https://liberty.ioCurated star highlights with fresh energy.en-usTue, 18 Aug 2026 11:25:19 -0400https://liberty.io/how-to-find-the-n-th-term-of-the-series-1-22-333-4444-55555.htmlhttps://liberty.io/how-to-find-the-n-th-term-of-the-series-1-22-333-4444-55555.htmlHow to find the $n$-th term of the series $1, 22, 333, 4444, 55555..........?$ Here the $i^{\text{th}}$ term is the concatenation of $i$ for $i$ times. for ef $13$ the term = $13$ concatenated $13$......Tue, 18 Aug 2026 15:22:57 -0400https://liberty.io/on-when-and-when-not-to-divide-by-tan-theta.htmlhttps://liberty.io/on-when-and-when-not-to-divide-by-tan-theta.htmlThe textbook I'm using has this problem involving trig equations which goes $\tan(x) \sin^2(x)=2 \tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $\tan(x)$ ......Tue, 18 Aug 2026 15:22:28 -0400https://liberty.io/having-a-unique-solution-for-a-rectangular-matrix.htmlhttps://liberty.io/having-a-unique-solution-for-a-rectangular-matrix.htmlSuppose we are given a $4\times3$ matrix $A$ and a column $b$ such that $Ax=b$ always has a unique solution. What is the reduced row echelon form of $A$? I think that since the solution is unique,......Tue, 18 Aug 2026 15:18:55 -0400https://liberty.io/how-or-why-are-implicit-functions-actually-functions.htmlhttps://liberty.io/how-or-why-are-implicit-functions-actually-functions.htmlI'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function. I didn't quite get it at that time so I decided to ...Tue, 18 Aug 2026 15:13:54 -0400https://liberty.io/jacobian-calculation-with-homographic-coordinates.htmlhttps://liberty.io/jacobian-calculation-with-homographic-coordinates.htmlI'm trying to work my way through an image processing paper (Creating Full View Panoramic Image Mosaics and Environment Maps) and am stuck on how the Jacobian is calculated, especially terms ($J_{1......Tue, 18 Aug 2026 15:03:33 -0400https://liberty.io/third-order-differential-equations.htmlhttps://liberty.io/third-order-differential-equations.htmlI am having trouble solving the third order differential equation $y'''+y'=0$ It was given to me in a quiz (which I got wrong) with boundary conditions $y(0) = 0$ $y'(0)=2$ $y(\pi)=6$ I know t......Tue, 18 Aug 2026 15:03:15 -0400https://liberty.io/why-do-we-need-prime-ideals-in-the-spectrum-of-a-ring.htmlhttps://liberty.io/why-do-we-need-prime-ideals-in-the-spectrum-of-a-ring.htmlI'm reading Atiyah Macdonald, where they introduce in the exercises of chapter one a topological space $\operatorname{Spec}(A)$ associated to a ring $A$, which is defined as $\operatorname{Spec}(A) \...Tue, 18 Aug 2026 15:01:00 -0400https://liberty.io/cauchy-039-s-formula-for-repeated-integrals-proof-by-induction.htmlhttps://liberty.io/cauchy-039-s-formula-for-repeated-integrals-proof-by-induction.htmlI was trying to follow along with the proof on Wikipedia for Cauchy's formula for repeated integrals and I'm stuck on the last step. How do you go from $$\int_a^x \frac {d}{dy} \left( \int_a^y (y-t......Tue, 18 Aug 2026 14:59:31 -0400https://liberty.io/is-there-a-way-to-reverse-factorials-duplicate.htmlhttps://liberty.io/is-there-a-way-to-reverse-factorials-duplicate.htmlIs there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do factorials and factorial roots (for lack of a better term)? Here is an example: 5!......Tue, 18 Aug 2026 14:53:48 -0400https://liberty.io/given-a-point-p-outside-equilateral-delta-abc-but-inside-angle-abc-if-.htmlhttps://liberty.io/given-a-point-p-outside-equilateral-delta-abc-but-inside-angle-abc-if-.htmlGiven a point $P$ outside equilateral $\Delta ABC$ but inside $\angle ABC$, if the distance between $P$ to $BC,CA,AB$ are $h_1,h_2,h_3$ respectively, where $h_1 - h_2 + h_3 = 6$, find $[\Delta ABC]......Tue, 18 Aug 2026 14:50:35 -0400https://liberty.io/show-that-sin45-sin15-sin75.htmlhttps://liberty.io/show-that-sin45-sin15-sin75.htmlSteps I took: 1) Finding the value of the left hand side $$\sin45=\sin\frac { 90 }{ 2 } =\sqrt { \frac { 1-\cos90 }{ 2 } } =\sqrt { \frac { 1 }{ 2 } } =\frac { \sqrt { 2 } }{ 2 } $$ $$\sin15=......Tue, 18 Aug 2026 14:29:04 -0400https://liberty.io/how-do-we-write-the-derivation-of-distance-formula-of-two-points-from-.htmlhttps://liberty.io/how-do-we-write-the-derivation-of-distance-formula-of-two-points-from-.htmlI learnt the derivation of the distance formula of two points in first quadrant I.e., $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ where it is easy to find the legs of the hypotenuse (distance betwee......Tue, 18 Aug 2026 14:24:09 -0400https://liberty.io/asymptotic-and-exact-confidence-interval.htmlhttps://liberty.io/asymptotic-and-exact-confidence-interval.htmlAssume we toss a thumbtack 300 times. After every time, we note $1$ if it lands point up or $0$ if it lands point down. In summary, we get 124 times $1$. So we know that the number of the rounds w......Tue, 18 Aug 2026 14:17:41 -0400https://liberty.io/how-to-calculate-distance-run-on-athletics-track.htmlhttps://liberty.io/how-to-calculate-distance-run-on-athletics-track.htmlfirst time poster and definitely no maths expert. I am trying to solve a basic problem using an athletics track. The total distance around a standard athletics track is 400m: If you run in the first ...Tue, 18 Aug 2026 14:17:04 -0400https://liberty.io/how-to-calculate-a-weighted-gpa.htmlhttps://liberty.io/how-to-calculate-a-weighted-gpa.htmlHow would I be able to calculate my overall average grade (or grade point average) for these courses? Course Course Credit (weight) Course Grade Biology 101 45 86 Psychology 210 30 74 Chemistry......Tue, 18 Aug 2026 14:05:11 -0400https://liberty.io/set-of-algebraic-integers-is-closed-under-addition-and-multiplication.htmlhttps://liberty.io/set-of-algebraic-integers-is-closed-under-addition-and-multiplication.htmlIf $\alpha$ and $\beta$ are algebraic integers, then show $\alpha + \beta$ and $\alpha \times \beta$ are both algebraic integers. I know that an algebraic integer is a root of some monic polynomial ...Tue, 18 Aug 2026 14:05:06 -0400https://liberty.io/what-is-the-ratio-of-the-area-of-a-circle-to-its-radius-closed.htmlhttps://liberty.io/what-is-the-ratio-of-the-area-of-a-circle-to-its-radius-closed.htmlI solved this. $$πr^2/r=π r$$ In my book, they said it can not be determined. Am I right or wrong?...Tue, 18 Aug 2026 14:03:11 -0400https://liberty.io/how-were-the-sine-cosine-and-tangent-tables-originally-calculated.htmlhttps://liberty.io/how-were-the-sine-cosine-and-tangent-tables-originally-calculated.htmlAs I understand it... ahem... the (cosine, sine) vector was calculated for (30 degrees, PI/6), (45 degrees, PI/4) and (60 degrees, PI/3) angles etcetera, however, I would like know the original ...Tue, 18 Aug 2026 14:00:57 -0400https://liberty.io/learning-how-to-flip-equations.htmlhttps://liberty.io/learning-how-to-flip-equations.htmlI took Algebra and Geometry in high school, never thought I'd use them, then became a programmer. I guess I was wrong. To date, I have the hardest time taking equations and "flipping them," ie: ...Tue, 18 Aug 2026 13:56:04 -0400https://liberty.io/calculating-running-time-in-seconds-of-algorithms-of-a-given-complexit.htmlhttps://liberty.io/calculating-running-time-in-seconds-of-algorithms-of-a-given-complexit.htmlI've tried to find answers on this but a lot of the questions seem focused on finding out the time complexity in Big O notation, I want to find the actual time. I was wondering how to find the run......Tue, 18 Aug 2026 13:51:51 -0400https://liberty.io/difference-between-continuity-and-uniform-continuity.htmlhttps://liberty.io/difference-between-continuity-and-uniform-continuity.htmlI understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book ...Tue, 18 Aug 2026 13:50:49 -0400https://liberty.io/is-the-present-king-of-france-is-bald-studied-by-maths.htmlhttps://liberty.io/is-the-present-king-of-france-is-bald-studied-by-maths.htmlIntuitively, "The present King of France is bald." is false. But Bertrand Russell said it would mean that "The present King of France is not bald.", which seems to be false. This apparently leads......Tue, 18 Aug 2026 13:50:30 -0400https://liberty.io/symbol-of-a-partition.htmlhttps://liberty.io/symbol-of-a-partition.htmlI have a series of subsets that form a set however each subset is actually a partition. Currently I use the conventional subset symbol when writing the appropriate notation and then note the subse......Tue, 18 Aug 2026 13:50:01 -0400https://liberty.io/buchberger-039-s-algorithm.htmlhttps://liberty.io/buchberger-039-s-algorithm.htmlI am trying to calculate a Gröbner basis for $I=\langle \mathcal{B}\rangle$, where $\mathcal{B}=\{f=x_3-x_1^5, g=x_2-x_1^3\}$, with respect both lexicographic and graded reverse lexicographic order......Tue, 18 Aug 2026 13:43:28 -0400https://liberty.io/min-and-max-of-a-product.htmlhttps://liberty.io/min-and-max-of-a-product.htmlLet $x_i\in X_i\subset \mathbb{R}$ so that each $X_i$ is a compact set with no isolated points for all $1\leq i\leq n$. Let: $a_i\in X_i$ so that $|x_i|\leq|a_i|$ for all $x_i\in X_i$. $b_i = \max\...Tue, 18 Aug 2026 13:35:40 -0400https://liberty.io/a-proportionality-puzzle-if-half-of-5-is-3-then-what-039-s-one-third-o.htmlhttps://liberty.io/a-proportionality-puzzle-if-half-of-5-is-3-then-what-039-s-one-third-o.htmlMy professor gave us this problem. In a foreign country, half of 5 is 3. Based on that same proportion, what's one-third of 10? I removed my try because it's wrong....Tue, 18 Aug 2026 13:34:08 -0400https://liberty.io/probability-of-of-an-event-happening-at-least-once-in-a-sequence-of-in.htmlhttps://liberty.io/probability-of-of-an-event-happening-at-least-once-in-a-sequence-of-in.htmlI want to find the probability of flipping heads at least once if you flip a coin two times. The possible outcomes (we don't care about the order) are (each equally likely) TT, TH, HT, HH. Three o......Tue, 18 Aug 2026 13:31:53 -0400https://liberty.io/integration-over-a-triangle.htmlhttps://liberty.io/integration-over-a-triangle.htmlLet $\Delta$ be a triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$ in ${\bf R}^2$. I want to compute $$I=\int\limits_\Delta x^2\mathrm{e}^{y^2}\;\mathrm{d}A.$$ This is what I've done so far: Not......Tue, 18 Aug 2026 13:31:13 -0400https://liberty.io/find-the-nth-power-of-a-matrix.htmlhttps://liberty.io/find-the-nth-power-of-a-matrix.htmlLet matrix A is $$A=\left (\begin{array}{rrr} 1&0&0 \\ 1&1&0\\ 0&0&1 \end{array}\right)$$ How can I find the 30 th power of A.. Is diagonalization possible? I found the......Tue, 18 Aug 2026 13:20:38 -0400https://liberty.io/dodecahedron-proof-on-number-of-vertices-edges-and-faces.htmlhttps://liberty.io/dodecahedron-proof-on-number-of-vertices-edges-and-faces.htmlFind, with a proof, the number of vertices, edges, and faces of a dodecahedron. Its is clear that their are $20$ vertices, $30$ edges, and $12$ faces. I am not sure how to prove this though....Tue, 18 Aug 2026 13:08:55 -0400https://liberty.io/can-the-poisson-distribution-be-used-to-find-the-expected-value-of-tim.htmlhttps://liberty.io/can-the-poisson-distribution-be-used-to-find-the-expected-value-of-tim.htmlMy understanding of the Poisson Distribution is that its PMF $P(x=k) = \dfrac {\lambda^k e^{-\lambda}} {k!}$ refers to the probability of finding k events given an expected arrival expectancy $\lam......Tue, 18 Aug 2026 13:08:41 -0400https://liberty.io/matrix-representations-of-parabola.htmlhttps://liberty.io/matrix-representations-of-parabola.htmlContinuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...Tue, 18 Aug 2026 12:59:22 -0400https://liberty.io/how-to-evaluate-the-following-integral.htmlhttps://liberty.io/how-to-evaluate-the-following-integral.html$$P(S)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(a^2+c^2+ \lambda ~ b^2)} \delta[S-\sqrt{4b^2+(a-c)^2}] ~da~db~bc.$$ where $\lambda$ is a constant. How to Evaluate......Tue, 18 Aug 2026 12:54:35 -0400https://liberty.io/what-is-the-equation-of-a-3d-cone-with-generalised-tilt.htmlhttps://liberty.io/what-is-the-equation-of-a-3d-cone-with-generalised-tilt.htmlWhat is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that......Tue, 18 Aug 2026 12:49:06 -0400https://liberty.io/the-definition-of-ln-x.htmlhttps://liberty.io/the-definition-of-ln-x.htmlWhen I taught my student the logarithm, he asked me about the historical definition of $\ln(x)$. The first definition I found is that $$\ln(x)=\int_{1}^{x}{ \frac{dt}{t} } $$ Defined as the loga......Tue, 18 Aug 2026 12:43:58 -0400https://liberty.io/how-many-different-arrangements-are-possible-with-3-digits-numbers-fro.htmlhttps://liberty.io/how-many-different-arrangements-are-possible-with-3-digits-numbers-fro.htmlif there are numbers 1,2,3,4,5,6,7,8,9 how many different three digit values we can make such that the digits are in ascending order? since there are numbers from 1-9, I understand that the first ......Tue, 18 Aug 2026 12:43:38 -0400https://liberty.io/get-the-inverse-of-ratio-what-is-the-mathematical-term.htmlhttps://liberty.io/get-the-inverse-of-ratio-what-is-the-mathematical-term.htmlI need to create kind of the inversed ratio. What i mean: Let's say there are the numbers 1,1,2 the ratio is 0.25,0.25,0.5. The sum of the ratio is of course 1. But i want the "inversed" ratio so ......Tue, 18 Aug 2026 12:34:58 -0400https://liberty.io/operation-that-is-commutative-but-not-associative-duplicate.htmlhttps://liberty.io/operation-that-is-commutative-but-not-associative-duplicate.htmlDoes there exist an operation that is commutative but not associative? do you know what is operation?...Tue, 18 Aug 2026 12:32:32 -0400https://liberty.io/how-to-calculate-the-surface-area-of-parametric-surface.htmlhttps://liberty.io/how-to-calculate-the-surface-area-of-parametric-surface.htmlSuppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2\leq 1$. My homework is forcing me to use the parameterization $$\textbf{r}_1(s,t)= <......Tue, 18 Aug 2026 12:25:53 -0400https://liberty.io/rational-solutions-for-x-2-y-2-3.htmlhttps://liberty.io/rational-solutions-for-x-2-y-2-3.htmlAre there any rational numbers $x$ and $y$ such that $x^2+y^2=3$. I think there are no rational solutions, but I haven't been able to prove it....Tue, 18 Aug 2026 12:23:35 -0400https://liberty.io/why-is-sphere-non-euclidean-space.htmlhttps://liberty.io/why-is-sphere-non-euclidean-space.htmlThere are 5 axioms that define euclidean space and I believe that all hold also for a sphere. The definition of axioms from wikipedia: "To draw a straight line from any point to any point." Defi......Tue, 18 Aug 2026 12:13:25 -0400https://liberty.io/size-of-eigenvalues-and-diagonal-dominance.htmlhttps://liberty.io/size-of-eigenvalues-and-diagonal-dominance.htmlIf $X$ is a diagonally dominant matrix and if $Y$ is a diagonal matrix with the diagonal elements of $X$, then how can one show that $ Y^{-1}(Y - X)$ has eigenvalues whose magnitudes are all strictly ...Tue, 18 Aug 2026 12:03:01 -0400https://liberty.io/finding-roots-of-z-3-8.htmlhttps://liberty.io/finding-roots-of-z-3-8.htmlI am having trouble finding the cubed roots of $8$ as a complex number. $$z^3 = 8+0i$$ $$z^3 = r^3 e^{3\theta i}=8e^{2i\pi k},\quad k\in \Bbb Z$$ $$\implies r=2,3\theta = 2\pi k\implies \theta = ......Tue, 18 Aug 2026 11:56:00 -0400https://liberty.io/when-to-stop-checking-if-a-transition-matrix-is-regular.htmlhttps://liberty.io/when-to-stop-checking-if-a-transition-matrix-is-regular.htmlThe definition that I have of a Transition Matrix for a Markov Chain is: A transition matrix is regular if some power of it is positive. Doesn't this mean though that in theory, you could keep ...Tue, 18 Aug 2026 11:53:03 -0400https://liberty.io/how-to-show-that-if-f-2-x-is-uniformly-continous-function-then-f-is-un.htmlhttps://liberty.io/how-to-show-that-if-f-2-x-is-uniformly-continous-function-then-f-is-un.html$f:R\to [0,\infty)$ is function such that $f^2(x)$ is uniformly continuous on R then I have to show that f is uniformly continuous ? My attempt : $|f^2(x)-f^2(y)|<\epsilon$ for $|x-y|<\del......Tue, 18 Aug 2026 11:45:35 -0400https://liberty.io/what-is-the-limit-of-lim-x-1-x-1-x-when-x-approaches-infinity-without-.htmlhttps://liberty.io/what-is-the-limit-of-lim-x-1-x-1-x-when-x-approaches-infinity-without-.htmlthis is my first post. I've been struggling whit this limit for too long (without using l'Hôpital's rule): $$\lim_{x\to {\infty}} \left(\frac{x-1}{x+1}\right)^x$$ My answer is $\frac1e$, but the ...Tue, 18 Aug 2026 11:35:18 -0400https://liberty.io/how-to-find-net-torque-closed.htmlhttps://liberty.io/how-to-find-net-torque-closed.htmlFor the meter stick in Figure 10-4, what is the magnitude of the net torque due to both forces $F_1$ and $F_2$ about an axis perpendicular to the page through point $A$? Is it clockwise or ...Tue, 18 Aug 2026 11:31:13 -0400https://liberty.io/proving-lim-x-n-a-n-as-x-goes-to-a-from-definition.htmlhttps://liberty.io/proving-lim-x-n-a-n-as-x-goes-to-a-from-definition.htmlUsing the $\varepsilon-\delta$ definition show that $\displaystyle \lim_{x \to a} x^n = a^n$ What I've done? Pretty much stuck on the step where one usually constructs delta: $\displaystyle |x^......Tue, 18 Aug 2026 11:28:56 -0400https://liberty.io/proof-that-there-are-infinitely-many-primes-congruent-to-1-mod-3.htmlhttps://liberty.io/proof-that-there-are-infinitely-many-primes-congruent-to-1-mod-3.htmlNeed a quick check on this proof. Show that there are infinitely many primes congruent to 1 mod 3. Proof: Suppose by contradiction finitely many primes $p≡1\pmod{3}$ say $p_1,...,p_k$. Define $m......Tue, 18 Aug 2026 11:28:22 -0400https://liberty.io/notation-convention-for-integer-in-a-certain-range.htmlhttps://liberty.io/notation-convention-for-integer-in-a-certain-range.htmlI