https://liberty.ioCurated star highlights with fresh energy.en-usSun, 23 Aug 2026 10:36:14 -0400https://liberty.io/is-a-sequence-of-all-the-same-numbers-monotonic.htmlhttps://liberty.io/is-a-sequence-of-all-the-same-numbers-monotonic.htmlI'm wondering based on the definition of monotonicity: A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. So given that the sequence $a_n = 3$ is all the same numbers an......Sun, 23 Aug 2026 14:30:08 -0400https://liberty.io/can-a-piece-of-a4-paper-be-folded-so-that-it-039-s-thick-enough-to-rea.htmlhttps://liberty.io/can-a-piece-of-a4-paper-be-folded-so-that-it-039-s-thick-enough-to-rea.htmlWhile procrastinating around the web I stumbled on a page that contained the image below, from cracked.com. I can't help but believe that this is false… Even though the article header says......Sun, 23 Aug 2026 14:24:03 -0400https://liberty.io/derivative-over-the-function-itself.htmlhttps://liberty.io/derivative-over-the-function-itself.htmlI am new to calculus. Let $f(X)$ be continuous and differentiable over a set of variables $X$ where $x \in X$ and $x \in (0,\infty)$. Is there any general solution or rules to solve $\frac{f'(X)}{f......Sun, 23 Aug 2026 14:23:15 -0400https://liberty.io/how-to-find-in-which-number-base-the-operation-was-done-by-looking-at-.htmlhttps://liberty.io/how-to-find-in-which-number-base-the-operation-was-done-by-looking-at-.html$$23 + 25 = 51 $$ What base is used in the above addition operation ? I have 2 methods to do this Method 1 : Through equations assume base be a $$23_a + 25_a = 51_a $$ $$2a + 3 + 2a + 5 = 5a +......Sun, 23 Aug 2026 14:12:34 -0400https://liberty.io/how-to-explain-what-it-means-to-say-a-function-is-defined-on-an-interv.htmlhttps://liberty.io/how-to-explain-what-it-means-to-say-a-function-is-defined-on-an-interv.htmlI am having difficulty in explaining the terminology "defined" to the students I am assisting. Here is the sentence: If a real-valued function $f$ is defined and continuous on the closed interval $......Sun, 23 Aug 2026 14:07:29 -0400https://liberty.io/error-of-taylor-series.htmlhttps://liberty.io/error-of-taylor-series.htmlPart of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75. Finding the series was easy eno......Sun, 23 Aug 2026 14:07:04 -0400https://liberty.io/how-to-find-the-average-value-of-y-e-x-between-x-e-and-x-2e-closed.htmlhttps://liberty.io/how-to-find-the-average-value-of-y-e-x-between-x-e-and-x-2e-closed.htmlWhat approach would be ideal in finding the average value of $y = e^x$ between $x = e$ and $x = 2e$?...Sun, 23 Aug 2026 14:04:03 -0400https://liberty.io/which-of-these-event-can-be-represented-in-either-a-venn-diagram-and-t.htmlhttps://liberty.io/which-of-these-event-can-be-represented-in-either-a-venn-diagram-and-t.htmlWhich of these events can be represented in a Venn Diagram. Independent events Dependent events Mutually exclusive events Mutually independent events Basic reasoning with example would be nice......Sun, 23 Aug 2026 14:03:10 -0400https://liberty.io/how-do-you-find-the-number-that-corresponds-to-quartile-1-when-given-a.htmlhttps://liberty.io/how-do-you-find-the-number-that-corresponds-to-quartile-1-when-given-a.htmlIf you have an odd number of data values {39, 40, 42, 44, 47, 48, 49, 51, 53}, what is the score that value that corresponds to Quartile 1? In general, I would like to know if you include the medi......Sun, 23 Aug 2026 14:01:46 -0400https://liberty.io/general-solution-to-wave-equation-algebraic-approach.htmlhttps://liberty.io/general-solution-to-wave-equation-algebraic-approach.htmlConsider the Wave Equation $$\frac{\partial^{2} u}{\partial x^{2}} - c^{2}\frac{\partial^{2} u}{\partial y^{2}} = 0$$ The algebraic approach from wikipedia is: Take $\alpha = x - cy$ and $\bet......Sun, 23 Aug 2026 13:46:51 -0400https://liberty.io/prove-that-if-x-and-y-are-sequences-such-that-x-converges-to-x-neq-0-a.htmlhttps://liberty.io/prove-that-if-x-and-y-are-sequences-such-that-x-converges-to-x-neq-0-a.htmlI want to prove that given convergent sequences $X$ and $XY$ then $Y$ converges. Call arbitrarily $z_n=(x_n)(y_n)$, as by hypothesis $\lim(x_n)=x≠0$, then it must happen that $(x_n)≠0$, thus we can ...Sun, 23 Aug 2026 13:45:59 -0400https://liberty.io/proof-by-induction-binomial-theorem.htmlhttps://liberty.io/proof-by-induction-binomial-theorem.htmlLet $n\in N$, $k\in Z$, $o\leq k \leq n$. Define $C^{n}_k$ as the coefficient of $x^{n-k}y^k$ in the expansion of $(x+y)^n$ $$(x+y)^n= \sum^{n}_{k=0} C^{n}_k x^{n-k}y^k$$ Use