https://liberty.ioCurated star highlights with fresh energy.en-usSat, 05 Sep 2026 10:40:07 -0400https://liberty.io/why-aren-039-t-repeating-decimals-irrational-but-something-like-pi-is.htmlhttps://liberty.io/why-aren-039-t-repeating-decimals-irrational-but-something-like-pi-is.htmlWe use closest representations for both of them, but they are not completely true. $\frac{22}7$ and $3.14$ are not exactly $\pi$ but we use them as the best option available. $\frac13$ is $0.\bar......Sat, 05 Sep 2026 14:38:34 -0400https://liberty.io/int-e-mean-what-does-int-mean.htmlhttps://liberty.io/int-e-mean-what-does-int-mean.htmlIn real analysis what is Int symbol mean? like Int(E) , int (A U B) ?? I want to know what is Int mean in real analysis a few example will be also good thank you...Sat, 05 Sep 2026 14:36:31 -0400https://liberty.io/arbitrary-matrices-solving.htmlhttps://liberty.io/arbitrary-matrices-solving.htmlI'm trying to solve some equations with arbitrary matrices. My Problem is that I don't know what way to solve the equation should be taken. Examples: Solve for $X$. $X,A$ are arbitrary matrices. ......Sat, 05 Sep 2026 14:26:19 -0400https://liberty.io/find-the-derivative-at-the-point-1-ln-pi-4-of-x-arctan-x-e-y.htmlhttps://liberty.io/find-the-derivative-at-the-point-1-ln-pi-4-of-x-arctan-x-e-y.htmlFind the derivative at the point $(1, \ln(\pi/4)$ of $x \arctan x = e^y$. So I think this may be an implicit differentiation problem. I'm not exactly sure where to start. I tried differentiating......Sat, 05 Sep 2026 14:25:26 -0400https://liberty.io/uniqueness-proof-discrete-math-help.htmlhttps://liberty.io/uniqueness-proof-discrete-math-help.htmlWhile reading I came across Uniqueness Proofs. Where a theorem asserts the existence of a unique element with a particular property. In order to prove this two steps are needed, Prove existence and......Sat, 05 Sep 2026 14:24:05 -0400https://liberty.io/proving-the-volume-of-a-cone.htmlhttps://liberty.io/proving-the-volume-of-a-cone.htmlAbstract definition of a cone: locus of points of line segments join a single point $v$, called vertex with set of coplanar points called base. Height of cone is distance from vertex to plane. If I ...Sat, 05 Sep 2026 14:20:19 -0400https://liberty.io/calculating-the-number-of-subintervals-required-for-the-difference-bet.htmlhttps://liberty.io/calculating-the-number-of-subintervals-required-for-the-difference-bet.htmlI'm having trouble with calculating the minimum number of subintervals required for the difference between the upper and lower Riemann sums to be a particular value. So say I have the following ...Sat, 05 Sep 2026 14:12:26 -0400https://liberty.io/derivation-of-cubic-formula-closed.htmlhttps://liberty.io/derivation-of-cubic-formula-closed.htmlHow do we derive the so called cubic formula without using Cardano's method or substitution? I would like to see a step by step proof of where wolfram alpha derives this answer. And also explain wh......Sat, 05 Sep 2026 14:09:20 -0400https://liberty.io/topological-invariants.htmlhttps://liberty.io/topological-invariants.htmlDo continuous maps necessarily preserve topological invariants? Or is it necessary for the maps to be homeomorphisms? Are there simple examples where continuous maps do not preserve these invariants?...Sat, 05 Sep 2026 14:05:56 -0400https://liberty.io/what-is-the-integral-of-function-f-x-sin-x-x.htmlhttps://liberty.io/what-is-the-integral-of-function-f-x-sin-x-x.htmlI want to know if the function sinx/x is integrable and if it is, then what's its integral? My high school book says its a non-integrable function while WolframAlpha says its integral is Si(x) + ...Sat, 05 Sep 2026 14:03:47 -0400https://liberty.io/question-from-putnam-039-89-primes-of-the-form-101-ldots01.htmlhttps://liberty.io/question-from-putnam-039-89-primes-of-the-form-101-ldots01.htmlI'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general in......Sat, 05 Sep 2026 13:59:16 -0400https://liberty.io/math-conversion-question-1km-0-6214mi-and-1gal-3-78l.htmlhttps://liberty.io/math-conversion-question-1km-0-6214mi-and-1gal-3-78l.htmlIn Europe, gasoline efficiency is measured in km/L. If your car's gas mileage is 35.0mi/gal , how many liters of gasoline would you need to buy to complete a 142-km trip in Europe? Use the following ...Sat, 05 Sep 2026 13:47:59 -0400https://liberty.io/finding-the-parallel-sides-of-a-trapezoid-given-all-side-lengths-and-h.htmlhttps://liberty.io/finding-the-parallel-sides-of-a-trapezoid-given-all-side-lengths-and-h.htmlSuppose that we are given side lengths $a, b, c, d$ of a trapezoid. We know that two of them are parallel, and all values are different. Moreover, we are given the height $h$ from the base (dista......Sat, 05 Sep 2026 13:38:22 -0400https://liberty.io/how-can-num2str-function-cut-off-some-digits-in-matlab.htmlhttps://liberty.io/how-can-num2str-function-cut-off-some-digits-in-matlab.htmlI am reading the book " MATLAB: A practical introductoin programming and problem solving", written by Stormy Attaway and getting curious with the function "num2str". On page 256, the author used the ...Sat, 05 Sep 2026 13:29:53 -0400https://liberty.io/obtain-equation-of-right-circular-cylinder.htmlhttps://liberty.io/obtain-equation-of-right-circular-cylinder.htmlObtain equation of right circular cylinder on the circle through points $(a,0,0),(0,b,0)$ and $(0,0,c)$ as the guiding curve. I proceeded by considering this circle as intersection of sphere from the ...Sat, 05 Sep 2026 13:23:15 -0400https://liberty.io/find-the-area-of-a-triangle-given-the-coordinates-of-its-vertices.htmlhttps://liberty.io/find-the-area-of-a-triangle-given-the-coordinates-of-its-vertices.htmlThe following is a GRE practice test question which I don't understand. According to this general argument, it appears that any arbitrary point $(x, 7)$ would have "height" equal to 10 but then if......Sat, 05 Sep 2026 13:11:03 -0400https://liberty.io/conic-sections-parabola-what-is-p.htmlhttps://liberty.io/conic-sections-parabola-what-is-p.htmlHelp my teacher says $p$ can't be negative because it's distance. I watched TOCT's tutorial (The Organic Chemistry Tutor) in YouTube about parabola and he said "$(x-h)^2 = 4p(y-k)$ if $p$ is posi......Sat, 05 Sep 2026 12:50:59 -0400https://liberty.io/matrix-addition-multiplication-with-different-sizes.htmlhttps://liberty.io/matrix-addition-multiplication-with-different-sizes.htmlI have the following two matrices: $$A=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ and }B=\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$ So I have two matrixes with ...Sat, 05 Sep 2026 12:47:26 -0400https://liberty.io/what-is-the-exact-definition-of-an-inverse-function.htmlhttps://liberty.io/what-is-the-exact-definition-of-an-inverse-function.htmlI recently started reading a book which has challenged my understanding of functions and inverse functions quite a bit (mostly the format of the notations that are used); I know that we write a fun......Sat, 05 Sep 2026 12:47:09 -0400https://liberty.io/trigonometric-quadratic-formula-and-other-trig-solutions-for-roots-of-.htmlhttps://liberty.io/trigonometric-quadratic-formula-and-other-trig-solutions-for-roots-of-.htmlI was experimenting with trig functions and their connections to roots of polynomials when I derived a sort of quadratic formula: $$ax^2+bx+c=0\implies x=n\cos\left(\frac12\arccos\left(\frac{b^2......Sat, 05 Sep 2026 12:41:47 -0400https://liberty.io/is-there-a-power-series-with-0-radius-of-convergence.htmlhttps://liberty.io/is-there-a-power-series-with-0-radius-of-convergence.htmlQuestion: Is there a power series with $0$ radius of convergence?(doesn't converge anywhere) I asked my teacher this question and he replied in negative, however he didn't mention any proof for......Sat, 05 Sep 2026 12:39:33 -0400https://liberty.io/order-of-integration-in-triple-integral.htmlhttps://liberty.io/order-of-integration-in-triple-integral.htmlIs there any hard and fast rule for what order you integrate for triple integrals. I know of Fubini's theorem but surely this doesn't cover all cases of triple integrals. Say for example I have, $$\...Sat, 05 Sep 2026 12:39:13 -0400https://liberty.io/explanation-of-line-element-formula-dl-2-dx-2-dy-2.htmlhttps://liberty.io/explanation-of-line-element-formula-dl-2-dx-2-dy-2.htmlI found this in a physics textbook without justification: $$dl^2 = dx^2 +dy^2,$$ where I presume that $l = \sqrt{x^2+y^2}$. Why is this so? By my calculations I obtain $$ dl = \dfrac{\partial l}{\...Sat, 05 Sep 2026 12:28:39 -0400https://liberty.io/grouping-numbers-that-are-close-to-each-other.htmlhttps://liberty.io/grouping-numbers-that-are-close-to-each-other.htmlLet's say you have a set of numbers S and you want to obtain subsets of $S$, $s_1,s_2,\ldots, s_i$, where $i < N$. Is there a particular operation that will group the numbers that are "close to......Sat, 05 Sep 2026 12:28:17 -0400https://liberty.io/using-only-the-field-axioms-of-real-numbers-prove-that-1-1-1.htmlhttps://liberty.io/using-only-the-field-axioms-of-real-numbers-prove-that-1-1-1.htmlUsing only the field axioms of real numbers prove that $(-1)(-1) = 1$ (1) I start with an obvious fact:$$0 = 0$$ (2) Add $(-1)$ to both sides of the equation: $$0 + (-1) = 0+ (-1)$$ (3) Zero is the ...Sat, 05 Sep 2026 12:25:08 -0400https://liberty.io/how-do-we-know-the-ratio-between-circumference-and-diameter-is-the-sam.htmlhttps://liberty.io/how-do-we-know-the-ratio-between-circumference-and-diameter-is-the-sam.htmlThe number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same......Sat, 05 Sep 2026 12:18:50 -0400https://liberty.io/notation-for-a-constant-vs-a-variable.htmlhttps://liberty.io/notation-for-a-constant-vs-a-variable.htmlI have a straight line $y=f(x)$, i.e. $$ y=kx+m $$ $k, m$ are constants. $x$ is a variable. But is it correct to write $$ k,m\in\mathbb R $$ and $$ x\in \mathbb R \qquad? $$ Is this notation corre......Sat, 05 Sep 2026 12:17:51 -0400https://liberty.io/what-does-it-really-mean-for-a-metric-space-to-be-compact-duplicate.htmlhttps://liberty.io/what-does-it-really-mean-for-a-metric-space-to-be-compact-duplicate.htmlI've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset ......Sat, 05 Sep 2026 12:14:40 -0400https://liberty.io/how-many-numbers-are-in-this-set.htmlhttps://liberty.io/how-many-numbers-are-in-this-set.htmlLet M a positive integer is called “Minos” if in its binary representation, any two consecutive appearance of digit 1 are separated by 2 or more 0. Example 36= 100100 (binary) is “Minos” number, b......Sat, 05 Sep 2026 12:14:23 -0400https://liberty.io/graphing-a-tangent-function-with-an-amplitude.htmlhttps://liberty.io/graphing-a-tangent-function-with-an-amplitude.htmlBecause of the all real values for the range of a tangent graph, the amplitude seems to have no effect on the maximum or minimum of the graph, but to change its steepness. i just want to seek reass......Sat, 05 Sep 2026 12:08:21 -0400https://liberty.io/ratio-of-angles-in-a-triangle-given-lengths-of-triangle-039-s-sides-cl.htmlhttps://liberty.io/ratio-of-angles-in-a-triangle-given-lengths-of-triangle-039-s-sides-cl.htmlIf I have a triangle $\,\triangle ABC,\,$ with sides of lengths $\,AB=6, \;BC=4, \;CA=5,\,$ then what can I know about the ratio of $\,\dfrac{\angle ACB}{\angle BAC}\,$?...Sat, 05 Sep 2026 12:03:56 -0400https://liberty.io/why-zn-integers-modulo-n-under-multiplication-is-a-group-if-and-only-i.htmlhttps://liberty.io/why-zn-integers-modulo-n-under-multiplication-is-a-group-if-and-only-i.htmlIs it true that $(\Bbb Z_n,\cdot)$, integers modulo $n$ under multiplication, is a group if and only if $n$ is prime? If it's true, why? How can I prove it?...Sat, 05 Sep 2026 11:57:20 -0400https://liberty.io/use-the-sum-identity-and-double-identity-for-sine-to-find-sin-3x.htmlhttps://liberty.io/use-the-sum-identity-and-double-identity-for-sine-to-find-sin-3x.htmlQ. Use the sum identity and double identity for sine to find $\sin 3x$. $$ \begin{align} \sin 3x &= \sin (2x + x)\\ &=\sin 2x \cos x + \cos 2x \sin x \\ &= (2\sin x \cos x) \cos x + (1 ......Sat, 05 Sep 2026 11:57:16 -0400https://liberty.io/lim-x-to-0-sin-x-tan-x.htmlhttps://liberty.io/lim-x-to-0-sin-x-tan-x.htmlI have stumbled on this question in one of my problem sets from Cal I and I'm not sure how to proceed after the last step. $$\lim_{x\to 0^+} (\sin x)^ {\tan x}$$ // Applying exponential rule $$x=......Sat, 05 Sep 2026 11:41:52 -0400https://liberty.io/find-the-subgroups-of-a4.htmlhttps://liberty.io/find-the-subgroups-of-a4.htmlI had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even permutati......Sat, 05 Sep 2026 11:37:47 -0400https://liberty.io/how-find-this-equation-solution-2-sqrt-3-2y-1-y-3-1.htmlhttps://liberty.io/how-find-this-equation-solution-2-sqrt-3-2y-1-y-3-1.htmlfind this equation roots: $$2\sqrt[3]{2y-1}=y^3+1$$ My try: since $$8(2y-1)=(y^3+1)^3=y^9+1+3y^3(y^3+1)$$ then $$y^9+3y^6+3y^3-16y+9=0$$ Then I can't.Thank you someone can take hand find the equ......Sat, 05 Sep 2026 11:33:34 -0400https://liberty.io/what-is-transcendental-equation-function.htmlhttps://liberty.io/what-is-transcendental-equation-function.htmlI looked up several sources