am wondering what notation I should use when writing that some variable is an integer within some range. What is the most common way to do this? Here are some ideas I have but I'm not sure what the ...Tue, 18 Aug 2026 11:12:06 -0400https://liberty.io/calculating-the-divergence-of-a-tensor.htmlhttps://liberty.io/calculating-the-divergence-of-a-tensor.htmlI am working through a fluid dynamics paper and came across this equation: $$ \frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}=\nabla\cdot T - \frac{1}{\rho}\nabla \phi\tag1$$ where ......Tue, 18 Aug 2026 11:09:36 -0400https://liberty.io/find-the-derivative-of-y-2-x-as-a-function-of-y.htmlhttps://liberty.io/find-the-derivative-of-y-2-x-as-a-function-of-y.htmlFind the derivative of $$y^2=x$$ as a function of y. i have found for the function of x, it will be $$\pm\frac{1}{2\sqrt x}$$ however for the function of y will be $$\frac{dy}{dx}=2y$$ ? it loo......Tue, 18 Aug 2026 11:06:19 -0400https://liberty.io/where-does-b-2a-come-from.htmlhttps://liberty.io/where-does-b-2a-come-from.htmlIve came across the idea to find the minimum/maximum of a parabola , I keep running into the formula $-b/2a$ i wanted to know where does this actually come from ? I know that the parabola has to be ...Tue, 18 Aug 2026 11:03:44 -0400https://liberty.io/how-can-you-prove-that-1-5-9-cdots-4n-3-2n-2-n-without-using-induction.htmlhttps://liberty.io/how-can-you-prove-that-1-5-9-cdots-4n-3-2n-2-n-without-using-induction.htmlUsing mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result wi......Tue, 18 Aug 2026 10:50:29 -0400https://liberty.io/how-to-simplify-a-sum-of-exponential-equation.htmlhttps://liberty.io/how-to-simplify-a-sum-of-exponential-equation.htmlSuppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ......Tue, 18 Aug 2026 10:49:36 -0400https://liberty.io/prove-that-if-f-n-in-o-g-n-then-g-n-in-omega-f-n.htmlhttps://liberty.io/prove-that-if-f-n-in-o-g-n-then-g-n-in-omega-f-n.htmlProve that if $f(n) \in O(g(n))$ then $g(n) \in \Omega(f(n))$. So I know that $$O(g(n)) \Rightarrow f(n) \leq c\cdot g(n)$$ and that $$\Omega(g(n)) \Rightarrow f(n) \geq c\cdot g(n)$$ but how ......Tue, 18 Aug 2026 10:47:54 -0400https://liberty.io/what-is-the-practical-application-of-factorials.htmlhttps://liberty.io/what-is-the-practical-application-of-factorials.htmlI'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer. I understand that a factorial of n items......Tue, 18 Aug 2026 10:33:18 -0400https://liberty.io/natural-deduction-proof-problem-with-existential-elimination.htmlhttps://liberty.io/natural-deduction-proof-problem-with-existential-elimination.htmlI have problems with the following proof: $$ \forall x \exists y (Rxy \land Py), \forall x \neg Rxx \vdash \neg \forall x \forall y (Px \implies (Py \implies x=y)) $$ The problem is with the applic......Tue, 18 Aug 2026 10:30:16 -0400https://liberty.io/25-chance-of-happening-but-has-to-happen-4-times-in-a-row-what-is-the-.htmlhttps://liberty.io/25-chance-of-happening-but-has-to-happen-4-times-in-a-row-what-is-the-.htmlSay you need to win 4 times in a row at a 25% success chance for each event, how do I calculate the chance of winning? Overall, how does this calculation goes? given that 1st is a win, what is the ......Tue, 18 Aug 2026 10:16:05 -0400https://liberty.io/difference-between-state-space-and-sample-space.htmlhttps://liberty.io/difference-between-state-space-and-sample-space.htmlAren't they both the same? According to me they are just different terminologies used for different types of systems. What I understand is that sample space is used when we talk about static system......Tue, 18 Aug 2026 10:08:55 -0400https://liberty.io/which-value-goes-on-which-axis-in-plot-a-against-b.htmlhttps://liberty.io/which-value-goes-on-which-axis-in-plot-a-against-b.htmlWhen the question asks to plot $a$ against $b$, which value should go on which axis?...Tue, 18 Aug 2026 10:07:49 -0400https://liberty.io/can-039-t-understand-a-recursive-definition-of-concatenation-of-two-st.htmlhttps://liberty.io/can-039-t-understand-a-recursive-definition-of-concatenation-of-two-st.htmlI'm reading Rosen's Discrete Mathematics and its applications(6ed), but I can't understand a recursive definition about concatenation of two strings: Two strings can be combined via the operatio......Tue, 18 Aug 2026 09:44:44 -0400https://liberty.io/probability-of-getting-a-triple-when-rolling-12-fair-12-sided-dice-onc.htmlhttps://liberty.io/probability-of-getting-a-triple-when-rolling-12-fair-12-sided-dice-onc.htmlThere are $12^{12}$ = 8,916,100,448,256 total possible outcomes. I decided to approach it by calculating the probability of rolling the $12$ dice WITHOUT getting a triple and subtract from $1$. I f......Tue, 18 Aug 2026 09:27:04 -0400https://liberty.io/what-is-x-times-sqrt-x.htmlhttps://liberty.io/what-is-x-times-sqrt-x.htmlDoes $x \times \sqrt x = x$? I thought it was correct because sqrt is the opposite of multiplying by a number, so I figured by multiplying by a number it would balance out and be that number norm......Tue, 18 Aug 2026 09:26:15 -0400https://liberty.io/representing-ln-x-as-a-power-series-centered-at-2-without-computing-an.htmlhttps://liberty.io/representing-ln-x-as-a-power-series-centered-at-2-without-computing-an.htmlI am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old ser......Tue, 18 Aug 2026 09:11:34 -0400https://liberty.io/what-is-the-number-of-numbers.htmlhttps://liberty.io/what-is-the-number-of-numbers.htmlFor a positive integer $n≥2$, let $f(n)$ denote the smallest prime divisor of $n$. Find the number of integers $n$, $2≤n≤1000$, such that $f(n)=5$. All the numbers I could think of were $5,25,35,55......Tue, 18 Aug 2026 09:07:29 -0400https://liberty.io/prove-the-trigonometric-identities-cot-x-cot-2x-csc-2x.htmlhttps://liberty.io/prove-the-trigonometric-identities-cot-x-cot-2x-csc-2x.htmlProve $\cot(x) - \cot(2x) =\csc(2x)$. I start to solve from LHS, and change all the terms into $\sin$ and $\cos$, but I could not prove it into $\csc(2x)$....Tue, 18 Aug 2026 09:03:45 -0400https://liberty.io/express-the-following-complex-numbers-in-the-form-of-a-bi.htmlhttps://liberty.io/express-the-following-complex-numbers-in-the-form-of-a-bi.htmlI'm struggling with the following exercises: I tried to use the reasoning as follows: $$(a+bi)^n=(re^{\theta i})^n=r^ne^{\theta in}=r^n(\cos(\theta n)+i\sin(\theta n))$$ So for the first one I d......Tue, 18 Aug 2026 09:00:07 -0400https://liberty.io/the-difference-of-two-consecutive-perfect-squares-is-always-odd.htmlhttps://liberty.io/the-difference-of-two-consecutive-perfect-squares-is-always-odd.htmlI am working on another homework assignment about proofs. The question is: Prove or find counterexample: the difference of two consecutive perfect squares is odd? There is no counterexample corre......Tue, 18 Aug 2026 08:35:54 -0400https://liberty.io/i-ab-be-invertible-leftrightarrow-i-ba-is-invertible-duplicate.htmlhttps://liberty.io/i-ab-be-invertible-leftrightarrow-i-ba-is-invertible-duplicate.htmlassume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance...Tue, 18 Aug 2026 08:34:19 -0400https://liberty.io/do-the-taylor-series-of-sin-x-and-cos-x-depend-on-the-identity-sin-2-x.htmlhttps://liberty.io/do-the-taylor-series-of-sin-x-and-cos-x-depend-on-the-identity-sin-2-x.htmlI had this crazy idea trying to prove the Pythagorean trigonometric identity;$$\sin^2x+\cos^2x=1$$by squaring the infinite Taylor series of $\sin x$ and $\cos x$. But it came out quite beautiful, ...Tue, 18 Aug 2026 08:30:23 -0400https://liberty.io/help-with-proof-of-injection-and-surjection.htmlhttps://liberty.io/help-with-proof-of-injection-and-surjection.htmlFor the record, I am sorry, I haven't yet learnt how to use LaTeX I have a function $f(x) = 2x^3 - 1$ My proof of injection is as follows: $f$ is one to one for all $x_1,x_2$ element $X$, if $f(x......Tue, 18 Aug 2026 08:14:51 -0400https://liberty.io/relationship-between-primes-and-fibonacci-sequence.htmlhttps://liberty.io/relationship-between-primes-and-fibonacci-sequence.htmlI recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connect......Tue, 18 Aug 2026 07:58:37 -0400https://liberty.io/integral-definition-of-e.htmlhttps://liberty.io/integral-definition-of-e.htmlI know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...Tue, 18 Aug 2026 07:54:49 -0400https://liberty.io/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlhttps://liberty.io/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlProblem : How many zeros are in all natural numbers from $1$ to $2019$ ? My attempt : From $1$ to $100$ we have $11$ zeros Now from $101$ to $110$ we have $10$ zeros So from $20$ zeros ...Tue, 18 Aug 2026 07:43:23 -0400https://liberty.io/optimizing-svd-factors.htmlhttps://liberty.io/optimizing-svd-factors.htmlProblem Description My aim is to find the SVD decomposition of the minimizer of $F:\mathbb{R}^{m\times n}\to\mathbb{R}$ $$ U_*\Sigma_* V^\top_* = A_*\qquad \text{where} \qquad A_* = \arg\min_A F(A)......Tue, 18 Aug 2026 07:37:53 -0400https://liberty.io/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlhttps://liberty.io/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlIf $A$ triangle's side length are $25,25$ and $30$ and $B$ triangle's length are $25,25$ and $40$ what is the ratio between the two areas of the triangles? #math...Tue, 18 Aug 2026 07:36:12 -0400https://liberty.io/i-have-a-doubt-in-roman-numericals.htmlhttps://liberty.io/i-have-a-doubt-in-roman-numericals.htmlThe below question, I am looking at the 80, They translated it as LXXX, but I can also write it as XXC right?...Tue, 18 Aug 2026 07:20:21 -0400https://liberty.io/how-do-inverse-functions-exist-for-exponential-functions.htmlhttps://liberty.io/how-do-inverse-functions-exist-for-exponential-functions.htmlI know that they exist for exponential functions (we currently have them in class), but to me it doesn't seem "reasonable" when I look at the definition of what an inverse function is. The inverse is ...Tue, 18 Aug 2026 07:14:30 -0400https://liberty.io/deriving-the-variance-of-the-bernoulli-distribution.htmlhttps://liberty.io/deriving-the-variance-of-the-bernoulli-distribution.htmlFor a Bernoulli distribution, $\mu_X = p$. I can easily derive this from the general equation for mean of a discrete random variable: $$ \mu_X=\sum_{i=1}^kx_iPr(X=x) $$ $$ \mu_X=1(p)+0(1-p)=p $$ I ......Tue, 18 Aug 2026 07:09:06 -0400https://liberty.io/why-do-hyperbolic-trig-functions-seem-to-be-encountered-rarely.htmlhttps://liberty.io/why-do-hyperbolic-trig-functions-seem-to-be-encountered-rarely.htmlHyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarel......Tue, 18 Aug 2026 07:07:26 -0400https://liberty.io/mathematical-symbol-for-and.htmlhttps://liberty.io/mathematical-symbol-for-and.htmlI have found some pretty complete lists (I think) of mathematical symbols here and here, but I don't see a symbol for the word "and" on either list. A person could easily just write the word "and"......Tue, 18 Aug 2026 06:58:17 -0400https://liberty.io/geometrical-interpretation-of-trigonometric-antiderivative.htmlhttps://liberty.io/geometrical-interpretation-of-trigonometric-antiderivative.htmlI know about geometrical explanation of [definite] integral as an area under the curve, and I wonder if there are some ideas, which may give similar insight in taking antiderivatives [indefinite ...Tue, 18 Aug 2026 06:56:00 -0400https://liberty.io/are-the-graphs-of-inverse-functions-always-reflections-of-the-function.htmlhttps://liberty.io/are-the-graphs-of-inverse-functions-always-reflections-of-the-function.htmlFor an invertible function, is the graph of its inverse always the mirror image of the function's graph in the line $y=x$? And, if yes, then why is it so?...Tue, 18 Aug 2026 06:53:27 -0400https://liberty.io/what-books-are-prerequisites-for-spivak-039-s-calculus.htmlhttps://liberty.io/what-books-are-prerequisites-for-spivak-039-s-calculus.htmlFor financial reasons, I dropped out my senior year of college as a piano performance major. I will be returning to college to dual major in mathematics and computer science. I've taught myself to ...Tue, 18 Aug 2026 06:48:21 -0400https://liberty.io/what-calculation-shortcuts-exist-to-help-or-speed-up-mental-or-paper-c.htmlhttps://liberty.io/what-calculation-shortcuts-exist-to-help-or-speed-up-mental-or-paper-c.htmlAnything to speed up or simplify calculations. A simple example might be to get a multiple of $19$, for instance, $38 \cdot 19 = 38 \cdot 20 - 38$. (This is hard to tag with so few tags in play!) ...Tue, 18 Aug 2026 06:44:32 -0400https://liberty.io/using-operators-and-4-4-4-4-digits-find-all-formulas-that-would-resolv.htmlhttps://liberty.io/using-operators-and-4-4-4-4-digits-find-all-formulas-that-would-resolv.htmlI had a conversation with a colleague of mine and he brought up an interesting problem. Using the + - * / operators and four 4 4 4 4 digits, create an algorithm that will output all the formulas that ...Tue, 18 Aug 2026 06:44:31 -0400https://liberty.io/help-to-clarify-proof-of-euler-039-s-theorem-on-homogenous-equations.htmlhttps://liberty.io/help-to-clarify-proof-of-euler-039-s-theorem-on-homogenous-equations.htmlWhy