induction to prove......Sun, 23 Aug 2026 13:37:34 -0400https://liberty.io/cofree-construction-thought-process.htmlhttps://liberty.io/cofree-construction-thought-process.htmlLet $F: \mathbf{A} \to \mathbf{B}$ be any functor and $B$ an object of $\mathbf{B}$. We say the pair $(A,\epsilon)$, with $A$ and object of $\mathbf{A}$ and $\epsilon: FA \to B$ is a morphism of $\......Sun, 23 Aug 2026 13:34:00 -0400https://liberty.io/expressing-5-8-sqrt-40-56-4-5-8-sqrt-10-56-4-in-simplified-radical-for.htmlhttps://liberty.io/expressing-5-8-sqrt-40-56-4-5-8-sqrt-10-56-4-in-simplified-radical-for.htmlI am trying to complete the question below, but I am not sure how to simplify the radical. What I have so far is $$(5.8\sqrt{40} + 56.4) - (5.8\sqrt{10} + 56.4) \;=\; \text{the difference}$$ How do......Sun, 23 Aug 2026 13:33:57 -0400https://liberty.io/how-many-solutions-does-riemann-039-s-p-symbol-describe.htmlhttps://liberty.io/how-many-solutions-does-riemann-039-s-p-symbol-describe.htmlThe Papperitz-Riemann P-symbol $$ \tag 1 y(z) = P \left\{ \begin{matrix} z_1 & z_2 & z_3 & \; \\ \alpha_1 & \alpha_2 & \alpha_3 & z \\ \beta_1 & \beta_2 & \be......Sun, 23 Aug 2026 13:21:15 -0400https://liberty.io/improper-integral-of-sin-x-x-from-zero-to-infinity-duplicate.htmlhttps://liberty.io/improper-integral-of-sin-x-x-from-zero-to-infinity-duplicate.htmlI was having trouble with the following integral: $\int_{0}^\infty \frac{\sin(x)}{x}dx$. My question is, how does one go about evaluating this, since its existence seems fairly intuitive, while its ...Sun, 23 Aug 2026 13:13:40 -0400https://liberty.io/why-use-the-letter-k-in-the-function-transformation-formula-f-x-h-k.htmlhttps://liberty.io/why-use-the-letter-k-in-the-function-transformation-formula-f-x-h-k.htmlThis is strictly a historical "why is it the letter k rather than say v for vertical" question -- is it the initial letter of something from a specific language? Is it arbitrary? While we're at it......Sun, 23 Aug 2026 13:10:28 -0400https://liberty.io/how-to-prove-that-two-lines-in-3d-are-not-parallel-and-do-not-intersec.htmlhttps://liberty.io/how-to-prove-that-two-lines-in-3d-are-not-parallel-and-do-not-intersec.htmlProblem Given two lines: $$(l_1)=(x,y,z)=(3,-1,2)+t(1,1,0)$$ $$(l_2)=(x,y,z)=(0,5,2)+t(1,-2,1)$$ Explain why these two lines are not parallel and why they do not intersect each other. Also, find the ...Sun, 23 Aug 2026 13:10:06 -0400https://liberty.io/the-degree-of-a-polynomial-which-also-has-negative-exponents.htmlhttps://liberty.io/the-degree-of-a-polynomial-which-also-has-negative-exponents.htmlIn theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that......Sun, 23 Aug 2026 12:49:42 -0400https://liberty.io/hausdorff-topological-space.htmlhttps://liberty.io/hausdorff-topological-space.htmlWanted to explain what I think a Hausdorff is in my own words because maybe that is the root of the problem. A Hausdorff Space is one in which for every x and y in X with x does not equal y, there ...Sun, 23 Aug 2026 12:48:07 -0400https://liberty.io/what-is-arctan-9-in-terms-of-pi.htmlhttps://liberty.io/what-is-arctan-9-in-terms-of-pi.htmlWhat is the pi fraction of $\arctan (9)$? I converted it to degrees and decimal, but how is it expressed in terms of $\pi$?...Sun, 23 Aug 2026 12:46:00 -0400https://liberty.io/almost-complex-manifolds-are-orientable.htmlhttps://liberty.io/almost-complex-manifolds-are-orientable.htmlI want to verify the fact that every almost complex manifold is orientable. By definition, an almost complex manifold is an even-dimensional smooth manifold $M^{2n}$ with a complex structure,......Sun, 23 Aug 2026 12:35:15 -0400https://liberty.io/is-the-square-root-of-negative-1-equal-to-i-or-is-it-equal-to-plus-or-.htmlhttps://liberty.io/is-the-square-root-of-negative-1-equal-to-i-or-is-it-equal-to-plus-or-.htmlI didn't see a duplicate.. The motivation is you tube. Tanton lectures of which one is titled " The Complex number i is NOT the square root of negative one". Does anyone have a clue why this may......Sun, 23 Aug 2026 12:32:58 -0400https://liberty.io/user-jiayin-pan-mathematics-stack-exchange.htmlhttps://liberty.io/user-jiayin-pan-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 23 Aug 2026 12:30:56 -0400https://liberty.io/probability-of-getting-a-pair-in-texas-hold-039-em-poker.htmlhttps://liberty.io/probability-of-getting-a-pair-in-texas-hold-039-em-poker.htmlI want to know the probability of getting a pair after all 5 cards are dealt on the table and assuming that I haven't got a pocket