on the internet. A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not h......Sat, 05 Sep 2026 11:29:20 -0400https://liberty.io/find-the-area-and-perimeter-of-a-segment-of-a-circle.htmlhttps://liberty.io/find-the-area-and-perimeter-of-a-segment-of-a-circle.htmlFind the area and perimeter of the shaded region in the figure My work $$ \begin{align} \text{Area} &= \frac{r^2}{2} \theta - \frac12 r^2 \sin(\theta)\\ &= \frac{8^2}{2} \frac{37 \pi}{1......Sat, 05 Sep 2026 11:23:44 -0400https://liberty.io/understanding-rubik-039-s-cube-duplicate.htmlhttps://liberty.io/understanding-rubik-039-s-cube-duplicate.htmlI learnt how to solve a 3x3 Rubik's cube 10 years ago. Every now and then, I picked up a cube, scrambled it, and solved it for fun. I used to work on speed-solving, and memorised lots of formulae f......Sat, 05 Sep 2026 11:14:17 -0400https://liberty.io/when-a-simple-ideal-contained-in-the-sum-of-ideals.htmlhttps://liberty.io/when-a-simple-ideal-contained-in-the-sum-of-ideals.htmlI want to know if a simple ( minimal) ideal of a commutative ring with ideantity is contained in the sum of some ideal we can deduced it is contained in one of them....Sat, 05 Sep 2026 11:12:28 -0400https://liberty.io/finding-the-joint-probability-density-function-of-two-independent-rand.htmlhttps://liberty.io/finding-the-joint-probability-density-function-of-two-independent-rand.htmlIs there a way of determining the joint probability density function of two random variables? If we have two independent random variables, $X$ and $Y$ that both are uniform on [0,1], then how do one ...Sat, 05 Sep 2026 11:08:56 -0400https://liberty.io/tangent-plane-and-normal-vector.htmlhttps://liberty.io/tangent-plane-and-normal-vector.htmlFind a normal vector and tangent plane to the surface $x^2+4y^2=z^2$ at $(3,2,5)$ So, this is what I have done: $f(x,y,z) = x^2+4y^2-z^2$ Partial Differentiation gives: $\nabla f(x,y,z) = (2x,......Sat, 05 Sep 2026 10:57:27 -0400https://liberty.io/linearization-of-a-function.htmlhttps://liberty.io/linearization-of-a-function.htmlI am suppose to find the linearization of a function at a, I don't know what an a is. $f(x) = x^4 + 3x^2, a= -1$ is an a suppose to be like a y?...Sat, 05 Sep 2026 10:50:51 -0400https://liberty.io/tetrahedral-dice-throw.htmlhttps://liberty.io/tetrahedral-dice-throw.htmlIf we threw $30$ tetrahedral dice and summed the outcomes, how many values in the distribution would have a non-zero probability? a) If we calculate the distribution of the sum of two thrown ...Sat, 05 Sep 2026 10:28:35 -0400https://liberty.io/from-the-graph-of-the-derivative-f-039-x-make-a-sketch-of-the-original.htmlhttps://liberty.io/from-the-graph-of-the-derivative-f-039-x-make-a-sketch-of-the-original.htmlI haven't started finding the derivatives of functions yet, so at the moment this is strictly about finding the right derivative graph to an original graph. The task was this: Look at the graph be......Sat, 05 Sep 2026 10:11:16 -0400https://liberty.io/rigorous-definition-of-geometric-similarity.htmlhttps://liberty.io/rigorous-definition-of-geometric-similarity.htmlOn wikipedia it says : " Two geometrical objects are called similar if one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotatio......Sat, 05 Sep 2026 10:10:29 -0400https://liberty.io/how-do-we-know-what-the-integral-of-sin-x-is.htmlhttps://liberty.io/how-do-we-know-what-the-integral-of-sin-x-is.htmlSince I started more or less formally learning the foundations of calculus, I naturally came across the Riemann definition of the integral. At first I considered this definition intuitive but not r......Sat, 05 Sep 2026 10:09:30 -0400https://liberty.io/why-two-vectors-cannot-span-bf-r-3.htmlhttps://liberty.io/why-two-vectors-cannot-span-bf-r-3.htmlConsider two vectors $$ u_{1} = \begin{bmatrix} -1 \\ 3 \\ 2 \\ \end{bmatrix},\quad u_{2} = \begin{bmatrix} 6 \\ 1 \\ 1 \\ \end{bmatrix} $$ I can tell they don't span $R^3$ because $R^3$ re......Sat, 05 Sep 2026 10:03:19 -0400https://liberty.io/how-many-sets-of-5-can-you-make-from-22-elements.htmlhttps://liberty.io/how-many-sets-of-5-can-you-make-from-22-elements.htmlI know you can just say $\large\frac{22!}{(22!-5!)5!