https://liberty.ioCurated star highlights with fresh energy.en-usTue, 01 Sep 2026 23:34:44 -0400https://liberty.io/why-do-we-add-a-zero-to-dividend-during-long-division.htmlhttps://liberty.io/why-do-we-add-a-zero-to-dividend-during-long-division.htmlSuppose we want to divide 3 by 2. By Long Division we first write 1 as the first digit as the quotient, then we subtract 2 from 3, then we add a zero to the remainder 1, and then add a decimal point ...Wed, 02 Sep 2026 03:33:49 -0400https://liberty.io/find-the-gradient-and-equation-of-the-normal-to-the-curve-y-x-3-x-5-at.htmlhttps://liberty.io/find-the-gradient-and-equation-of-the-normal-to-the-curve-y-x-3-x-5-at.htmlI was wondering if anyone can help with this question.I'm just not sure how to find the gradients and equations of these questions. If you could help it would be greatly appreciated. $$y=x^3+x+5$$...Wed, 02 Sep 2026 03:30:31 -0400https://liberty.io/limit-points-and-interior-points.htmlhttps://liberty.io/limit-points-and-interior-points.htmlI am reading Rudin's book on real analysis and am stuck on a few definitions. First, here is the definition of a limit/interior point (not word to word from Rudin) but these definitions are worde......Wed, 02 Sep 2026 03:29:15 -0400https://liberty.io/calculus-maximum-volume-equation.htmlhttps://liberty.io/calculus-maximum-volume-equation.htmlI tried my best to come up with an answer but I failed. I would really appreciate if someone can help me with this: Question: To carry a suitcase on an airplace, the length + width + height must......Wed, 02 Sep 2026 03:27:17 -0400https://liberty.io/is-there-a-name-for-the-point-of-a-exponential-curve-where-the-y-axis-.htmlhttps://liberty.io/is-there-a-name-for-the-point-of-a-exponential-curve-where-the-y-axis-.htmlIt's been hard to come up with a question title that makes sense so please bear with me. On an exponential curve there's a point on the x axis where the y axis starts increasing significantly. The......Wed, 02 Sep 2026 03:20:36 -0400https://liberty.io/derivative-of-determinant-of-a-matrix.htmlhttps://liberty.io/derivative-of-determinant-of-a-matrix.htmlI would like to know how to calculate: $$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big).$$...Wed, 02 Sep 2026 02:55:48 -0400https://liberty.io/evaluating-lim-x-to0-frac-2-cot-2x-cot-x-sin-2x.htmlhttps://liberty.io/evaluating-lim-x-to0-frac-2-cot-2x-cot-x-sin-2x.htmlWhat is the value of $\lim_{x\to0}\frac{2\cot 2x-\cot x}{\sin 2x}$ ? $1)\text{zero}\qquad\qquad2)\frac12\qquad\qquad3)-\frac12\qquad\qquad4)-1$ Here is my work: $$\lim_{x\to0}\frac{2\cot 2x-\cot x......Wed, 02 Sep 2026 02:35:36 -0400https://liberty.io/is-zero-positive-or-negative.htmlhttps://liberty.io/is-zero-positive-or-negative.htmlFollow up to this question. Is $0$ a positive number?...Wed, 02 Sep 2026 02:32:40 -0400https://liberty.io/why-is-mathbb-f-3-x-x-2-1-cong-mathbb-f-9.htmlhttps://liberty.io/why-is-mathbb-f-3-x-x-2-1-cong-mathbb-f-9.htmlI want to show that $\mathbb Z[i]/(3) \cong \mathbb F_9$, and I know that $\mathbb Z[i]/(3) \cong \mathbb Z[x]/(3, x^2 + 1) \cong \mathbb F_3[x]/(x^2 + 1)$. How do we know that $\mathbb F_3[x]/(x^......Wed, 02 Sep 2026 02:28:10 -0400https://liberty.io/given-system-of-equations-a-b-2-ab-4-solve-a-2-b-2-and-a-3.htmlhttps://liberty.io/given-system-of-equations-a-b-2-ab-4-solve-a-2-b-2-and-a-3.htmlI am trying to solve $a^2+b^2$ and $a^3$ given $a+b = 2, ab=4$. I have the key with the answers $a^2+b^2=-4$ and $a^3=-8$ but am wondering which steps to take to get to that answer. My understand......Wed, 02 Sep 2026 02:09:47 -0400https://liberty.io/set-of-positive-integers-with-unique-sums.htmlhttps://liberty.io/set-of-positive-integers-with-unique-sums.htmlWhat I'm looking for is the name of a type of number set. Given a number T (for total) and a set of positive integers S, I want to uniquely identify the subset of S that sums to T. All sets contain......Wed, 02 Sep 2026 02:05:19 -0400https://liberty.io/list-of-ways-to-tell-if-degree-sequence-is-impossible-for-a-simple-gra.htmlhttps://liberty.io/list-of-ways-to-tell-if-degree-sequence-is-impossible-for-a-simple-gra.htmlI'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$ is not possible because there are only 3 vertices in total so one can't have degree 3. The ......Wed, 02 Sep 2026 02:02:50 -0400https://liberty.io/in-a-venn-diagram-where-are-other-number-sets-located.htmlhttps://liberty.io/in-a-venn-diagram-where-are-other-number-sets-located.htmlI