My question is whether ......Sat, 15 Aug 2026 00:12:03 -0400https://liberty.io/definition-of-overlapping-sets.htmlhttps://liberty.io/definition-of-overlapping-sets.htmlIs this definition of overlapping sets correct? If the intersection of two sets is non-empty set but neither is a subset of the other, the sets are called overlapping sets It is written in ......Fri, 14 Aug 2026 23:50:30 -0400https://liberty.io/what-039-s-the-chance-of-a-random-whole-number-between-1-999-having-th.htmlhttps://liberty.io/what-039-s-the-chance-of-a-random-whole-number-between-1-999-having-th.htmlI've thought about this problem the following way. If this was represented by 3 dials which have number 0-9 in them ([0-9][0-9][0-9]) then the chance of hitting 9 in one of them is 1/10. Since ther......Fri, 14 Aug 2026 23:38:26 -0400https://liberty.io/using-exchange-argument-in-proving-greedy-algorithm.htmlhttps://liberty.io/using-exchange-argument-in-proving-greedy-algorithm.htmlHere's a problem solvable by greedy algorithm: You are a company and you have list of tasks that still need to be done (but you're late with them already). For a given task we have information about ...Fri, 14 Aug 2026 23:38:26 -0400https://liberty.io/negative-exponents-in-binomial-theorem.htmlhttps://liberty.io/negative-exponents-in-binomial-theorem.htmlI'm looking at extensions of the binomial formula to negative powers. I've figured out how to do $n \choose k$ when $n < 0 $ and $k \geq 0$: $${n \choose k} = (-1)^k {-n + k - 1 \choose k}$$ So......Fri, 14 Aug 2026 23:30:10 -0400https://liberty.io/super-hard-system-of-equations.htmlhttps://liberty.io/super-hard-system-of-equations.htmlSolve the system of equation for real numbers \begin{split} (a+b) &(c+d) &= 1 & \qquad (1)\\ (a^2+b^2)&(c^2+d^2) &= 9 & \qquad (2)\\ (a^3+b^3)&(c^3+d^3) &= 7......Fri, 14 Aug 2026 23:29:49 -0400https://liberty.io/partial-fraction-decomp-why-repeated-factors-and-why-the-cx-d-for-quad.htmlhttps://liberty.io/partial-fraction-decomp-why-repeated-factors-and-why-the-cx-d-for-quad.html1) First, why are powers of linear factors repeated? For example, $(x+1)^2$ gets $\frac{A}{(x+1)}$ and $\frac{B}{(x+1)^2}$ 2) Why does a quad factor like $x^2+1$ get a term $\frac{Cx+d}{x^2+1}$ ...Fri, 14 Aug 2026 23:15:20 -0400https://liberty.io/how-to-use-the-lagrange-039-s-remainder-to-prove-that-log-1-x-sum.htmlhttps://liberty.io/how-to-use-the-lagrange-039-s-remainder-to-prove-that-log-1-x-sum.htmlUsing Lagrange's remainder, I have to prove that: $\log(1+x) = \sum\limits_{n=1}^\infty (-1)^{n+1} \cdot \frac{x^n}{n}, \; \forall |x| < 1$ I am not quite sure how to do this. I started with ......Fri, 14 Aug 2026 23:14:26 -0400https://liberty.io/number-of-ways-to-divide-n-people-into-k-groups-with-at-least-2-people.htmlhttps://liberty.io/number-of-ways-to-divide-n-people-into-k-groups-with-at-least-2-people.htmlI'm trying to figure out the number of ways to divide n people into k groups with at least 2 people in each group. Should I first decide a recurrence relation of the number? I don't know how I could ...Fri, 14 Aug 2026 23:07:34 -0400https://liberty.io/do-two-right-triangles-with-the-same-length-hypotenuse-have-the-same-a.htmlhttps://liberty.io/do-two-right-triangles-with-the-same-length-hypotenuse-have-the-same-a.htmlI watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with......Fri, 14 Aug 2026 22:57:45 -0400https://liberty.io/does-0-chance-mean-impossible-duplicate.htmlhttps://liberty.io/does-0-chance-mean-impossible-duplicate.htmlSuppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in......Fri, 14 Aug 2026 22:46:07 -0400https://liberty.io/why-do-we-multiply-by-100-for-finding-percentage.htmlhttps://liberty.io/why-do-we-multiply-by-100-for-finding-percentage.htmlFor