pair. Let's assume that I hold an ace and a king and 50 cards are......Sun, 23 Aug 2026 12:16:52 -0400https://liberty.io/expected-value-of-square-of-euclidean-norm-of-a-gaussian-random-vector.htmlhttps://liberty.io/expected-value-of-square-of-euclidean-norm-of-a-gaussian-random-vector.htmlIf $X$ is a $p \times 1$ gaussian random vector with such that $X \sim \mathcal{N}(0,\Sigma)$. What is the expected value of the square of the euclidean norm i.e $E[\|AX\|_2^2]$? Here $A$ is a $n \......Sun, 23 Aug 2026 11:40:26 -0400https://liberty.io/difference-between-omitting-the-multiplication-sign-and-keeping-it-dup.htmlhttps://liberty.io/difference-between-omitting-the-multiplication-sign-and-keeping-it-dup.htmlWhat is the difference between 2*a and 2a? What is the difference between 2(3+4) and 2*(3+4)? We all know that omitting the multiplication sign still means multiplication so nothing changed! This ...Sun, 23 Aug 2026 11:33:42 -0400https://liberty.io/show-that-the-equation-represents-a-circle-and-find-the-centre.htmlhttps://liberty.io/show-that-the-equation-represents-a-circle-and-find-the-centre.htmlI have the following equation: ${x^2 + y^2 - 4x -6y + 9 = 0}$ And I am asked to show that the equation represents a circle and find the centre. In order to show that the equation represents a ci......Sun, 23 Aug 2026 11:25:08 -0400https://liberty.io/what-is-infinite-horizon-problem.htmlhttps://liberty.io/what-is-infinite-horizon-problem.htmlIn optimal control, What is infinite horizon problem? What is the difference between finite and infinite horizon? What are their real life examples (finite & infinite)?...Sun, 23 Aug 2026 11:25:01 -0400https://liberty.io/how-to-show-that-derivative-of-phi-v-with-respect-to-v-is-phi-039-v-a-.htmlhttps://liberty.io/how-to-show-that-derivative-of-phi-v-with-respect-to-v-is-phi-039-v-a-.htmlHow to show that derivative of $\phi(v)$ with respect to $v$ is $$\frac{d \phi}{d v}= \frac{a}{2}(1-\phi^2(v)),$$ where $$\phi(v) = \frac{1-\exp(-av)}{1-\exp(-av)}=\tanh(av/2).$$ What is the v......Sun, 23 Aug 2026 11:23:42 -0400https://liberty.io/how-can-i-define-an-infinitely-small-positive-value.htmlhttps://liberty.io/how-can-i-define-an-infinitely-small-positive-value.htmlHow do you define an infinitely small value that is greater than zero? $1/\infty$ is apparently invalid because you can't divide by infinity. And $0.\bar{0}1$ is invalid because you can't put a 1 a......Sun, 23 Aug 2026 11:22:22 -0400https://liberty.io/three-dimensional-random-walk.htmlhttps://liberty.io/three-dimensional-random-walk.htmlA particle starts at an origin $O$ in three-space. Thinking of point $O$ as the center of a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight ...Sun, 23 Aug 2026 11:19:04 -0400https://liberty.io/the-product-of-two-positive-definite-matrices-has-real-and-positive-ei.htmlhttps://liberty.io/the-product-of-two-positive-definite-matrices-has-real-and-positive-ei.htmlGiven two real positive definite (and therefore, symmetric) matrices $A$ and $B$, are all the eigenvalues of $AB$ real and positive? Wikipedia says $AB$ is positive definite if $A$ and $B$ are pos......Sun, 23 Aug 2026 11:14:03 -0400https://liberty.io/how-to-find-distance-between-two-different-circles.htmlhttps://liberty.io/how-to-find-distance-between-two-different-circles.htmlI am trying the find the distance between two different sized circles, both centred on the horizontal plane. I know the diameter of each circle, and the length around both circles if wrapped like a......Sun, 23 Aug 2026 11:04:00 -0400https://liberty.io/find-the-number-of-roots-of-a-polynomial-using-rouche-039-s-theorem.htmlhttps://liberty.io/find-the-number-of-roots-of-a-polynomial-using-rouche-039-s-theorem.htmlUse Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that was ...Sun, 23 Aug 2026 11:03:01 -0400https://liberty.io/how-to-calculate-log-x-1-2-log-16-1-3-log-8-1.htmlhttps://liberty.io/how-to-calculate-log-x-1-2-log-16-1-3-log-8-1.htmlThis is my first question. This is basic math, but what I get does not match the alternatives I have. So I was wondering if I did something wrong. Step 1: $\log(x) = 1/2\log(16) - 1/3\log(8) + 1$ ...Sun, 23 Aug 2026 10:57:56 -0400https://liberty.io/how-do-you-work-out-the-angle-in-this-square.htmlhttps://liberty.io/how-do-you-work-out-the-angle-in-this-square.htmlI have labelled all the angles that I can work out. But I can't think of any other way to find the other angles without being 100% sure. Thank you! P.S. I have attached the official question - I......Sun, 23 Aug 2026 10:55:14 -0400https://liberty.io/interpreting-meaning-of-cauchy-series-not-sequence.htmlhttps://liberty.io/interpreting-meaning-of-cauchy-series-not-sequence.htmlI am trying to prove that some particular infinite series is Cauchy. Of course, I know what it means for a sequence to be Cauchy, but I can't seem to find anything related to Cauchy series. Could s......Sun, 23 Aug 2026 10:49:32 -0400https://liberty.io/the-difference-between-a-set-and-a-subset.htmlhttps://liberty.io/the-difference-between-a-set-and-a-subset.htmlI am currently taking real analysis, and I'm afraid this might be a very elementary question, but the further I go in the course the more confused I become about the difference between a set in $\m......Sun, 23 Aug 2026 10:45:54 -0400https://liberty.io/is-there-a-mathematical-notation-for-repeat-x-until-y.htmlhttps://liberty.io/is-there-a-mathematical-notation-for-repeat-x-until-y.htmlThe repeat X until Y methodology is a key component of many algorithms. Yet, I haven't come across a good mathematical notation for it. (Perhaps because it is of little use in manipulating expressi......Sun, 23 Aug 2026 10:45:52 -0400https://liberty.io/meaning-of-topology-and-topological-space.htmlhttps://liberty.io/meaning-of-topology-and-topological-space.htmlAfter looking at the Wikipedia article on topological space, I still cannot grasp intuitively what topological space is. For example, if we are to define topology on real numbers, can there be many ...Sun, 23 Aug 2026 10:39:23 -0400https://liberty.io/number-of-special-ordered-set.htmlhttps://liberty.io/number-of-special-ordered-set.htmlAn ordered set is a set such that the order in which the objects appear in the set is significant. For example, (1, 2, 3) and (2, 1, 3) are two different ordered sets of integers. An ordered set of ...Sun, 23 Aug 2026 10:28:29 -0400https://liberty.io/determine-whether-the-solutions-are-stable-or-unstable.htmlhttps://liberty.io/determine-whether-the-solutions-are-stable-or-unstable.htmlDetermine whether the solutions x(t)=0 and x(t)=1 of the single scalar equation $dx/dt=-x(1-x)$ are stable or unstable. So far in the book I have just done problems like this except with dx/dt=(some ...Sun, 23 Aug 2026 10:24:55 -0400https://liberty.io/calculate-this-triple-integral-in-cylindrical-coordinates-the-result-i.htmlhttps://liberty.io/calculate-this-triple-integral-in-cylindrical-coordinates-the-result-i.htmlI want to calculate triple integral \begin{equation}\int\limits_{-1}^{1}\int\limits_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int\limits_{x^2+y^2}^1 2z dzdydx.\end{equation} (the surface is $z=x^2+y^2$, $0\le......Sun, 23 Aug 2026 10:19:17 -0400https://liberty.io/what-exactly-is-a-scalar-closed.htmlhttps://liberty.io/what-exactly-is-a-scalar-closed.htmlParticularly, referring to scalars I am only implying to individual numbers, for this instance, let's take the set R(real numbers). These numbers do lie of the real line, i.e. the x axis, which is ...Sun, 23 Aug 2026 10:19:09 -0400https://liberty.io/notation-for-the-first-element-in-a-set.htmlhttps://liberty.io/notation-for-the-first-element-in-a-set.htmlI'm looking for a widely used notation to describe the first element of a set. E.g.: S = {5, 7,...,123} is the set, and obviously 5 is the first element. S = {345, 123,...,33} is the set, and ...Sun, 23 Aug 2026 10:10:42 -0400https://liberty.io/is-there-some-consensus-on-the-dimensions-of-a-jacobian-matrix-and-of-.htmlhttps://liberty.io/is-there-some-consensus-on-the-dimensions-of-a-jacobian-matrix-and-of-.htmlAccording to Wikipedia, given a differentiable mapping $F: \mathbb{R}^n \to \mathbb{R}^m$, its Jacobian matrix is a $m \times n$ matrix defined as: $$ J_F=\begin{bmatrix} \dfrac{\partial y_1}{\par......Sun, 23 Aug 2026 10:07:46 -0400https://liberty.io/dividing-an-obtuse-triangle-into-acute-triangles.htmlhttps://liberty.io/dividing-an-obtuse-triangle-into-acute-triangles.htmlCan an obtuse triangle be subdivided into only acute triangles (right triangles are not allowed)? Any number of subdivisions can be made as long as all of the angles in all resulting triangles ar......Sun, 23 Aug 2026 10:00:18 -0400https://liberty.io/oblong-or-rectangle-oval-or-ellipse-why-do-we-have-different-names-for.htmlhttps://liberty.io/oblong-or-rectangle-oval-or-ellipse-why-do-we-have-different-names-for.htmlI am wondering why, sometimes, we have two different names for the same shape (for example, oblong and rectangle or oval and ellipse)? Is there