https://liberty.ioCurated star highlights with fresh energy.en-usFri, 14 Aug 2026 18:58:41 -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 -0400https://liberty.io/logarithmic-differentiation-when-to-use.htmlhttps://liberty.io/logarithmic-differentiation-when-to-use.htmlSorry if this is an ignorant or uninformed question, but I would like to know when I can (or should use) logarithmic differentiation. I haven't taken calculus in a while so I'm quite rusty. So, le......Fri, 14 Aug 2026 17:52:51 -0400https://liberty.io/proof-of-additivity-of-domain-for-definite-integrals.htmlhttps://liberty.io/proof-of-additivity-of-domain-for-definite-integrals.htmlI would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ......Fri, 14 Aug 2026 17:51:45 -0400https://liberty.io/finding-which-sets-are-subspaces-of-r3.htmlhttps://liberty.io/finding-which-sets-are-subspaces-of-r3.htmlHello. I have attached an image of the question I am having trouble with. I thought that it was 1,2 and 6 that were subspaces of $\mathbb R^3$. Here is my working: Rearranged equation ---> $x+y......Fri, 14 Aug 2026 17:50:09 -0400https://liberty.io/what-is-the-name-of-the-4d-shape-that-is-equivalent-to-a-cylinder-exce.htmlhttps://liberty.io/what-is-the-name-of-the-4d-shape-that-is-equivalent-to-a-cylinder-exce.htmlSorry if my question is poorly explained. Basically I want the name of the shape that always looks like a sphere in 3 dimensions when fed into 3d space standing up in the 4th dimension; just like h......Fri, 14 Aug 2026 17:45:09 -0400https://liberty.io/sign-language-and-deaf-mathematicians.htmlhttps://liberty.io/sign-language-and-deaf-mathematicians.htmlSomething I've often wondered (and I suppose this goes for all kinds of technical terminology, not just that of mathematics) is what kind of sign language exists for practising professional ...Fri, 14 Aug 2026 17:35:38 -0400https://liberty.io/the-meaning-of-spacecraft-attitude-represented-in-quaternion.htmlhttps://liberty.io/the-meaning-of-spacecraft-attitude-represented-in-quaternion.htmlI am reading the following paper about the attitude control of aircraft: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1271671 The quaternion represents the relative orientation of two ...Fri, 14 Aug 2026 17:26:58 -0400https://liberty.io/using-rouche-039-s-theorem-to-find-no-of-roots-of-p-z-2z-4-2z-3-2z-2-7.htmlhttps://liberty.io/using-rouche-039-s-theorem-to-find-no-of-roots-of-p-z-2z-4-2z-3-2z-2-7.htmlLet $p(z)=2z^4-2z^3+2z^2+7$. I'm trying to determine the number of roots of $p(z)$ that lie inside $\{z\in\mathbb{C}:|z|<2\}$ I used Rouche's theorem and I'm trying to find find $f(z)$ such tha......Fri, 14 Aug 2026 17:22:00 -0400https://liberty.io/what-is-sine-of-a-real-number.htmlhttps://liberty.io/what-is-sine-of-a-real-number.htmlI never understand what the trigonometric function sine is.. We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. T......Fri, 14 Aug 2026 17:10:26 -0400https://liberty.io/repeated-roots-of-a-polynomial.htmlhttps://liberty.io/repeated-roots-of-a-polynomial.htmlWhenever a part of the graph of a polynomial is in the form of a parabola whose vertex touches the x axis we conclude that a root is repeated at that point. Why is a root said to be repeated at that ...Fri, 14 Aug 2026 17:08:10 -0400https://liberty.io/what-is-sin-2-x-equal-to.htmlhttps://liberty.io/what-is-sin-2-x-equal-to.htmlLet's take the sine of $30^\circ$ which is one-half. If you take $\sin^2(30^\circ)$, would that just be the sine of $900$? Or would it be equal to one-quarter, or would it be equal to something ...Fri, 14 Aug 2026 16:58:41 -0400https://liberty.io/sphere-eversion-and-inversion.htmlhttps://liberty.io/sphere-eversion-and-inversion.htmlthe following things are asked because of a research I am doing for a short novel that I am writing. It contains only very little mathematics, however I want it to