https://liberty.ioCurated star highlights with fresh energy.en-usSat, 15 Aug 2026 16:49:20 -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 -0400https://liberty.io/gps-coordinates-processing.htmlhttps://liberty.io/gps-coordinates-processing.htmlI want to check a circle intercepts with a line or not. Lets say i have two points with knowing their longitutudes and latitudes. And i also know the circle's longitude, latitude and radius as mete......Sat, 15 Aug 2026 15:58:10 -0400https://liberty.io/proof-of-a-ramanujan-integral.htmlhttps://liberty.io/proof-of-a-ramanujan-integral.htmlWhile studying Ramanujan's Collected Papers I came across a paper titled "Some Definite Integrals" which appeared in Messenger of Mathematics, ${\tt XLIV}, 1915, \mbox{10-18}$. It contain......Sat, 15 Aug 2026 15:48:38 -0400https://liberty.io/show-that-the-surface-area-of-a-cone-is-pi-a-sqrt-a-2-h-2-by-consideri.htmlhttps://liberty.io/show-that-the-surface-area-of-a-cone-is-pi-a-sqrt-a-2-h-2-by-consideri.htmlShow that the surface area of a cone in terms of it’s height and radius is $\pi a \sqrt {a^2+h^2}$ by considering how a sector of a disc radius r angle $\phi $ can be folded to make a cone. I cannot ...Sat, 15 Aug 2026 15:42:30 -0400https://liberty.io/limit-of-geometric-series-sum-when-r-1.htmlhttps://liberty.io/limit-of-geometric-series-sum-when-r-1.htmlI'm currently learning the prove of sum of geometric series on Khan Academy. I understand the behaviour of the function when $|r| > 1$, when $|r| < 1$, when $r = 0$ and when $r = -1$, but I......Sat, 15 Aug 2026 15:39:08 -0400https://liberty.io/why-the-negative-gradient-gives-the-direction-of-the-steepest-decrease.htmlhttps://liberty.io/why-the-negative-gradient-gives-the-direction-of-the-steepest-decrease.htmlI understand that the gradient vector gives the direction of the maximum growth. What I don't get is why going the exact opposite direction is going to get the maximum decrease? By sure that hold......Sat, 15 Aug 2026 15:38:49 -0400https://liberty.io/symmetric-difference-using-venn-diagrams-discrete-math.htmlhttps://liberty.io/symmetric-difference-using-venn-diagrams-discrete-math.htmlMy question is about that image: Is this the correct Venn diagram for part c ? Why is there a red region in the middle that is painted ? Where did this red region came from? And about the relat......Sat, 15 Aug 2026 15:37:23 -0400https://liberty.io/cos-5-x-as-a-sum.htmlhttps://liberty.io/cos-5-x-as-a-sum.htmlHow can I write the $$ \cos^{5}(x) $$ as a sum of first order terms? I have started noticing that: $$ \cos^{5}(x) = \cos^{4}(x) \cos(x),$$ and then: $$ \cos^{4}(x)=\cos^{2}(x)\cos^{2}(x). $$ Finall......Sat, 15 Aug 2026 15:29:08 -0400https://liberty.io/graph-theory-how-do-we-know-hamiltonian-path-exists-in-graph-where-eve.htmlhttps://liberty.io/graph-theory-how-do-we-know-hamiltonian-path-exists-in-graph-where-eve.htmlI am trying to prove that if every node of a graph G has degree of at least 3 then G contains a cycle and a chord. My current approach is as follows: Separate the graph G into connected component......Sat, 15 Aug 2026 15:28:20 -0400https://liberty.io/how-to-find-the-surface-area-of-a-open-top-rectangular-container-when-.htmlhttps://liberty.io/how-to-find-the-surface-area-of-a-open-top-rectangular-container-when-.htmlI'm a computer science major, and I was asked to write a program with the following question. A manufacturer wishes to compare the cost of producing an open-top cylindrical container and an open......Sat, 15 Aug 2026 15:16:16 -0400https://liberty.io/parametric-equation-of-an-arc-with-given-radius-and-two-points.htmlhttps://liberty.io/parametric-equation-of-an-arc-with-given-radius-and-two-points.htmlso I need the parametric equation of the arc. So, arc is a sector of a circle. Parametric circle equation is: $$ c \equiv f(t) = (\cos(t), \sin(t)),\quad 0\le t < 2\pi $$ So, we just need to find ...Sat, 15 Aug 2026 15:15:57 -0400https://liberty.io/definition-of-the-weight-k-hyperbolic-laplacian.htmlhttps://liberty.io/definition-of-the-weight-k-hyperbolic-laplacian.htmlI saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mat......Sat, 15 Aug 2026 15:07:20 -0400https://liberty.io/completing-the-square-with-multiple-variables.htmlhttps://liberty.io/completing-the-square-with-multiple-variables.htmlI'm trying to understand a solution to a PDE problem, and it involves reducing an expression by completing the square. I'm not sure how to go about the steps. The expression is: $$-x^2+2xy-y^2+4kt......Sat, 15 Aug 2026 14:44:41 -0400https://liberty.io/divergence-in-spherical-coordinates-vs-cartesian-coordinates.htmlhttps://liberty.io/divergence-in-spherical-coordinates-vs-cartesian-coordinates.htmlThe divergence in spherical coordinates is derived as $$\nabla\vec A=\frac{1}{r^2}\frac{\partial(r^2A_r)}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(A_\theta\sin\theta)+\frac......Sat, 15 Aug 2026 14:41:54 -0400https://liberty.io/to-find-the-modulus-of-a-complex-number.htmlhttps://liberty.io/to-find-the-modulus-of-a-complex-number.htmlI was given the following expression , $|\frac{(3+i)(2-i)}{(1+i)}|$ , and was asked to find its value This is how I proceeded , On solving the numerator , the given expression transforms to $|\...Sat, 15 Aug 2026 14:40:53 -0400https://liberty.io/prove-that-4-is-the-only-solution-to-2-2-duplicate.htmlhttps://liberty.io/prove-that-4-is-the-only-solution-to-2-2-duplicate.htmlThis question was featured on Saturday Morning Breakfast Cereal and I haven't been able to find a proof. Can anyone help?...Sat, 15 Aug 2026 14:21:28 -0400https://liberty.io/how-to-find-the-perimeter-of-the-triangle-inscribed-within-the-square.htmlhttps://liberty.io/how-to-find-the-perimeter-of-the-triangle-inscribed-within-the-square.htmlIn square $ABCD$, the length of its sides is $5$. $E, F$ are two points on $AB$ and $AD$ in such a way so that $∠ECF = 45^\circ$, find the maximum value of the perimeter of $ΔAEF$....Sat, 15 Aug 2026 14:14:56 -0400https://liberty.io/verification-of-induction-proof-for-handshake-lemma.htmlhttps://liberty.io/verification-of-induction-proof-for-handshake-lemma.htmlI did the following proof which seems correct to me but does not match the approach of the answer provided by my professor, and seems pretty different from the question here in terms of notation and ...Sat, 15 Aug 2026 14:13:58 -0400https://liberty.io/trick-fake-proof-of-2x2-5-algebraic-form-closed.htmlhttps://liberty.io/trick-fake-proof-of-2x2-5-algebraic-form-closed.htmlJust ran into "algebraic proof" that 2x2=5 on youtube. They didn't hide that it was a trick. Can't find where is the mistake. Let a=4, b=5, c=1 c=b-a /multiply by (b-a)......Sat, 15 Aug 2026 14:13:23 -0400https://liberty.io/finding-connected-components-in-a-graph-using-bfs.htmlhttps://liberty.io/finding-connected-components-in-a-graph-using-bfs.htmlI'd like to know how do I change the known BFS algorithm in order to find all of the connected components of a given graph. The original algorithm stops whenever we've colored an entire component in ...Sat, 15 Aug 2026 14:05:16 -0400https://liberty.io/how-to-verify-if-a-curve-is-exponential-by-eyeballing.htmlhttps://liberty.io/how-to-verify-if-a-curve-is-exponential-by-eyeballing.htmlA plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitat......Sat, 15 Aug 2026 14:03:15 -0400https://liberty.io/prove-that-an-upper-triangular-matrix-is-invertible-if-and-only-if-eve.htmlhttps://liberty.io/prove-that-an-upper-triangular-matrix-is-invertible-if-and-only-if-eve.htmlProve that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero. I have proved that if every diagonal entry is non-zero, then the matrix is invertible by showing......Sat, 15 Aug 2026 13:47:54 -0400https://liberty.io/properties-of-the-number-50.htmlhttps://liberty.io/properties-of-the-number-50.htmlI will shortly be engaging with my 50th (!) birthday. 