}$. My question is deeper than this, however. What if the set of 5 cannot be used again? For instance, I have 22 coloured balls, my first set of 5 ...Sat, 05 Sep 2026 09:55:32 -0400https://liberty.io/sum-of-independent-gamma-distributions-is-a-gamma-distribution.htmlhttps://liberty.io/sum-of-independent-gamma-distributions-is-a-gamma-distribution.htmlIf $X\sim \Gamma(a_1,b)$ and $Y \sim \Gamma(a_2,b)$, I need to prove $X+Y\sim\Gamma(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and just tryin......Sat, 05 Sep 2026 09:53:10 -0400https://liberty.io/why-non-real-means-only-the-square-root-of-negative.htmlhttps://liberty.io/why-non-real-means-only-the-square-root-of-negative.htmlOnce in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...Sat, 05 Sep 2026 09:46:15 -0400https://liberty.io/explanation-circular-permutation.htmlhttps://liberty.io/explanation-circular-permutation.htmlWell if one looks at the formula of circular permutations $Pc = (n-1)!$ But as we come to that formula, I need a concrete example and an explanation. Another thing, I have seen that when working ......Sat, 05 Sep 2026 09:45:34 -0400https://liberty.io/trace-of-a-diagonalized-matrix.htmlhttps://liberty.io/trace-of-a-diagonalized-matrix.htmlWhy do I have: $Tr(SDS^{-1})=Tr(D)$?...Sat, 05 Sep 2026 09:42:52 -0400https://liberty.io/what-do-mathematicians-do.htmlhttps://liberty.io/what-do-mathematicians-do.htmlThe American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: Wha......Sat, 05 Sep 2026 09:41:51 -0400https://liberty.io/f-ax-f-x-2-1-what-is-f.htmlhttps://liberty.io/f-ax-f-x-2-1-what-is-f.htmlSuppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm cons......Sat, 05 Sep 2026 09:40:53 -0400https://liberty.io/what-are-the-interesting-applications-of-hyperbolic-geometry.htmlhttps://liberty.io/what-are-the-interesting-applications-of-hyperbolic-geometry.htmlI am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...Sat, 05 Sep 2026 09:35:30 -0400https://liberty.io/proving-a-reduction-formula-for-the-antiderivative-of-cos-n-x-duplicat.htmlhttps://liberty.io/proving-a-reduction-formula-for-the-antiderivative-of-cos-n-x-duplicat.htmlI want to show that for all $n\ge 2$, it holds that $$ \int \cos^n x\ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n}\int \cos^{n-2} x\ dx. $$ I'm not even getting the result for the induction......Sat, 05 Sep 2026 09:24:44 -0400https://liberty.io/how-far-is-the-list-of-known-primes-known-to-be-complete.htmlhttps://liberty.io/how-far-is-the-list-of-known-primes-known-to-be-complete.htmlSo there is always the search for the next "biggest known prime number". The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twenty million digits. Wikipedia also lists the ...Sat, 05 Sep 2026 09:10:41 -0400https://liberty.io/how-to-prove-the-closed-map-lemma.htmlhttps://liberty.io/how-to-prove-the-closed-map-lemma.htmlThe closed map lemma says that if $f : X \to Y$ is a continuous function, $X$ is compact and $Y$ is Hausdorff, then $f$ is a closed map. How can I prove this ? Here is my attempt so far: Suppose......Sat, 05 Sep 2026 08:32:35 -0400https://liberty.io/quaternions-why-is-the-angle-frac-theta-2-duplicate.htmlhttps://liberty.io/quaternions-why-is-the-angle-frac-theta-2-duplicate.htmlThe equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta 2\ri......Sat, 05 Sep 2026 08:29:28 -0400https://liberty.io/how-do-i-find-the-equation-of-a-parabola-given-the-max-and-two-points.htmlhttps://liberty.io/how-do-i-find-the-equation-of-a-parabola-given-the-max-and-two-points.htmlI was given the points (2, -1) and (10,-1) and also a max of 4. How would I go about finding the equation of the parabola given this info?...Sat, 05 Sep 2026 08:24:21 -0400https://liberty.io/prove-the-sum-to-product-formulas-with-complex-numbers.htmlhttps://liberty.io/prove-the-sum-to-product-formulas-with-complex-numbers.htmlFirst of all, I don't know if it's possible but I'm just wondering about it. I'm pretty standard when it comes to complex numbers knowledge. So my goal: $$\sin x+\sin y=2\sin\left(\frac{x+y}{2}\rig......Sat, 05 Sep 2026 08:15:22 -0400https://liberty.io/the-relationship-between-spectral-decomposition-eigendecomposition-and.htmlhttps://liberty.io/the-relationship-between-spectral-decomposition-eigendecomposition-and.htmlI am trying to clarify the relationship between the spectral decomposition / eigendecomposition of a matrix and projection operators. I understand that there is a connection between diagonalizabil......Sat, 05 Sep 2026 08:14:36 -0400https://liberty.io/a-detail-in-the-proof-of-auslander-buchsbaum-theorem.htmlhttps://liberty.io/a-detail-in-the-proof-of-auslander-buchsbaum-theorem.htmlI'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m)$, where $m$ is the maximal ideal, and a finitely generated non-zero $R$-module $M$ such ...Sat, 05 Sep 2026 07:51:32 -0400https://liberty.io/find-the-volume-v-of-the-described-solid-s.htmlhttps://liberty.io/find-the-volume-v-of-the-described-solid-s.htmlFind the volume $V$ of the described solid $S$. The base of $S$ is the triangular region with vertices $(0, 0), (4,0)$, and $(0, 4)$.Cross-sections perpendicular to the x−axis are squares. My an......Sat, 05 Sep 2026 07:46:35 -0400https://liberty.io/antiderivative-of-sin-x-pi-3.htmlhttps://liberty.io/antiderivative-of-sin-x-pi-3.htmlI used $u$-substitution to break down the problem to be $-\cos(u)$ and then got my final answer to be $-\cos(x-\pi/3)$ (re-insertion for my $u$) and it is not completely correct. What am I missing ......Sat, 05 Sep 2026 07:43:23 -0400https://liberty.io/why-is-this-series-of-square-root-of-twos-equal-pi.htmlhttps://liberty.io/why-is-this-series-of-square-root-of-twos-equal-pi.htmlWikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, ...Sat, 05 Sep 2026 07:42:54 -0400https://liberty.io/what-is-the-integral-of-a-cumulative-distribution-function.htmlhttps://liberty.io/what-is-the-integral-of-a-cumulative-distribution-function.htmlI cannot find what is the integral of a cumulative distribution function $$\int G(\xi)d\xi$$ I think it should be simple, but I have no idea where else to look for it....Sat, 05 Sep 2026 07:40:34 -0400https://liberty.io/log-or-antilog-tables-which-ones-are-more-useful.htmlhttps://liberty.io/log-or-antilog-tables-which-ones-are-more-useful.htmlI'm trying to make a Log or Antilog table small enough to fit in the back of a wallet calendar (or a business card). My intend is to build a mathematically useful gift that can be used by anybody e......Sat, 05 Sep 2026 07:36:09 -0400https://liberty.io/solve-x-3-3x-4-0-in-real-numbers.htmlhttps://liberty.io/solve-x-3-3x-4-0-in-real-numbers.htmlSolve $x^3 +3x -4 =0$ in Real Numbers. I know the answer is 1 but i want the solution. if you solve it with decomposition, explain how did you find what to do to decompose this expression. (I know......Sat, 05 Sep 2026 07:34:49 -0400https://liberty.io/find-the-smallest-distance-between-point-and-ellipsoid.htmlhttps://liberty.io/find-the-smallest-distance-between-point-and-ellipsoid.htmlFind the smallest distance between the points on the ellipsoid $x^2+2y^2+z^2=16$ and the point $(0,0,1)$. Answer: Let $(x,y,z)$ be the closest point of the ellipsoid. The distance is $$\sqrt{x^2+......Sat, 05 Sep 2026 07:34:48 -0400https://liberty.io/definition-of-identical-in-distribution.htmlhttps://liberty.io/definition-of-identical-in-distribution.htmlWhile one can find various online references (and in books) for the definition of "convergence in distribution", I have several times today been stumped (as a non-probabilist) by the notion of "ide......Sat, 05 Sep 2026 07:22:03 -0400https://liberty.io/surface-area-and-volume-of-cone.htmlhttps://liberty.io/surface-area-and-volume-of-cone.htmlConsider a cone with height $H$ and radius $R$. Let $dh$ and $dr$ be the respective changes in $H$ and $R$. Therefore $$\text{Volume}=\int H \pi r^2 dh$$ Now $r$ is a function of $h$, so $$h=H−H/R*......Sat, 05 Sep 2026 07:16:05 -0400https://liberty.io/which-is-bigger-a-googolplex-or-10-100.htmlhttps://liberty.io/which-is-bigger-a-googolplex-or-10-100.htmlA googol is defined as $ 10^{100}$ Let x = $10^{100}$ A googolplex is defined as $10^{x}$ Which is bigger: a googolplex or $10^{100!}$ I only know that: $100! = 1×2×3×...×98×99×100$ $10^{100} = 1......Sat, 05 Sep 2026 07:11:33 -0400https://liberty.io/an-ill-conditioned-matrix.htmlhttps://liberty.io/an-ill-conditioned-matrix.htmlIf C is an ill-conditioned matrix and I want to get the inverse, one way is to take a pseudo-inverse of some sort. Instead, is the following, which uses the (normal) inverse, also a way to deal wit......Sat, 05 Sep 2026 07:11:04 -0400https://liberty.io/axiom-of-double-induction.htmlhttps://liberty.io/axiom-of-double-induction.htmlWhat would the set-theoretical axiom of induction look like for double induction* when stated in the mathematical language of first- or second-order logic? *References as to What Double Inductio......Sat, 05 Sep 2026 07:10:04 -0400https://liberty.io/extending-039-guess-2-3-of-the-average-039-game.htmlhttps://liberty.io/extending-039-guess-2-3-of-the-average-039-game.htmlIn game theory, 'Guess $\frac{2}3$ of the Average' is a game where $n$ people are asked to choose a real number between $0$ and $100$ inclusive. The person with the closest answer to $\frac{2}3$ of......Sat, 05 Sep 2026 07:04:56 -0400https://liberty.io/ogf-for-partitions-of-integer-r-n-into-even-parts-less-than-or-equal-t.htmlhttps://liberty.io/ogf-for-partitions-of-integer-r-n-into-even-parts-less-than-or-equal-t.htmlLet $a_r$ be the number of partitions of the integer r-N into even parts, the largest of which is less than or equal to 2m, where N is some integer that may depend on m. Find the ordinary generative ...Sat, 05 Sep 2026 06:55:52 -0400https://liberty.io/cauchy-039-s-residue-theorem-on-a-singularity-outside-a-contour.htmlhttps://liberty.io/cauchy-039-s-residue-theorem-on-a-singularity-outside-a-contour.htmlI recently ran into the following exercise: Evaluate $$\oint_\Gamma\frac{\cos z}{(z-\pi)^2}dz,$$where $\Gamma$ is a complete circuit of the circle $|z|=1$. Clearly, the singularity lies outsi......Sat, 05 Sep 2026 06:55:20 -0400https://liberty.io/minimum-number-of-attempts-to-guess-a-pin-code-given-constraints.htmlhttps://liberty.io/minimum-number-of-attempts-to-guess-a-pin-code-given-constraints.htmlI'm playing a video game at the moment called Sleeping Dogs, in which some of the mini-missions are to 'hack' a security camera, by guessing a four-digit PIN code. Here are the rules: 1) You are ...Sat, 05 Sep 2026 06:50:42 -0400https://liberty.io/a-consequence-of-the-mean-value-theorem-for-integrals.htmlhttps://liberty.io/a-consequence-of-the-mean-value-theorem-for-integrals.htmlSuppose that $f,g: [a,b] \rightarrow R$ are continuous functions. Its true that exists $a \leq c \leq b$, $$ \int _a^b f(x)g(x) dx = f(c) \int_a^b g(x) dx ? $$ And if $g \geq 0$ what happens? Thi......Sat, 05 Sep 2026 06:42:21 -0400https://liberty.io/proving-the-splitting-lemma-on-pg-147-in-at.htmlhttps://liberty.io/proving-the-splitting-lemma-on-pg-147-in-at.htmlThe Lemma and a sketch of its proof are given below: My questions are: 1- why the diagram that must be commutative looks specifically like this? My professor actually drew a diagram without the ...Sat, 05 Sep 2026 06:37:55 -0400https://liberty.io/is-calculus-necessary-for-computer-science-student-closed.htmlhttps://liberty.io/is-calculus-necessary-for-computer-science-student-closed.htmlI'm a freshman in university and I'm studying Computer science and engineering. This will be my second year of studying. We don't have Calculus as a mandatory class but I can take it from elective ...Sat, 05 Sep 2026 06:35:52 -0400https://liberty.io/after-adding-10-cups-of-water-to-a-container-1-8-full-it-becomes-3-4-f.htmlhttps://liberty.io/after-adding-10-cups-of-water-to-a-container-1-8-full-it-becomes-3-4-f.htmlA container is $1/8$ full of water. After $10$ cups of water are added, the container is $3/4$ full. What is the volume of the container, in cups? Ok, I wrote out an equation: $$\frac{1}{8}V + 10C......Sat, 05 Sep 2026 06:29:53 -0400https://liberty.io/on-the-jacobian-determinant-for-conversion-to-cylindrical-coordinates.htmlhttps://liberty.io/on-the-jacobian-determinant-for-conversion-to-cylindrical-coordinates.htmlI have an exercise that asks to compute the triple integral over a region $E$ (a paraboloid with a horizontal slice), which I know to be $$\iiint_E x\,dV=\int_{-1}^1\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^......Sat, 05 Sep 2026 06:27:22 -0400https://liberty.io/what-is-the-interpretation-of-the-fundamental-theorem-of-line-integral.htmlhttps://liberty.io/what-is-the-interpretation-of-the-fundamental-theorem-of-line-integral.htmlThe fundamental theorem of line integrals is: $$\int_{a}^{b} \nabla F \cdot \vec{dr} = F(r(b)) - F(r(a))$$ for some curve traced by $r$. What is the intuition for why this is true? The proof is ...Sat, 05 Sep 2026 06:15:25 -0400https://liberty.io/describe-the-span-of-the-given-vectors-geometrically-and-algebraically.htmlhttps://liberty.io/describe-the-span-of-the-given-vectors-geometrically-and-algebraically.htmlDescribe the span of the given vectors geometrically and algebraically: $\pmatrix{1\\0\\-1}$, $\pmatrix{-1\\1\\0}$, $\pmatrix{0\\-1\\1}$. I have figured out that these vectors are linearly depend......Sat, 05 Sep 