https://liberty.ioCurated star highlights with fresh energy.en-usSat, 15 Aug 2026 22:59:50 -0400https://liberty.io/recursive-vs-inductive-definition.htmlhttps://liberty.io/recursive-vs-inductive-definition.htmlI once had an argument with a professor of mine, if the following definition was a recursive or inductive definition: Suppose you have sequence of real numbers. Define $a_0:=2$ and $a_{i+1}:=\frac......Sun, 16 Aug 2026 02:30:44 -0400https://liberty.io/use-depth-first-search-to-produce-a-spanning-tree-for-the-given-simple.htmlhttps://liberty.io/use-depth-first-search-to-produce-a-spanning-tree-for-the-given-simple.htmlUse depth-first search to produce a spanning tree for the given simple graph. Choose $a$ as the root of this spanning tree. Here is answer and I am not sure whether is correct or not Can anyone ......Sun, 16 Aug 2026 02:22:20 -0400https://liberty.io/notation-is-there-a-symbol-for-not-a-function-of.htmlhttps://liberty.io/notation-is-there-a-symbol-for-not-a-function-of.htmlFor example, let's say a term $A(x,y)$, a function of two random variables $x$ and $y$, is the argument of an expectation over $y$. The resulting term is no longer a function of $y$. Is there a ...Sun, 16 Aug 2026 02:19:26 -0400https://liberty.io/integrate-x-e-x-2-from-infty-to-infty.htmlhttps://liberty.io/integrate-x-e-x-2-from-infty-to-infty.htmlI suppose this question is very simple, but I still don't get where I am wrong. I try to proof that the integral: $$\int_{-\infty}^{\infty} x e^{-x^2} dx$$ is equal to zero. I suppose that by ...Sun, 16 Aug 2026 02:14:48 -0400https://liberty.io/moment-of-inertia-of-a-disc.htmlhttps://liberty.io/moment-of-inertia-of-a-disc.htmlIn my mechanics textbook there is a derivation of the moment of inertia of a disc of mass $m$ and radius $r$ about an axis through its centre and perpendicular to its plane surface, which goes some......Sun, 16 Aug 2026 02:10:07 -0400https://liberty.io/how-to-properly-take-derivatives-in-calculus-of-variations-euler-lagra.htmlhttps://liberty.io/how-to-properly-take-derivatives-in-calculus-of-variations-euler-lagra.htmlWhy is it that, in calculus of variations (specifically Euler-Lagrange), we can take the derivative of a function with respect to a function $f$ and set this derivative to $0$ if only $f'$ appears ......Sun, 16 Aug 2026 02:10:04 -0400https://liberty.io/derivative-of-e-ln-1-x.htmlhttps://liberty.io/derivative-of-e-ln-1-x.htmlThis question looks so simple, yet it confused me. If $f(x) = e^{\ln(1/x)}$, then $f'(x) =$ ? I got $e^{\ln(1/x)} \cdot \ln(1/x) \cdot (-1/x^2)$. And the correct answer is just the plain $-1/x......Sun, 16 Aug 2026 02:06:47 -0400https://liberty.io/what-is-formally-to-define-a-function-by-induction.htmlhttps://liberty.io/what-is-formally-to-define-a-function-by-induction.htmlLet me be more specific: Given the context of a function as a relation between 2 sets with specific properties, what is formally happening when I define a function by induction? EXAMPLE: (from La......Sun, 16 Aug 2026 02:01:08 -0400https://liberty.io/square-root-of-height-of-a-paraboloid-equal-to-radius.htmlhttps://liberty.io/square-root-of-height-of-a-paraboloid-equal-to-radius.htmlI'm reading the book Mathematics: It's Content, Methods, and Meanings and I'm unsure as to how one of the variables in an example was derived. The question is about the volume of a paraboloid and h......Sun, 16 Aug 2026 01:42:42 -0400https://liberty.io/numbers-that-cannot-be-expressed-as-fractions.htmlhttps://liberty.io/numbers-that-cannot-be-expressed-as-fractions.htmlWhat are Numbers that cannot be expressed as Fractions called?...Sun, 16 Aug 2026 01:27:25 -0400https://liberty.io/is-kruskal-rank-and-rank-of-a-matrix-the-same.htmlhttps://liberty.io/is-kruskal-rank-and-rank-of-a-matrix-the-same.htmlThe definition of Kruskal rank : maximum value of $k$ such that any $k$ columns of a matrix $\textbf{A}$ are linearly independent, then $k$ is the Kruskal rank of matrix $\textbf{A}$. How is it ...Sun, 16 Aug 2026 01:25:34 -0400https://liberty.io/infinite-amount-of-vertical-asymptotes.htmlhttps://liberty.io/infinite-amount-of-vertical-asymptotes.htmlIs it possible that the graph of function has infinitely many vertical asymptotes? I suppose, that it is not possible, because such function would not exist. But I need to prove it in a math-fashi......Sun, 16 Aug 2026 01:15:22 -0400https://liberty.io/the-most-common-theorems-taught-in-abstract-algebra.htmlhttps://liberty.io/the-most-common-theorems-taught-in-abstract-algebra.htmlI