https://liberty.ioCurated star highlights with fresh energy.en-usTue, 25 Aug 2026 13:12:07 -0400https://liberty.io/differential-equation-x-039-039-kx.htmlhttps://liberty.io/differential-equation-x-039-039-kx.htmlInitial condition $x(0)=1$, $k>0$ \begin{eqnarray} \lambda^2=-k \end{eqnarray} \begin{eqnarray} \lambda=\pm\sqrt{-k} \end{eqnarray} I have no idea how to erase this $k$ what should be the nex......Tue, 25 Aug 2026 16:58:34 -0400https://liberty.io/find-the-probability-distribution-of-x.htmlhttps://liberty.io/find-the-probability-distribution-of-x.htmlLet $X$ be a random variable with cumulative distribution function: $$F(x)=\begin{cases} 0 & : x < -2 \\ 1/8 & : -2 \leq x < -1 \\ 3/8 & : -1 \leq x < 0 \\ 5/8 & : 0......Tue, 25 Aug 2026 16:35:31 -0400https://liberty.io/what-is-a-diameter-of-smaller-circle.htmlhttps://liberty.io/what-is-a-diameter-of-smaller-circle.htmlI have two coins $A$ and $B$. The diameter of coins A is $d_A=18$ mm. The smaller coin $B$ rolls around $A$ without sliding. Both coins have marks on the edge. At the beginning of the movement, the......Tue, 25 Aug 2026 16:31:45 -0400https://liberty.io/volume-of-regular-octahedron.htmlhttps://liberty.io/volume-of-regular-octahedron.htmlAccording to several sources on the internet, the volume of a regular octahedron with unit edge lengths is approximately 0.47. I haven't seen an explanation for why this is so yet. When I tried to ......Tue, 25 Aug 2026 16:31:38 -0400https://liberty.io/calculating-logs-in-your-head.htmlhttps://liberty.io/calculating-logs-in-your-head.htmlI quite often need to estimate $\log_2(x)$ for positive integer values of $x$. I find it frustrating that I have to rely on a calculator/computer to do this. By way of contrast, I can happily com......Tue, 25 Aug 2026 16:30:37 -0400https://liberty.io/mutually-exclusive-events-can-be-independent-flipping-a-coin-example-d.htmlhttps://liberty.io/mutually-exclusive-events-can-be-independent-flipping-a-coin-example-d.htmlFlipping a coin we get either head or tail can't get both so events are mutually exclusive. i.e P(A and B) =0. but flipping the same coin twice may result in either head or tail and result of flipp......Tue, 25 Aug 2026 16:28:58 -0400https://liberty.io/proving-associativity-of-a-certain-binary-aperation-in-any-complemente.htmlhttps://liberty.io/proving-associativity-of-a-certain-binary-aperation-in-any-complemente.htmlIf, in a Boolean lattice $(X,\vee,\wedge,0,1,')$ (i.e. a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+z)$ ra......Tue, 25 Aug 2026 16:24:55 -0400https://liberty.io/hamiltonian-graph-and-2-connected-graph.htmlhttps://liberty.io/hamiltonian-graph-and-2-connected-graph.htmlA non-complete graph is called 2-connected if it stays connected after removing a vertex (and all edges which are incident to that vertex). Show that a Hamiltonian graph is 2-connected. I'm having ...Tue, 25 Aug 2026 16:13:25 -0400https://liberty.io/definition-of-the-mathematical-proof.htmlhttps://liberty.io/definition-of-the-mathematical-proof.htmlHow do we define a mathematical proof? Is it a series of arguments? Is it a series of conclusions? Is it manipulation of formulas? Is it a mixture of laws of logic and axioms,theorems or defini......Tue, 25 Aug 2026 16:12:38 -0400https://liberty.io/count-the-number-of-solutions-of-the-inequality-x-y-z-leq-n.htmlhttps://liberty.io/count-the-number-of-solutions-of-the-inequality-x-y-z-leq-n.htmlProblem Given $A, B, C $ and $N$. How many integer solutions are there of the following inequality: $$x + y + z \leq N$$ where $0 \leq x \leq A, 0 \leq y \leq B, 0 \leq z \leq C$? When $A + ......Tue, 25 Aug 2026 16:01:49 -0400https://liberty.io/inversion-of-rotation-matrix.htmlhttps://liberty.io/inversion-of-rotation-matrix.htmlFor example, I have a two-dimensional rotation matrix $$ \begin{bmatrix} 0.5091 & -0.8607 \\ 0.8607 & \phantom{-}0.5091 \end{bmatrix} $$ and I have a vector I'd like to ...Tue, 25 Aug 2026 16:00:50 -0400https://liberty.io/how-to-factorize-x-3-7x-6.htmlhttps://liberty.io/how-to-factorize-x-3-7x-6.htmlHow do you factorize this polynomial: $${x^3 - 7x + 6}$$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a pre-......Tue, 25 Aug 2026 15:56:02 -0400https://liberty.io/how-to-simplify-a-n-b-n.htmlhttps://liberty.io/how-to-simplify-a-n-b-n.htmlHow to simplify $a^n - b^n$? If it would be $(a+b)^n$, then I could use the binomial theorem, but