remember of this image I've learned at school: I've heard about other number (which I'm not really sure if they belong to a new set) such as quaternions, p-adic numbers. Then I got three questio......Wed, 02 Sep 2026 01:44:29 -0400https://liberty.io/uniqueness-of-adjoint-functors-up-to-isomorphism.htmlhttps://liberty.io/uniqueness-of-adjoint-functors-up-to-isomorphism.htmlSuppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...Wed, 02 Sep 2026 01:31:24 -0400https://liberty.io/completeness-of-probability-space-and-p-null-set.htmlhttps://liberty.io/completeness-of-probability-space-and-p-null-set.htmlCompleteness of probability space is often equivalent to the sigma algebra of the space contains all $P$-null sets. Why does this definition make intuitive sense? Why would it be "incomplete" if the ...Wed, 02 Sep 2026 01:29:21 -0400https://liberty.io/how-can-a-gradient-be-thought-of-as-a-function.htmlhttps://liberty.io/how-can-a-gradient-be-thought-of-as-a-function.htmlIf a function $f(x,y)$ would output a value in a third dimension, $z=f(x,y)$ for example. How can we treat the gradient of $f$ as a function in $x$ and $y$, when the output of the gradient is a vec......Wed, 02 Sep 2026 01:20:27 -0400https://liberty.io/determine-if-the-sequence-an-ln-n-n-1-n-is-convergent.htmlhttps://liberty.io/determine-if-the-sequence-an-ln-n-n-1-n-is-convergent.htmlI have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!...Wed, 02 Sep 2026 01:17:24 -0400https://liberty.io/notation-for-every-odd-integer-number.htmlhttps://liberty.io/notation-for-every-odd-integer-number.htmlI have this equation: $$f(x)=\tan(x)$$ I found the vertical asymptotes to be: $$x=\frac{\pi}{2}k$$ What is the proper notation for that k is equal to every odd number integer(negative,posi......Wed, 02 Sep 2026 01:14:42 -0400https://liberty.io/tangent-normal-and-binormal-vectors.htmlhttps://liberty.io/tangent-normal-and-binormal-vectors.htmlLet $C$ be the graph of vector function $r(s)$ parametrized by arc length with $r'(0)= ({2\over3},{1\over3},{2\over3})$ and $r''(0)=(-3,12,-3)$. I need to find tangent vector $T(0)$, normal vec......Wed, 02 Sep 2026 01:13:39 -0400https://liberty.io/geometric-meanings-of-hyperbolic-cosine-and-sine.htmlhttps://liberty.io/geometric-meanings-of-hyperbolic-cosine-and-sine.htmlIn euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry. Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosine and sine ?...Wed, 02 Sep 2026 01:08:00 -0400https://liberty.io/how-many-5-digit-zip-codes-can-there-be-with-exactly-two-digits-the-sa.htmlhttps://liberty.io/how-many-5-digit-zip-codes-can-there-be-with-exactly-two-digits-the-sa.htmlHow many 5-digit United States zip codes can there be with exactly one pair of digits? This includes both zip codes that correspond to an actual city, town, incorporated community, ghost town, etc.......Wed, 02 Sep 2026 01:06:40 -0400https://liberty.io/proof-that-the-three-interior-angles-of-a-triangle-is-congruent-to-a-s.htmlhttps://liberty.io/proof-that-the-three-interior-angles-of-a-triangle-is-congruent-to-a-s.htmlI'm trying to essentially prove that the interior angles of a triangle add up to 180 degrees. However, I'm trying to prove it without mentioning measurements of an angle. I think I understand the p......Wed, 02 Sep 2026 01:06:22 -0400https://liberty.io/a-constant-function-on-matlab-closed.htmlhttps://liberty.io/a-constant-function-on-matlab-closed.htmlclear clc eta = 100; a= 0; xi=0.016; axis([0 0.01 0 1.1]); x = 0:0.0001:0.01; N = -a+1.6*(1/(xi*eta)); y=N; plot(x,y,'LineWidth',2); set(gca,'FontSize',10); Hi. I'd like to plot the function $N(x......Wed, 02 Sep 2026 01:04:09 -0400https://liberty.io/applications-of-the-law-of-the-iterated-logarithm.htmlhttps://liberty.io/applications-of-the-law-of-the-iterated-logarithm.htmlThe law of the iterated logarithm says that if $X_n$ is a sequence of iid random variables with zero expectation and unit variance, then the partial sums sequence $S_n = \sum_{i = 1}^n X_i$ satisfies ...Wed, 02 Sep 2026 01:03:01 -0400https://liberty.io/adding-exponents-with-unknowns-with-the-same-base.htmlhttps://liberty.io/adding-exponents-with-unknowns-with-the-same-base.htmlHow do you add numbers with the same base but with unknown exponents? For example, $2^{x+2} + 2^{x+2}$ I understand that you do $2^x \cdot 2^2$ but I get stuck here. I don't know what to do from he......Wed, 02 Sep 2026 01:01:13 -0400https://liberty.io/function-map-notation.htmlhttps://liberty.io/function-map-notation.htmlPlease forgive my ignorance, if I've phrased my question improperly. I'm not sure what the appropriate terminology is; that's the basis of my question. So, I'm not sure if I'm even remotely close i......Wed, 02 Sep 2026 00:47:35 -0400https://liberty.io/find-the-equation-of-the-normal-to-the-curve.htmlhttps://liberty.io/find-the-equation-of-the-normal-to-the-curve.htmlGood day everyone. The normal to the curve $y=x^2-2x$ at a point P is parallel to the straight line $y=-2x+5$ .Find the equation of the normal to the curve at point P. What does it question mean? ...Wed, 02 Sep 2026 00:38:28 -0400https://liberty.io/how-to-simplify-frac-sin-alpha-sin-beta-cos-beta-cos-alpha-cdot-cos-al.htmlhttps://liberty.io/how-to-simplify-frac-sin-alpha-sin-beta-cos-beta-cos-alpha-cdot-cos-al.htmlThere is this problem: $$\left(\frac{\sin\alpha - \sin\beta}{\cos\beta -\cos\alpha}\right)\cdot \cos\alpha \cdot \cos\beta = \frac{1}{\tan\alpha -\tan\beta}$$ I started as $$\left(\frac{\sin\alpha ......Wed, 02 Sep 2026 00:36:00 -0400https://liberty.io/uniqueness-of-payout-of-nash-equilibrium-in-a-2-player-zero-sum-game.htmlhttps://liberty.io/uniqueness-of-payout-of-nash-equilibrium-in-a-2-player-zero-sum-game.htmlI'm convinced that in a 2 player zero sum game, any two NE strategies must have the same expected payout for both players. I'm even more convinced that this is not true for 3 or more players. Where......Wed, 02 Sep 2026 00:16:46 -0400https://liberty.io/how-to-show-that-this-matrix-is-positive-semidefinite.htmlhttps://liberty.io/how-to-show-that-this-matrix-is-positive-semidefinite.htmlUsing the definition, show that the following matrix is positive semidefinite. $$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$ In other words, if the ...Wed, 02 Sep 2026 00:11:20 -0400https://liberty.io/creating-a-tridiagonal-matrix-in-matlab.htmlhttps://liberty.io/creating-a-tridiagonal-matrix-in-matlab.htmlHow can I create a tridiagonal matrix that I can use for Crout factorization? And, I don't have any codes on how to create one since I am new to matlab. Ok, please help me understand what does the ...Wed, 02 Sep 2026 00:05:31 -0400https://liberty.io/approximation-vs-interpolation.htmlhttps://liberty.io/approximation-vs-interpolation.htmlSorry if this is a silly question, i'm just getting back into math after a long time away. My question is regarding approximation and interpolation. In which cases is it appropriate for one technique ...Wed, 02 Sep 2026 00:05:10 -0400https://liberty.io/must-a-comeager-set-be-dense.htmlhttps://liberty.io/must-a-comeager-set-be-dense.htmlSay, on a complete separable metric space. Separable probably doesn't matter. It's easy to see the opposite; if $D$ is a dense $G_\delta$ set, it's a countable intersection of open sets which con......Wed, 02 Sep 2026 00:03:11 -0400https://liberty.io/is-there-a-general-matrix-to-reflect-about-the-line-y-mx-c.htmlhttps://liberty.io/is-there-a-general-matrix-to-reflect-about-the-line-y-mx-c.htmlI have been looking into matrix transformations and found the following matrix to reflect about the line $y=(\tan\theta)x$. $$R = \begin{bmatrix} \cos(2\theta)& \sin(2\theta)\\ \s......Tue, 01 Sep 2026 23:58:16 -0400https://liberty.io/whats-the-rule-for-putting-up-a-plus-minus-sign-when-taking-under-root.htmlhttps://liberty.io/whats-the-rule-for-putting-up-a-plus-minus-sign-when-taking-under-root.htmlGiven a simple equation.... $\ (x+1)^2 =21 $ if we take the under root of both sides , we get $\ x+1 = \pm \sqrt{21} $ why dont we get a $ \pm $ on the left hand side ?...Tue, 01 Sep 2026 23:52:49 -0400https://liberty.io/how-do-you-find-the-altitude-in-a-pyramid-sat-math-question.htmlhttps://liberty.io/how-do-you-find-the-altitude-in-a-pyramid-sat-math-question.htmlThe pyramid shown above has altitude h and a square base of side m. The four edges that meet at V, the vertex of the pyramid, each have length e. If e = m, what is the value of h in terms of m? A) $\...Tue, 01 Sep 2026 23:49:13 -0400https://liberty.io/mean-expected-value-or-expectation-of-a-constant.htmlhttps://liberty.io/mean-expected-value-or-expectation-of-a-constant.htmlAccording to Wikipedia, "the expected value of a constant is equal to the constant itself; i.e., if $c$ is a constant, then $\mathbb E[c] = c$." I am currently having a hard time picturing what this ...Tue, 01 Sep 2026 23:42:42 -0400https://liberty.io/how-to-obtain-the-gradient-in-polar-coordinates.htmlhttps://liberty.io/how-to-obtain-the-gradient-in-polar-coordinates.htmlI'm not sure on how to find the gradient in polar coordinates. The thing that troubles me the most is how to find the unit vectors $\hat{r}$ and $\hat{\theta}$. My approach for the rest is expressi......Tue, 01 Sep 2026 23:40:23 -0400https://liberty.io/calculate-the-tangent-plane-of-x-2y-xy-2-at-a-b-1-1.htmlhttps://liberty.io/calculate-the-tangent-plane-of-x-2y-xy-2-at-a-b-1-1.htmlWrite the equation of the tangent plane at the point $(a, b, f(a, b))$ on the graph of $f$. The equation to the tangent plane for partial derivatives is given as: $$z = f(a.b)+(x-a)\frac{\partial f}{\...Tue, 01 Sep 2026 23:16:45 -0400https://liberty.io/if-a-point-has-no-dimension-and-no-area-how-can-there-be-space.htmlhttps://liberty.io/if-a-point-has-no-dimension-and-no-area-how-can-there-be-space.htmlEveryone knows that a point on any plane has zero magnitude, height, width, or volume. A line is a collection of points, and a plane is a collection of lines, while a cube is a collection of planes......Tue, 01 Sep 2026 23:16:24 -0400https://liberty.io/symbolic-rule-help.htmlhttps://liberty.io/symbolic-rule-help.htmlThis is a basic algebra question, I could use a little assistant with I just want guidance with symbolic rule. A photocopier was bought for 4000 dollars and depreciates at a rate of $500 per year. ...Tue, 01 Sep 2026 23:15:29 -0400https://liberty.io/matlab-too-many-output-arguments-error-need-help-closed.htmlhttps://liberty.io/matlab-too-many-output-arguments-error-need-help-closed.htmlI don't really know how to explain what I'm after without an example. I guess the title could be better but I'm not sure how to describe the issue. I've got the following function function [x,y] ......Tue, 01 Sep 2026 23:11:16 -0400https://liberty.io/how-many-m-ms-do-you-have-to-draw-from-an-infinite-box-to-ensure-you-h.htmlhttps://liberty.io/how-many-m-ms-do-you-have-to-draw-from-an-infinite-box-to-ensure-you-h.htmlI recently ran across this interview screening question: You have a bag that contains 10 different colors of M&Ms. There is an equal amount of each color of M&M. You do not know how ......Tue, 01 Sep 2026 23:08:11 -0400https://liberty.io/how-to-identify-which-is-transverse-and-conjugate-axis-of-hyperbola.htmlhttps://liberty.io/how-to-identify-which-is-transverse-and-conjugate-axis-of-hyperbola.htmlGiven Equation of Hyperbola as $$\frac{(x-2y+3)^2}{20}-\frac{(2x+y-1)^2}{\frac{80}{9}}=1$$ we have axes equations as $x-2y+3=0$ and $2x+y-1=0$. But how to identify which is Transverse and which is ...Tue, 01 Sep 2026 23:03:25 -0400https://liberty.io/what-is-the-base-measure-in-measure-theory.htmlhttps://liberty.io/what-is-the-base-measure-in-measure-theory.htmlI see the term "base measure" used frequently about measures. I do not completely get what that exactly means: Some examples are: Let $\cal F$ be the space of all probability density functions ......Tue, 01 Sep 2026 23:00:18 -0400https://liberty.io/what-is-gcd-0-a-where-a-is-a-positive-integer.htmlhttps://liberty.io/what-is-gcd-0-a-where-a-is-a-positive-integer.htmlI have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is ...Tue, 01 Sep 2026 22:58:35 -0400https://liberty.io/what-is-the-summation-of-2-4-8-16-1024.htmlhttps://liberty.io/what-is-the-summation-of-2-4-8-16-1024.htmlI am trying to figure out how to write the summation for 2+4+8+16+...+1024. Clearly my answer below is wrong by 1: I tried to apply the following summation formula but I guess I did not do it prop......Tue, 01 Sep 2026 22:57:10 -0400https://liberty.io/trace-of-a-matrix-product.htmlhttps://liberty.io/trace-of-a-matrix-product.htmlLet $A \in \mathbb{R}^{n\times m}$, and $B\in \mathbb{R}^{m\times m}$. Let $A'$ denote the transpose of $A$. From this we know that : $A'\in \mathbb{R}^{m\times n}$ and $AB\in \mathbb{R}^{n\tim......Tue, 01 Sep 2026 22:57:01 -0400https://liberty.io/boolean-algebra-simplifying-multiple-xor-and-xnor.htmlhttps://liberty.io/boolean-algebra-simplifying-multiple-xor-and-xnor.htmlIs there any way to simplify a combination of XOR and XNOR gates in the following expression? I have tried multiple boolean theorems and I have not been able to simplify this any further: The simp......Tue, 01 Sep 2026 22:50:08 -0400https://liberty.io/intersection-between-line-segment-and-a-plane.htmlhttps://liberty.io/intersection-between-line-segment-and-a-plane.htmlIs there a very simple test to check if a