am self learning abstract algebra. I want to know which theorems are a must to understand. Now these are limits I have to deal with (please consider when answering): I have limited inter......Sun, 16 Aug 2026 01:15:19 -0400https://liberty.io/using-chain-rule-to-find-frac-dz-dt.htmlhttps://liberty.io/using-chain-rule-to-find-frac-dz-dt.htmlI know what the chain rule is but i am unsure how to apply it to this case. $$z=x^3\cdot e^{2y}$$ $$x=2t$$ $$y=t^2$$ Edit: Fixed title to $\frac{dz}{dt}$....Sun, 16 Aug 2026 01:11:47 -0400https://liberty.io/cos-2-cos-1-x-x-2-how.htmlhttps://liberty.io/cos-2-cos-1-x-x-2-how.html$$\cos^2 (\cos^{-1} x)$$ $$=x^2$$ Which trigonometric rule was used here?...Sun, 16 Aug 2026 01:09:04 -0400https://liberty.io/prove-that-sinh-2x-2sinh-x-cosh-x.htmlhttps://liberty.io/prove-that-sinh-2x-2sinh-x-cosh-x.htmlWhen trying to solve a textbook question I'm getting a wrong result but I'm pretty sure I'm not doing anything wrong. Can somebody spot my mistake? Question: Prove that $$sinh(2x) = 2sinh(x)cosh......Sun, 16 Aug 2026 01:08:48 -0400https://liberty.io/term-for-sum-difference-product-quotient.htmlhttps://liberty.io/term-for-sum-difference-product-quotient.htmlFeel free to let me know if this is better suited for: https://english.stackexchange.com/ https://stackoverflow.com/ That said, I'm wondering if there's a math term for the nature of "sum", "...Sun, 16 Aug 2026 01:03:09 -0400https://liberty.io/what-sets-does-the-shades-on-the-venn-diagram-represent.htmlhttps://liberty.io/what-sets-does-the-shades-on-the-venn-diagram-represent.htmlI need help with this Venn Diagram. I think I got the first (please check if they are right). The last one is the one that I'm missing. Would appreciate your help. a. $A'∩B$ b. $(A∪B)'=A'∩B'$ c.......Sun, 16 Aug 2026 00:55:16 -0400https://liberty.io/how-many-essentially-different-drive-ya-nuts-puzzles-exist.htmlhttps://liberty.io/how-many-essentially-different-drive-ya-nuts-puzzles-exist.htmlIn the Drive Ya Nuts puzzle, we are given $7$ hexagonal nuts, whose edges have been numbered from $1$ to $6$. Each nut uses all $6$ numbers. The aim of the puzzle is to place all the nuts in a ...Sun, 16 Aug 2026 00:54:37 -0400https://liberty.io/finding-the-lengths-of-the-sides-of-a-triangle-given-3-angles-only.htmlhttps://liberty.io/finding-the-lengths-of-the-sides-of-a-triangle-given-3-angles-only.htmlIf a right $\triangle ABC$ has $\angle A= 90^\circ$, $\angle B=45^\circ$, $\angle C=45^\circ$. Is there a way of finding the lengths of the sides $a$, $b$ & $c$ without knowing any of their len......Sun, 16 Aug 2026 00:49:52 -0400https://liberty.io/references-to-supplement-goldblatt-039-s-topoi.htmlhttps://liberty.io/references-to-supplement-goldblatt-039-s-topoi.htmlI've been using the book "Topoi: A Categorial Analysis of Logic" by Robert Goldblatt. I found it intuitive and fun, but I feel like some parts are missing. For instance, I struggled to prove $a^1 \......Sun, 16 Aug 2026 00:39:45 -0400https://liberty.io/a-complete-k-partite-graph.htmlhttps://liberty.io/a-complete-k-partite-graph.htmlA complete $k$-partite graph is a graph with disjoint sets of nodes where there is no edges between the nodes in same set, and there is an edge between any node and other nodes in the other subsets......Sun, 16 Aug 2026 00:28:58 -0400https://liberty.io/simplification-or-alternate-form.htmlhttps://liberty.io/simplification-or-alternate-form.htmlWhen given an algrebraic expression of the form $$ \ 6A + 7{A^2}\ $$ and asked to simplify, would the correct answer be a) $$ \ A(6 + 7A)\ $$ or b) That this cannot be simplified further...Sun, 16 Aug 2026 00:27:36 -0400https://liberty.io/calculating-the-area-of-an-irregular-polygon.htmlhttps://liberty.io/calculating-the-area-of-an-irregular-polygon.htmlGiven the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? Thanks in advance...Sun, 16 Aug 2026 00:17:10 -0400https://liberty.io/why-must-triangle-inscribed-in-circle-have-a-right-angle.htmlhttps://liberty.io/why-must-triangle-inscribed-in-circle-have-a-right-angle.htmlYou should also notice that since $CP$ is the diameter of the circle, angle $CBP$ is a right angle. says the solution to this question. But what is the reasoning or proof behind the claim that......Sat, 15 Aug 2026 23:55:12 -0400https://liberty.io/how-to-tell-if-a-differential-equation-is-homogeneous-or-inhomogeneous.htmlhttps://liberty.io/how-to-tell-if-a-differential-equation-is-homogeneous-or-inhomogeneous.htmlSometimes