2026 05:54:58 -0400https://liberty.io/precedence-between-implication-and-bi-implication.htmlhttps://liberty.io/precedence-between-implication-and-bi-implication.htmlI came across this question: Let p, q, and r be the propositions p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these ...Sat, 05 Sep 2026 05:33:48 -0400https://liberty.io/what-are-the-rules-for-a-tetartoid-pentagon.htmlhttps://liberty.io/what-are-the-rules-for-a-tetartoid-pentagon.htmlThe tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry. It has twelve identical ...Sat, 05 Sep 2026 05:28:46 -0400https://liberty.io/cartesian-coordinates-of-lemniscate.htmlhttps://liberty.io/cartesian-coordinates-of-lemniscate.htmlA problem from Spivak's Calculus: Sketch the graph of the lemniscate $$r^2 = 2a^2\cos 2 \theta.$$ Find an equation for its cartesian coordinates. Show that it is the collection of ......Sat, 05 Sep 2026 05:26:26 -0400https://liberty.io/determining-the-area-of-a-right-triangle-perimeter-given-hypotenuse-va.htmlhttps://liberty.io/determining-the-area-of-a-right-triangle-perimeter-given-hypotenuse-va.htmlThe problem states: Right Triangle- perimeter of $84$, and the hypotenuse is $2$ greater than the other leg. Find the area of this triangle. I have tried different methods of solving this pro......Sat, 05 Sep 2026 05:26:24 -0400https://liberty.io/find-the-partial-fraction-for-frac-1-x-2-1-2-x-1.htmlhttps://liberty.io/find-the-partial-fraction-for-frac-1-x-2-1-2-x-1.htmlI want to find the partial fraction of: $$\frac{1}{(x^2+1)^2(x-1)}$$ Although I want this expressed through the format of finding the complex number that goes into $x$ so that I can get this equati......Sat, 05 Sep 2026 05:18:07 -0400https://liberty.io/a-box-contains-10-balls-numbered-from-1-to-10.htmlhttps://liberty.io/a-box-contains-10-balls-numbered-from-1-to-10.htmlAn urn contains balls numbered $1, \ldots, 10$. Five balls are drawn without replacement. What is the probability that the second largest of the five numbers drawn will be 8? I believe the number of ...Sat, 05 Sep 2026 05:12:31 -0400https://liberty.io/why-can-039-t-you-flatten-a-sphere.htmlhttps://liberty.io/why-can-039-t-you-flatten-a-sphere.htmlIt's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old? As soon as an explanation starts using terms like "Gauss......Sat, 05 Sep 2026 05:11:40 -0400https://liberty.io/extended-euclidean-algorithm-for-modular-inverse.htmlhttps://liberty.io/extended-euclidean-algorithm-for-modular-inverse.htmlI'm currently learning how to find the inverse of a modulo with the Extended Euclid Algorithm and I stumbled upon a problem when finding an inverse when the $m>p$ as for $m \equiv 1 \pmod{p}$ For ...Sat, 05 Sep 2026 05:10:37 -0400https://liberty.io/converting-augmented-matrix-to-reduced-row-echelon-form.htmlhttps://liberty.io/converting-augmented-matrix-to-reduced-row-echelon-form.htmlI am struggling with reducing an augmented matrix to reduced row echelon form. The given linear system is \begin{cases} 3x-2z=-3 \\ -2x+z=-2 \\ -z=2 \\ \end{cases} I can write this as an augmen......Sat, 05 Sep 2026 04:57:52 -0400https://liberty.io/find-the-indicated-probability.htmlhttps://liberty.io/find-the-indicated-probability.htmlQuestion: On one tropical island, hurricanes occur with a mean of 2.74 per year. Assuming that the number of hurricanes can be modeled by a Poisson distribution, find the probability that during th......Sat, 05 Sep 2026 04:54:31 -0400https://liberty.io/find-the-arclength-of-the-curve-defined-by-r-t-i-9t-2j-t-3k-for-0-leq-.htmlhttps://liberty.io/find-the-arclength-of-the-curve-defined-by-r-t-i-9t-2j-t-3k-for-0-leq-.htmlFirst I found $r'(t)=\langle 1,18t^2,3t^2\rangle$ and so the magnitude of $r'(t)= \sqrt{1+(18t)^2+(3t^2)^2}$ thus the integral from $0$ to $\sqrt{28}$ of $\sqrt{1+324t^2+9t^4} dt$. When I plugged $......Sat, 05 Sep 2026 04:54:23 -0400https://liberty.io/classifying-map.htmlhttps://liberty.io/classifying-map.htmlLet $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f......Sat, 05 Sep 2026 04:48:51 -0400https://liberty.io/sum-of-multiplication-of-two-binomial-coefficients.htmlhttps://liberty.io/sum-of-multiplication-of-two-binomial-coefficients.htmlIs there any formula for calculating $\sum_{k=0}^n {n\choose k} {2n\choose 2k}$ ? One possible way is to use Stirling's approximation, but couldn't reach a reasonable answer....Sat, 05 Sep 2026 04:47:35 -0400