one which is preferred over the other? Do they have ......Sun, 23 Aug 2026 09:59:22 -0400https://liberty.io/linear-least-squares-with-linear-inequality-constraints.htmlhttps://liberty.io/linear-least-squares-with-linear-inequality-constraints.htmlI'm trying to follow this older paper, page 19. The goal is to solve: $\min \|Ax-b\|^2 s.t. Gx \ge h$ given $A, G, b, h$ By combining the equations into a single LCP of the form: $Mz + q = w$ s.t......Sun, 23 Aug 2026 09:57:27 -0400https://liberty.io/difference-between-equilibrium-point-and-unique-equilibrium-point.htmlhttps://liberty.io/difference-between-equilibrium-point-and-unique-equilibrium-point.htmlIs there any difference between unique equilibrium point and equilibrium point? If yes, please tell me what it is and How is it used to solve a dynamical system. You can consider a dynamical syste......Sun, 23 Aug 2026 09:49:52 -0400https://liberty.io/density-of-measure.htmlhttps://liberty.io/density-of-measure.htmlLet $dx$ be the Lebesgue measure on $\mathbb R^d$. Let $u:\mathbb R^d\to{\mathbb R}\cup\{\infty\}$ is a non-negative and measurable function. The question is that, what are the conditions on $u$ so......Sun, 23 Aug 2026 09:44:24 -0400https://liberty.io/how-does-summation-index-shifting-work.htmlhttps://liberty.io/how-does-summation-index-shifting-work.htmlI'm looking over the solution to a test problem on recurrence relations, and I am stuck. I can't figure out the jump that was made. All that it says is "To simplify the algebra, do an index shift:......Sun, 23 Aug 2026 09:44:21 -0400https://liberty.io/arithmetic-overflow-and-underflowing.htmlhttps://liberty.io/arithmetic-overflow-and-underflowing.htmlI am a bit unclear about underflowing in terms of binary representation. Let's say that an unsigned 8-bit variable gets overflown from the addition of $150+150$. A signed 8-bit variable gets ...Sun, 23 Aug 2026 09:39:54 -0400https://liberty.io/proof-of-the-asa-triangle-congruence-criterium.htmlhttps://liberty.io/proof-of-the-asa-triangle-congruence-criterium.htmlSo we studied the triangle congruence criteria and we proved the ASA criterium: If two triangles have respectively congruent two angles and the side included in them, then the two triangles are con......Sun, 23 Aug 2026 09:30:13 -0400https://liberty.io/deriving-the-center-of-mass-of-a-semi-circular-disk-with-cylindrical-c.htmlhttps://liberty.io/deriving-the-center-of-mass-of-a-semi-circular-disk-with-cylindrical-c.htmlProblem: Derive the Center of Mass of a semi-circular disk of mass $M$ and radius $R$. My attempt: $$Y_{CM}=\int ydm$$ Now, $$dm=\sigma dA$$ where $\sigma$ is mass per unit area. Convertin......Sun, 23 Aug 2026 09:29:57 -0400https://liberty.io/what-is-the-magnitude-of-dot-product-of-two-vectors.htmlhttps://liberty.io/what-is-the-magnitude-of-dot-product-of-two-vectors.htmlThis was the question (or rather to prove) given by our professor: $|(a.c)| \leq |a||c|$ where $a$ and $c$ are vectors in $\mathbb{R}^n$. My friend and I solved it in two different ways and now we ......Sun, 23 Aug 2026 09:26:00 -0400https://liberty.io/why-are-homeomorphisms-important.htmlhttps://liberty.io/why-are-homeomorphisms-important.htmlI attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of alg......Sun, 23 Aug 2026 09:20:09 -0400https://liberty.io/volume-of-a-pyramid-with-equilateral-triangle-as-base.htmlhttps://liberty.io/volume-of-a-pyramid-with-equilateral-triangle-as-base.htmlThe base of a right pyramid is an equilateral triangle of side 4cm each. Each slant edge is 5cm long. The volume of pyramid is For this question, answer provided by sscadda.com is $\dfrac{20......Sun, 23 Aug 2026 09:17:58 -0400https://liberty.io/how-to-show-that-the-product-of-two-irrational-numbers-may-be-irration.htmlhttps://liberty.io/how-to-show-that-the-product-of-two-irrational-numbers-may-be-irration.htmlShow that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot \sqrt......Sun, 23 Aug 2026 09:01:28 -0400https://liberty.io/egorov-039-s-theorem-on-intervals.htmlhttps://liberty.io/egorov-039-s-theorem-on-intervals.htmlFrm Folland's Real Analysis exercise 2.43 part (b): Suppose that $(X, \mathcal M, \mu)$ is a measure space, with $\mu(X) < \infty$. Let $f: X × [0,1] \to \mathbb C$ is a function such that $f......Sun, 23 Aug 2026 08:58:39 -0400https://liberty.io/intuition-degeneracy-maps.htmlhttps://liberty.io/intuition-degeneracy-maps.htmlI'm trying to get an intuitive understanding for the notion of a simplicial set. Roughly, a simplicial set consists