it's a bit different, and I have no idea how to solve it. Thanks in advance....Tue, 25 Aug 2026 15:50:35 -0400https://liberty.io/finding-domain-and-range-of-inverse-function-without-finding-inverse.htmlhttps://liberty.io/finding-domain-and-range-of-inverse-function-without-finding-inverse.htmlSo i found a question which states: Without finding the inverse, state the domain and range of $f^{-1}$ $f(x) = (x-1)/(x-4)$ where $ x≠4$ how can i find the domain and range of the function's ...Tue, 25 Aug 2026 15:32:38 -0400https://liberty.io/how-do-i-calculate-derivative-of-sgn-x.htmlhttps://liberty.io/how-do-i-calculate-derivative-of-sgn-x.htmlWe know $|x| = \sqrt(x^2)$, determine the second derivative $\frac{d^2}{dx^2}|x|$, So the first derivative is sgn(x), but how do I get the second?...Tue, 25 Aug 2026 15:29:38 -0400https://liberty.io/proving-aaa-sas.htmlhttps://liberty.io/proving-aaa-sas.htmlDoes anyone have a clue how I can prove that (Angle - Angle - Angle) <-> (Side - Angle - Side) for the similarity of triangles? I don´t know where to start? /Alex...Tue, 25 Aug 2026 15:25:44 -0400https://liberty.io/find-the-global-maximum-and-global-minimum.htmlhttps://liberty.io/find-the-global-maximum-and-global-minimum.html$ f(x) = x^3 - 6x^2 - 15x + 8 $ for the interval $ [-2,6] $ My answer was : $$ f'(x) = 3(x^2-4x-5) $$ $$ 3(x-5)(x+1) $$ so the critical points are $ x =5\; and \;x=-1 $ plugging them back with ......Tue, 25 Aug 2026 15:24:13 -0400https://liberty.io/why-is-the-divergence-of-curl-expected-to-be-zero.htmlhttps://liberty.io/why-is-the-divergence-of-curl-expected-to-be-zero.htmlI was reading the proof of Helmholtz decomposition theorem where I found the relation between the rotational and the irrotational fields are not symmetric. And by that I mean if the divergence of the ...Tue, 25 Aug 2026 15:23:28 -0400https://liberty.io/compute-the-limit-as-x-approaches-infinity.htmlhttps://liberty.io/compute-the-limit-as-x-approaches-infinity.html$$\lim_{x\to \infty} \frac{6x^4+4}{(x^2-2)(6x^2-1)}$$ Based on the way the function behaves I can say it will reach 1, but I can't seem to be able to calculate the limit. Thanks for the help!...Tue, 25 Aug 2026 15:17:50 -0400https://liberty.io/find-the-area-enclosed-by-circles-r-3-cos-theta-and-r-3-sin-theta.htmlhttps://liberty.io/find-the-area-enclosed-by-circles-r-3-cos-theta-and-r-3-sin-theta.htmlFind the area enclosed by circles $r=3\cos\theta$ and $r=3\sin\theta$ I know that the graph of $r=3\sin\theta$ intersects at $\theta=0$ and at $\theta=\pi$ why aren't these two bounds used and why......Tue, 25 Aug 2026 15:15:52 -0400https://liberty.io/prove-pascal-039-s-rule-algebraically.htmlhttps://liberty.io/prove-pascal-039-s-rule-algebraically.htmlI am trying to prove Pascal's Rule algebraically but I'm stuck on simplifying the numerator. This is the last step that I have, but I'm not sure where to go from here $$=\frac{\left[(k-1)(n-k)!+k(......Tue, 25 Aug 2026 15:15:24 -0400https://liberty.io/why-are-proofs-not-written-as-collections-of-logic-symbols-but-are-ins.htmlhttps://liberty.io/why-are-proofs-not-written-as-collections-of-logic-symbols-but-are-ins.htmlMathematical proofs are written as sentences and not as collections of logic symbols. Through logical operations, it is much easier for me to visualize what the symbols are trying to tell us rather......Tue, 25 Aug 2026 15:10:49 -0400https://liberty.io/proof-of-the-inclusion-exclusion-formula-in-probability.htmlhttps://liberty.io/proof-of-the-inclusion-exclusion-formula-in-probability.htmlLet $(\Omega,\mathcal F, P)$ be a probability space and let $A_1,A_2,...,A_n$ be events in $\mathcal F$. Prove the following inclusion-exclusion formula $P(\bigcup_{i=1}^nA_i)=\sum_{k=1}^n$ $\sum_{\...Tue, 25 Aug 2026 15:03:25 -0400https://liberty.io/show-that-the-eigenvalues-of-a-unitary-matrix-have-modulus-1.htmlhttps://liberty.io/show-that-the-eigenvalues-of-a-unitary-matrix-have-modulus-1.htmlShow that the eigenvalues of a unitary matrix have modulus $1$. I know that a unitary matrix can be defined as a square complex matrix $A$, such that $$AA^*=A^*A=I$$ where $A^*$ is the conjugate ...Tue, 25 Aug 2026 14:58:37 -0400https://liberty.io/surface-area-of-a-cube-as-a-function-of-its-volume.htmlhttps://liberty.io/surface-area-of-a-cube-as-a-function-of-its-volume.htmlExpressing the surface area of a cube as a function of its volume is like plugging in the surface area into volume? Analyzing $\text{S.A.