example, if there are 50 boys in a school and the total number of students is $200$ then, for finding percentage of boys why do we do like this, $\left(\, 50/200\, \right)100\ \% = 25\ \%$&nbsp......Fri, 14 Aug 2026 22:38:35 -0400https://liberty.io/proof-of-the-dkw-inequality.htmlhttps://liberty.io/proof-of-the-dkw-inequality.htmlMy goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= \frac{1}{......Fri, 14 Aug 2026 22:32:44 -0400https://liberty.io/how-to-solve-x-6-6x-5-0.htmlhttps://liberty.io/how-to-solve-x-6-6x-5-0.htmlIs there any elementary way to solve (i.e. not involving derivatives etc.) $x^6 - 6x + 5 > 0$ ? I can divide it two times by $(x - 1)$ to get $x^6 - 6x + 5 = (x - 1)^2 (x^4 + 2 x^3 + 3 x^2 + 4 x......Fri, 14 Aug 2026 22:31:39 -0400https://liberty.io/how-to-understand-intuitively-the-stolz-cesaro-theorem-for-sequences.htmlhttps://liberty.io/how-to-understand-intuitively-the-stolz-cesaro-theorem-for-sequences.htmlI have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the proof and verified that, yes, step B does indeed follow logically from step A, etc. and ...Fri, 14 Aug 2026 22:27:15 -0400https://liberty.io/check-whether-a-point-is-within-a-3d-triangle.htmlhttps://liberty.io/check-whether-a-point-is-within-a-3d-triangle.htmlI have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on and inside the 3D triangle? So, for example, if I have a plane defined by $......Fri, 14 Aug 2026 22:20:47 -0400https://liberty.io/trying-to-understand-what-a-polynomial-ring-is.htmlhttps://liberty.io/trying-to-understand-what-a-polynomial-ring-is.htmlI don' really understand what a polynomial ring is, maybe because the lack of examples. Consider for example the polynomial ring $\mathbb{Z}[x]$. Can you please tell me how this polynomial ring (its ...Fri, 14 Aug 2026 22:20:35 -0400https://liberty.io/what-are-autonomous-and-non-autonomous-systems.htmlhttps://liberty.io/what-are-autonomous-and-non-autonomous-systems.htmlWhat are autonomous and non-autonomous systems and how are they different from each other. What are the differences between the types of systems they are describing? Do both autonomous and non-auto......Fri, 14 Aug 2026 22:11:06 -0400https://liberty.io/maximum-area-perimeter-ratio.htmlhttps://liberty.io/maximum-area-perimeter-ratio.htmlWith no limitation, to achieve the maximum area with a fixed perimeter, the shape is a circle, and the area / perimeter ratio would be $\frac{L}{4\pi}$ where $L$ is the perimeter lenght. However, ......Fri, 14 Aug 2026 22:10:37 -0400https://liberty.io/position-of-a-particle-moving-along-the-x-axis-question.htmlhttps://liberty.io/position-of-a-particle-moving-along-the-x-axis-question.htmlI am in need of help solving this question. The position of a particle moving along the $x$-axis is given by the function $s(t) = t^3 -6t^2 +9t$ where $s$ is in meters and $t$ is in seconds. When......Fri, 14 Aug 2026 22:06:24 -0400https://liberty.io/visualizing-why-rotations-preserve-orientation.htmlhttps://liberty.io/visualizing-why-rotations-preserve-orientation.htmlIt's clear geometrically that if you have two vectors in $\mathbb{R}^3$ a rotation will preserve their lengths and the angle between them. But how do you visualize that a rotation preserves orienta......Fri, 14 Aug 2026 22:02:00 -0400https://liberty.io/solving-x-2-2x-3-0.htmlhttps://liberty.io/solving-x-2-2x-3-0.htmlIf i have to solve $x^2-2x-3<0$ I would do $$x+1 < 0, \quad x-3<0$$ and end up getting $x<-1$ and $x<3$. Why is it wrong to use the same inequality sign? Shouldn’t both of the li......Fri, 14 Aug 2026 22:01:19 -0400https://liberty.io/functional-minimization-with-constraints.htmlhttps://liberty.io/functional-minimization-with-constraints.htmlSuppose you