of a set $S_n$ of $n$-simplices for each non-negative $n$, and families of face m......Sun, 23 Aug 2026 08:46:36 -0400https://liberty.io/why-are-all-circles-similar-why-is-pi-a-constant-duplicate.htmlhttps://liberty.io/why-are-all-circles-similar-why-is-pi-a-constant-duplicate.htmlI just know that I'm going to look like a crackpot, but here goes. The number $\pi$ is defined as the ratio of the circumference of a circle to its diameter. So there is an assumption here that all ...Sun, 23 Aug 2026 08:46:08 -0400https://liberty.io/product-between-a-column-vector-and-a-row-vector.htmlhttps://liberty.io/product-between-a-column-vector-and-a-row-vector.htmlI know that matrices product is correct when the number of the columns of the first matrix is equal to the number of rows of the second matrix. Why I can't do the product between a column vector a......Sun, 23 Aug 2026 08:43:03 -0400https://liberty.io/mclaurins-with-e-sin-x.htmlhttps://liberty.io/mclaurins-with-e-sin-x.htmlTo evaluate $e^{\sin(x)}$ I use the standard series $e^t$ and $\sin(t)$, combining them gives me: $e^t = 1+t+\dfrac{t^2}{2!}+\dfrac{t^3}{3!}+\dfrac{t^4}{4!}+O(t^5)$ $\sin(t) = t-\dfrac{t^3}{3!}+O......Sun, 23 Aug 2026 08:32:45 -0400https://liberty.io/origin-of-the-dot-and-cross-product.htmlhttps://liberty.io/origin-of-the-dot-and-cross-product.htmlMost questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...Sun, 23 Aug 2026 08:30:49 -0400https://liberty.io/understanding-the-proof-for-the-divergence-of-the-harmonic-series.htmlhttps://liberty.io/understanding-the-proof-for-the-divergence-of-the-harmonic-series.htmlI'm trying to understand the following proof for the divergence of the harmonic series: The parts which I can't seem to understand are: $$1. \sum_{i=0}^{n-1} \sum_{r=2^i+1}^{2^{i+1}} \frac{1}{r} ......Sun, 23 Aug 2026 08:22:36 -0400https://liberty.io/prove-sin-pi-2-1-using-taylor-series.htmlhttps://liberty.io/prove-sin-pi-2-1-using-taylor-series.htmlProve $\sin(\pi/2)=1$ using the Taylor series definition of $\sin x$, $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ It seems rather messy to substitute in $\pi/2$ for $x$. So we have $$\sin......Sun, 23 Aug 2026 08:20:32 -0400https://liberty.io/why-is-a-linear-combination-of-solutions-also-a-solution.htmlhttps://liberty.io/why-is-a-linear-combination-of-solutions-also-a-solution.htmlI'm working on DEs and I've come across some speedbumps. I'm not very familiar with linear algebra and that might be the problem. However, I'm sure that there's a non-linear algebra explanation t......Sun, 23 Aug 2026 08:06:51 -0400https://liberty.io/3d-coordinate-rotation-using-roll-pitch-yaw.htmlhttps://liberty.io/3d-coordinate-rotation-using-roll-pitch-yaw.htmlGiven an orthogonal unit vector triad (i,j,k) described in the 3-dimensional coordinate space (X,Y,Z), as shown below. I need to rotate the XYZ coordinate system such that X is collinear with i, Y......Sun, 23 Aug 2026 08:02:31 -0400https://liberty.io/area-of-octagon-constructed-in-a-square.htmlhttps://liberty.io/area-of-octagon-constructed-in-a-square.htmlThe following picture is constructed by connecting each corner of a square with the midpoint of a side from the square that is not adjacent to the corner. These lines create the following red octag......Sun, 23 Aug 2026 07:57:10 -0400https://liberty.io/must-a-complete-subgraph-be-induced.htmlhttps://liberty.io/must-a-complete-subgraph-be-induced.htmlIf we had some $K_n$ subgraph where $K_n \subseteq G$, must the complete subgraph $K_n$ be an induced subgraph from $G$? In other words, can we create a situation where we remove vertices from a si......Sun, 23 Aug 2026 07:49:50 -0400https://liberty.io/arc-length-using-integration.htmlhttps://liberty.io/arc-length-using-integration.htmlIn Thomas's calculus $10^{th}$ edition, chapter $5$, exercises $5.3$, problem $9$, I am asked to find the length of an arc with the equation: $$x=\int^y_0\sqrt{\sec^4(t)-1} dt$$ and $\displaystyl......Sun, 23 Aug 2026 07:42:25 -0400https://liberty.io/express-18x-2-12x-48-in-mathbb-q-x-as-the-product-of-a-constant-polyno.htmlhttps://liberty.io/express-18x-2-12x-48-in-mathbb-q-x-as-the-product-of-a-constant-polyno.htmlI believe I get the correct answer for the following problem but the solution says it is wrong. I have no idea why. Express $18X^2-12X+48$ in $\mathbb{Z}[X]$ and in $\mathbb{Q}[X]$ as the produc......Sun, 23 Aug 2026 07:41:32 -0400https://liberty.io/given-a-matrix-b-what-is-det-b-4.htmlhttps://liberty.io/given-a-matrix-b-what-is-det-b-4.htmlMy