}=6s^2$, and $V=s^3$ I would assume that the Surface area is ...Tue, 25 Aug 2026 14:51:23 -0400https://liberty.io/how-to-prove-that-frac-pi-2-sin-pi-z-2-sum-n-infty-n-neq-0-n-infty-1-z.htmlhttps://liberty.io/how-to-prove-that-frac-pi-2-sin-pi-z-2-sum-n-infty-n-neq-0-n-infty-1-z.htmlThe following question was part of my complex analysis assignment and I am not able to prove it. How to prove that $$\frac {\pi^2 } { \sin(\pi z)^2 } = \sum_{n=-\infty , n \neq0 }^{n=\infty} \fra......Tue, 25 Aug 2026 14:40:28 -0400https://liberty.io/how-to-find-the-maximum-value-of-f-x-y-39-x-2-y-2-on-the-line-x-6y-37.htmlhttps://liberty.io/how-to-find-the-maximum-value-of-f-x-y-39-x-2-y-2-on-the-line-x-6y-37.htmlso I'm totally lost how to find this question find the maximum value of $f(x,y) = 39 - x^2 - y^2$ on the line $x+6y=37$ I know how to find local max and min of multivariable equations, but now I ......Tue, 25 Aug 2026 14:35:39 -0400https://liberty.io/density-of-the-sum-of-two-exponential-random-variable.htmlhttps://liberty.io/density-of-the-sum-of-two-exponential-random-variable.htmlFor independent random variables X ∼ Exp(1) and Y ∼ Exp(2), find the density of the random variable Z = X + Y . My work: For any exponential distribution with parameter $\lambda$ the function is $......Tue, 25 Aug 2026 14:21:25 -0400https://liberty.io/dividing-an-infinite-power-series-by-another-infinite-power-series.htmlhttps://liberty.io/dividing-an-infinite-power-series-by-another-infinite-power-series.htmlLet's say I have two power series $\,\mathrm{F}\left(x\right) = \sum_{n = 0}^{\infty}\,a_{n}\,x^{n}$ and $\,\mathrm{G}\left(x\right) = \sum_{n = 0}^{\infty}\,b_{n}\,x^{n}$. If I define the functio......Tue, 25 Aug 2026 14:18:17 -0400https://liberty.io/graham-039-s-number-why-so-big.htmlhttps://liberty.io/graham-039-s-number-why-so-big.htmlCan someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !...Tue, 25 Aug 2026 14:16:06 -0400https://liberty.io/what-is-the-simplest-proof-of-the-pythagorean-theorem-you-know-duplica.htmlhttps://liberty.io/what-is-the-simplest-proof-of-the-pythagorean-theorem-you-know-duplica.htmlMaybe enough so to explain it to children....Tue, 25 Aug 2026 14:14:23 -0400https://liberty.io/why-doesn-039-t-my-ti-84-simplify-fractions.htmlhttps://liberty.io/why-doesn-039-t-my-ti-84-simplify-fractions.htmlSay we have: $$\frac 12 + \frac 13 + \frac 14$$ When I calculate this on my TI-84 Plus CE-T, it gives $$\frac {13}{12}$$ Instead of $$1\frac {1}{12}$$ Is there any reason for it doing so an......Tue, 25 Aug 2026 14:14:17 -0400https://liberty.io/proof-the-levi-civita-symbol-is-a-tensor.htmlhttps://liberty.io/proof-the-levi-civita-symbol-is-a-tensor.htmlA tensor of rank $n$ has components $T_{ij\cdots k}$ (with $n$ indices) with respect to each basis $\{\mathbf{e}_i\}$ or coordinate system $\{x_i\}$, and satisfies the following rule of change of b......Tue, 25 Aug 2026 14:10:23 -0400https://liberty.io/what-is-sqrt-36.htmlhttps://liberty.io/what-is-sqrt-36.htmlI am working on a math problem $(x^{2} - 6x + 45)$ where I must utilize the "completing the square" method, and was able to break down the problem to $(x-3)^{2} = -36.$ I know I have to ......Tue, 25 Aug 2026 13:55:05 -0400https://liberty.io/definition-of-exp-x.htmlhttps://liberty.io/definition-of-exp-x.htmlI've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia ......Tue, 25 Aug 2026 13:54:25 -0400https://liberty.io/expressing-logarithms-as-ratios-of-natural-logarithms.htmlhttps://liberty.io/expressing-logarithms-as-ratios-of-natural-logarithms.html$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an abstract ...Tue, 25 Aug 2026 13:42:14 -0400https://liberty.io/explanation-of-the-rational-root-theorem.htmlhttps://liberty.io/explanation-of-the-rational-root-theorem.htmlI just discovered the rational root theorem and I feel like I can understand it if I can get past the notational jargon presented in Wikipedia. Here's what I think I understood. If $r$ is a rational ...Tue, 25 Aug 2026 13:39:12 -0400https://liberty.io/limited-function.htmlhttps://liberty.io/limited-function.htmlLet $f:[0,+\infty) \rightarrow \mathbb{R}$ be a function bounded on each bounded interval. Prove that if $\lim_{x \rightarrow +\infty } [f(x+1)-f(x)] = L$, then $\lim_{x \rightarrow +\infty} \frac......Tue, 25 Aug 2026 13:34:51 -0400https://liberty.io/formula-for-probability-of-specific-die-outcomes.htmlhttps://liberty.io/formula-for-probability-of-specific-die-outcomes.htmlRoll some number, N, of the same kind of dice, say d6. Picking a target outcome, say 1, what is the FORMULA to determine the chance of getting 1s as a result, which is to say, how many chances of ......Tue, 25 Aug 2026 13:32:39 -0400https://liberty.io/why-balanced-ternary-exists-what-is-the-problem-with-ternary.htmlhttps://liberty.io/why-balanced-ternary-exists-what-is-the-problem-with-ternary.htmlRecently, I countered a problem of representing data in ternary. I came to know there exists ternary and balanced ternary representation. It is my best understanding that balanced ternary helps in ...Tue, 25 Aug 2026 13:30:54 -0400https://liberty.io/prove-that-a-subseteq-b-and-b-is-an-ideal-of-r.htmlhttps://liberty.io/prove-that-a-subseteq-b-and-b-is-an-ideal-of-r.htmlLet $R$ be a commutative ring with unity, let $A$ be an ideal of $R$ and let $a \in R$. Define $B = \{b \in R ~|~ ab \in A \}$. Prove that $A \subseteq B$ (where $A$ does not have to be the proper ......Tue, 25 Aug 2026 13:03:25 -0400https://liberty.io/probability-of-duplicate-guid.htmlhttps://liberty.io/probability-of-duplicate-guid.htmlA GUID (globally unique identifier) is a 32 character hexadecimal string: http://en.wikipedia.org/wiki/Globally_Unique_Identifier If you randomly generate 2, the chance of them being the same is ...Tue, 25 Aug 2026 13:02:18 -0400https://liberty.io/how-do-you-solve-an-equation-involving-the-weierstrass-p-function.htmlhttps://liberty.io/how-do-you-solve-an-equation-involving-the-weierstrass-p-function.htmlWhat $z\in \mathbb{C}$ satisfy \begin{eqnarray*} \wp(z) = \wp(nz) \end{eqnarray*} if $n\in \mathbb{N}$? The Weierstrass p-function is defined as \begin{eqnarray*} \wp(z)=\frac{1}{z^2}+\sum_{\omega......Tue, 25 Aug 2026 12:56:30 -0400https://liberty.io/sufficient-conditions-on-the-isomorphism-of-two-groups.htmlhttps://liberty.io/sufficient-conditions-on-the-isomorphism-of-two-groups.htmlLet $G_1, G_2$ be two groups with at least one nontrivial proper subgroup each. Let $S_1, S_2$ be the sets of proper subgroups of, respectively $G_1, G_2$. Suppose there exists a bijective functi......Tue, 25 Aug 2026 12:43:18 -0400https://liberty.io/can-the-entropy-of-a-random-variable-with-countably-many-outcomes-be-i.htmlhttps://liberty.io/can-the-entropy-of-a-random-variable-with-countably-many-outcomes-be-i.htmlConsider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by $$H(X) = \sum_i -p_i \log p_i.$$ Is it possibl......Tue, 25 Aug 2026 12:15:31 -0400https://liberty.io/where-can-i-find-a-list-of-large-prime-numbers-closed.htmlhttps://liberty.io/where-can-i-find-a-list-of-large-prime-numbers-closed.htmlA repository of say 13 digit prime, 15 digit primes etc....Tue, 25 Aug 2026 11:58:54 -0400https://liberty.io/is-basis-change-ever-useful-in-practical-linear-algebra.htmlhttps://liberty.io/is-basis-change-ever-useful-in-practical-linear-algebra.htmlIn layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?...Tue, 25 Aug 2026 11:55:21 -0400https://liberty.io/how-to-factor-x-2-2xy-y-2.htmlhttps://liberty.io/how-to-factor-x-2-2xy-y-2.htmlI know the formula $x^2 + 2xy + y^2 = (x + y)^2$ by heart and in order to understand it better I am trying to solve the formula myself; however, I don't understand how step 4 is derived from step 3......Tue, 25 Aug 2026 11:47:22 -0400https://liberty.io/integral-of-1-x-2-without-power-rule.htmlhttps://liberty.io/integral-of-1-x-2-without-power-rule.htmlI was wondering if it was possible to evaluate the following integral without using the power rule for negative exponents \begin{equation*} \int \frac{1}{x^2} \; dx \end{equation*} When using ...Tue, 25 Aug 2026 11:39:09 -0400https://liberty.io/math-logic-symbol-for-instance-of.htmlhttps://liberty.io/math-logic-symbol-for-instance-of.htmlI saw a discussion of a possible symbol for "example," but I need "example/instance of." There is of course $a \in S$ which is a is a member of/in S, but is there a specific way of stating $a "...Tue, 25 Aug 2026 11:32:29 -0400https://liberty.io/i-want-an-intutive-theoretical-understanding-of-the-sin-omega-t-f-clos.htmlhttps://liberty.io/i-want-an-intutive-theoretical-understanding-of-the-sin-omega-t-f-clos.htmlWhat is the meaning of $x(t)=x_0\cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ phase constant? I know how