line segment in $3D$ space cuts a plane? It is assumed we have the coordinates of the endpoints of the line segment, so $p_1,p_2$ and that we have the equ......Tue, 01 Sep 2026 22:45:07 -0400https://liberty.io/how-to-derive-an-equation-for-terminal-velocity-assuming-air-resistanc.htmlhttps://liberty.io/how-to-derive-an-equation-for-terminal-velocity-assuming-air-resistanc.htmlSo for my latest physics homework question, I had to derive an equation for the terminal velocity of a ball falling in some gravitational field assuming that the air resistance force was equal to s......Tue, 01 Sep 2026 22:44:14 -0400https://liberty.io/determining-if-a-relation-is-reflexive-symmetric-or-transitive-closed.htmlhttps://liberty.io/determining-if-a-relation-is-reflexive-symmetric-or-transitive-closed.htmlLet $A = \{0,1,2,3\}$ Define a relation $T$ on $A$ as follows: $T = \{(0,1),(2,3)\}$ Is $T$ reflexive? symmetric? transitive?...Tue, 01 Sep 2026 22:40:18 -0400https://liberty.io/find-volume-on-a-shape-with-base-of-an-ellipse.htmlhttps://liberty.io/find-volume-on-a-shape-with-base-of-an-ellipse.htmlI have an ellipse with area $\pi ab$ $a = 6$, $b = 4$ these are the axis lengths. I am suppose to compute the volume of a cone of height 12. I tried many solutions but none of them worked and I do......Tue, 01 Sep 2026 22:31:37 -0400https://liberty.io/p-series-convergence.htmlhttps://liberty.io/p-series-convergence.htmlShow that if $p>1$, $\sum\frac{1}{n^{p}}$ converges and if $p<1$ it diverges for $p\in\mathbb{R}^{+}$. Is there any way to show another series converges or diverges and then use the Comparison ...Tue, 01 Sep 2026 22:30:44 -0400https://liberty.io/how-would-one-be-able-to-prove-mathematically-that-1-1-2.htmlhttps://liberty.io/how-would-one-be-able-to-prove-mathematically-that-1-1-2.htmlIs it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?...Tue, 01 Sep 2026 22:26:28 -0400https://liberty.io/finding-the-virtual-center-of-a-cloud-of-points.htmlhttps://liberty.io/finding-the-virtual-center-of-a-cloud-of-points.htmlGiven: (latitude, longitude) points $P_1, P_2,\ldots, P_n$. Presumably, all the points should form a dense cloud. However, noise is possible. Needed: The virtual center of the points. For inst......Tue, 01 Sep 2026 22:25:53 -0400https://liberty.io/finding-eigenvalues-and-eigenfunctions-of-this-boundary-value-problem.htmlhttps://liberty.io/finding-eigenvalues-and-eigenfunctions-of-this-boundary-value-problem.htmlConsider the boundary value problem $$ -y'' + 2 y' = \lambda y $$ with boundary conditions $y(0) = y(L) = 0$. I need to determine the eigenvalues and eigenfunctions $y_n$. So we have the second or......Tue, 01 Sep 2026 22:20:15 -0400https://liberty.io/suggestions-for-studying-for-real-analysis-linear-algebra.htmlhttps://liberty.io/suggestions-for-studying-for-real-analysis-linear-algebra.htmlI apologize if this question is inappropriate for this site, but I'm new here and am not entirely sure where to direct it. I've just begun a course on real analysis and linear algebra at my univer......Tue, 01 Sep 2026 22:17:21 -0400https://liberty.io/how-to-find-a-generator-of-a-cyclic-group.htmlhttps://liberty.io/how-to-find-a-generator-of-a-cyclic-group.htmlA cyclic group is a group that is generated by a single element. That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$. This ...Tue, 01 Sep 2026 22:16:39 -0400https://liberty.io/newton-039-s-method-for-estimating-square-roots.htmlhttps://liberty.io/newton-039-s-method-for-estimating-square-roots.htmlSometime ago I wrote a program that used Newtons Method and derivatives to approximate unknown square roots (say $\sqrt 5$) from known square roots like $\sqrt 4$.I have since lost the calculator ......Tue, 01 Sep 2026 22:12:49 -0400https://liberty.io/how-to-solve-trigonometric-equations-of-the-form-tan-x-m-where-m-is-a-.htmlhttps://liberty.io/how-to-solve-trigonometric-equations-of-the-form-tan-x-m-where-m-is-a-.htmlHow to solve trigonometric equations of the form $\tan(x)=m$ where $m$ is a number? I know how to solve $\sin(x)=m$ where $m$ is a real number and $\cos(x)=m$ where $m$ is a real number but I don'......Tue, 01 Sep 2026 22:01:35 -0400https://liberty.io/formula-for-natural-logarithm-of-pi.htmlhttps://liberty.io/formula-for-natural-logarithm-of-pi.htmlDoes any formula or expansion exist that gives $\ln \pi$ ? The expansion should not just be any formula of $\pi$ with a $\ln$ before it. For example $\ln \pi$ = k + $\sum f(x)$ or