task is this: Compute det$(B^4)$, where $$B =\begin{pmatrix} 1 & 0 & 1\\ 1 & 1 & 2\\ 1 & 2 & 1 \end{pmatrix}$$ My work so far: It can be shown that $B = \begin{pmat......Sun, 23 Aug 2026 07:25:17 -0400https://liberty.io/finding-the-solution-set-for-a-quadratic-inequality-x-2-2-frac-7-2-x.htmlhttps://liberty.io/finding-the-solution-set-for-a-quadratic-inequality-x-2-2-frac-7-2-x.htmlIf $x^2-2<\frac{7}{2}x$ then what is the solution set for $x$? I have most of the problem done, I just don't know how to lay out my answer. The answer is supposed to be $$\boxed{-\frac12<x&l......Sun, 23 Aug 2026 07:15:22 -0400https://liberty.io/find-the-differential-equation-of-all-circles-of-radius-a-closed.htmlhttps://liberty.io/find-the-differential-equation-of-all-circles-of-radius-a-closed.htmlCan someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle....Sun, 23 Aug 2026 07:15:14 -0400https://liberty.io/fourier-cosine-expansion-of-sin-x.htmlhttps://liberty.io/fourier-cosine-expansion-of-sin-x.htmlInstructor mentioned something in class that got me thinking... Why is it possible for $\sin x$ to have a Fourier Cosine expansion in $[0, \pi]$? It seems strange to me that you're able to express......Sun, 23 Aug 2026 07:05:24 -0400https://liberty.io/a-cardinality-of-a-graph.htmlhttps://liberty.io/a-cardinality-of-a-graph.htmlIf I have graph $G=(V,E)\\$ What is the meaning of $|G|$? (The cardinality of G). I'd like to know few words about it. Thank you!...Sun, 23 Aug 2026 07:01:08 -0400https://liberty.io/why-do-4-digit-binary-numbers-have-16-solutions-and-3-digits-8-closed.htmlhttps://liberty.io/why-do-4-digit-binary-numbers-have-16-solutions-and-3-digits-8-closed.htmlIf I have a 4 digit binary number, how come there are 16 possible "numbers", and similarly, if I have a 4 digit binary number, how come there are 8 possible "numbers?"...Sun, 23 Aug 2026 06:51:25 -0400https://liberty.io/how-do-computers-draw-function-graphs.htmlhttps://liberty.io/how-do-computers-draw-function-graphs.htmlCould anyone give me a general overview of the algorithm computers use to draw graphs? I'm guessing it is either by graphing many, many points and just connecting them or by doing a general study ......Sun, 23 Aug 2026 06:48:41 -0400https://liberty.io/does-liouville-039-s-theorem-work-in-real-analysis.htmlhttps://liberty.io/does-liouville-039-s-theorem-work-in-real-analysis.htmlLiouville's Theorem states that every bounded entire function must be constant. Does it work in real analysis? Justify your answer! I asked it because Liouville's Theorem is proved by complex analy......Sun, 23 Aug 2026 06:40:50 -0400https://liberty.io/linearity-of-determinant.htmlhttps://liberty.io/linearity-of-determinant.htmlLinearity property of Determinant function: For an $n\times n$ matrix $A$, we can consider det$A$ as a function of the $n$ column vectors in $A$. We will show that if all columns except one are held ...Sun, 23 Aug 2026 06:35:15 -0400https://liberty.io/what-is-symmetric-about-set-symmetric-difference.htmlhttps://liberty.io/what-is-symmetric-about-set-symmetric-difference.htmlUsing $\Delta$ for set symmetric difference, $A \Delta B$ is all the elements in exactly one of the sets but not all of them. $A \Delta B \Delta C $ is all the elements in exactly one of the sets......Sun, 23 Aug 2026 06:33:52 -0400https://liberty.io/how-to-express-the-given-disk-d-in-polar-coordinates.htmlhttps://liberty.io/how-to-express-the-given-disk-d-in-polar-coordinates.htmlGiven $D=\{(x,y)|0\le x\le2,0\le y\le\sqrt{4-x^2}\}$ Express $D$ in polar coordinates. I have the answer as $D=\{(r,\theta)|0\le\theta\le\dfrac{\pi}{2},0\le r\le2\}$ One more example is $D=\{(x,y......Sun, 23 Aug 2026 06:15:49 -0400https://liberty.io/on-integration-when-solving-differential-equations-specifically-separa.htmlhttps://liberty.io/on-integration-when-solving-differential-equations-specifically-separa.htmlSo here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a so......Sun, 23 Aug 2026 06:05:20 -0400https://liberty.io/height-of-an-isosceles-trapezoid.htmlhttps://liberty.io/height-of-an-isosceles-trapezoid.htmlSuppose we have an isosceles trapezoid whose length of the associated square is $20$ and the length of the hypotenuse of the triangles is $30$. I want to determine if it is possible to find the hei......Sun, 23 Aug 2026 06:05:04 -0400https://liberty.io/learning-mathematics-using-only-audio.htmlhttps://liberty.io/learning-mathematics-using-only-audio.htmlAre there any mathematics audio books or other audio