50 = 1+49 = 25+25 can perhaps be described as a "sub-Ramanujan" number. I'm trying to put together a quiz including some mathematical content. ...Sat, 15 Aug 2026 13:46:37 -0400https://liberty.io/calculating-the-modular-multiplicative-inverse-without-all-those-stran.htmlhttps://liberty.io/calculating-the-modular-multiplicative-inverse-without-all-those-stran.htmlI am sure all those symbols are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me. How could I do this on a basic calculator? or with......Sat, 15 Aug 2026 13:44:35 -0400https://liberty.io/littlewood-039-s-principles-and-counterexamples.htmlhttps://liberty.io/littlewood-039-s-principles-and-counterexamples.htmlLittlewood’s first principle says that each measurable set E of finite measure is nearly a union of finitely many open intervals. However, if the G for which the symmetric difference with E is les......Sat, 15 Aug 2026 13:29:29 -0400https://liberty.io/help-getting-a-zero-out-of-the-denominator-of-a-limit.htmlhttps://liberty.io/help-getting-a-zero-out-of-the-denominator-of-a-limit.htmlWhy is the following limit equal to $1/2$. I get undefined. :-( $$\lim_{x\to 0}\frac{(x+1)^{1/2}+1}x$$ this should $= 1/2$. When I multiply the top and bottom by $(x+1)^{1/2} - 1$, I end up with......Sat, 15 Aug 2026 13:28:16 -0400https://liberty.io/what-is-the-standard-deviation-of-dice-rolling.htmlhttps://liberty.io/what-is-the-standard-deviation-of-dice-rolling.htmlWhen trying to find how to simulate rolling a variable amount of dice with a variable but unique number of sides, I read that the mean is $\dfrac{sides+1}{2}$, and that the standard deviation is $\......Sat, 15 Aug 2026 13:22:30 -0400https://liberty.io/growth-of-the-riemann-siegel-theta-function.htmlhttps://liberty.io/growth-of-the-riemann-siegel-theta-function.htmlThe Riemann-Siegel theta function is defined by: $ \theta(t) = arg\left(\Gamma(\frac{0.5+it}{2}) \right) - \frac{t}{2}\log{\pi}$ As in the following: https://en.wikipedia.org/wiki/Riemann%E2%80%...Sat, 15 Aug 2026 13:08:07 -0400https://liberty.io/calculating-the-variance-using-the-probability-mass-function-pmf.htmlhttps://liberty.io/calculating-the-variance-using-the-probability-mass-function-pmf.htmlMy Problem: The random variables X and Y have the simultaneous probability function pX,Y(x, y). pX,Y(x, y) X = 0 X = 1 X = 2 Y = 0 0.1 (Z = 0) 0.2 (Z = 1) 0.2 (Z = 2) Y = 1 0.1 (Z = 1) 0.1 (Z = ......Sat, 15 Aug 2026 13:05:49 -0400https://liberty.io/6-team-round-robin.htmlhttps://liberty.io/6-team-round-robin.htmlIs it possible to create a schedule where we have 6 teams play 5 different games with each team playing each game and each team only once? Or do we have to increase the amount of games to make this ...Sat, 15 Aug 2026 13:04:50 -0400https://liberty.io/what-is-the-difference-between-infinite-and-transfinite.htmlhttps://liberty.io/what-is-the-difference-between-infinite-and-transfinite.htmlI have never properly got my head round exactly what the difference is between "transfinite" and "infinite". Wikipedia says: transfinite numbers are numbers that are "infinite" in the sense that t......Sat, 15 Aug 2026 13:01:12 -0400https://liberty.io/how-to-construct-a-transcendental-number.htmlhttps://liberty.io/how-to-construct-a-transcendental-number.htmlI am doing a project about irrational and transcendental numbers and I was wondering how could I construct a "new" transcendental number. I know that all Liouville numbers are transcendental so this ...Sat, 15 Aug 2026 12:59:17 -0400https://liberty.io/a-rectangle-is-twice-as-long-as-it-is-wide.htmlhttps://liberty.io/a-rectangle-is-twice-as-long-as-it-is-wide.htmlA Farmer fenced a rectangular area with $75$ metres of chicken wire. The rectangle is twice as long as it is wide. What are its dimensions?...Sat, 15 Aug 2026 12:58:04 -0400https://liberty.io/the-well-ordering-principle.htmlhttps://liberty.io/the-well-ordering-principle.htmlHere