have a multi-variable functional given as $$ \int_{t_0}^{t_1} L(t,x,x', x'' ..., y, y', y'', ... ) \ dt $$ That you wish to to optimize [i.e find an $x$ and $y$ that maximize] but subje......Fri, 14 Aug 2026 21:55:07 -0400https://liberty.io/why-can-039-t-i-use-the-chain-rule-to-solve-this-trigonometric-integra.htmlhttps://liberty.io/why-can-039-t-i-use-the-chain-rule-to-solve-this-trigonometric-integra.htmlThe question is: what is the indefinite integral: $\int \sin^2(kx) \, \mathrm dx$? I get the correct answer using trig identities to change the $(\sin(kx))^2$ into $\dfrac{1}{2} - \dfrac{(\cos(2kx......Fri, 14 Aug 2026 21:46:43 -0400https://liberty.io/different-definitions-for-semisimple-lie-group.htmlhttps://liberty.io/different-definitions-for-semisimple-lie-group.htmlI am confused about two definitions for the notion of a semisimple Lie group i found. Lets say for simplicity i am only interested in matrix groups. In this case, do the following two object-classes ...Fri, 14 Aug 2026 21:46:22 -0400https://liberty.io/there-is-a-formal-definition-of-mathematical-space.htmlhttps://liberty.io/there-is-a-formal-definition-of-mathematical-space.htmlYesterday I was searching for a formal definition of mathematical space, then I found it wikipedia article, where the definition rely in the concept of mathematical structure. But I didnt found a ......Fri, 14 Aug 2026 21:45:46 -0400https://liberty.io/how-to-determine-and-explain-the-sum-of-angles-without-measuring.htmlhttps://liberty.io/how-to-determine-and-explain-the-sum-of-angles-without-measuring.htmlBelow is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f. I understand that angles are formed my intersecting ...Fri, 14 Aug 2026 21:40:23 -0400https://liberty.io/is-brute-force-the-only-answer-to-int-u-2-1-u-2-4du.htmlhttps://liberty.io/is-brute-force-the-only-answer-to-int-u-2-1-u-2-4du.htmlFeedback: After some initial interest, it seems this question has been down voted. Jyrki Lahtonen may be right that spending time on unsolvable problems is pointless, but, to a student, at what po......Fri, 14 Aug 2026 21:38:19 -0400https://liberty.io/find-derivative-of-e-3-sqrt-x-using-chain-rule.htmlhttps://liberty.io/find-derivative-of-e-3-sqrt-x-using-chain-rule.htmlim asked to find the derivative of this function using the chain rule. $$e^{3\sqrt{x}}$$ here are my steps. step 1 - identify the inner and outer functions. therefore I identified outer functio......Fri, 14 Aug 2026 21:36:43 -0400https://liberty.io/real-analysis-folland-problem-5-5-54-hilbert-spaces.htmlhttps://liberty.io/real-analysis-folland-problem-5-5-54-hilbert-spaces.htmlproblem 5.5.54 - For any nonempty set $A$, $l^{2}(A)$ is complete. Attempted proof (inspired by the answer below): Let $x\in A$, and $\{T_n(x)\}_{n\in\mathbb{N}}$ be a Cauchy sequence in $l^{2}(A)$. ...Fri, 14 Aug 2026 21:34:48 -0400https://liberty.io/very-difficult-calculus-related-rates-question.htmlhttps://liberty.io/very-difficult-calculus-related-rates-question.htmlI was looking at a math problem from a few years ago that I could not solve. I was wondering if anyone knows where to even begin. I have the answer along with the question, however, I do not know h......Fri, 14 Aug 2026 21:32:21 -0400https://liberty.io/rotating-a-cone.htmlhttps://liberty.io/rotating-a-cone.htmlI learned that, given a point $(x,y)$ in the plane, $\sigma_{\phi}(x,y) = (x\cos(\phi)-y\sin(phi),x\sin(\phi)+x\cos(\phi))$ is the point corresponding to rotating $(x,y)$ by an angle $\phi$ counter-...Fri, 14 Aug 2026 21:30:10 -0400https://liberty.io/why-is-1-raised-to-infinity-not-defined-and-not-1-duplicate.htmlhttps://liberty.io/why-is-1-raised-to-infinity-not-defined-and-not-1-duplicate.html$1$ square is $1$, so is raised $1$ to $123434234$. My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this? I know that any number raised to ...Fri, 14 Aug 2026 21:28:57 -0400https://liberty.io/how-was-the-integral-for-zeta-function-created.htmlhttps://liberty.io/how-was-the-integral-for-zeta-function-created.htmlHow was the zeta function integrated from $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ To $$\zeta(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx$$ I've tried googling......Fri, 14 Aug 2026 21:23:11 -0400https://liberty.io/if-the-square-root-of-two-is-irrational-why-can-it-be-created-by-divid.htmlhttps://liberty.io/if-the-square-root-of-two-is-irrational-why-can-it-be-created-by-divid.htmlI'm doing pre-calculus and got a bit caught on this question… I looked online and it said that a number is rational if it can be the quotient of two integers. So I did this: I know that it is ...Fri, 14 Aug 2026 21:12:36 -0400https://liberty.io/what-is-the-dot-product-of-complex-vectors.htmlhttps://liberty.io/what-is-the-dot-product-of-complex-vectors.htmlWhen I run dot(a,b) in MATLAB, I get a very different number than $\sum a_i \times b_i$, but is that not how one takes a dot product? For instance: A = [1+i 1-i -1+i -1-i]; B = [3-4i 6-2i 1+2i 4+3......Fri, 14 Aug 2026 20:54:37 -0400https://liberty.io/why-is-log-a-b-frac-ln-b-ln-a-closed.htmlhttps://liberty.io/why-is-log-a-b-frac-ln-b-ln-a-closed.htmlI know that formula, but I don't understand it. $\log_a(b)=\frac{\ln(b)}{\ln(a)}$ Thanks....Fri, 14 Aug 2026 20:53:58 -0400https://liberty.io/which-statements-about-a-linear-transformation-would-be-correct.htmlhttps://liberty.io/which-statements-about-a-linear-transformation-would-be-correct.htmlLet T be a linear transformation. If we have this then what statements would hold true? T contains $\vec{0}$ (i.e $\vec{0} \in T$) T is a 3 x 4 matrix The columns of T are linearly independent T(......Fri, 14 Aug 2026 20:38:17 -0400https://liberty.io/calculate-this-triple-integral-over-d.htmlhttps://liberty.io/calculate-this-triple-integral-over-d.html$$\iiint \sqrt{10+y^2-x^2} \,dx \,dy \, dz$$ over the region bounded by $D$: $D=\{(x,y,z)\mid x^2+z^2\le y^2+10,x^2+y^2\le 9\}$ So It's a cylinder intersecting a hyperboloid. Is It correct to sa......Fri, 14 Aug 2026 20:31:34 -0400https://liberty.io/why-do-we-only-solve-for-0-on-the-numerator-of-this-fraction.htmlhttps://liberty.io/why-do-we-only-solve-for-0-on-the-numerator-of-this-fraction.htmlI have this function where I need to find the $x$-intercepts I see they are $x = 0, 4$. And know that this is found by doing: $x=0, (x-4) = 0$ [Solve]. But why is this only done for the numerator?...Fri, 14 Aug 2026 20:19:09 -0400https://liberty.io/computing-a-revenue-for-vcg-auction.htmlhttps://liberty.io/computing-a-revenue-for-vcg-auction.htmlI would like your help with the following question regarding computing a revenue for a seller of an VCG (vickrey clarke groves) auction, I'm really new to this auctions\game theory so I'd really ...Fri, 14 Aug 2026 20:12:11 -0400https://liberty.io/probability-of-getting-a-7-in-minesweeper.htmlhttps://liberty.io/probability-of-getting-a-7-in-minesweeper.htmlLet's say you have a $30$ by $16$ grid and $99$ mines. What is the probability of having at least one empty block surrounded by exactly $7$ mines? For the sake of this question, assume that the mi......Fri, 14 Aug 2026 20:08:59 -0400https://liberty.io/how-to-solve-the-equation-x-2-2x-sqrt-2x-2-4x-3.htmlhttps://liberty.io/how-to-solve-the-equation-x-2-2x-sqrt-2x-2-4x-3.htmlHow to solve the equation $x^2-2x=\sqrt{ 2x^2-4x+3} $ Will squaring both sides introduce a $x^4$ term that may complicate the equation?...Fri, 14 Aug 2026 20:06:12 -0400https://liberty.io/is-sin-2-x-the-same-as-sin-x-2.htmlhttps://liberty.io/is-sin-2-x-the-same-as-sin-x-2.htmlI'm working with derivatives and need to know if $\sin^2(x)$ the same as $\sin(x^2)$? I almost don't want to ask because my last question was closed. It was a valid question and so is this one. I've ...Fri, 14 Aug 2026 20:04:45 -0400https://liberty.io/sum-of-series-1-11-111.htmlhttps://liberty.io/sum-of-series-1-11-111.htmlSum of series $1+11+111+\cdots+11\cdots11$ ($n$ digits) We have: $1=\frac {10-1}9,$ $11=\frac {10^2-1}9$ . . . $11...11= \frac {10^n-1}9$ (number with $n$ digits) and summing them we find ......Fri, 14 Aug 2026 19:55:06 -0400https://liberty.io/prove-10n-8-9n-6-n-2-is-divisible-by-45.htmlhttps://liberty.io/prove-10n-8-9n-6-n-2-is-divisible-by-45.htmlBasically, I have to use Euler theorem to prove that $10n^8 -9n^6 -n^2$ is divisible by $45$ So my approach so far is to say $10n^8 - 9n^6 - n^2 = 0 \bmod 45$ Now $45$ can be factored into $5$ ......Fri, 14 Aug 2026 19:45:33 -0400https://liberty.io/how-many-squares-are-in-this-image-is-there-a-method-to-check.htmlhttps://liberty.io/how-many-squares-are-in-this-image-is-there-a-method-to-check.htmlIn this image I have counted 14 but others say 18. Is there a method to check exactly?...Fri, 14 Aug 2026 19:33:19 -0400https://liberty.io/how-to-find-the-greatest-value-of-this-expression.htmlhttps://liberty.io/how-to-find-the-greatest-value-of-this-expression.html"This expression" is $$\frac{5\sin^2\alpha+4\cos^2\alpha}{4\cos^2\beta+5\sin^2\beta}.$$ The answer is $1.25$ I used simple steps to simplify this, but couldn't find the greatest value, since......Fri, 14 Aug 2026 19:31:32 -0400https://liberty.io/formula-for-calculating-residue-at-a-simple-pole.htmlhttps://liberty.io/formula-for-calculating-residue-at-a-simple-pole.htmlSuppose $f=P/Q$ is a rational function and suppose $f$ has a simple pole at $a$. Then a formula for calculating the residue of $f$ at $a$ is $$ \text{Res}(f(z),a)=\lim_{z\to a}(z-a)f(z)=\lim_{z\t......Fri, 14 Aug 2026 19:28:04 -0400https://liberty.io/finding-out-if-a-number-is-rational-or-irrational-duplicate.htmlhttps://liberty.io/finding-out-if-a-number-is-rational-or-irrational-duplicate.htmlHow does one find out if a given number is a rational number or irrational number?...Fri, 14 Aug 2026 19:24:20 -0400https://liberty.io/finding-vertex-cut-using-menger-039-s-theorem.htmlhttps://liberty.io/finding-vertex-cut-using-menger-039-s-theorem.htmlMenger's theorem states that A Graph, $G$, is $k$-connected if and only if for every $x,y \in V(G)$, there exists $k$ pairwise internally disjoint paths. If I need to find size of vertex cut f......Fri, 14 Aug 2026 19:23:44 -0400https://liberty.io/user-or-eisenberg-mathematics-stack-exchange.htmlhttps://liberty.io/user-or-eisenberg-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 14 Aug 2026 19:17:54 -0400https://liberty.io/how-to-calculate-cumulative-probabilities-for-a-discrete-random-variab.htmlhttps://liberty.io/how-to-calculate-cumulative-probabilities-for-a-discrete-random-variab.htmlI want to understand the difference between the cumulative distribution function and probability density function for a discrete random variable. I thought I understood the difference, but then I ...Fri, 14 Aug 2026 19:07:59 -0400https://liberty.io/how-to-find-unit-tangent-normal-and-binormal-vectors.htmlhttps://liberty.io/how-to-find-unit-tangent-normal-and-binormal-vectors.htmlI was given that $$p(t)=(1+2\cos t)\mathbf i + 2(1+\sin t)\mathbf j + (9+4\cos t+8 \sin t)\mathbf k$$ and that I needed to find the tangent, normal, and binormal vectors. The curvature and the ...Fri, 14 Aug 2026 18:53:11 -0400https://liberty.io/negating-statements-with-quantifiers.htmlhttps://liberty.io/negating-statements-with-quantifiers.htmlI