sources for learning mathematics, like for example math podcasts which really go into detail. I ask this because I make about 1 hour from my hou......Sun, 23 Aug 2026 05:59:05 -0400https://liberty.io/surface-integral-over-graph-of-a-function.htmlhttps://liberty.io/surface-integral-over-graph-of-a-function.htmlQ: Evaluate the surface integral $\iint_S \mathbf{F}\cdot d\mathbf{S}$, where $\mathbf{F}(x,y,z)= 3x^2\mathbf{i}-2xy\mathbf{j}+8\mathbf{k}$, and $S$ is the graph of the function $z=f(x,y)=2x-y$ fo......Sun, 23 Aug 2026 05:57:12 -0400https://liberty.io/help-solving-2-second-order-differential-equation.htmlhttps://liberty.io/help-solving-2-second-order-differential-equation.html$y'' + (y')^2 = y$ and $y*y''= (y')^2 - (y')^3$ I know that both of these equations can be solved by using $p(y) = y'$, but Im getting stuck in the middle, and can't seem to solve the thing aft......Sun, 23 Aug 2026 05:56:42 -0400https://liberty.io/convert-expression-to-nand-only.htmlhttps://liberty.io/convert-expression-to-nand-only.htmlI have to convert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot \bar{C} + A \cdot B\cdot C$$ I've looked at the following website: http://en....Sun, 23 Aug 2026 05:48:23 -0400https://liberty.io/if-sin-theta-cos-theta-tan-theta-cot-theta-sec-theta-csc-theta-7-then-.htmlhttps://liberty.io/if-sin-theta-cos-theta-tan-theta-cot-theta-sec-theta-csc-theta-7-then-.html$$ 0<\theta<\pi/2$$ and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$ then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$ I tried to use ......Sun, 23 Aug 2026 05:41:04 -0400https://liberty.io/which-of-the-following-statements-are-true-about-finite-cyclic-groups.htmlhttps://liberty.io/which-of-the-following-statements-are-true-about-finite-cyclic-groups.html$1.$ If $G$ is a finite cyclic group, then for every subgroup $H,K$ of $G,$ we have either $H\subset K$ or $K\subset H.$ $2.$ If for every subgroup $H,K$ of a finite group $G,$ we have either $H\su......Sun, 23 Aug 2026 05:34:00 -0400https://liberty.io/da-in-polar-coordinates.htmlhttps://liberty.io/da-in-polar-coordinates.htmlI have seen a picture for $dV$ so that $dV = r^{2} \sin(\theta)\,dr\,d\theta\,d\phi$. But how can I deduce things like $dA$ and $dV$? In a simpler coordinate (not sure about the name), $dA = r \,dr......Sun, 23 Aug 2026 05:30:08 -0400https://liberty.io/are-the-domain-and-range-of-a-function-and-its-inverse-always-exactly-.htmlhttps://liberty.io/are-the-domain-and-range-of-a-function-and-its-inverse-always-exactly-.htmlThe function $ f(x)=\sqrt{x+4} $ has domain $[-4,\infty)$ and range $[0,\infty)$. Its inverse function $ f^{-1}(x)=x^2-4 $ by itself seems to have domain $\Bbb{R}$ and range $[-4,\infty)$. Howeve......Sun, 23 Aug 2026 05:26:53 -0400https://liberty.io/proving-the-inverse-of-the-invertible-matrix-theorem.htmlhttps://liberty.io/proving-the-inverse-of-the-invertible-matrix-theorem.htmlLet $A$ be an $n \times n$ invertible matrix and $v_1,v_2,...,v_m$ an element of $\mathbb{R}^{n}$. If $\{v_1,v_2,...,v_m\}$ spans $\mathbb{R}^{n}$ then $\{Av_1, Av_2,...,Av_m\}$ also spans $\mathbb......Sun, 23 Aug 2026 05:26:39 -0400https://liberty.io/diagonal-terms-of-the-inverse-of-positive-definite-matrix.htmlhttps://liberty.io/diagonal-terms-of-the-inverse-of-positive-definite-matrix.htmlI have a matrix $A = I + G$ where $G_{ii} = 0$ for all $i$, and $G_{ij} \in \{0, a\}$ with $a \in [0, 1[$, $G$ is symmetric, and such that $A$ is positive definite. We know that the inverse of $A$,......Sun, 23 Aug 2026 05:23:46 -0400https://liberty.io/arc-length-of-y-1-3-sqrt-x-x-3.htmlhttps://liberty.io/arc-length-of-y-1-3-sqrt-x-x-3.html$y = 1/3 \sqrt{x} (x-3)$ from 1-9 In my book it is actually $x = 1/3 \sqrt{y} (y-3)$ but I prefer to work with y so I just swap the two variables and I think everything should be the same. The fi......Sun, 23 Aug 2026 05:18:57 -0400https://liberty.io/what-is-the-difference-between-l-2-norm-and-ell-2-norm.htmlhttps://liberty.io/what-is-the-difference-between-l-2-norm-and-ell-2-norm.htmlI can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?...Sun, 23 Aug 2026 05:15:41 -0400https://liberty.io/how-to-solve-trigonometric-equations-without-a-calculator-duplicate.htmlhttps://liberty.io/how-to-solve-trigonometric-equations-without-a-calculator-duplicate.htmlHow does one solve for sine, cosine, or tangent (and inverse sine, cosine, tangent) without the use of a calculatour? For example: cosine of 131 degrees — how must one calculate this?...Sun, 23 Aug 2026 04:56:01 -0400