is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered. In the book, it says that the chief advan......Sat, 15 Aug 2026 12:55:07 -0400https://liberty.io/what-are-the-rules-of-powers-of-powers-duplicate.htmlhttps://liberty.io/what-are-the-rules-of-powers-of-powers-duplicate.htmlWhat would $2^{3^4}$ equate to? I can think of two rules that may apply: $a^{b^c} = a^{(b^c)}$ (Making $2^{3^4} = 2^{81}\approx2.417\cdot10^{24}$) or $a^{b^c} = (a^b)^c = a^{bc}$ (Making $2^{3^4}......Sat, 15 Aug 2026 12:52:37 -0400https://liberty.io/what-does-a-colon-mean-as-a-mathematical-symbol-in-the-context-of-mapp.htmlhttps://liberty.io/what-does-a-colon-mean-as-a-mathematical-symbol-in-the-context-of-mapp.htmlI am trying to understand the principle of recursion: Let ℕ be the natural numbers. Let T be a set. Let a ∈ T. Let g: T → T be a mapping. $$\forall x \in N: f \left({x}\r......Sat, 15 Aug 2026 12:46:59 -0400https://liberty.io/why-does-the-minimum-value-of-x-x-equal-1-e.htmlhttps://liberty.io/why-does-the-minimum-value-of-x-x-equal-1-e.htmlThe graph of $y=x^x$ looks like this: As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$. I know that $e$ is the number for expon......Sat, 15 Aug 2026 12:43:31 -0400https://liberty.io/why-algebraic-topology-is-also-called-combinatorial-topology.htmlhttps://liberty.io/why-algebraic-topology-is-also-called-combinatorial-topology.htmlI remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...Sat, 15 Aug 2026 12:41:55 -0400https://liberty.io/largest-inscribed-rectangle-in-sector.htmlhttps://liberty.io/largest-inscribed-rectangle-in-sector.htmlWhat is the largest (by area) rectangle that can be inscribed in a circular sector wiith radius $r$ and central angle $\alpha$? I think I have an answer to this question, but would like it confirm......Sat, 15 Aug 2026 12:39:55 -0400https://liberty.io/every-subspace-of-mathbb-r-n-has-an-orthonormal-basis.htmlhttps://liberty.io/every-subspace-of-mathbb-r-n-has-an-orthonormal-basis.htmlIs there a non-constructive proof of this statement, i.e., one that avoids Gram-Schmidt?...Sat, 15 Aug 2026 12:36:07 -0400https://liberty.io/what-is-the-role-of-conjectures-in-modern-mathematics.htmlhttps://liberty.io/what-is-the-role-of-conjectures-in-modern-mathematics.htmlToday, I heard of something so called Goldbach's conjecture from my mathematics teacher in the class. This was one of the most interesting things that I have ever heard in mathematics. This made me ...Sat, 15 Aug 2026 12:32:47 -0400https://liberty.io/l-039-hospital-039-s-rule-for-the-indeterminate-form-0-0.htmlhttps://liberty.io/l-039-hospital-039-s-rule-for-the-indeterminate-form-0-0.htmlI'm teaching an introductory real analysis course for the first time this year and one of the textbook problems asks to prove the following: Let $f$ be differentiable on $(a,b)$ and let $c \in (a,......Sat, 15 Aug 2026 12:17:24 -0400https://liberty.io/limit-of-infinite-sequence-from-partial-sum.htmlhttps://liberty.io/limit-of-infinite-sequence-from-partial-sum.htmlI think there was a rule in Calculus that mentions this, but I am not sure. If I need to find $\lim_{n \to \infty} a_n$ and I am only given the nth partial sum: $S_n =\sum_{k=1}^{n} a_k = f(n)$ ......Sat, 15 Aug 2026 12:11:41 -0400https://liberty.io/two-lines-that-are-perpendicular-to-a-third-line-do-not-have-to-be-par.htmlhttps://liberty.io/two-lines-that-are-perpendicular-to-a-third-line-do-not-have-to-be-par.htmlIn 3-space, two lines that are perpendicular to the same plane must be parallel. However, two lines that are perpendicular to a third line do not have to be parallel. Explain why. I am having a very ...Sat, 15 Aug 2026 12:07:52 -0400https://liberty.io/getting-probability-from-z-score-on-ti-83.htmlhttps://liberty.io/getting-probability-from-z-score-on-ti-83.htmlI'm able to use a $z$-table, but would like to be able to use my ti-$83+$ to get value. For example, $P(z \leq 2.5) = 0.9938$ , given $2.5$ I would like to get $0.9938$. Any help