it arrives to me that I try to solve a linear differential equation for a long time and in the end it turn out that it is not homogeneous in the first place. Is there a way to see direc......Sat, 15 Aug 2026 23:54:46 -0400https://liberty.io/infinite-alternating-series-problem.htmlhttps://liberty.io/infinite-alternating-series-problem.htmlWhile studying Leibniz test for infinite alternating series I am having an intuition, consider a series $$ \sum_{n=1}^{\infty}(-1)^{n+1}u_{n} $$ If $$ \left \{ u_{n} \right \} $$ is a non-decreasing ...Sat, 15 Aug 2026 23:52:23 -0400https://liberty.io/non-linear-congruence-equation.htmlhttps://liberty.io/non-linear-congruence-equation.htmlI want to find solutions to the equation $x^3 + 2x - 3 \equiv 0 (mod 45)$. I have already found solutions to $x^3 + 2x - 3 \equiv 0 (mod 5)$ and $x^3 + 2x - 3 \equiv 0 (mod 9)$, simply by brute......Sat, 15 Aug 2026 23:42:28 -0400https://liberty.io/double-integral-with-polar-substitution.htmlhttps://liberty.io/double-integral-with-polar-substitution.htmlThis is the first time I write in this forum so hope I get everything right! I'm trying to resolve this double integral but I'm having trouble with it :/. First thing I thought about was a polar ...Sat, 15 Aug 2026 23:40:42 -0400https://liberty.io/linear-algebra-terminology-unique-trivial-non-trivial-inconsistent-and.htmlhttps://liberty.io/linear-algebra-terminology-unique-trivial-non-trivial-inconsistent-and.htmlLet me get this straight: Unique solution- has an exact solution (such as a POI of 2 intersecting lines) Trivial solution- when 0 needs to equal the zero vector in $Ax=0 vector$ Non-Trivial-? Per......Sat, 15 Aug 2026 23:36:23 -0400https://liberty.io/user-deven-ware-mathematics-stack-exchange.htmlhttps://liberty.io/user-deven-ware-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 15 Aug 2026 23:27:34 -0400https://liberty.io/topology-of-matrices.htmlhttps://liberty.io/topology-of-matrices.html1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all mat......Sat, 15 Aug 2026 23:16:31 -0400https://liberty.io/prove-tan-84-circ-tan-78-circ-tan-72-circ-tan-60-circ.htmlhttps://liberty.io/prove-tan-84-circ-tan-78-circ-tan-72-circ-tan-60-circ.htmlProve that: $$\tan 84^{\circ}=\tan 78^{\circ}+\tan 72^{\circ}+\tan 60^{\circ}$$ First I simplified $\tan 78^{\circ}+\tan 72^{\circ}$ and then I simplified $\tan 84^{\circ}-\tan 60^{\circ}$. Both ......Sat, 15 Aug 2026 23:10:58 -0400https://liberty.io/understanding-the-useness-of-classification-of-finite-simple-groups.htmlhttps://liberty.io/understanding-the-useness-of-classification-of-finite-simple-groups.htmlI know that we are able to classify all the finite simple group, but I don't understand how to use this classification to prove results like that every simple group has two generators. The diffic......Sat, 15 Aug 2026 22:56:26 -0400https://liberty.io/graphically-representing-vectors-with-polar-unit-vectors-without-conve.htmlhttps://liberty.io/graphically-representing-vectors-with-polar-unit-vectors-without-conve.htmlShort version : How do you graphically represent a vector(without converting to Cartesian) given components in direction of $\hat r$ and $\hat \theta$ (unit vectors in polar coordinates)? and what......Sat, 15 Aug 2026 22:55:30 -0400https://liberty.io/how-to-restrict-only-top-range-of-ellipse-function-and-what-is-its-dom.htmlhttps://liberty.io/how-to-restrict-only-top-range-of-ellipse-function-and-what-is-its-dom.htmlI am trying to graph the function of an ellipse that is: $$1=\frac{x^2}{49}+\frac{(y+1)^2}{9}$$. I want to make the horizontal ellipse's range $y \leq 0.838$. So, when I also have to write the doma......Sat, 15 Aug 2026 22:52:26 -0400https://liberty.io/finding-complex-solutions-of-an-equation.htmlhttps://liberty.io/finding-complex-solutions-of-an-equation.htmlHow does one solve this equation. I would like to see the solution of this problem in steps. $z\cdot\bar{z}=\left|3\cdot z \right|$ EDIT: Is it possible to solve this by converting to the form $z......Sat, 15 Aug 2026 22:50:58 -0400https://liberty.io/all-numbers-that-are-less-than-four-units-from-zero.htmlhttps://liberty.io/all-numbers-that-are-less-than-four-units-from-zero.htmlLooking at a simple algebra question which is to graph "All numbers that are less than four units from zero." The knee jerk response is to draw a number line with an open circle at -4 and a ...Sat, 15 Aug 2026 22:42:50 -0400https://liberty.io/is-the-function-f-x-tan-x-odd-even-or-neither.htmlhttps://liberty.io/is-the-function-f-x-tan-x-odd-even-or-neither.htmlIs the function $f(x) = \tan(x)$ odd, even, or neither? Here is what I have so far: I know the function is not even because $f(x) ≠ f(-x):$ $$f(-x) = \tan(-x)$$ $$\tan(-x) ≠ \tan(x)$$ Now I wa......Sat, 15 Aug 2026 22:22:08 -0400https://liberty.io/can-all-trigonometric-expressions-be-written-in-terms-of-sine-and-cosi.htmlhttps://liberty.io/can-all-trigonometric-expressions-be-written-in-terms-of-sine-and-cosi.htmlI know that sine and cosine can be rewritten in terms of the real and complex parts of the exponential function as a result of Euler's formula. My question is, can every trigonometric expression be ...Sat, 15 Aug 2026 22:22:07 -0400https://liberty.io/general-solution-to-a-differential-eq-with-complex-eigenvalues.htmlhttps://liberty.io/general-solution-to-a-differential-eq-with-complex-eigenvalues.htmlI need a little explanation here the general solution is $$x(t)=c_1u(t)+c_2v(t)$$ where $u(t)=e^{\lambda t}(\textbf{a} \cos \mu t-\textbf{b} \sin \mu t$ and $v(t)=e^{\lambda t}(\textbf{a} \sin \mu ......Sat, 15 Aug 2026 22:21:04 -0400https://liberty.io/user-logan-stokols-mathematics-stack-exchange.htmlhttps://liberty.io/user-logan-stokols-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 15 Aug 2026 22:19:28 -0400https://liberty.io/guessing-a-number-between-1-and-100-with-10-distinct-questions-and-9-h.htmlhttps://liberty.io/guessing-a-number-between-1-and-100-with-10-distinct-questions-and-9-h.htmlI've been trying to solve this riddle: Say I'm thinking of a number between 1 and 100. You can ask me 10 distinct yes/no questions to guess the number, but exactly one of my answers will be a li......Sat, 15 Aug 2026 22:15:16 -0400https://liberty.io/deriving-the-formula-for-the-height-of-a-trapezoid.htmlhttps://liberty.io/deriving-the-formula-for-the-height-of-a-trapezoid.htmlThe bases of a trapezoid have lengths $a$ and $b$, and its legs have lengths $c$ and $d$. A formula for the height is \begin{equation*} h = \frac{ \sqrt{(-a + b + c + d)(a - b + c + d)(a - b - c + ......Sat, 15 Aug 2026 22:11:23 -0400https://liberty.io/algebraic-signatures-as-quivers-is-there-somewhere-i-can-learn-more-ab.htmlhttps://liberty.io/algebraic-signatures-as-quivers-is-there-somewhere-i-can-learn-more-ab.htmlIn my opinion, a cool definition of "algebraic signature" is as follows: An algebraic signature on the sort symbols $\mathcal{X} = \{X_0,...,X_{n-1}\}$ is precisely a quiver whose underlying s......Sat, 15 Aug 2026 22:05:12 -0400https://liberty.io/remember-trig-identities-like-cos-pi-3-1-2.htmlhttps://liberty.io/remember-trig-identities-like-cos-pi-3-1-2.htmlI have started doing complex analysis and I keep having to switch between rectangular co-ordinates and polar form and I keep running into stuff like - $\cos(\pi/3) = 1/2$. I keep having to look th......Sat, 15 Aug 2026 22:04:05 -0400https://liberty.io/find-all-complex-and-real-roots-of-higher-degree-polynomials-given-one.htmlhttps://liberty.io/find-all-complex-and-real-roots-of-higher-degree-polynomials-given-one.html$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!...Sat, 15 Aug 2026 22:01:44 -0400https://liberty.io/how-many-digits-are-in-the-integer-representation-of-2-to-the-30th-pow.htmlhttps://liberty.io/how-many-digits-are-in-the-integer-representation-of-2-to-the-30th-pow.htmlHow many digits are in the integer representation of 2 to the 30th power? Since I didn't really know any 'expert' way to approach this, I just started out by listing the powers of 2, like 2, 4, 8,......Sat, 15 Aug 2026 21:35:58 -0400https://liberty.io/what-is-rank-nullity-range-and-kernel-in-relation-to-each-other.htmlhttps://liberty.io/what-is-rank-nullity-range-and-kernel-in-relation-to-each-other.htmlSo from my understanding of Rank, that its the row space of of a Matrix, count how many rows there are, that are not the same if you multiply them by a constant. My understanding that a Kernel me......Sat, 15 Aug 2026 21:33:17 -0400https://liberty.io/finding-all-tangent-lines-that-pass-through-a-specific-point-not-the-o.htmlhttps://liberty.io/finding-all-tangent-lines-that-pass-through-a-specific-point-not-the-o.htmlI was given the function $y = x^3-x$ and told to find all tangent lines that pass through $(-2,2)$. Not sure what steps to take past finding the derivative....Sat, 15 Aug 2026 21:25:48 -0400https://liberty.io/proving-trigonometric-cofunction-identities-for-angles-larger-than-pi-.htmlhttps://liberty.io/proving-trigonometric-cofunction-identities-for-angles-larger-than-pi-.htmlBackground For the sine and cosine functions, we have $$\sin(\frac{\pi}{2} - x) = \cos(x)$$ $$\cos(\frac{\pi}{2} - x) = \sin(x)$$ For $x\in[0, \pi/2]$, this can be easily shown using a right tr......Sat, 15 Aug 2026 21:24:40 -0400https://liberty.io/find-radius-of-convergence-of-power-series.htmlhttps://liberty.io/find-radius-of-convergence-of-power-series.htmlSince we know that given $\sum_{n=0}^{\infty }C_nz^n$, if $\lim_{n\rightarrow \infty }|C_n|^{1/n}$ exists then $R^{-1}=\lim_{n\rightarrow \infty }|C_n|^{1/n}$ where $R$ is the radius of convergen......Sat, 15 Aug 2026 21:18:46 -0400https://liberty.io/calculate-width-and-height-of-rectangle-containing-given-area-and-conf.htmlhttps://liberty.io/calculate-width-and-height-of-rectangle-containing-given-area-and-conf.htmlMy algebra is so rusty, this should be an easy one! The area of a rectangle is $$A = W \times H$$ If you're given $A$ (say $150$) and a ratio of $W:H$ (say $3:2$) how can you calculate $W$ and $H$?...Sat, 15 Aug 2026 21:13:37 -0400https://liberty.io/how-to-calculate-the-expected-value-of-the-powerball-lottery.htmlhttps://liberty.io/how-to-calculate-the-expected-value-of-the-powerball-lottery.htmlThe current Powerball jackpot is at roughly 675 million USD and the chances of winning with one random ticket is 1 in 292.2 million. Each ticket costs 2 USD. From a general perspective, it appears......Sat, 15 Aug 2026 21:06:38 -0400https://liberty.io/doubly-stochastic-matrix-proof.htmlhttps://liberty.io/doubly-stochastic-matrix-proof.htmlA transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and consis......Sat, 15 Aug 2026 20:57:29 -0400https://liberty.io/formula-for-the-sum-of-fractions-duplicate.htmlhttps://liberty.io/formula-for-the-sum-of-fractions-duplicate.htmlHow to find the formula for the sum of fractions like this? $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n\times (n+1)}=$$...Sat, 15 Aug 2026 20:45:55 -0400https://liberty.io/find-a-second-partial-derivative-of-the-implicitly-given-function-3y-z.htmlhttps://liberty.io/find-a-second-partial-derivative-of-the-implicitly-given-function-3y-z.htmlI am stuck at this partial derivative question. Someone earlier helped me with the first part of the question but i still cant understand the 2nd part of the question involving second order partial ...Sat, 15 Aug 2026 20:40:10 -0400https://liberty.io/how-to-see-the-logarithm-as-the-inverse-function-of-the-exponential.htmlhttps://liberty.io/how-to-see-the-logarithm-as-the-inverse-function-of-the-exponential.htmlI saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connect......Sat, 15 Aug 2026 20:39:49 -0400https://liberty.io/proof-that-a-matrix-is-nonsingular.htmlhttps://liberty.io/proof-that-a-matrix-is-nonsingular.htmlLet $A$ be $n \times n$ matrix. Show that if $A^2 = 0$, then $I - A$ is non-singular and $(I-A)^{-1} = I+A$. The second part is easy for me, but how can I show that if $A^2 = 0$, then $I - A$ is non-...Sat, 15 Aug 2026 20:34:56 -0400https://liberty.io/element-wise-or-pointwise-operations-notation.htmlhttps://liberty.io/element-wise-or-pointwise-operations-notation.htmlIs there a notation for element-wise (or pointwise) operations? For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same ......Sat, 15 Aug 2026 20:30:24 -0400https://liberty.io/1-2-4-7-11-16-22-closed.htmlhttps://liberty.io/1-2-4-7-11-16-22-closed.htmlI try to distribute beams on an horizontal array, with increasing spacings according to this series. 1+1=2 2+2=4 4+3=7 7+4=11 11+5=16 16+6=22 I'd like to feel what it looks like and learn how to ...Sat, 15 Aug 2026 20:25:22 -0400https://liberty.io/motivation-behind-the-definition-of-complete-metric-space.htmlhttps://liberty.io/motivation-behind-the-definition-of-complete-metric-space.htmlWhat is motivation behind the definition of a complete metric space? Intuitively,a complete metric is complete if they are no points missing from it. How does the definition of completeness (in t......Sat, 15 Aug 2026 20:20:43 -0400https://liberty.io/how-do-i-prove-the-uniqueness-of-additive-identity.htmlhttps://liberty.io/how-do-i-prove-the-uniqueness-of-additive-identity.htmlFirst, suppose $i_1$ and $i_2$ are additive identity in ring R. From the definition of "additive identity" $a+i_1=a$ such that there is $a$ $\in$ $R$, including for $a=i_2$, so $i_2+i_1=i_2$. But $......Sat, 15 Aug 2026 20:11:31 -0400https://liberty.io/reflecting-coordinates-over-the-line-x-1.htmlhttps://liberty.io/reflecting-coordinates-over-the-line-x-1.htmlI know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: ......Sat, 15 Aug 2026 20:00:15 -0400https://liberty.io/find-the-length-of-the-chord-given-that-the-circle-039-s-diameter-and-.htmlhttps://liberty.io/find-the-length-of-the-chord-given-that-the-circle-039-s-diameter-and-.htmlA chord of a circle subtends an angle of 89 degrees at its centre. Find the length of the chord given that the circle's diameter is 11.4 cm. The problem I have here is that I can't visualise this ...Sat, 15 Aug 2026 19:55:21 -0400https://liberty.io/use-three-11-039-s-and-various-math-symbols-to-make-an-equation-equal-.htmlhttps://liberty.io/use-three-11-039-s-and-various-math-symbols-to-make-an-equation-equal-.htmlThe puzzle is to use the following symbols $$+,\;-,\;*,\;/,\;(\;,\;),\;!, \;\sqrt(\cdot)$$ in order to make a valid equation out of $$11~~~~~~11~~~~~~~11 = 6.$$ (There are three elevens with space......Sat, 15 Aug 2026 19:49:45 -0400https://liberty.io/calculate-the-normal-cone-of-a-convex-set-at-a-point.htmlhttps://liberty.io/calculate-the-normal-cone-of-a-convex-set-at-a-point.htmlLet $C$ be a convex set in $\mathbb{R}^d$ and $\overline{x}\in C$. We define the normal cone of $C$ at $\overline{x}$ by \begin{equation} N_C(\overline{x}) = \{ y \in \mathbb{R}^d \ <y ,c-\ove......Sat, 15 Aug 2026 19:33:34 -0400https://liberty.io/what-is-the-probablity-that-the-sum-of-two-dice-is-4-or-6.htmlhttps://liberty.io/what-is-the-probablity-that-the-sum-of-two-dice-is-4-or-6.htmlWhat is the probablity that the sum of two dice is 4 or 6? The explanation I found is as follows: Total number of outcomes $6 \times6 = 36$ Number of outcomes where the event occurs: $1+3, 2+2, 3+......Sat, 15 Aug 2026 19:24:43 -0400https://liberty.io/the-number-of-divisors.htmlhttps://liberty.io/the-number-of-divisors.htmlI am reading a math book. It states the following rule: For an integer $n$ greater than 1, let the prime factorization of $n$ be $$n=p^a_1p^b_2p^c_3...p^m_k$$ Here $a, b, c,..., m$ are nonne......Sat, 15 Aug 2026 19:24:16 -0400https://liberty.io/abbreviated-notation-for-vector-normalized-by-its-length-unit-vector.htmlhttps://liberty.io/abbreviated-notation-for-vector-normalized-by-its-length-unit-vector.htmlIs there any established notation to abbreviate $\vec{x}/||\vec{x}|| = \frac{\vec{x}}{||\vec{x}||}$? Sure, this expression looks quite short already, but with a long argument such as $\vec{x} = a\c......Sat, 15 Aug 2026 19:23:55 -0400https://liberty.io/what-are-the-eigenvalues-of-x-xx-t-x-in-mathbb-r-d-duplicate.htmlhttps://liberty.io/what-are-the-eigenvalues-of-x-xx-t-x-in-mathbb-r-d-duplicate.htmlI'm given the matrix $X = xx^{T}\in\mathbb{R}^{d \ x \ d}, x\in\mathbb{R}^{d}$. Does somebody know how to compute $\lambda_{max}(X)$ or $\lambda_{min}(X)$? I only want to know these two eigenvalues......Sat, 15 Aug 2026 19:15:04 -0400https://liberty.io/show-that-the-three-medians-of-a-triangle-are-concurrent-at-a-point.htmlhttps://liberty.io/show-that-the-three-medians-of-a-triangle-are-concurrent-at-a-point.htmlI am working through Paul Lockhart's Measurements. It's one of the most engaging and insightful introductions to elementary mathematics I've read. In the opening pages, he presents a simple chall......Sat, 15 Aug 2026 18:54:32 -0400https://liberty.io/understanding-conditional-entropy-intuitively-h-y-x-x-vs-h-y-x.htmlhttps://liberty.io/understanding-conditional-entropy-intuitively-h-y-x-x-vs-h-y-x.htmlI was trying to understand conditional entropy better. The part that was confusing me exactly was the difference between $H[Y|X=x]$ vs $H[Y|X]$. $E[Y|X=x]$ makes some sense to me intuitively beca......Sat, 15 Aug 2026 18:53:23 -0400https://liberty.io/what-039-s-the-opposite-of-a-cross-product.htmlhttps://liberty.io/what-039-s-the-opposite-of-a-cross-product.htmlFor example, $a \times b = c$ If you only know $a$ and $c$, what method can you use to find $b$?...Sat, 15 Aug 2026 18:22:24 -0400https://liberty.io/how-to-tell-if-two-3d-vectors-are-in-the-same-direction.htmlhttps://liberty.io/how-to-tell-if-two-3d-vectors-are-in-the-same-direction.htmlGiven: $$AB=\left( \begin{array}{ccc}2\\1\\3\end{array} \right) \;\;\;\; \text{and}\;\;\;\; CD=\left( \begin{array}{ccc}4\\3\\6\end{array} \right).