https://liberty.ioCurated star highlights with fresh energy.en-usMon, 24 Aug 2026 07:37:15 -0400https://liberty.io/proving-3-n-geq-3n-using-mathematical-induction.htmlhttps://liberty.io/proving-3-n-geq-3n-using-mathematical-induction.htmlSo I have to prove that $3^n$ is greater than or equal to $3n$ using induction. The base case is a not a problem, but I can't seem to figure out where to go for $(n-1)$. I've tried saying: $$3^n=3\......Mon, 24 Aug 2026 11:27:52 -0400https://liberty.io/example-of-hypergraph.htmlhttps://liberty.io/example-of-hypergraph.htmlDefinition: A hypergraph $\Gamma=(V,\mathcal{E})$ is a set of vertices $V$ and a collection $\mathcal{E}$ of subsets of $V$ such that for every $E\in \mathcal{E}$, we have $|E|\geq 2$. The members ......Mon, 24 Aug 2026 11:26:37 -0400https://liberty.io/identifying-self-intersection-points-in-one-parametric-graph.htmlhttps://liberty.io/identifying-self-intersection-points-in-one-parametric-graph.htmlMy question for you is how to identify self-intersection points in a parametric curve of the form x = f(t), y = g(t). The specific problem asks for the t values of the intersection where $x = \sin(......Mon, 24 Aug 2026 11:25:45 -0400https://liberty.io/generalized-inverse-of-transpose-closed.htmlhttps://liberty.io/generalized-inverse-of-transpose-closed.htmlHow can we show that if $G$ is generalized Inverse of $A$ , then $G^T$ is generalized Inverse of $A^T$...Mon, 24 Aug 2026 11:09:46 -0400https://liberty.io/notion-of-cusp-in-algebraic-geometry.htmlhttps://liberty.io/notion-of-cusp-in-algebraic-geometry.htmlThis relates to problem 3.14(b) of Hartshorne, wherein I am required to show that the projection of the twisted cubic curve (viewed as a subset of $\mathbb P^3$) from the point $(0,0,1,0)$ onto the ...Mon, 24 Aug 2026 10:56:58 -0400https://liberty.io/write-csc-x-in-terms-of-sec-x.htmlhttps://liberty.io/write-csc-x-in-terms-of-sec-x.htmlI was working on trig homework and came across a question that I didn't understand how to even begin to approach. It asked us to use trigonometric identities to write $\csc(x)$ in terms of $\sec(x......Mon, 24 Aug 2026 10:52:08 -0400https://liberty.io/the-composite-function-limit-theorem-proof.htmlhttps://liberty.io/the-composite-function-limit-theorem-proof.htmlLet us define two functions:- $$f(x)$$ $$g(x)$$ And now let us try to find the limit :- $$\lim_{x \to c} f(g(x))$$ Assuming that all the limits mentioned below exist we can say:- $$ \lim_{x \to c}g......Mon, 24 Aug 2026 10:42:54 -0400https://liberty.io/a-quadratic-trinomial-that-generates-only-prime-numbers-of-the-form-4m.htmlhttps://liberty.io/a-quadratic-trinomial-that-generates-only-prime-numbers-of-the-form-4m.htmlIt is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$. The best trino......Mon, 24 Aug 2026 10:42:23 -0400https://liberty.io/probability-to-guess-at-least-one-answer-correctly.htmlhttps://liberty.io/probability-to-guess-at-least-one-answer-correctly.htmlA multiple choice test consists of three problems. For each problem, there are five choices , one of which is correct .One student comes totally unprepared and decides to answer by sheer guessing .......Mon, 24 Aug 2026 10:39:44 -0400https://liberty.io/why-do-we-need-span-in-linear-algebra.htmlhttps://liberty.io/why-do-we-need-span-in-linear-algebra.htmlIn my linear algebra course in university we started learning about span and I was curious what is it good for? and if someone know, how does it relate to 3D graphics? Thank you....Mon, 24 Aug 2026 10:38:48 -0400https://liberty.io/logarithms-with-a-fraction-as-base.htmlhttps://liberty.io/logarithms-with-a-fraction-as-base.htmlHow does one solve a logarithmic expression where the base is a fraction? In my example I am trying to solve the following: $$ n^{\log_\frac{3}{2}(1)} \tag{1} $$ This is related to using the "ma......Mon, 24 Aug 2026 10:36:58 -0400https://liberty.io/find-height-of-irregular-trapezoid-with-known-angles-and-surface-area.htmlhttps://liberty.io/find-height-of-irregular-trapezoid-with-known-angles-and-surface-area.htmlKNOWN: Length DC Alpha Beta Surface S NEEDED: Height h For an algorithm, I require a way to solve this for any trapezoid. Sort of like this question (Given a known isosceles Trapezoid find heig......Mon, 24 Aug 2026 10:36:44 -0400https://liberty.io/write-an-equation-as-a-single-power-grade-11-math-function.htmlhttps://liberty.io/write-an-equation-as-a-single-power-grade-11-math-function.html$$\frac{10^{-4/5} \cdot 