would be appreci......Sat, 15 Aug 2026 12:07:21 -0400https://liberty.io/square-root-inside-a-square-root.htmlhttps://liberty.io/square-root-inside-a-square-root.htmlHi guys I just want to ask how to solve this. The given is: $$f(x)=\sqrt{x}$$ then solve for the $$(f \circ f)(x)$$ then it becomes. $$(f \circ f)(x)=f(f(x))$$ $$f(x)=\sqrt{\sqrt{x}}$$ Can the ...Sat, 15 Aug 2026 12:05:51 -0400https://liberty.io/what-is-the-difference-between-3-choose-2-and-3-choose-1-2-choose-1.htmlhttps://liberty.io/what-is-the-difference-between-3-choose-2-and-3-choose-1-2-choose-1.htmlWhile choosing 2 from 3, we do ${3 \choose 2}$. But what would happen if we do ${3 \choose 1}{2 \choose 1}$? [Choosing 1 from 3 and again 1 from the remaining two)? I know the latter is incorrect b......Sat, 15 Aug 2026 11:57:15 -0400https://liberty.io/value-of-summation-of-log-n.htmlhttps://liberty.io/value-of-summation-of-log-n.htmlContext: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: For each ...Sat, 15 Aug 2026 11:57:09 -0400https://liberty.io/linear-algebra-identity-matrix-and-its-relation-to-eigenvalues-and-eig.htmlhttps://liberty.io/linear-algebra-identity-matrix-and-its-relation-to-eigenvalues-and-eig.htmlKindly help me understand this statement made by my prof. "The identity matrix I has the property that any non zero vector V is an eigenvector of eigenvalue 1." My assumption of this statement is......Sat, 15 Aug 2026 11:41:49 -0400https://liberty.io/what-precisely-is-the-difference-between-euclidean-geometry-and-non-eu.htmlhttps://liberty.io/what-precisely-is-the-difference-between-euclidean-geometry-and-non-eu.htmlI was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that non-Eucl......Sat, 15 Aug 2026 11:41:09 -0400https://liberty.io/can-someone-please-help-me-convert-this-triple-integral-to-spherical-c.htmlhttps://liberty.io/can-someone-please-help-me-convert-this-triple-integral-to-spherical-c.htmlI'm not sure how to approach the problem, though I know it needs to be broken up into two integrals before it can be evaluated based on the way the answer input is set up. I do not know how to start ...Sat, 15 Aug 2026 11:13:51 -0400https://liberty.io/why-are-exact-differential-equations-called-so.htmlhttps://liberty.io/why-are-exact-differential-equations-called-so.htmlAs the question says, why are exact differential equations called so? From Wikipedia, I got "The nomenclature of "exact differential equation" refers to the exact differential of a function". That ...Sat, 15 Aug 2026 11:13:51 -0400https://liberty.io/closure-of-the-set-of-all-polynomial-with-variable-x-in-0-1.htmlhttps://liberty.io/closure-of-the-set-of-all-polynomial-with-variable-x-in-0-1.htmlLet ${P}$ denote the set of all polynomial with variable $x\in [0,1]$, I need to know what is the closure of ${P}$ in $C[0,1]$? Well, Stone-Weierstrass theorem says: If $f\in C[0,1]$ then there exi......Sat, 15 Aug 2026 11:10:29 -0400https://liberty.io/what-are-the-mean-and-variance-of-the-log-of-a-random-variable.htmlhttps://liberty.io/what-are-the-mean-and-variance-of-the-log-of-a-random-variable.htmlHere's the problem. We have a random variable X that follows a Poisson law. If we take the log of this variable, what are the first two moments (mean and variance) of the law it follows? This looks ...Sat, 15 Aug 2026 11:02:54 -0400https://liberty.io/a-double-limit-question.htmlhttps://liberty.io/a-double-limit-question.htmlMaybe it's easy, but: Is it true that $$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$ If it is, could you help me prove it? Thanks...Sat, 15 Aug 2026 10:51:50 -0400https://liberty.io/recognizing-the-sequence-1-16-1-8-3-16-1-4-5-16.htmlhttps://liberty.io/recognizing-the-sequence-1-16-1-8-3-16-1-4-5-16.htmlWhat is the missing number? $$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \ \ \ [?]$$ $$A. \frac{5}{4}\quad B. \frac{3}{4}\quad C. \frac{5}{8}\quad D. \frac{3}{8}$$ Spo......Sat, 15 Aug 2026 10:45:05 -0400