be correct. Do I understand corr......Fri, 14 Aug 2026 16:56:58 -0400https://liberty.io/show-that-int-0-1-times-0-1-frac-dx-dy-1-xy-frac-pi-2-6.htmlhttps://liberty.io/show-that-int-0-1-times-0-1-frac-dx-dy-1-xy-frac-pi-2-6.html$$ \int_{[0,1] \times [0,1]} \frac{dx \, dy}{1-xy}=\frac{\pi^2}{6}, $$ there is a hint for substitution given but I can't seem to get anywhere with it. $$ \begin{cases} x = \frac{u-v}{\sqrt{2}} \\ ......Fri, 14 Aug 2026 16:48:18 -0400https://liberty.io/questions-concerning-smallest-fraction-between-two-given-fractions.htmlhttps://liberty.io/questions-concerning-smallest-fraction-between-two-given-fractions.htmlI recently encountered this on a practice test Find the smallest positive integer $n$ such that there exists an integer $m$ satisfying $0.33 < \frac{m}{n} < \frac{1}{3}$ My answer: $$\begin{...Fri, 14 Aug 2026 16:19:07 -0400https://liberty.io/how-to-solve-this-simple-algebraic-equation-wolfram-symbolab-both-stum.htmlhttps://liberty.io/how-to-solve-this-simple-algebraic-equation-wolfram-symbolab-both-stum.htmlI need to solve the following equation for a hobby coding project I'm working on and I can't figure out the solution. Wolfram and SymboLab solvers are both stumped. $$0 = d - (r^{p-1} * (p-1) * c)......Fri, 14 Aug 2026 16:09:35 -0400https://liberty.io/discriminant-of-elliptic-curves.htmlhttps://liberty.io/discriminant-of-elliptic-curves.htmlIn the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = ......Fri, 14 Aug 2026 15:29:58 -0400https://liberty.io/two-independent-events-a-and-b-find-b-given-only-a-and-a-cup-b.htmlhttps://liberty.io/two-independent-events-a-and-b-find-b-given-only-a-and-a-cup-b.htmlSorry to be a burden but my maths exam is tomorrow and I have no idea how to do this at all. Here is the exact question : The two independent events $A$ and $B$ are such that $P(A) = 0.2$, $P(A\cu......Fri, 14 Aug 2026 15:25:23 -0400https://liberty.io/find-the-3-angles-of-the-hyperbolic-triangle.htmlhttps://liberty.io/find-the-3-angles-of-the-hyperbolic-triangle.htmlA(0,5) B(0,2) C(4,2) In Euclidean geometry the three points given are the vertices of a right-angled triangle. Find the three angles of the hyperbolic triangle with vertices A,B,C. Find the hyperb......Fri, 14 Aug 2026 15:22:41 -0400https://liberty.io/definition-of-minor-for-a-general-m-times-n-matrix.htmlhttps://liberty.io/definition-of-minor-for-a-general-m-times-n-matrix.htmlFollowing is the definition of minor given in the text : For a matrix $A$, consider its $k × k$ submatrix, obtained by taking $k$ rows and $k$ columns. The determinant of this matrix is calle......Fri, 14 Aug 2026 15:20:51 -0400https://liberty.io/what-is-a-null-set.htmlhttps://liberty.io/what-is-a-null-set.htmlI am very confused with null sets. I get that a set which has no elements will be called a null set but I am not getting the examples given below. Please help me by explaining how $P,Q,R$ are all......Fri, 14 Aug 2026 15:15:26 -0400https://liberty.io/absolute-value-graph-sketching-x-1-1.htmlhttps://liberty.io/absolute-value-graph-sketching-x-1-1.htmlWhere would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then......Fri, 14 Aug 2026 15:12:11 -0400https://liberty.io/how-can-i-prove-that-the-angles-of-the-tetrahedral-structure-is-109-5-.htmlhttps://liberty.io/how-can-i-prove-that-the-angles-of-the-tetrahedral-structure-is-109-5-.htmlHow can I prove that the dihedral angles of the regular tetrahedral structure is $109.5^\circ$ with calculus or any other technique. I could do it with geometry by considering a cube....Fri, 14 Aug 2026 15:08:55 -0400https://liberty.io/what-is-the-vertex-of-y-x-x.htmlhttps://liberty.io/what-is-the-vertex-of-y-x-x.htmlFor $x > 0$, the graph takes on a parabola-like shape. Is there a method to algebraically and not graphically determine the vertex of the graph? Desmos rendering of $x^x$...Fri, 14 Aug 2026 