10^{1/15}}{10^{2/3}}$$ The answer is $10^{-7/5}$, which seems impossible to me. I get: $10^{-4/5} \cdot 10^{-11/15}$. I see where the numerator $7$ comes from but the ...Mon, 24 Aug 2026 10:33:14 -0400https://liberty.io/what-will-be-the-formula-for-trace-t.htmlhttps://liberty.io/what-will-be-the-formula-for-trace-t.htmlLet $M_n(K)$ denote the space of all $n\times n$ matrices with entries in a field $K$. Fix a non-singular matrix $A=(A_{ij})\in M_n(K)$, and consider the linear map $T:M_n(K)\to M_n(K)$ given by: $......Mon, 24 Aug 2026 10:26:22 -0400https://liberty.io/interchange-of-expected-value-and-summation.htmlhttps://liberty.io/interchange-of-expected-value-and-summation.htmlHey guys I'm studying Statistics and I'm currently at unbiased estimators. I'm trying to find the estimators for different problems. My question is : I know that the following formula holds $$E (\......Mon, 24 Aug 2026 10:25:54 -0400https://liberty.io/what-is-the-mathematical-symbol-for-the-sum-of-numbers.htmlhttps://liberty.io/what-is-the-mathematical-symbol-for-the-sum-of-numbers.htmlFor example, when $n=5$, what is the symbol for $5+4+3+2+1$?...Mon, 24 Aug 2026 10:13:14 -0400https://liberty.io/long-division-differs-from-calculator.htmlhttps://liberty.io/long-division-differs-from-calculator.htmli have been helping my daughter with her long division homework. One of the questions was 172/6 she worked out the answer using the long division tree, and upon checking it found the answer differ......Mon, 24 Aug 2026 10:08:50 -0400https://liberty.io/what-is-so-special-about-the-special-orthogonal-group.htmlhttps://liberty.io/what-is-so-special-about-the-special-orthogonal-group.htmlI am trying to learn a bit about (geometrical Euclidean) group theory and am now wondering "Wat is so special about the Special Orthogonal group?" Or what is the difference between the groups $O(......Mon, 24 Aug 2026 10:04:04 -0400https://liberty.io/clopen-subsets-of-the-cantor-set.htmlhttps://liberty.io/clopen-subsets-of-the-cantor-set.htmlIn our Topology course, we have been studying the anti-equivalance of categories between zero-dimensional compact Hausdorff spaces and Boolean algebras. The Cantor set has come up a lot, and I hav......Mon, 24 Aug 2026 10:00:05 -0400https://liberty.io/what-are-some-good-rigorous-precalculus-textbooks-for-self-study.htmlhttps://liberty.io/what-are-some-good-rigorous-precalculus-textbooks-for-self-study.htmlBooks that: has an introduction to proof, logic, and topics like sets and groups. Books can prepare you for rigorous calculus texts like Spivak and Apostol. Question 1. I've been looking at the......Mon, 24 Aug 2026 09:58:26 -0400https://liberty.io/limit-of-1-x-as-x-approaches-infinity-duplicate.htmlhttps://liberty.io/limit-of-1-x-as-x-approaches-infinity-duplicate.htmlWhy is $\lim_{x\to\infty} \frac{1}{x}$ equal to $0$?...Mon, 24 Aug 2026 09:52:31 -0400https://liberty.io/volume-of-an-ellipsoid-using-cylindrical-polar-coordinates.htmlhttps://liberty.io/volume-of-an-ellipsoid-using-cylindrical-polar-coordinates.htmlI've been working on a question about finding the volume of an ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$$ This is fine if I consider rescaling the axes to give a spher......Mon, 24 Aug 2026 09:50:06 -0400https://liberty.io/suppose-t-is-a-linear-transformation-such-that-t-1-1-1-0-1-2-t-1-0-1-1.htmlhttps://liberty.io/suppose-t-is-a-linear-transformation-such-that-t-1-1-1-0-1-2-t-1-0-1-1.htmlWhat is $T(x,y,z)$ What I tried: $$T(x,y,z)= \begin{pmatrix} 1&1&1|0&1&2\\ 1&0&1|1&1&1\\ 0&0&1|1&2&3\\ \end{pmatrix} $$ Which reduces to \begin{pm......Mon, 24 Aug 2026 09:42:11 -0400https://liberty.io/approximating-logs-and-antilogs-by-hand.htmlhttps://liberty.io/approximating-logs-and-antilogs-by-hand.htmlI have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful......Mon, 24 Aug 2026 09:41:22 -0400https://liberty.io/what-is-the-definition-of-double-sequence-a-mn-being-convergent-to-l.htmlhttps://liberty.io/what-is-the-definition-of-double-sequence-a-mn-being-convergent-to-l.htmlWhat is the definition of double sequence $a_{mn}$ being convergent to $l$? I have this definition. Definition: The double sequence $(a_{m,n})^∞_{m,n=1}$ is said to Converge to the real number $......Mon, 24 Aug 2026 09:39:15 -0400https://liberty.io/what-is-the-column-space-in-a-5-times-5-invertible-matrix.htmlhttps://liberty.io/what-is-the-column-space-in-a-5-times-5-invertible-matrix.htmlIf $A$ is any $5 \times 5$ invertible matrix, then what is its column space? Why? I'm totally lost with column space. Any ideas?...Mon, 24 Aug 2026 09:36:49 -0400https://liberty.io/oriented-atlas-on-a-circle.htmlhttps://liberty.io/oriented-atlas-on-a-circle.htmlI'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d......Mon, 24 Aug 2026 09:35:58 -0400https://liberty.io/dual-of-h-1-0-h-1-or-h-0-1.htmlhttps://liberty.io/dual-of-h-1-0-h-1-or-h-0-1.htmlI have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...Mon, 24 Aug 2026 09:28:45 -0400https://liberty.io/is-this-the-correct-way-to-say-an-integral.htmlhttps://liberty.io/is-this-the-correct-way-to-say-an-integral.htmlFor this structure: Is the correct way to verbally say this (such that a screen reader would read it aloud correctly), "The integral over the interval from negative infinity to positive infinity o......Mon, 24 Aug 2026 09:27:12 -0400https://liberty.io/partial-fractions-proper-fraction-with-a-perfect-square-factor.htmlhttps://liberty.io/partial-fractions-proper-fraction-with-a-perfect-square-factor.htmlI think I have figured why the following is the case, I need however a confirmation that it is indeed so. Consider: $\displaystyle{\frac{x-1}{(x+1)(x-2)^2} \equiv \frac{A}{x+1} + \frac{Bx+C}{(x-2......Mon, 24 Aug 2026 09:25:57 -0400https://liberty.io/domain-and-range-of-a-multivariable-function.htmlhttps://liberty.io/domain-and-range-of-a-multivariable-function.htmlI have this exercise $$z={2x\over y+5}$$, and I am supposed to obtain the domain and range. I understand that the domain is all the pair of $(x,y)$ except $y=-5$ , then the exercise said that the r......Mon, 24 Aug 2026 09:25:43 -0400https://liberty.io/show-that-4mn-m-n-can-never-be-a-square.htmlhttps://liberty.io/show-that-4mn-m-n-can-never-be-a-square.htmlLet $m$ and $n$ be positive integers. Show that $$4mn-m-n$$ can never be a square. In my attempt I started by assuming for the sake of contradiction that $$4mn-m-n=k^2$$ for some $k \in \mathbb{N_......Mon, 24 Aug 2026 09:25:30 -0400https://liberty.io/degenerate-critical-points-and-higher-order-derivatives-intuition.htmlhttps://liberty.io/degenerate-critical-points-and-higher-order-derivatives-intuition.htmlThe question I'd like to ask is about the relevance of higher order derivatives in determining the behaviour of a function around a critical point in the multivariate case. The question occurs towa......Mon, 24 Aug 2026 09:19:08 -0400https://liberty.io/indeterminate-form-of-infinity-over-0.htmlhttps://liberty.io/indeterminate-form-of-infinity-over-0.htmlI know that indeterminate forms exist in limits, such as $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0^0$, $\infty^0$, $1^\infty$. Then, if $\lim\limits_{x \to a} p(x)=\infty$ and $\lim\limits_{x \t......Mon, 24 Aug 2026 09:17:05 -0400https://liberty.io/lawn-mowing-problem-solving.htmlhttps://liberty.io/lawn-mowing-problem-solving.htmlKate can mow the lawn in 45 minutes. Kate's sister takes twice as long to mow the same lawn. If they both have a mower and mow the lawn together, how many minutes will it take them? I know the ans......Mon, 24 Aug 2026 09:13:53 -0400https://liberty.io/ti-nspire-cas-implicit-differentiation-not-returning-correct-result-re.htmlhttps://liberty.io/ti-nspire-cas-implicit-differentiation-not-returning-correct-result-re.htmlTaking the derivative of $y^3-xy=2$, the correct result should be $y'=y/(3y^2-x)$. However, when I use the impDif() function on TI Nspire CAS, the returned result is 0, as shown in the screenshot ......Mon, 24 Aug 2026 09:11:30 -0400https://liberty.io/understanding-of-the-kronecker-delta.htmlhttps://liberty.io/understanding-of-the-kronecker-delta.htmlFrom my understanding $e_a\cdotp e_b$ = $\delta_{ab}$ (for $a,b = 1,2,3$) equals $1$ when $a=b$ and $0$ when $a \neq b $ where $e_a$ and $e_b$ are vectors with entries 1 in the $a$'th and $b$'th row, ...Mon, 24 Aug 2026 09:11:20 -0400https://liberty.io/primitive-elements-of-gf-8.htmlhttps://liberty.io/primitive-elements-of-gf-8.htmlI'm trying to find the primitive elements of $GF(8),$ the minimal polynomials of all elements of $GF(8)$ and their roots, and calculate the powers of $\alpha^i$ for $x^3 + x + 1.