is the last step (setting $\lambda = 1$ allowed? I have trouble accepting this because if I set $\lambda =1 $ at the very start, then: $f(\lambda x , \lambda y)=\lambda^r f(x,y)$ becomes $f......Tue, 18 Aug 2026 06:34:24 -0400https://liberty.io/how-do-i-find-the-composition-of-two-functions.htmlhttps://liberty.io/how-do-i-find-the-composition-of-two-functions.htmlIn a question sheet for my university course I am given the sets: $A = \{1, 2, 3, 4\}, B = \{x, y, z, w\}$ and $C = \{a, b, c\}$. I am then told to work out the functions: $f: A → B$ whi......Tue, 18 Aug 2026 06:33:22 -0400https://liberty.io/how-to-approximate-partial-derivatives-from-contour-map.htmlhttps://liberty.io/how-to-approximate-partial-derivatives-from-contour-map.htmlA problem of approximating partial derivatives from a given contour map Hi, I would like to know the connection between a contour map and partial derivatives. I understand that partial derivatives ......Tue, 18 Aug 2026 06:26:58 -0400https://liberty.io/creating-link-matrix.htmlhttps://liberty.io/creating-link-matrix.htmlI have a matrix called $ A $ with elements equal to $ 0 $ and $ 1 $. if element $ A _ { i j } $ is equal to $ 1 $, it means that member $ i $ is connected to member $ j $ directly. if elements $ A ......Tue, 18 Aug 2026 06:21:01 -0400https://liberty.io/i-can-039-t-find-a-way-to-simplify-this-boolean-algebra.htmlhttps://liberty.io/i-can-039-t-find-a-way-to-simplify-this-boolean-algebra.htmlxyz' + x'y + x'z the answer from boolean expression calculator on the internet is x'z + yz' I know the end x'z same like the answer. but I can't find a way to simplify xyz' + x'y = yz'...Tue, 18 Aug 2026 06:18:10 -0400https://liberty.io/how-many-integers-between-1-and-1000-use-exactly-three-digits.htmlhttps://liberty.io/how-many-integers-between-1-and-1000-use-exactly-three-digits.htmlThere is a problem in which I am obtaining a different answer than my professor. The problem is as follows: How many integers between $1$ and $1000$ use exactly three digits? The professor shows......Tue, 18 Aug 2026 06:15:36 -0400https://liberty.io/what-should-i-do-if-no-one-answers-my-question.htmlhttps://liberty.io/what-should-i-do-if-no-one-answers-my-question.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 18 Aug 2026 06:14:34 -0400https://liberty.io/sum-of-total-scores-odds-will-end-even-or-odd.htmlhttps://liberty.io/sum-of-total-scores-odds-will-end-even-or-odd.htmlFriend and i are betting on the ending total scores of basketball game, he says odds are greater it will end with an even sum, i say the odds are equal. Who is right?...Tue, 18 Aug 2026 06:09:46 -0400https://liberty.io/this-projection-is-a-covering-map.htmlhttps://liberty.io/this-projection-is-a-covering-map.htmlIf $\tilde X$ is the product of $X$ with a discrete space, the projection $\tilde X \to X$ is a covering map. This question seems really easy, but as I'm a beginner there are some things a little ......Tue, 18 Aug 2026 06:05:03 -0400https://liberty.io/what-is-so-special-about-klein-4-group.htmlhttps://liberty.io/what-is-so-special-about-klein-4-group.htmlThis is my first course in abstract algebra and so far I am only learning about groups. So is there anyone who can explain to me why Klein 4-group is so special that it warrants a category of its o......Tue, 18 Aug 2026 05:59:44 -0400https://liberty.io/wrong-wolfram-alpha-result.htmlhttps://liberty.io/wrong-wolfram-alpha-result.htmlI have this function $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2} & \mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y) = (0,0) \end{array} \right. $$ And I want to ......Tue, 18 Aug 2026 05:44:07 -0400https://liberty.io/find-a-vector-function-that-represents-the-curve-of-intersection-of-th.htmlhttps://liberty.io/find-a-vector-function-that-represents-the-curve-of-intersection-of-th.htmlThe cone $z=\sqrt{x^2+y^2}$ and the plane $z=1+y$ I parameterized the plane and put it into vector form: $t=1+y \to y=t-1$ $z= 1+t-1 \to z = t$ $y= t-1$, $z=t$ Since I'm finding the intersecti......Tue, 18 Aug 2026 05:32:38 -0400https://liberty.io/what-is-the-length-of-ef.htmlhttps://liberty.io/what-is-the-length-of-ef.htmlIn triangle $ABC$ , $AD$ is perpendicular to $BC$. It is given that $BC$=8 cm and $AD$=6 cm. If $E$ and $F$ are the mid points of $BD$ and $AC$ respectively, find $EF$. I tried using the mid point ...Tue, 18 Aug 2026 05:21:55 -0400