to solve the mathematical problems whic......Tue, 25 Aug 2026 11:22:58 -0400https://liberty.io/translation-of-weber-039-s-lehrbuch-der-algebra-vol-1-2-3.htmlhttps://liberty.io/translation-of-weber-039-s-lehrbuch-der-algebra-vol-1-2-3.htmlI have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would ......Tue, 25 Aug 2026 11:17:26 -0400https://liberty.io/what-039-s-the-difference-between-a-b-and-a-b.htmlhttps://liberty.io/what-039-s-the-difference-between-a-b-and-a-b.htmlIs there any difference between $]a,b[$ and $(a,b)$? If there is no difference, what would be the motivation of using $]a,b[$ over $(a,b)$?...Tue, 25 Aug 2026 11:16:57 -0400https://liberty.io/axiomatic-definition-of-groups.htmlhttps://liberty.io/axiomatic-definition-of-groups.htmlTo define natural numbers one can either: use the Peano axioms in second-order logic; encode them in set theory as von Neumann ordinals. The relation between those two definitions of natural num......Tue, 25 Aug 2026 11:15:22 -0400https://liberty.io/use-laplace-transform-to-solve-the-following-initial-value-problems.htmlhttps://liberty.io/use-laplace-transform-to-solve-the-following-initial-value-problems.htmlUse Laplace transform to solve the following initial–value problem. $y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$ Answer $s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(......Tue, 25 Aug 2026 11:06:40 -0400https://liberty.io/how-much-does-the-owner-lose.htmlhttps://liberty.io/how-much-does-the-owner-lose.htmlA thief stole a $\$100$ from the owner without him knowing. The same thief comes back inside the store $5$ minutes later and spends $\$70$ of the money. The owner then gives him back $\$30$ in chan......Tue, 25 Aug 2026 11:04:44 -0400https://liberty.io/simple-double-summation.htmlhttps://liberty.io/simple-double-summation.htmlI've seen how nesting works with a simple $(i+j)$ but this problem below is tripping me up. It's either because of the multipliers or because they each start at zero but I get 60, and the answer I ...Tue, 25 Aug 2026 10:59:18 -0400https://liberty.io/probability-density-of-the-maximum-of-samples-from-a-uniform-distribut.htmlhttps://liberty.io/probability-density-of-the-maximum-of-samples-from-a-uniform-distribut.htmlSuppose $$X_1, X_2, \dots, X_n\sim Unif(0, \theta), iid$$ and suppose $$\hat\theta = \max\{X_1, X_2, \dots, X_n\}$$ How would I find the probability density of $\hat\theta$? Thank you!...Tue, 25 Aug 2026 10:55:53 -0400https://liberty.io/how-to-quickly-tell-if-a-discontinuity-is-removable-or-non-removable.htmlhttps://liberty.io/how-to-quickly-tell-if-a-discontinuity-is-removable-or-non-removable.htmlGiven any function where you're asked to find all points of discontinuity and to classify those points, is there a quick way to look at it and tell if that discontinuity is removable or not?...Tue, 25 Aug 2026 10:52:43 -0400https://liberty.io/reverse-of-modulo-operator.htmlhttps://liberty.io/reverse-of-modulo-operator.htmlI have posted this question originally in Stack Overflow. The question is, "Is there a mathematical approach in getting the reverse of the Modulo Operator with given result $r$ and divisor $d$?" ......Tue, 25 Aug 2026 10:45:02 -0400https://liberty.io/tan-2x-sec-2x-express-in-terms-of-sin-cos.htmlhttps://liberty.io/tan-2x-sec-2x-express-in-terms-of-sin-cos.htmlI am trying to express this problem in terms of sin/cos and simplify. I couldn't figure out where to go, I tried as best I could. I know the answer is -1 but I am more interested to know how to do ......Tue, 25 Aug 2026 10:39:40 -0400https://liberty.io/meaning-of-occurs-for-infinitely-many-and-all-but-finitely-many.htmlhttps://liberty.io/meaning-of-occurs-for-infinitely-many-and-all-but-finitely-many.htmlSuppose $P(n)$ is certain statement. I hear all the time that my teachers say $$ P(n) \; \; \text{occurs for infinitely many} \; \; \;n $$ $$ P(n) \; \; \text{for all but finitely many} \; \; n ......Tue, 25 Aug 2026 10:32:08 -0400https://liberty.io/rewrite-double-integral-in-polar-coordinates-int-0-3-int-x-3-1-fdydx.htmlhttps://liberty.io/rewrite-double-integral-in-polar-coordinates-int-0-3-int-x-3-1-fdydx.htmlI have the following... $$\int_{0}^{3}\int_{x/3}^{1}fdydx$$ I need to rewrite this in polar coordinates. I graphed the triangle and can see that... $$\tan(\theta) = \frac{1}{3}$$ But I do not kn......Tue, 25 Aug 2026 10:26:33 -0400https://liberty.io/is-a-decimal-with-a-predictable-pattern-a-rational-number.htmlhttps://liberty.io/is-a-decimal-with-a-predictable-pattern-a-rational-number.htmlI'm starting as a private Math tutor for a high school kid; in one of his Math Laboratories (that came with an answer sheet) I was stumped by an answer I encountered in the True or False section (I'm ...Tue, 25 Aug 2026 10:24:46 -0400https://liberty.io/non-brute-force-proof-of-multiplicative-property-of-determinants.htmlhttps://liberty.io/non-brute-force-proof-of-multiplicative-property-of-determinants.htmlDoes anybody have a "non brute" force way to prove the following for non-singular matrices $A, B$: $$\det(AB) = \det(A) \det(B)$$...Tue, 25 Aug 2026 10:18:44 -0400https://liberty.io/what-does-it-mean-for-a-function-to-be-riemann-integrable.htmlhttps://liberty.io/what-does-it-mean-for-a-function-to-be-riemann-integrable.htmlI need to come up with a precise mathematical definition of what a Riemann integrable function is. I know what the Riemann integral is but when I look for definitions all I find are proofs of how to ...Tue, 25 Aug 2026 10:04:43 -0400https://liberty.io/step-2-in-sard-039-s-theorem-proof-countably-many-neighborhoods.htmlhttps://liberty.io/step-2-in-sard-039-s-theorem-proof-countably-many-neighborhoods.htmlI'm studying Sard's Theorem in prof john lee's 'smooth manifolds' The proof contains three step and there is a second step: F($C_k$ \ $C_{k+1}$) is measure zero for smooth map F and $C_k$ = {$x \i......Tue, 25 Aug 2026 10:01:45 -0400https://liberty.io/expected-value-for-blackjack.htmlhttps://liberty.io/expected-value-for-blackjack.htmlIn the game of blackjack, the odds of winning each hand are slightly less than 50 percent. As you play an infinite amount of hands, you would always lose money because you would win less than 50 pe......Tue, 25 Aug 2026 09:46:33 -0400https://liberty.io/continuous-function-on-0-1-f-0-f-1.htmlhttps://liberty.io/continuous-function-on-0-1-f-0-f-1.htmlI came across this very interesting question, which seems to be partially answered in a couple posts around here: Let $f:[0,1]\rightarrow\mathbb{R}$ continuous such that $f(0)=f(1)$. Then for all $......Tue, 25 Aug 2026 09:38:36 -0400https://liberty.io/integrating-cubic-spline.htmlhttps://liberty.io/integrating-cubic-spline.htmlSuppose that I have a finite number of samples of the function $f(x)$ for the range $a < x < b$. I don't have the expression for $f(x)$ but I can do a cubic spline interpolation. If I want to ...Tue, 25 Aug 2026 09:36:05 -0400https://liberty.io/math-symbol-for-where.htmlhttps://liberty.io/math-symbol-for-where.htmlPretty much any math text I've read introduces notation through a similar format of equation-notation or notation-equation form e.g ax+b+c where a = ..., b = ..., etc. or Let a = ..., b =......Tue, 25 Aug 2026 09:35:37 -0400https://liberty.io/finding-real-and-imaginary-part-of-exponential-function.htmlhttps://liberty.io/finding-real-and-imaginary-part-of-exponential-function.htmlCan someone explain to me how I find the real and the imaginary part of $e^{\theta i}$? I'm learning complex numbers but I don't quite understand how $e$ is intertwined in all this....Tue, 25 Aug 2026 09:34:47 -0400https://liberty.io/if-there-are-7-runners-in-a-race-how-many-different-orderings-can-ther.htmlhttps://liberty.io/if-there-are-7-runners-in-a-race-how-many-different-orderings-can-ther.htmlIf there are 7 runners in a race, how many different orderings of 1st, 2nd, and 3rd place can there be? I did 7*6*5 to get 210. I did this because there are 7 runners that can get first, after one ...Tue, 25 Aug 2026 09:18:13 -0400https://liberty.io/proving-standard-properties-of-sine-and-cosine-defined-by-their-power-.htmlhttps://liberty.io/proving-standard-properties-of-sine-and-cosine-defined-by-their-power-.htmlDefinition: We define $\displaystyle \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!