$ If I did my math ...Mon, 24 Aug 2026 09:00:49 -0400https://liberty.io/show-the-integration-of-legendre-polynomials-is-0.htmlhttps://liberty.io/show-the-integration-of-legendre-polynomials-is-0.htmlShow that for the Legendre polynomial $P_n$ with $n\neq0$, $$\int_{-1}^{+1} P_n (x) dx =0$$ I put this polynomial in the Legendre equation then got stuck. Can you help me find out what to do next?...Mon, 24 Aug 2026 08:50:06 -0400https://liberty.io/what-is-representation-theory.htmlhttps://liberty.io/what-is-representation-theory.htmlI'm beginning a course that uses representation theory, but I do not really understand what that is about. In the text I am following, I have the following definition: A representation of the Lie ...Mon, 24 Aug 2026 08:44:30 -0400https://liberty.io/how-to-select-convergence-criterion-in-numerical-analysis.htmlhttps://liberty.io/how-to-select-convergence-criterion-in-numerical-analysis.htmlWhen doing old exams in basic numerical analysis, I encountered this problem: Solution proposal from lecturer: My idea was to select $|f(x)| \le 0.5 \times 10^{-5}$ as the convergence criterion. ......Mon, 24 Aug 2026 08:43:37 -0400https://liberty.io/question-about-rational-functions-and-horizontal-asymptotes.htmlhttps://liberty.io/question-about-rational-functions-and-horizontal-asymptotes.htmlI am working on a math problem for class and am stumped by the nature of its horizontal asymptote. The equation is f(x) = x / (x+5)(x-2). Based on the rules for horizontal asymptotes given to me in ...Mon, 24 Aug 2026 08:37:29 -0400https://liberty.io/calculating-the-semi-major-axis-with-one-point-the-eccentricity-and-th.htmlhttps://liberty.io/calculating-the-semi-major-axis-with-one-point-the-eccentricity-and-th.htmlGiven is one "random" point on the ellipse, the eccentricity and the center of the ellipse. How can I calculate the semi-major axis? I'm not very math-savvy, so I hope the problem is easy to solve. ...Mon, 24 Aug 2026 08:34:21 -0400https://liberty.io/notation-for-general-n-simplex-number.htmlhttps://liberty.io/notation-for-general-n-simplex-number.htmlI'm looking to see if there is a standard notation for the following other than Binomial coefficient notation... The triangular numbers $1,3,6,10,15,21,...$ are notated as $T_n$ and represented as $\...Mon, 24 Aug 2026 08:33:24 -0400https://liberty.io/derivative-power-rule-conditions.htmlhttps://liberty.io/derivative-power-rule-conditions.htmlI'm kinda a newbie in calculus, but what are the conditions for the power rule to happen? For example, if we have the number $e$, with the property $\frac{d}{dt}{ {e^t} } = {e^t}$ we can get, using......Mon, 24 Aug 2026 08:30:40 -0400https://liberty.io/how-to-get-sine-cosine-value-out-of-tangens.htmlhttps://liberty.io/how-to-get-sine-cosine-value-out-of-tangens.htmlI know that: $\tan(\alpha) = 1/2$. How can I get clean values for sine / cosine without the calculator? Is there a relationship? I know that $\sin(\arctan(1/2))$ is a way ... But I hope you get ......Mon, 24 Aug 2026 08:24:49 -0400https://liberty.io/positive-matrix-and-positive-vector.htmlhttps://liberty.io/positive-matrix-and-positive-vector.htmlLet $A \in \mathbb{R}^{n \times n}$ be a non-negative matrix, i.e. $A_{i,j} \geq 0$ $\forall (i,j)$. Let $x \in \mathbb{R}^n \setminus \{0\}$ be a non-negative vector, i.e. $x_i \geq 0$ $\forall i......Mon, 24 Aug 2026 08:17:47 -0400https://liberty.io/pls-help-to-solve-this-question-from-probability-closed.htmlhttps://liberty.io/pls-help-to-solve-this-question-from-probability-closed.htmlOf the $40$ employees at a certain company, $20$ are available to meet on Monday $(M)$, $17$ are available to meet on wednesday $(W)$, and $8$ are available neither on Monday nor wednesday $(\overl......Mon, 24 Aug 2026 08:13:35 -0400https://liberty.io/what-is-the-distance-between-the-centre-of-the-sphere-and-the-vertex-o.htmlhttps://liberty.io/what-is-the-distance-between-the-centre-of-the-sphere-and-the-vertex-o.htmlA hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is $4$ metres and the radius of the base is $1$ metre. The volume of the sphere is same as that of the ......Mon, 24 Aug 2026 08:10:10 -0400https://liberty.io/solve-in-mathbb-z-the-equation-x-4-1-2y-2.htmlhttps://liberty.io/solve-in-mathbb-z-the-equation-x-4-1-2y-2.htmlFind all pairs of intergers $(x,y)$ such that $x^4 + 1 = 2y^2$. I'm thinking of Gaussian integers, since the LHS can be factored in $\mathbb{C}$. But I don't know how to continue here....Mon, 24 Aug 2026 08:09:12 -0400https://liberty.io/affine-transformation-applied-to-a-multivariate-gaussian-random-variab.htmlhttps://liberty.io/affine-transformation-applied-to-a-multivariate-gaussian-random-variab.htmlGiven a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of $\mathbf{......Mon, 24 Aug 2026 07:35:45 -0400https://liberty.io/finding-two-non-congruent-right-angle-triangles.htmlhttps://liberty.io/finding-two-non-congruent-right-angle-triangles.htmlThe map $g: B \to A, \ (x,y) \mapsto \left(\dfrac {x^2 - 25} y, \dfrac {10x} y, \dfrac {x^2 + 25} y \right)$ is a bijection where $A = \{ (a,b,c) \in \Bbb Q ^3 : a^2 + b^2 = c^2, \ ab = 10 \}$ and ......Mon, 24 Aug 2026 07:33:12 -0400https://liberty.io/what-does-it-mean-when-something-says-in-thousands.htmlhttps://liberty.io/what-does-it-mean-when-something-says-in-thousands.htmlI'm doing a research report, and I need to determine a companies assets. So I found their annual report online, and for the assets, it says (in thousands). One of the rows is: Net sales $ 26,234......Mon, 24 Aug 2026 07:30:12 -0400https://liberty.io/cross-product-for-vector-angular-position.htmlhttps://liberty.io/cross-product-for-vector-angular-position.htmlThe angular velocity of a particle $\omega = r \times v$ is a pseudovector because it is formed by the cross product of two vectors (position and linear velocity). Likewise the angular acceleratio......Mon, 24 Aug 2026 07:29:12 -0400https://liberty.io/is-this-a-valid-recursive-formula-for-the-geometric-sequence.htmlhttps://liberty.io/is-this-a-valid-recursive-formula-for-the-geometric-sequence.htmlYour friend lends you \$100. He adds a \$0.25 fee every single day, and then charges you 1% interest. Create a formula that represents this relationship. The first day: $T_1=(100+0.25)*1.01=\$101......Mon, 24 Aug 2026 07:28:07 -0400https://liberty.io/checking-the-singularity-of-a-matrix.htmlhttps://liberty.io/checking-the-singularity-of-a-matrix.htmlIs there any simple way of telling if a matrix $A$ is singular or not? I saw somewhere the following: If $rank(A)<n \Rightarrow$ $A$ is singular. If $rank(A)\geqslant n \Rightarrow$ A is non-...Mon, 24 Aug 2026 07:20:21 -0400https://liberty.io/what-are-the-laws-of-rational-exponents.htmlhttps://liberty.io/what-are-the-laws-of-rational-exponents.htmlOn Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a num......Mon, 24 Aug 2026 07:12:23 -0400https://liberty.io/prove-that-the-conjugate-of-a-j-cycle-is-a-j-cycle-and-is-given-by-the.htmlhttps://liberty.io/prove-that-the-conjugate-of-a-j-cycle-is-a-j-cycle-and-is-given-by-the.htmlProblem Statement Let $\tau \in S_n$ and suppose that $\sigma = (k_1 k_2 ... k_j)$ be a $j$-cycle. Prove that the conjugate of $\sigma$ by $\tau$ is also a $j$-cycle and is given by $$ \tau \sigma......Mon, 24 Aug 2026 07:10:40 -0400https://liberty.io/how-do-i-find-upper-triangular-form-of-a-given-3-by-3-matrix.htmlhttps://liberty.io/how-do-i-find-upper-triangular-form-of-a-given-3-by-3-matrix.htmlWe are asked to find an invertible matrix $P$ and an upper triangular matrix $U$ such that: $P^{-1}\begin{pmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{pmatrix}P=U$ ......Mon, 24 Aug 2026 07:08:55 -0400https://liberty.io/chebyshev-vs-euclidean-distance.htmlhttps://liberty.io/chebyshev-vs-euclidean-distance.htmlWhen calculating the distance in $\mathbb R^2$ with the euclidean and the chebyshev distance I would assume that the euclidean distance is always the shortest distance between two points. Consider......Mon, 24 Aug 2026 07:04:47 -0400https://liberty.io/minimizing-surface-area-for-a-given-volume.htmlhttps://liberty.io/minimizing-surface-area-for-a-given-volume.htmlMath question:An open-top box with a square base is to have a volume of 4 cubic ft. Find the dimensions of the box that can be be made with the smallest amount of material. This is the only thing ......Mon, 24 Aug 2026 07:02:31 -0400https://liberty.io/is-there-a-solution-for-x-for-x-2-y-sqrt-3-x-2-2-1.htmlhttps://liberty.io/is-there-a-solution-for-x-for-x-2-y-sqrt-3-x-2-2-1.htmlI tried to solve this equation (heart shaped function) for x, but cannot get a nice solution. Why? $x^2+(y-\sqrt[3]{x^2})^2=1$ The formula is taken from this image: I just wanted to rewrite the fo......Mon, 24 Aug 2026 06:48:52 -0400https://liberty.io/how-to-determine-the-exact-value-of-sin-585-circ.htmlhttps://liberty.io/how-to-determine-the-exact-value-of-sin-585-circ.htmlI'm clueless on this question. Could someone explain how to do it?...Mon, 24 Aug 2026 06:48:15 -0400https://liberty.io/how-do-you-type-sec-2-0-on-a-calculator.htmlhttps://liberty.io/how-do-you-type-sec-2-0-on-a-calculator.htmlI press cos^-1 then ^2 then brack (o) but then it comes up with syntax error...Mon, 24 Aug 2026 06:43:07 -0400https://liberty.io/calculating-angle-in-circle.htmlhttps://liberty.io/calculating-angle-in-circle.htmlHow to calculate angle in a circle. Please see the diagram to get the idea what I want to calculate? I have origin of circle that is $(x_1,x_2)$. I have a point on circumstance of circle that is $(......Mon, 24 Aug 2026 06:40:18 -0400https://liberty.io/why-is-negative-times-negative-positive.htmlhttps://liberty.io/why-is-negative-times-negative-positive.htmlSomeone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Ass......Mon, 24 Aug 2026 06:34:34 -0400https://liberty.io/find-the-subtotal-from-the-total-including-tax.htmlhttps://liberty.io/find-the-subtotal-from-the-total-including-tax.htmlIf I have the Grand Total (including VAT) amount of money paid and the VAT % which was paid, how can I figure out the sub total (total before VAT)? For example If Sub Total = 70 and VAT is 18% (70*......Mon, 24 Aug 2026 06:22:53 -0400https://liberty.io/write-the-product-of-two-trig-equations-equal-to-one.htmlhttps://liberty.io/write-the-product-of-two-trig-equations-equal-to-one.htmlSolve: Write the product of two trig equations that is equal to one. This one confuses me because I can think of trig equations that equal one, but I can't think of trig equations that I could mul......Mon, 24 Aug 2026 06:10:30 -0400https://liberty.io/why-doesn-039-t-wolframalpha-factor-my-equation.htmlhttps://liberty.io/why-doesn-039-t-wolframalpha-factor-my-equation.htmlI entered in a simple-looking quadratic equation, expecting WA to factorize it for me, but then it gives me all this other stuff without the actual factors. The result is exactly the same as my inp......Mon, 24 Aug 2026 06:08:26 -0400https://liberty.io/number-of-combinations-where-a-and-b-are-together.htmlhttps://liberty.io/number-of-combinations-where-a-and-b-are-together.htmlThis question has been answered before, but I want an elaboration as to why: If there are 10 people including A and B, how many arrangements in a line(I guess this is permutation since order does ......Mon, 24 Aug 2026 06:05:57 -0400https://liberty.io/is-it-true-that-a-2x2-matrix-is-diagonalizable-iff-it-has-two-distinct.htmlhttps://liberty.io/is-it-true-that-a-2x2-matrix-is-diagonalizable-iff-it-has-two-distinct.htmlI diagonal matrix is obviously diagonalizable since I can conjugate it with the identity. ...(1) Besides, a matrix 2x2 is diagonalizable iff it has two distinct eigenvalues....(2) For example the m......Mon, 24 Aug 2026 06:03:27 -0400https://liberty.io/the-domain-of-natural-logarithm-function.htmlhttps://liberty.io/the-domain-of-natural-logarithm-function.htmlCould somebody show me their method for finding the domain of this function? $\ln(1-(1/x))$ With the usual method I get that $1-(1/x)>0$ which results in $x>1$. however, the actual answer is a ...Mon, 24 Aug 2026 06:01:13 -0400https://liberty.io/for-all-n-2-there-exists-a-prime-number-between-n-and-n.htmlhttps://liberty.io/for-all-n-2-there-exists-a-prime-number-between-n-and-n.htmlHow to prove that there exists a prime number between $n$ and $ n!$, for all $ n> 2$? (Bertrand's postulate gives a much better bound, but this question is about obtaining a self-contained proof.)...Mon, 24 Aug 2026 06:00:02 -0400https://liberty.io/show-that-the-equation-represents-a-sphere-find-center-and-radius.htmlhttps://liberty.io/show-that-the-equation-represents-a-sphere-find-center-and-radius.html$2x^2 + 2y^2 + 2z^2 = 8x - 24z + 1$ So here's my guess: $2x^2 - 8x + 2y^2 + 2z^2 + 24z = 1$ Get all variables on one side $2x^2 - 8x + 16 + 2y^2 + 2z^2 + 24z + 144 = 1 + 16 + 144$ Complete ...Mon, 24 Aug 2026 05:56:50 -0400https://liberty.io/placing-some-sets-in-the-arithmetic-hierarchy.htmlhttps://liberty.io/placing-some-sets-in-the-arithmetic-hierarchy.htmlI'm working on the following problem: let $W_e$ be the computably enumerable set which is the domain of the $e$-th Turing program, and $K$ be the Halting problem, at which level of the arithmetic ...Mon, 24 Aug 2026 05:53:17 -0400https://liberty.io/what-is-the-proof-for-the-vortex-vector-field-equation.htmlhttps://liberty.io/what-is-the-proof-for-the-vortex-vector-field-equation.htmlI'm Struggling to