14:44:53 -0400https://liberty.io/construct-triangle-given-angle-and-centroid.htmlhttps://liberty.io/construct-triangle-given-angle-and-centroid.htmlI am stumped by Euclidea problem 8.11: From a triangle are given angle A and the centroid G Construct the points B and C. Please only a hint...Fri, 14 Aug 2026 14:44:42 -0400https://liberty.io/what-is-physical-interpretation-of-dot-product-duplicate.htmlhttps://liberty.io/what-is-physical-interpretation-of-dot-product-duplicate.htmlConsider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$. When we take their dot product we get a real number. How is that number related to the vectors? Is there any way we can visualize it?...Fri, 14 Aug 2026 14:43:37 -0400https://liberty.io/solve-this-logical-puzzle-only-using-logical-equivalencies-is-there-a-.htmlhttps://liberty.io/solve-this-logical-puzzle-only-using-logical-equivalencies-is-there-a-.htmlHere's a puzzle: Four friends have been identified as suspects for an authorized access into a computer system. They have made statements to the investigating authorities. Alice said "Carlos......Fri, 14 Aug 2026 14:31:11 -0400https://liberty.io/how-to-prove-that-the-altitudes-of-the-triangle-are-concurrent.htmlhttps://liberty.io/how-to-prove-that-the-altitudes-of-the-triangle-are-concurrent.htmlGiven a triangle of sides $A$ $B$ $C$ .how to prove that altitudes from sides $A$ $B$ $C$ are concurrent . There is a theorem called ceva's theorem.but i dont know how to use that theorem in this p......Fri, 14 Aug 2026 14:25:07 -0400https://liberty.io/the-position-of-a-ladder-leaning-against-a-wall-and-touching-a-box-und.htmlhttps://liberty.io/the-position-of-a-ladder-leaning-against-a-wall-and-touching-a-box-und.htmlI was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter c......Fri, 14 Aug 2026 14:23:58 -0400https://liberty.io/what-defines-a-chaotic-system.htmlhttps://liberty.io/what-defines-a-chaotic-system.htmlSuppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My que......Fri, 14 Aug 2026 14:19:22 -0400https://liberty.io/solving-the-equation-11x-2-6000x-27500-0-preferably-without-the-quadra.htmlhttps://liberty.io/solving-the-equation-11x-2-6000x-27500-0-preferably-without-the-quadra.htmlI obtained this form while solving an aptitude question. $$\frac{3000}{x-50} + \frac{3000}{x+50} = 11$$ I changed it into quadratic equation $$11x^2 -6000x - 27500 =0$$ but I don't know how to ......Fri, 14 Aug 2026 14:17:14 -0400https://liberty.io/simply-explain-the-general-number-field-sieve.htmlhttps://liberty.io/simply-explain-the-general-number-field-sieve.htmlAs a beginner to the world of integer factorization, my idea of factoring an integer is to generate a large list of prime numbers below this number and to repeatedly try to divide the integer by th......Fri, 14 Aug 2026 14:16:59 -0400https://liberty.io/how-does-one-orient-a-simplicial-complex.htmlhttps://liberty.io/how-does-one-orient-a-simplicial-complex.htmlI have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a ...Fri, 14 Aug 2026 14:13:30 -0400https://liberty.io/finding-the-area-of-an-ellipse-using-the-formula.htmlhttps://liberty.io/finding-the-area-of-an-ellipse-using-the-formula.htmlI'm preparing for the ACTM Sate contest, and I stumbled across this problem. I've never learned how to decipher equations like this, so could someone explain what these equations mean and how to so......Fri, 14 Aug 2026 14:13:14 -0400https://liberty.io/easiest-way-to-solve-this-integral-closed.htmlhttps://liberty.io/easiest-way-to-solve-this-integral-closed.htmlI was solving this problem from a calculus textbook and I got stuck at this particular problem. I tried to put it into Integral Calculator after I was unable to solve it, but now I wonder if there ......Fri, 14 Aug 2026 14:11:52 -0400https://liberty.io/trace-and-the-coefficients-of-the-characteristic-polynomial-of-a-matri.htmlhttps://liberty.io/trace-and-the-coefficients-of-the-characteristic-polynomial-of-a-matri.htmlLet $A\in M(\mathbb F)_{n \times n}$ Prove that the trace of A is minus the coefficient of $\lambda ^{n-1}$ in the characteristic polynomial of A. I had several ideas to approach this problem - ......Fri, 14 Aug 2026 14:07:18 -0400https://liberty.io/how-to-show-this-sequence-is-decreasing.htmlhttps://liberty.io/how-to-show-this-sequence-is-decreasing.htmlI was given a practice question in my math lecture today, it was to prove that the sequence below was decreasing: $a_n = \frac{2^n}{(n+1)!