}, \; x \in \mathbb{R} $ and $ \displaystyle \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x......Tue, 25 Aug 2026 09:08:10 -0400https://liberty.io/what-allows-us-to-use-imaginary-numbers.htmlhttps://liberty.io/what-allows-us-to-use-imaginary-numbers.htmlWhat axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers? In the beginning, when there were just reals, these operations were defined for them. T......Tue, 25 Aug 2026 09:02:41 -0400https://liberty.io/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-d.htmlhttps://liberty.io/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-d.htmlWe add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers). Why this trick i......Tue, 25 Aug 2026 09:02:33 -0400https://liberty.io/why-is-the-period-of-sin-2x-is-pi.htmlhttps://liberty.io/why-is-the-period-of-sin-2x-is-pi.htmlIn my textbook, it is stated that the period of $\sin(2x)$ is $\pi$. The author justifies this using a mathematical statement which I cannot understand. He writes that, since $\sin(2x) = \sin(2x+2\......Tue, 25 Aug 2026 08:56:27 -0400https://liberty.io/concrete-category-and-abstract-category.htmlhttps://liberty.io/concrete-category-and-abstract-category.htmlFrom Analysis and Its Foundations By Eric Schechter: A concrete category consists of a collection of objects and a collection of morphisms. I am curious what an "abstract category" is? I foun......Tue, 25 Aug 2026 08:56:08 -0400https://liberty.io/given-the-exponential-equation-4-x-64-what-is-the-logarithmic-form-of-.htmlhttps://liberty.io/given-the-exponential-equation-4-x-64-what-is-the-logarithmic-form-of-.htmlGiven the exponential equation $4^x = 64$, what is the logarithmic form of the equation in base $10$? Would it be $\frac{\log_{10}64}{\log_{10}4}$?...Tue, 25 Aug 2026 08:50:58 -0400https://liberty.io/why-is-a-constant-sequence-a-cauchy-sequence.htmlhttps://liberty.io/why-is-a-constant-sequence-a-cauchy-sequence.htmlI think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence: The sequence $\{q_n\}$ of rationals is a Cauchy sequence if, for every $\epsilon>0$, where $\e......Tue, 25 Aug 2026 08:40:16 -0400https://liberty.io/proving-two-lines-don-039-t-intersect.htmlhttps://liberty.io/proving-two-lines-don-039-t-intersect.htmlThe line $l_1$ passes through the position vector $i-j-2k$ and is parallel to the vector $3i-4j-2k$. The line $l_2$ passes through the position vector $(1+5\cos t)i-(1+5\sin t )j-14k$ where $0≤ t≤......Tue, 25 Aug 2026 08:36:30 -0400https://liberty.io/all-girls-are-liked-by-some-boys-no-boy-likes-a-sulk-no-girl-is-a-sulk.htmlhttps://liberty.io/all-girls-are-liked-by-some-boys-no-boy-likes-a-sulk-no-girl-is-a-sulk.htmlAll girls are liked by some boys. No boy likes a sulk. No girl is a sulk. How do you write the above in predicate logic statements without starting with a negation? My guess is below. Let me kno......Tue, 25 Aug 2026 08:36:14 -0400https://liberty.io/maclaurin-series-of-ln-1-z.htmlhttps://liberty.io/maclaurin-series-of-ln-1-z.htmlI'm not sure how to start on this problem other than the definition of $\ln(z)$, which is: $$\ln(1-z) = \ln|1-z| + i(\arg z)$$ or I don't know what to make of $\ln(1-z)$....Tue, 25 Aug 2026 08:29:09 -0400https://liberty.io/exact-values-of-hyperbolic-trig-functions.htmlhttps://liberty.io/exact-values-of-hyperbolic-trig-functions.htmlThere are some well-known exact values for trig functions, such as $$\sin\frac{\pi}{6}=\frac{1}{2},\quad \tan\frac{\pi}{3}=\sqrt 3, \quad\text{etc.}$$ Are there comparable special values for the ...Tue, 25 Aug 2026 08:29:01 -0400https://liberty.io/what-is-the-sum-of-all-real-numbers-from-0-to-1-closed.htmlhttps://liberty.io/what-is-the-sum-of-all-real-numbers-from-0-to-1-closed.htmlI wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it....Tue, 25 Aug 2026 08:26:44 -0400https://liberty.io/consistent-matlab-plot-size-w-different-axis-labels.htmlhttps://liberty.io/consistent-matlab-plot-size-w-different-axis-labels.htmlI'm trying to specify the sizes of two MATLAB figures in such a way that the graphs appear to be the same size, regardless of extra space used outside the axes. Specifically, I'm want the white re......Tue, 25 Aug 2026 08:17:01 -0400https://liberty.io/unary-predicates-theory-with-no-principal-types.htmlhttps://liberty.io/unary-predicates-theory-with-no-principal-types.htmlDefinition Take a satisfiable set of $L$-sentences $\Sigma$ and variables $x = x_1, \ldots, x_n$. Denote by $S_x(\Sigma)$ the set of all $\Sigma$-realizable $x$-types in $L$. A type $p(x) \in S_x(......Tue, 25 Aug 2026 08:14:18 -0400https://liberty.io/compound-interest-formula-with-regular-deposits-solve-for-time.htmlhttps://liberty.io/compound-interest-formula-with-regular-deposits-solve-for-time.htmlI cant seem solve the following problem due to my bad memory of logarithms if someone opens an account with initial balance of $B$ interest rate of $R$ and