understand why the vortex vector field is given by: Vortex vector field equation $\vec F(x,y) = (\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2})$ If anyone could explain why this is, I ......Mon, 24 Aug 2026 05:49:27 -0400https://liberty.io/is-the-orthocenter-and-incenter-of-a-triangle-the-same-point.htmlhttps://liberty.io/is-the-orthocenter-and-incenter-of-a-triangle-the-same-point.htmlAlthough the orthoceneter and the incenter of a triangle are technically different things: The point in which the three altitudes of a triangle meet is called the orthocenter of the triangle. ......Mon, 24 Aug 2026 05:39:52 -0400https://liberty.io/do-we-actually-define-implications-using-an-implication-itself.htmlhttps://liberty.io/do-we-actually-define-implications-using-an-implication-itself.htmlEverything in math stems from definitions. Eg: Let an 'implication' be defined as ... But any such 'let' actually means 'if it be true that'. So what we're really saying is 'If an implication be ...Mon, 24 Aug 2026 05:37:33 -0400https://liberty.io/solving-inequalities-with-absolute-values-on-both-sides.htmlhttps://liberty.io/solving-inequalities-with-absolute-values-on-both-sides.htmlI need to find the solution sets for the following inequalities: $$|3+2x|\leq|4-x|$$$$|2x-1|+|1-x|\geq3$$ After a bit of tinkering with the first one, I think the solution set is $[-7, \frac13]$,......Mon, 24 Aug 2026 05:30:43 -0400https://liberty.io/diagonalizable-matrix-a-invertible-also.htmlhttps://liberty.io/diagonalizable-matrix-a-invertible-also.htmlIf a matrix $A$ is diagonalizable, is $A$ invertible? I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself....Mon, 24 Aug 2026 05:27:26 -0400https://liberty.io/characteristic-of-a-ring-intuitive-explanation.htmlhttps://liberty.io/characteristic-of-a-ring-intuitive-explanation.htmlI know the following definition of characteristic of a ring: it is the smallest positive $n$ such that $$\underbrace{a+\cdots+a}_{n \text{ summands}} = 0$$ for every element a of the ring, if $n$ e......Mon, 24 Aug 2026 05:09:52 -0400https://liberty.io/how-do-you-construct-a-lattice-from-its-basis-or-its-gram-matrix.htmlhttps://liberty.io/how-do-you-construct-a-lattice-from-its-basis-or-its-gram-matrix.htmlI'm really having trouble trying to understand this. A few weeks back, I got pretty interested in sphere packing and I'm trying to grasp the idea of using a matrix to represent the basis of a latti......Mon, 24 Aug 2026 05:06:27 -0400https://liberty.io/solving-the-diophantine-equation-y-2-x-3-2.htmlhttps://liberty.io/solving-the-diophantine-equation-y-2-x-3-2.htmlIt is known that the diophantine equation $y^{2}=x^{3}-2$ has only one positive integer solution $(x,y)=(3,5)$. The proof of it can be read from the book "About Indeterminate Equation" (in Chinese,......Mon, 24 Aug 2026 05:05:21 -0400https://liberty.io/isometry-of-mathbb-r-2-that-preserves-orientation-is-either-a-rotation.htmlhttps://liberty.io/isometry-of-mathbb-r-2-that-preserves-orientation-is-either-a-rotation.htmlI want to show that if $\gamma$ is an isometry of $\mathbb{R}^2$ that preserves orientation, then it is either a translation or a rotation about a point. If $\gamma$ is an isometry of $\mathbb{R}......Mon, 24 Aug 2026 05:04:58 -0400https://liberty.io/prove-that-x-x-for-all-real-numbers-x-duplicate.htmlhttps://liberty.io/prove-that-x-x-for-all-real-numbers-x-duplicate.htmlIs the following a complete proof? It is from the book, $Prealgebra$ by R. Rusczyk, D. Patrick, and R. Boppana Consider the sum $$x + (-x) + (-(-x)).$$ That is, we are adding x, its negation -x......Mon, 24 Aug 2026 05:02:07 -0400https://liberty.io/why-do-we-use-this-definition-of-algebraic-integer.htmlhttps://liberty.io/why-do-we-use-this-definition-of-algebraic-integer.htmlA number is an "algebraic integer" if it is the root to a monic polynomial with integer coefficients. Artin says (Algebra, p. 411): The concept of algebraic integer was one of the most ...Mon, 24 Aug 2026 05:01:22 -0400https://liberty.io/surface-area-of-twisted-prism.htmlhttps://liberty.io/surface-area-of-twisted-prism.htmlTake a right prism whose bases are regular $n$ polygons and twist it uniformly by some angle $\phi<\pi$. This surface can be seen as the trajectories of the sides of the polygon as it translates......Mon, 24 Aug 2026 04:51:08 -0400https://liberty.io/find-a-second-order-ode-given-the-solution.htmlhttps://liberty.io/find-a-second-order-ode-given-the-solution.htmlFind