understand that when we want to negate a statement with universal quantifier, that quantifier changes to existential quantifier, and vice versa. For example, negation of $(\exists x\in\Bbb N)(x......Fri, 14 Aug 2026 18:36:14 -0400https://liberty.io/finding-the-equation-of-vertical-and-horizontal-asymptotes.htmlhttps://liberty.io/finding-the-equation-of-vertical-and-horizontal-asymptotes.htmlI am having some trouble understanding these two questions. Any help is appreciated. Scanned questions are included at the end. 6) We are given the function $ f(x) =\frac{1 - 2x} {2x^2 - 3x - 2}......Fri, 14 Aug 2026 18:35:10 -0400https://liberty.io/given-sin-x-3-4-and-pi-2-x-pi-find-sin-x-pi-6.htmlhttps://liberty.io/given-sin-x-3-4-and-pi-2-x-pi-find-sin-x-pi-6.htmlGiven $\sin x=3/4$ and $\pi/2<x<\pi$, find $\sin(x+\pi/6)$. I have been trying to calculate this question. My current working out is as follows: $\sin\theta = 3/4$ $\pi/2 < \theta < ......Fri, 14 Aug 2026 18:30:31 -0400https://liberty.io/what-is-the-cardinality.htmlhttps://liberty.io/what-is-the-cardinality.htmlWhat is the cardinality of the set A,B where My answer was 4 for set A and 6 for set B. When i looked at the text book for answer it is not provided in the text book and now i'm wondering what the ...Fri, 14 Aug 2026 18:28:17 -0400https://liberty.io/origin-of-the-terminology-connected-algebra.htmlhttps://liberty.io/origin-of-the-terminology-connected-algebra.htmlI was wondering what is the origin of the word "connected" for a connected algebra ? To be more precise, why is a graded $R$-algebra $A_{\ast}$ with an augmentation $A_{\ast} \to R$ that restricts......Fri, 14 Aug 2026 18:14:16 -0400https://liberty.io/geometry-book-recommendation.htmlhttps://liberty.io/geometry-book-recommendation.htmlCan someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...Fri, 14 Aug 2026 18:13:32 -0400https://liberty.io/proof-of-theorem-the-principle-of-mathematical-induction.htmlhttps://liberty.io/proof-of-theorem-the-principle-of-mathematical-induction.htmlI have a question from my textbook and wanted to make sure that I understood it. I have marked in a green box the question that I have....Fri, 14 Aug 2026 18:13:27 -0400https://liberty.io/how-to-use-stars-and-bars-for-solving-number-of-integral-solutions-to-.htmlhttps://liberty.io/how-to-use-stars-and-bars-for-solving-number-of-integral-solutions-to-.htmlHow to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$, where $x_i\in\mathbb{N}$. Is this the correct time to apply the method? This......Fri, 14 Aug 2026 18:12:02 -0400https://liberty.io/iff-if-and-only-if-vs-tfae-the-following-are-equivalent.htmlhttps://liberty.io/iff-if-and-only-if-vs-tfae-the-following-are-equivalent.htmlIf $P$ and $Q$ are statements, $P \iff Q$ and The following are equivalent: $(\text{i}) \ P$ $(\text{ii}) \ Q$ Is there a difference between the two? I ask because formulations of certain theor......Fri, 14 Aug 2026 18:03:39 -0400https://liberty.io/finding-extrema-non-critical-points-and-saddle-points-given-a-contour-.htmlhttps://liberty.io/finding-extrema-non-critical-points-and-saddle-points-given-a-contour-.htmlI'm trying to interpret the contour plot as like a mountain, where all the segregated areas are plains that are higher when it proceeds inwards. What is the significance of the numbers on x-axis an......Fri, 14 Aug 2026 18:03:01 -0400https://liberty.io/regular-octagons-and-diagonals-proof.htmlhttps://liberty.io/regular-octagons-and-diagonals-proof.htmlDefinition: A diagonal of a octagon is a line segment connecting any two non-adjacent vertices. Prove: Every vertex of the regular octagon will produce 2 diagonals that are parallel to at ...Fri, 14 Aug 2026 17:56:57 -0400