does a constant monthly deposit of $D$ ......Tue, 25 Aug 2026 08:09:07 -0400https://liberty.io/how-to-calculate-the-perceived-frequency-of-two-sinusoidal-waves-added.htmlhttps://liberty.io/how-to-calculate-the-perceived-frequency-of-two-sinusoidal-waves-added.htmlIf you add together two sinusoidal waves of different frequencies, how do you calculate the frequency of the resulting function as perceived by a human?...Tue, 25 Aug 2026 08:08:22 -0400https://liberty.io/differences-between-real-and-complex-analysis.htmlhttps://liberty.io/differences-between-real-and-complex-analysis.htmlTo start with, real analysis deals with numbers along the (one dimensional) number line, while complex analysis deals with numbers along two dimensions, real and imaginary, Cartesian style. Could t......Tue, 25 Aug 2026 08:03:16 -0400https://liberty.io/l-039-hospital-039-s-rule-doesn-039-t-converge-for-a-function-with-squ.htmlhttps://liberty.io/l-039-hospital-039-s-rule-doesn-039-t-converge-for-a-function-with-squ.htmlI was trying to find the limit of a function of the form $\frac{x}{\sqrt{(x+a)(x+b)}}$. When I apply L'Hospital's rule and differentiate the numerator and denominator, after simplification I end up......Tue, 25 Aug 2026 07:56:44 -0400https://liberty.io/what-is-the-name-of-the-answer-to-exponentiation.htmlhttps://liberty.io/what-is-the-name-of-the-answer-to-exponentiation.htmlWhat is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to exponenti......Tue, 25 Aug 2026 07:55:56 -0400https://liberty.io/ferris-wheel-problem.htmlhttps://liberty.io/ferris-wheel-problem.htmlOne of the largest ferris wheel ever built is in the british airways london eye which was completed in 2000. The diameter is 135 m and passengers get on at the bottom 4 m above the ground. The wheel ...Tue, 25 Aug 2026 07:46:55 -0400https://liberty.io/quadratic-lyapunov-function.htmlhttps://liberty.io/quadratic-lyapunov-function.htmlConsider the nonlinear model of a chemical process: $$ \begin{align} \dot{x}_1 &= x_2\\ \dot{x}_2 &=-5x_1 -3x_2\left(1 - \frac{0.6x_2}{1 + x_2^2}\right). \end{align} $$ Writing this system in ...Tue, 25 Aug 2026 07:45:23 -0400https://liberty.io/which-of-the-following-statements-are-false.htmlhttps://liberty.io/which-of-the-following-statements-are-false.htmlAll of the following sets are subsets of positive integers. $A = \{x\mid x\ \text{is divisible by 2}\} \\ B = \{x\mid x\ \text{is divisible by 4}\} \\ C = \{x\mid x\ \text{is divisible by 6}\}$ B......Tue, 25 Aug 2026 07:37:22 -0400https://liberty.io/annualized-probability-of-loan-default.htmlhttps://liberty.io/annualized-probability-of-loan-default.htmlCan someone help with how to calculate the annualized probability of a loan default given: 70% probability of survival (30% default) over the next 20 months? Edit: I should have been more specif......Tue, 25 Aug 2026 07:16:14 -0400https://liberty.io/prove-disprove-that-if-two-sets-have-the-same-power-set-then-they-are-.htmlhttps://liberty.io/prove-disprove-that-if-two-sets-have-the-same-power-set-then-they-are-.htmlI am really sure that if two sets have the same power set, then they are the same set. I just am wondering how does one exactly go about proving/showing this? I'm usually wrong, so if anyone can s......Tue, 25 Aug 2026 07:06:37 -0400https://liberty.io/continuity-essential-supremum-and-supremum.htmlhttps://liberty.io/continuity-essential-supremum-and-supremum.htmlTake $f: \mathbb{R}\to \mathbb{R}$ be continuous, then w.r.t. Lebesgue measure $L$, we have $$\operatorname{ess sup} |f|=\sup |f|$$ w.r.t. $L$. I have been provided a proof in a Measure theory co......Tue, 25 Aug 2026 06:54:15 -0400https://liberty.io/runge-kutta-and-butcher-table.htmlhttps://liberty.io/runge-kutta-and-butcher-table.htmlIn the Wikipedia article on Runge-Kutta methods, there is a notation explained using a Butcher table with a $c_{i}$ vector (nodes), a $b_{i}$ vector (weights) and a runge-kutta matrix $a_{ij}$. My ...Tue, 25 Aug 2026 06:52:22 -0400https://liberty.io/when-rolling-two-dice-which-of-the-following-events-are-independent-of.htmlhttps://liberty.io/when-rolling-two-dice-which-of-the-following-events-are-independent-of.htmlWhen rolling two dice, which of the following events are independent of the event that the first die is 4: (a) The second is 2 (b) The sum is 6 (c) The sum is 7 (d) The sum is even I thought only o......Tue, 25 Aug 2026 06:49:59 -0400