a second order differential equation so that $$y=C_1e^{-3x}\cos(4x)+C_2e^{-3x}\sin(4x)+4e^{3x}$$ solves the differential equation for any choice of $C_1$and $C_2$. The answer should b......Mon, 24 Aug 2026 04:51:06 -0400https://liberty.io/why-does-svd-provide-the-least-squares-and-least-norm-solution-to-a-x-.htmlhttps://liberty.io/why-does-svd-provide-the-least-squares-and-least-norm-solution-to-a-x-.htmlI am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=......Mon, 24 Aug 2026 04:41:06 -0400https://liberty.io/what-is-the-equation-for-plotting-points-on-a-curve-with-fixed-end-poi.htmlhttps://liberty.io/what-is-the-equation-for-plotting-points-on-a-curve-with-fixed-end-poi.htmlWhat is the equation for plotting points on an exponential curve with fixed end points? For example, if I want to plot 10 point along a curve that starts with 10,000 (x=1, y=10000) and ends with 3......Mon, 24 Aug 2026 04:37:10 -0400https://liberty.io/geometric-interpretation-of-derivative-of-a-function-of-more-than-one-.htmlhttps://liberty.io/geometric-interpretation-of-derivative-of-a-function-of-more-than-one-.htmlA function $f$ is defined on an open set $D$ of $\mathbb R^{2}$ is called a differentiable at a point $x\in D$ if there is a vector $m \in \mathbb R^{2} $ such that $$\lim_{h\to 0} \frac{f(x+h)-f(x......Mon, 24 Aug 2026 04:33:35 -0400https://liberty.io/difference-between-minimizing-and-maximizing-functions.htmlhttps://liberty.io/difference-between-minimizing-and-maximizing-functions.htmlCould someone please explain the difference between minimizing and maximizing functions or give me some links to explain the difference in very very very simple terms? I have searched online and I ...Mon, 24 Aug 2026 04:29:53 -0400https://liberty.io/defining-a-principal-square-root-in-the-complex-space-wolfram-case.htmlhttps://liberty.io/defining-a-principal-square-root-in-the-complex-space-wolfram-case.htmlIn Wikipedia(https://en.wikipedia.org/wiki/Square_root), we define the square root as In mathematics, a square root of a number $a$ is a number y such that $y^2 = a$; in other words, a number $y$ ......Mon, 24 Aug 2026 04:16:13 -0400https://liberty.io/probability-to-get-all-6-sides-of-dice.htmlhttps://liberty.io/probability-to-get-all-6-sides-of-dice.htmlBy rolling a dice 7 times, what is the probability that we could get all the sides to come up at least once? My attempt: Let the first role be any number. 2nd roll: 5/6 chance to get different nu......Mon, 24 Aug 2026 04:07:13 -0400https://liberty.io/what-is-the-exact-definition-of-an-injective-function.htmlhttps://liberty.io/what-is-the-exact-definition-of-an-injective-function.htmlAm I right to believe that a function is injective, if some elements of the first set are mapped to some elements of the second set? It is also possible to 4 elements of the first set, are mapped ......Mon, 24 Aug 2026 04:02:46 -0400https://liberty.io/changing-modulus-in-modular-arithmetic.htmlhttps://liberty.io/changing-modulus-in-modular-arithmetic.htmlIs it true that $$a\equiv b\pmod{m}\implies\frac{a}{n}\equiv\frac{b}{n} \pmod{\frac{m}{n}},$$ where $a, b, m, n, \frac{a}{n}, \frac{b}{n}, \frac{m}{n}\in\mathbb{N}$? If so, how do I prove it?...Mon, 24 Aug 2026 03:54:34 -0400https://liberty.io/n-4-4-n-is-a-not-a-prime-duplicate.htmlhttps://liberty.io/n-4-4-n-is-a-not-a-prime-duplicate.htmlProve that $n^4 + 4^n$ is not a prime for all $n > 1$ and $n \in \mathbb{N}$. This question appeared in the undergrad entrance exam of the Indian Statistical institute. When $n$ is even the p......Mon, 24 Aug 2026 03:52:58 -0400https://liberty.io/appearance-of-formal-derivative-in-algebra.htmlhttps://liberty.io/appearance-of-formal-derivative-in-algebra.htmlWhen studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are co......Mon, 24 Aug 2026 03:51:05 -0400https://liberty.io/laplace-transform-of-tf-t.htmlhttps://liberty.io/laplace-transform-of-tf-t.htmlQ. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: $$\mathfrak{L}\{tf(x)\}=\int_0^\...Mon, 24 Aug 2026 03:47:27 -0400https://liberty.io/probability-of-winning-fantasy-5-lotto.htmlhttps://liberty.io/probability-of-winning-fantasy-5-lotto.htmlIn Fantasy 5, you choose 5 numbers, each 1-39. You win if you choose match all 5 numbers, but you can also win if you match 4, 3, or even 2 of the numbers. The lotto website lists the probabiliti......Mon, 24 Aug 2026 03:35:44 -0400