something of thi......Tue, 01 Sep 2026 22:01:17 -0400https://liberty.io/evaluate-int-cos-cos-x-dx.htmlhttps://liberty.io/evaluate-int-cos-cos-x-dx.htmlEvaluate $\int \cos(\cos x)~dx$ I tried to use chain rule but failed. Can anyone help me please?...Tue, 01 Sep 2026 21:51:11 -0400https://liberty.io/how-would-you-find-and-classify-all-of-the-critical-points-of-a-functi.htmlhttps://liberty.io/how-would-you-find-and-classify-all-of-the-critical-points-of-a-functi.htmlSuppose you had the function $f(x,y)=x^{3}-y^{3}+9xy$. The critical points of this function would be the values of $(x,y)$, such that $f_{x}=f_{y}=0$. We have $f_{x}=3x^{2}+9y$ and $f_{y}=-3y^{2}+......Tue, 01 Sep 2026 21:37:12 -0400https://liberty.io/what-is-the-equation-of-a-straight-line.htmlhttps://liberty.io/what-is-the-equation-of-a-straight-line.htmlI'm reading a book named "Geometry, A comprehensive course" by Dan Pedoe at the moment. But I don't understand when they talk about the equation of a straight line... Here is what is written: The ...Tue, 01 Sep 2026 21:25:51 -0400https://liberty.io/use-pohlig-hellman-to-solve-discrete-log.htmlhttps://liberty.io/use-pohlig-hellman-to-solve-discrete-log.htmlWe have $$7^x = 166 \pmod{433}$$ I need to find $x$ using the Pohlig-Hellman algorithm....Tue, 01 Sep 2026 21:20:05 -0400https://liberty.io/solving-given-initial-value-problem-using-the-method-of-laplace-transf.htmlhttps://liberty.io/solving-given-initial-value-problem-using-the-method-of-laplace-transf.htmlI have the problem asking me to solve the initial value problem using the method of Laplace Transforms given $$ y''+y=f(t); \ \ \ \ \ y(0) = 0, y'(0)=1 $$ where $$ f(t)=\begin{cases} 0,& 0<......Tue, 01 Sep 2026 21:16:31 -0400https://liberty.io/slant-asymptote-and-limit-problem.htmlhttps://liberty.io/slant-asymptote-and-limit-problem.htmlThe graph $y = f(x)$ has a slant asymptote along the line $y = mx + b$ (with $m \not= 0$) if $$ \displaystyle\lim_{x \rightarrow \infty} |f(x) - (mx+b)| = 0. $$ Describe algebraically the condition......Tue, 01 Sep 2026 21:13:45 -0400https://liberty.io/how-many-positive-integers-n-satisfy-n-p-n-s-n.htmlhttps://liberty.io/how-many-positive-integers-n-satisfy-n-p-n-s-n.htmlLet $P(n)$ denote the product of digits of $n$ and let $S(n)$ denote the sum of digits of $n$. Then how many positive integers $n$ satisfy $$ n = P(n) + S(n) $$ I think I solved it, but I need......Tue, 01 Sep 2026 21:12:35 -0400https://liberty.io/is-x-5-2x-4-irreducible-in-mathbb-q-x.htmlhttps://liberty.io/is-x-5-2x-4-irreducible-in-mathbb-q-x.htmlI was asked this question on my exam but I didn't know how to solve it. Eisenstein criterion can't be applied, rational roots theorem shows that it doesn't have a linear factor because possible roo......Tue, 01 Sep 2026 21:11:05 -0400https://liberty.io/how-can-we-prove-that-the-field-of-complex-number-is-not-isomorphic-to.htmlhttps://liberty.io/how-can-we-prove-that-the-field-of-complex-number-is-not-isomorphic-to.htmlI want to prove this using contradiction, supposing that there is a ring isomorphism between the two and then finding a contradiction....Tue, 01 Sep 2026 21:07:39 -0400https://liberty.io/laplace-transform-of-cos-at-t.htmlhttps://liberty.io/laplace-transform-of-cos-at-t.htmlIf someone could help me solve for $$\mathcal{L}\left\{\frac{\cos(at)}{t}\right\}$$ it would be great. Step-by-step I have so far: $$\begin{align}\int_0^\infty \frac{\cos(at)\space e^{-st}}{t}\sp......Tue, 01 Sep 2026 20:53:48 -0400https://liberty.io/questions-tagged-improper-integrals.htmlhttps://liberty.io/questions-tagged-improper-integrals.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 20:52:49 -0400https://liberty.io/is-the-figure-the-circumference-of-a-unit-circle.htmlhttps://liberty.io/is-the-figure-the-circumference-of-a-unit-circle.htmlA friend of mine taught me the following question. I've never heard such a strange and interesting question! Qustion: Supposing that a figure $S$, which is constituted by points, satisfies the ...Tue, 01 Sep 2026 20:50:29 -0400https://liberty.io/inconsistencies-in-the-definition-of-derivative-of-a-polynomial-over-a.htmlhttps://liberty.io/inconsistencies-in-the-definition-of-derivative-of-a-polynomial-over-a.htmlA problem I came across defines a particular differentiation operator $D$ over the set of polynomials $\{P\}$ over a field $F$ with "the normal formula; that is $D(\sum_{i=0}^n a_nx^i) = \sum_{i=1}^n ...Tue, 01 Sep 2026 20:49:09 -0400https://liberty.io/how-do-i-calculate-a-weighted-average-from-two-averages.htmlhttps://liberty.io/how-do-i-calculate-a-weighted-average-from-two-averages.htmlI have two sets of averaged data I am working with, which account for a score and the average amount of users that achieved this. For example: Average Score $4$, Total Number of participants (wh......Tue, 01 Sep 2026 20:47:15 -0400https://liberty.io/what-does-asymptotically-proportional-mean.htmlhttps://liberty.io/what-does-asymptotically-proportional-mean.htmlIn an article about Metcalfe's law, I've read that $n(n-1)/2$ is asymptotically proportional to $n^2$. What does this mean? PS: I did find asymptotically optimal but I'm not sure if they mean the......Tue, 01 Sep 2026 20:39:34 -0400https://liberty.io/infinite-product-of-1-1-2n-terms.htmlhttps://liberty.io/infinite-product-of-1-1-2n-terms.htmlWhat is the value of the infinite product $$ \frac{1}{2} \frac{3}{4} \frac{5}{6}\cdots = \prod_{n=1}^{\infty} \frac{2n-1}{2n} = \prod_{n=1}^{\infty} \left(1-\frac{1}{2n}\right) = \lim_{n\to\infty}......Tue, 01 Sep 2026 20:38:15 -0400https://liberty.io/vector-field-using-green-039-s-theorem-other-integration.htmlhttps://liberty.io/vector-field-using-green-039-s-theorem-other-integration.htmlSo am I supposed to be using green's theorem for the first question, but where I'm confused is that there are three variables if I do, dx dy dz (I haven't learn how to use green's theorem for 3 ...Tue, 01 Sep 2026 20:37:10 -0400https://liberty.io/proving-that-an-irrational-number-exists-near-every-rational-number-du.htmlhttps://liberty.io/proving-that-an-irrational-number-exists-near-every-rational-number-du.htmlShow that arbitrarly close to any rational number there is a real (non-rational) number. In other words, show that to each real $\varepsilon>0$ and each rational $r\in\mathbb Q$ there exists $x\in\...Tue, 01 Sep 2026 20:30:24 -0400https://liberty.io/locally-vs-globally-lipschitz-confusion.htmlhttps://liberty.io/locally-vs-globally-lipschitz-confusion.htmlIs there any difference in a function being locally Lipschitz on $\mathbb{R^n}$ and being globally Lipschitz?...Tue, 01 Sep 2026 20:30:19 -0400https://liberty.io/how-do-the-floor-and-ceiling-functions-work-on-negative-numbers.htmlhttps://liberty.io/how-do-the-floor-and-ceiling-functions-work-on-negative-numbers.htmlIt's clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take $\,-0.8\,$ to $\,-1,\,$ then do you ta......Tue, 01 Sep 2026 20:24:02 -0400https://liberty.io/technique-to-draw-relative-velocities-vectors-and-diagrams.htmlhttps://liberty.io/technique-to-draw-relative-velocities-vectors-and-diagrams.htmlHow do you start drawing a diagram in a problem with relative velocities? I always see my coursemates and my lecturer drawing them in a second, as soon as they read the text, I asked them how they ......Tue, 01 Sep 2026 20:23:04 -0400https://liberty.io/why-does-tan-1-sqrt-x-2-1-sec-1-x.htmlhttps://liberty.io/why-does-tan-1-sqrt-x-2-1-sec-1-x.htmlWhy does $\tan^{-1}(\sqrt{x^2-1})=\sec^{-1}(x)$ How do you derive $\tan^{-1}(\sqrt{x^2-1})$ from $\sec^{-1}(x)$ ? (By $^{-1}$ I mean the inverse function)...Tue, 01 Sep 2026 20:21:26 -0400https://liberty.io/is-being-a-right-triangle-both-necessary-and-sufficient-for-the-pythag.htmlhttps://liberty.io/is-being-a-right-triangle-both-necessary-and-sufficient-for-the-pythag.htmlI recently encountered a Stack Overflow question (since closed) in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. ...Tue, 01 Sep 2026 20:20:22 -0400https://liberty.io/area-between-two-curves-with-absolute-value.htmlhttps://liberty.io/area-between-two-curves-with-absolute-value.htmlI have the two equations: y = 1 - 2x^2 y = abs(x) I solved the two equations using the absolute value for: abs(x) and -abs(x) Using these positive and negative absolute values I solved for the ...Tue, 01 Sep 2026 20:16:24 -0400https://liberty.io/idempotent-elements-of-z-4-times-z-8.htmlhttps://liberty.io/idempotent-elements-of-z-4-times-z-8.htmlI am trying to find idempotent elements of the direct product $Z_4 \times Z_8$. By the definition of direct product we have: $$Z_4 \times Z_8=\left\{(0,0),(0,1),.....