}$ using the condition that a sequence is decreasing if $......Fri, 14 Aug 2026 13:54:08 -0400https://liberty.io/probability-x-will-happen-out-of-two-tries.htmlhttps://liberty.io/probability-x-will-happen-out-of-two-tries.htmlSay you grab a card out a deck, check it, and put it back. If the probability of getting a non-club card is $\frac{3}{4}$, what is the probability it will happen at least once out of two tries? Y......Fri, 14 Aug 2026 13:49:51 -0400https://liberty.io/orthogonal-projection-onto-the-unit-simplex.htmlhttps://liberty.io/orthogonal-projection-onto-the-unit-simplex.htmlThe Unit Simplex is defined by: $$ \mathcal{S} = \left\{ x \in \mathbb{{R}^{n}} \mid x \succeq 0, \, \boldsymbol{1}^{T} x = 1 \right\} $$ Orthogonal Projection onto the Unit Simplex is defined by......Fri, 14 Aug 2026 13:48:20 -0400https://liberty.io/what-is-the-equation-of-a-half-sine-wave.htmlhttps://liberty.io/what-is-the-equation-of-a-half-sine-wave.htmlBasically, I am trying to generate a half sine wave using excel. The waveform that I am looking for is a wave which has a positive component from $0$ to $\pi$ and a zero component from $\pi$ to $2\......Fri, 14 Aug 2026 13:40:11 -0400https://liberty.io/find-the-partial-sum-of-the-series.htmlhttps://liberty.io/find-the-partial-sum-of-the-series.htmlCase: $$\sum _{n=1}^{\infty}(-1)^{n+1}(2n-1)$$ Question: $$S_n =\ ?$$ My partial solution and investigation: $$S_n = \sum _{k=1}^n (-1)^{n+1}(2n-1) = 1 -3+5-7+9-11+\cdots+(-1)^{n+1}(2n-1)$$ I see......Fri, 14 Aug 2026 13:38:23 -0400https://liberty.io/you-roll-a-6-sided-die-what-is-your-p-prime-closed.htmlhttps://liberty.io/you-roll-a-6-sided-die-what-is-your-p-prime-closed.htmlyou roll a 6 sided die. whqat is P(prime)? write your answer as a percentage...Fri, 14 Aug 2026 13:34:05 -0400https://liberty.io/proving-tan-20-circ-cdot-tan-30-circ-cdot-tan-40-circ-tan-10-circ.htmlhttps://liberty.io/proving-tan-20-circ-cdot-tan-30-circ-cdot-tan-40-circ-tan-10-circ.htmlhow to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$? I know how to prove $ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $ in this way: $ \tan{20^......Fri, 14 Aug 2026 13:32:02 -0400https://liberty.io/determining-a-branch-of-logarithm.htmlhttps://liberty.io/determining-a-branch-of-logarithm.htmlThe question I have is that what is the explicit mapping that takes the value $-i \pi/2$ at $-i$ where the mapping is a branch of the logarithm in the slit plane $\mathbb{C}- [0,\infty)$? I'm famil......Fri, 14 Aug 2026 13:24:20 -0400https://liberty.io/is-it-possible-in-a-group-of-seven-people-for-each-to-be-friends-with-.htmlhttps://liberty.io/is-it-possible-in-a-group-of-seven-people-for-each-to-be-friends-with-.htmlIs it possible in a group of seven people for each to be friends with exactly 3 others? I know that the sum of degrees of vertices in a graph must be even....Fri, 14 Aug 2026 13:20:39 -0400https://liberty.io/how-to-find-the-average-distance-from-a-point-to-a-all-the-points-on-a.htmlhttps://liberty.io/how-to-find-the-average-distance-from-a-point-to-a-all-the-points-on-a.htmlSuppose point $C$ is directly above point $B$ of $\overline{AB}$. $|\overline{AB}| =2$, and the distance from $C$ to $B$ is $1$. How can I find the average distance from point $C$ to every point o......Fri, 14 Aug 2026 13:19:16 -0400