$$ Justify if $AB$ has the same direction a......Sat, 15 Aug 2026 18:17:37 -0400https://liberty.io/what-exactly-is-approximation.htmlhttps://liberty.io/what-exactly-is-approximation.htmlThere are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what......Sat, 15 Aug 2026 18:13:34 -0400https://liberty.io/number-of-8-letter-words-with-at-least-3-vowels-why-is-my-result-wrong.htmlhttps://liberty.io/number-of-8-letter-words-with-at-least-3-vowels-why-is-my-result-wrong.htmlI have read and understood answer provided here Assume that there is $30$ letters, $5$ of which are vowels. My Result: There are $\binom{8}{3}$ combinations where you can write vowels. Each combin......Sat, 15 Aug 2026 18:11:32 -0400https://liberty.io/prove-that-the-number-of-common-tangents-to-two-circles-exterior-to-on.htmlhttps://liberty.io/prove-that-the-number-of-common-tangents-to-two-circles-exterior-to-on.htmlGiven two circles outside each other, the maximum number of common tangents to the two circles is 4 according to Wikipedia. How can this be proven? That's a picture above showing what I'm talking ......Sat, 15 Aug 2026 18:08:09 -0400https://liberty.io/looking-for-a-clear-definition-of-the-geometric-product.htmlhttps://liberty.io/looking-for-a-clear-definition-of-the-geometric-product.htmlIn brief: I'm looking for a clearly-worded definition1 of the geometric product of two arbitrary multivectors in $\mathbb{G}^n$. I'm having a hard time getting my bearings in the world of "...Sat, 15 Aug 2026 18:07:54 -0400https://liberty.io/what-039-s-the-probability-of-at-least-and-exactly-one-event-occurring.htmlhttps://liberty.io/what-039-s-the-probability-of-at-least-and-exactly-one-event-occurring.htmlIf I know the probability of event $A$ occurring and I also know the probability of $B$ occurring, how can I calculate the probability of "at least one of them" occurring? I was thinking that this......Sat, 15 Aug 2026 18:05:59 -0400https://liberty.io/how-do-we-define-the-sum-of-a-bi-infinite-series.htmlhttps://liberty.io/how-do-we-define-the-sum-of-a-bi-infinite-series.htmlIf we have a (usual) series of the form $S = \sum_{n=1}^{\infty} a_n$ then we define $S$ to be the limit as $N$ goes to infinity of the partial sums $S_N = \sum_{n=1}^{N} a_n,$ provided the limit e......Sat, 15 Aug 2026 17:53:04 -0400https://liberty.io/how-would-you-define-a-3d-angle.htmlhttps://liberty.io/how-would-you-define-a-3d-angle.htmlToday I somehow was wondering how could you define a 3d angle. First what I could think of was volume of a cone with side a=1. Then if it wasn't round, volume of a tetrahedron with side 1. Is any o......Sat, 15 Aug 2026 17:52:57 -0400https://liberty.io/summation-of-complex-numbers.htmlhttps://liberty.io/summation-of-complex-numbers.htmlThis is a series problem where the terms are complex numbers. I am looking for a better approach to solving this problem. If $\displaystyle z = \frac{1+i}{\sqrt2}$, Evaluate $1 + z + z^2 + ... +......Sat, 15 Aug 2026 17:48:38 -0400https://liberty.io/proving-that-arctan-x-arctan-1-x-pm-pi-2-could-this-line-of-reasoning-.htmlhttps://liberty.io/proving-that-arctan-x-arctan-1-x-pm-pi-2-could-this-line-of-reasoning-.htmlI know that two questions have already been asked about this exercise, but what I'm asking here is if this solution, which sounds rather strange to me, could possibly be correct. The problems is as ...Sat, 15 Aug 2026 17:47:45 -0400https://liberty.io/definition-of-epimorphism-in-category-theory.htmlhttps://liberty.io/definition-of-epimorphism-in-category-theory.htmlGiven a morphism $f : A \to B$, $f$ is en epimorphism iff $\forall g_1, g_2 : B \to C. f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ In words, a morphism $f : A \to B$ is epic if it has the proper......Sat, 15 Aug 2026 17:45:43 -0400https://liberty.io/proper-notation-for-coordinate-transformations.htmlhttps://liberty.io/proper-notation-for-coordinate-transformations.htmlLet's say I have a three dimensional cube that I am transforming by simply swapping the $x$ and $y$ axes (more accurately, a $90 ^{\circ}$ rotation o the geometry about the $z$ axis, where the valu......Sat, 15 Aug 2026 17:45:23 -0400https://liberty.io/how-to-calculate-a-group-percentage.htmlhttps://liberty.io/how-to-calculate-a-group-percentage.htmlIn a classroom, there are 8 desks equally arranged in two columns (i.e. each column has four desks). Each desk can accommodate 2 people and there are eight boys and eight girls in total. When two ...Sat, 15 Aug 2026 17:43:18 -0400https://liberty.io/solve-y-039-x-y.htmlhttps://liberty.io/solve-y-039-x-y.htmlI am suppose to use the substitution of $u = x + y$ $y' = x + y$ $u(x) = x + y(x)$ I actually forget the trick to this and it doesn't really make much sense to me. I know that I need to get ever......Sat, 15 Aug 2026 17:39:35 -0400https://liberty.io/existence-of-k-regular-trees-with-n-vertices.htmlhttps://liberty.io/existence-of-k-regular-trees-with-n-vertices.htmla $k$-regular tree is a tree for which all vertices have a degree of $k$ or $1$ (internal veritces have a degree of $k$ and the leaves have a degree of $1$). I've been asked to find for which valu......Sat, 15 Aug 2026 17:36:51 -0400https://liberty.io/l-hopital-s-rule-for-infinity-over-minus-infinity.htmlhttps://liberty.io/l-hopital-s-rule-for-infinity-over-minus-infinity.htmlCan I apply L’Hopital’s rule to this: $$ \lim_{x\to0}\frac{f(x)}{g(x)} $$ when $ \lim_{x\to0}f(x) = \infty $ and $ \lim_{x\to0}g(x) = -\infty $. Is this an indeterminate form?...Sat, 15 Aug 2026 17:33:21 -0400https://liberty.io/how-can-one-find-the-factorization-a-4-2a-3-3a-2-2a-1-a-2-a-1-2-from-s.htmlhttps://liberty.io/how-can-one-find-the-factorization-a-4-2a-3-3a-2-2a-1-a-2-a-1-2-from-s.htmlI want to prove $a^4 + 2a^3 + 3a^2 + 2a + 1$ always be perfect square for $a \in \mathbb{N}$. Using GeoGebra I found $a^4 + 2a^3 + 3a ² + 2a + 1 = (a^2 + a + 1)^2$ which is trivial to verify. Sinc......Sat, 15 Aug 2026 17:33:04 -0400https://liberty.io/primes-dividing-11-111-1111-duplicate.htmlhttps://liberty.io/primes-dividing-11-111-1111-duplicate.htmlHow can I prove that every prime except 2 and 5 divide infinitely many of the following integers $11, 111, 1111, ...$ ?...Sat, 15 Aug 2026 17:22:34 -0400https://liberty.io/multiplication-cancellation-property-by-peano-axioms.htmlhttps://liberty.io/multiplication-cancellation-property-by-peano-axioms.htmlI am trying to prove cancellation property of multiplication of natural numbers, $xy=xz$ implies $y=z$, with Peano axioms and arithmetic but not using or defining order on natural numbers. It can b......Sat, 15 Aug 2026 17:04:59 -0400https://liberty.io/using-set-builder-notation-on-a-set-that-jumps-in-intervals.htmlhttps://liberty.io/using-set-builder-notation-on-a-set-that-jumps-in-intervals.htmlI'm new to the world of Discrete Mathematics. I have been reviewing a little on Set Builder Notation and have looked over the following site thoroughly: http://www.mathsisfun.com/sets/set-builder-...Sat, 15 Aug 2026 16:53:54 -0400https://liberty.io/separating-the-variables-in-a-2nd-order-ode.htmlhttps://liberty.io/separating-the-variables-in-a-2nd-order-ode.htmlI am looking at the following ODE (obtained in order to use the similarity reduction method when solving the Heat Equation): $$ y'' + \frac{1}{2}xy'=0 $$ IC/BC: $y(0) = 1, y(\infty) = 0$ The text s......Sat, 15 Aug 2026 16:45:48 -0400https://liberty.io/how-many-significant-figures-in-this-negative-number.htmlhttps://liberty.io/how-many-significant-figures-in-this-negative-number.htmlI understand that $0.03$ would have $1$ significant figure. Does $-0.03$ also have one significant figure or $3$? Thanks....Sat, 15 Aug 2026 16:27:29 -0400https://liberty.io/what-is-the-rule-for-something-divided-by-itself-equaling-1.htmlhttps://liberty.io/what-is-the-rule-for-something-divided-by-itself-equaling-1.htmlIs there a name for the mathematical rule/axiom/property $x/x = 1?$ What are the conditions for it to apply? For instance, the rule does not apply where $x = 0$ or $x = \inf$. I saw one site that ...Sat, 15 Aug 2026 16:20:28 -0400https://liberty.io/prerequisites-books-for-a-first-course-in-linear-algebra.htmlhttps://liberty.io/prerequisites-books-for-a-first-course-in-linear-algebra.htmlWhat mathematical knowledge do I need to begin studying linear algebra? In particular, how much calculus do I need to know? Also, do you have a favorite linear algebra book you can recommend?...Sat, 15 Aug 2026 16:07:20 -0400https://liberty.io/solving-vector-equation.htmlhttps://liberty.io/solving-vector-equation.html$2$. a: For what vectors $$v = \begin{pmatrix}a \\ b \\ c\end{pmatrix}$$ does the equation $Ax = v$ have a solution if $$A = \left(\begin{array}{crc} 1 & 0 & 1 \\ 0 & -1 & 2 \\ 1......Sat, 15 Aug 2026 16:06:15 -0400https://liberty.io/from-discrete-dynamical-system-to-continuous-one.htmlhttps://liberty.io/from-discrete-dynamical-system-to-continuous-one.htmlTo me it seems that is not always the case that a discrete dynamical system of the form $$ x_{k+1} = f(x_k) $$ can be seen as the discretization of a continuous one like $$ \dot{x}(t) = f(x), $$ wh......Sat, 15 Aug 2026 16:01:52 -0400