(2,7),(3,7)\right\}$$ Now by the ...Tue, 01 Sep 2026 20:12:29 -0400https://liberty.io/why-is-decidable-included-in-turing-recognizable.htmlhttps://liberty.io/why-is-decidable-included-in-turing-recognizable.html(pic from "introduction to the theory of computation" - Michael Sipser) According to my understanding: Turing-recognizable languages are languages whice are accepted by a Turing machine; ...Tue, 01 Sep 2026 20:11:48 -0400https://liberty.io/what-is-ext-short-for-in-ext-functor-duplicate.htmlhttps://liberty.io/what-is-ext-short-for-in-ext-functor-duplicate.htmlStrangely, I've never heard Ext functors referred to by any other name, and so I'm not sure what "Ext" actually means. The only thing I can think of that "Ext" might stand for is "Exterior", which ......Tue, 01 Sep 2026 20:10:08 -0400https://liberty.io/find-all-possible-values-which-following-expression-can-take-sqrt-x-2-.htmlhttps://liberty.io/find-all-possible-values-which-following-expression-can-take-sqrt-x-2-.htmlGiven expression is: $\sqrt{x^2-7x+6}$. Now i first find values of x for which this is a valid question by putting guy inside square root equals to greater than 0. I get $x \in[-\infty,1]\cup[6,\......Tue, 01 Sep 2026 20:05:49 -0400https://liberty.io/fair-and-unfair-coin-probability.htmlhttps://liberty.io/fair-and-unfair-coin-probability.htmlI am stuck on this question. A coin with $P(H) = \frac{1}{2}$ is flipped $4$ times and then a coin with $P(H) = \frac{2}{3}$ is tossed twice. What is the probability that a total of $5$ heads o......Tue, 01 Sep 2026 19:54:49 -0400https://liberty.io/why-fourier-transform-uses-the-notation-x-jw-and-x-e-j-omega.htmlhttps://liberty.io/why-fourier-transform-uses-the-notation-x-jw-and-x-e-j-omega.htmlUsually books (e.g. Oppenheim) use the notation X(jw) for the continuous time Fourier Transform and X(e^j$\Omega$) for the discrete time Fourier transform. AFAIK the only parameter is w o $\Omega$, ...Tue, 01 Sep 2026 19:50:30 -0400https://liberty.io/normalizing-an-eigenvector-to-have-unit-length.htmlhttps://liberty.io/normalizing-an-eigenvector-to-have-unit-length.htmlI worked out a problem where one of the eigenvectors is $(1+ \sqrt{2}, 1)$. How do I normalize this vector? Should I use approximate values?...Tue, 01 Sep 2026 19:42:48 -0400https://liberty.io/minimum-possible-number-of-friendships.htmlhttps://liberty.io/minimum-possible-number-of-friendships.htmlHere is the problem: There are 2000 people on a social network. Each person sends 1000 friend requests. Two people are friends if they've sent a friend request to each other. What is the minimum ...Tue, 01 Sep 2026 19:40:12 -0400https://liberty.io/when-will-the-ball-hit-the-ground.htmlhttps://liberty.io/when-will-the-ball-hit-the-ground.htmlA girl is throwing a ball. If I have an equation $h= -4.9*t^2+20t+1.2$, where $t$ is number of second after the girl has thrown the ball, and $h$ is the height above ground. I want to find out w......Tue, 01 Sep 2026 19:30:00 -0400https://liberty.io/volume-and-lateral-surface-area-are-equal.htmlhttps://liberty.io/volume-and-lateral-surface-area-are-equal.htmlI need to express the radius $r$ of the right circular cone as a function of its height $h$ given that its volume equals to its lateral surface area. I know the two equations $\pi r \times \sqrt{......Tue, 01 Sep 2026 19:27:55 -0400https://liberty.io/permutations-vs-combinatorial-vs-factorials-vs-exponents.htmlhttps://liberty.io/permutations-vs-combinatorial-vs-factorials-vs-exponents.htmlI'm currently working on a probability course, and I am constantly having trouble figuring out when to use permutations vs. combinations vs. factorials vs. exponents in order to calculate sample si......Tue, 01 Sep 2026 19:27:54 -0400https://liberty.io/why-can-a-5x4-matrix-never-be-surjective.htmlhttps://liberty.io/why-can-a-5x4-matrix-never-be-surjective.htmlMy thought is that since a matrix can only be surjective if null(T)= n-m, no matrix can have a nullity of -1 therefore a 5x4 matrix cannot be surjective. Is this a valid proof?...Tue, 01 Sep 2026 19:27:49 -0400https://liberty.io/rgb-to-hsv-color-conversion-algorithm.htmlhttps://liberty.io/rgb-to-hsv-color-conversion-algorithm.htmlI'm a programmer looking to build an RGB to HSV color converter. I found an algorithm, but I have very little mathematical background and I'm not quite sure what's going on with it. A step-by-step ...Tue, 01 Sep 2026 19:26:59 -0400https://liberty.io/gaussian-elimination-with-4-variables.htmlhttps://liberty.io/gaussian-elimination-with-4-variables.htmlAfter this step however, I end up with 2 simultanious equations but still with four unknowns, I'm not sure if my gaussian elimination is wrong or not but how will i find the exact values of x,y,z a......Tue, 01 Sep 2026 19:21:48 -0400