https://liberty.ioCurated star highlights with fresh energy.en-usSat, 29 Aug 2026 02:18:54 -0400https://liberty.io/circular-definition.htmlhttps://liberty.io/circular-definition.htmlI am new to logic and studying it by Enderton's A Mathematical Introduction to Logic (2nd edition). For question 3.3.1 in the books, which states Show that in the structure $(\mathbb{N}; \cdot, ......Sat, 29 Aug 2026 06:13:18 -0400https://liberty.io/basis-for-the-solution-space.htmlhttps://liberty.io/basis-for-the-solution-space.htmlFind the basis for the solution space of the system and describe all solutions: $3x_1 - x_2 + x_4 = 0$ $x_1 + x_2 + x_3 + x_4 = 0$ I row reduce the matrix: $\begin{pmatrix} 1&1&1&1\......Sat, 29 Aug 2026 06:07:20 -0400https://liberty.io/proof-of-pinsker-039-s-inequality.htmlhttps://liberty.io/proof-of-pinsker-039-s-inequality.htmlHow to prove the following known (Pinsker's) inequality? For two strictly positive sequences $(p_i)^n_{i=l}$ and $(q_i)^n_{i=l}$ with $\sum_{i=1}^np_i=\sum_{i=1}^nq_i=1$ one has $$\sum_{i=1}^np......Sat, 29 Aug 2026 06:07:01 -0400https://liberty.io/range-of-a-piecewise-function.htmlhttps://liberty.io/range-of-a-piecewise-function.htmlApologies, this is a simple question but I seem to have some sort of brain freeze. I'm looking for the range of this piecewise function: $$f(x)=\begin{cases}x+9&\text{ if } x<-3\\ -2x&\......Sat, 29 Aug 2026 06:03:23 -0400https://liberty.io/shortest-path-between-three-nodes-in-a-graph.htmlhttps://liberty.io/shortest-path-between-three-nodes-in-a-graph.htmlI know Dijkstra's algorithm to find the shortest way between 2 nodes, but is there a way to find the shortest path between 3 nodes among $n$ nodes? Here are the details: I have $n$ nodes, some of ......Sat, 29 Aug 2026 06:02:07 -0400https://liberty.io/mean-value-theorem-find-c.htmlhttps://liberty.io/mean-value-theorem-find-c.htmlThe derivative of $\frac1x$ is $\frac{-1}{x^2}$. How do I find the $c$ if there is no zero in the derivative of the function? I started with $-1/x^2= -0.0625$ but I'm confused from here on....Sat, 29 Aug 2026 05:55:58 -0400https://liberty.io/why-is-the-ratio-of-circumference-to-diameter-pi-and-not-any-other-num.htmlhttps://liberty.io/why-is-the-ratio-of-circumference-to-diameter-pi-and-not-any-other-num.htmlHow do we specifically get to 3.14 from axioms through logic to that particular transcendental number being the case? I'm guessing it's the sum of an infinite series. But how do we go from "there'......Sat, 29 Aug 2026 05:53:46 -0400https://liberty.io/what-is-n-when-n-0-duplicate.htmlhttps://liberty.io/what-is-n-when-n-0-duplicate.htmlPossible Duplicate: Prove $0! = 1$ from first principles Why does 0! = 1? If I'm right, factorial $!$ means: $$n!=1 \cdot 2 \cdot 3 \cdot 4 \cdots n $$ so: $$ \begin{align} 5!&=1\cdot2\...Sat, 29 Aug 2026 05:44:39 -0400https://liberty.io/are-the-real-product-rule-and-quotient-rule-for-integration-already-kn.htmlhttps://liberty.io/are-the-real-product-rule-and-quotient-rule-for-integration-already-kn.htmlIn "A Quotient Rule Integration by Parts Formula", the authoress integrates the product rule of differentiation and gets the known formula for integration by parts: \begin{equation}\int f(x)g'(x)dx......Sat, 29 Aug 2026 05:41:55 -0400https://liberty.io/why-does-the-euclidean-algorithm-for-finding-gcd-work.htmlhttps://liberty.io/why-does-the-euclidean-algorithm-for-finding-gcd-work.htmlI am having trouble understanding why the Euclidean algorithm for finding the GCD of two numbers always works? I found some resources here (http://www.cut-the-knot.org/blue/Euclid.shtml), and her......Sat, 29 Aug 2026 05:39:26 -0400https://liberty.io/is-the-fibonacci-sequence-exponential.htmlhttps://liberty.io/is-the-fibonacci-sequence-exponential.htmlI could not find any information on this online so I thought I'd make a question about this. If we take the Fibonacci sequence $F_n = F_{n-1} + F_{n-2}$, is this growing exponentially? Or perhaps ......Sat, 29 Aug 2026 05:22:51 -0400https://liberty.io/adapting-a-picture-proof-of-trig-039-s-angle-addition-identities-to-an.htmlhttps://liberty.io/adapting-a-picture-proof-of-trig-039-s-angle-addition-identities-to-an.htmlI am currently learning about the Trig addition formula. The book I am reading has a proof for it. The proof is attached here: However, I have a question about it. What if the values of theta and......Sat, 29 Aug 2026 05:21:35 -0400https://liberty.io/category-of-all-categories-vs-set-of-all-sets.htmlhttps://liberty.io/category-of-all-categories-vs-set-of-all-sets.htmlIn naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...Sat, 29 Aug 2026 05:21:14 -0400https://liberty.io/the-determinant-of-adjugate-matrix.htmlhttps://liberty.io/the-determinant-of-adjugate-matrix.htmlI have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1......Sat, 29 Aug 2026 05:13:24 -0400https://liberty.io/on-the-behavior-of-transcendental-functions.htmlhttps://liberty.io/on-the-behavior-of-transcendental-functions.htmlGiven that single-variable algebraic functions take on algebraic values when the input is algebraic, and take on transcendental values when the input is transcendental, and knowing that transcenden......Sat, 29 Aug 2026 05:12:42 -0400https://liberty.io/finding-bases-for-row-space-and-null-space-of-matrix.htmlhttps://liberty.io/finding-bases-for-row-space-and-null-space-of-matrix.htmlMy problem is: For the matrix $$A = \begin{bmatrix} 1& 4& 5& 6& 9\\ 3& −2& 1& 4& −1\\ −1& 0& −1& −2& −1\\ 2& 3&......Sat, 29 Aug 2026 05:00:32 -0400https://liberty.io/trouble-understanding-sum-of-subspaces.htmlhttps://liberty.io/trouble-understanding-sum-of-subspaces.htmlI started reviewing linear algebra, from a different textbook (Axler's), after taking a fast paced summer class. Unfortunately, I've become confused with a concept that is introduced at the end of ...Sat, 29 Aug 2026 04:52:37 -0400https://liberty.io/painting-cube-with-3-different-colours-closed.htmlhttps://liberty.io/painting-cube-with-3-different-colours-closed.htmlPainting cube with 3 colours (using each colour at least one time) Find no. of ways to colour the cube. (Rotation don't change the colouring.) Solve it for class 9 student in a easier way. Please...Sat, 29 Aug 2026 04:49:03 -0400https://liberty.io/what-is-cot-pi-2.htmlhttps://liberty.io/what-is-cot-pi-2.htmlBase on the unit circle, I know $ \begin{align} &\cot\left(\frac{\pi}{2}\right) \\ =&\frac{0}{1}\\ =&0 \end{align} $ But it is also $ \begin{align} &\cot\left(\frac{\pi}{2}\righ......Sat, 29 Aug 2026 04:40:37 -0400https://liberty.io/an-explanation-of-the-kalman-filter.htmlhttps://liberty.io/an-explanation-of-the-kalman-filter.htmlIn the past 3 months, I've been trying to understand the Kalman Filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, ...Sat, 29 Aug 2026 04:37:52 -0400https://liberty.io/how-to-factorize-polynomials-to-the-5th-degree.htmlhttps://liberty.io/how-to-factorize-polynomials-to-the-5th-degree.htmlI have the polynomial: $$2x^5-x^4+10x^3-5x^2+8x-4$$ and I know that the final result is: $$(2x-1)(x^4+5x^2+4) = (2x-1)(x^2+1)(x^2+4)$$ But how would you do it step by step? I've seen a couple of vi......Sat, 29 Aug 2026 04:33:59 -0400https://liberty.io/quick-way-to-find-eigenvalues-of-anti-diagonal-matrix.htmlhttps://liberty.io/quick-way-to-find-eigenvalues-of-anti-diagonal-matrix.htmlIf $A \in M_n(\mathbb{R})$ is an anti-diagonal $n \times n$ matrix, is there a quick way to find its eigenvalues in a way similar to finding the eigenvalues of a diagonal matrix? The standard way for ...Sat, 29 Aug 2026 04:26:48 -0400https://liberty.io/how-to-determine-if-a-large-number-is-prime.htmlhttps://liberty.io/how-to-determine-if-a-large-number-is-prime.htmlFor a very large number n, how many divisibility tests are required to establish if its prime? I know this has something to do with the Golden Number, but I can't figure out what. I did try search......Sat, 29 Aug 2026 04:24:37 -0400https://liberty.io/how-does-sin2x-cos2x-1-2-sin4x-closed.htmlhttps://liberty.io/how-does-sin2x-cos2x-1-2-sin4x-closed.htmlI can't figure out how to simplify $\sin2\theta \cos2\theta$ to be $\frac12 \sin4\theta $. I assume double angle identities are involved, but I can't get it and would like some help....Sat, 29 Aug 2026 04:06:52 -0400https://liberty.io/questions-about-trigonometry.htmlhttps://liberty.io/questions-about-trigonometry.htmlI just learned the very basics of Trigonometry yesterday. I can answer all the questions on the homework but feel uneasy because I still have some question about it. Here are the questions: What do......Sat, 29 Aug 2026 03:49:50 -0400https://liberty.io/combination-with-repetitions.htmlhttps://liberty.io/combination-with-repetitions.htmlThe formula for computing a k-combination with repetitions from n elements is: $$\binom{n + k - 1}{k} = \binom{n + k - 1}{n - 1}$$ I would like if someone can give me a simple basic proof that a ...Sat, 29 Aug 2026 03:47:49 -0400https://liberty.io/difference-between-show-and-prove.htmlhttps://liberty.io/difference-between-show-and-prove.htmlIn many mathematics problems you see the phrase "prove that..." or "show that..." something is. What's the difference between these two phrases? Is "showing" something different from "proving" some......Sat, 29 Aug 2026 03:25:09 -0400https://liberty.io/the-right-way-of-proving-lhs-rhs.htmlhttps://liberty.io/the-right-way-of-proving-lhs-rhs.htmlI'm currently brushing up on my math before heading back to school for my bachelor's and am practicing on some questions. Given the following equation : $$\frac{x^2 - 3x}{x^2 - 9} = 1-\frac{3x-9}......Sat, 29 Aug 2026 03:24:21 -0400https://liberty.io/what-does-with-respect-to-x-mean-when-integrating.htmlhttps://liberty.io/what-does-with-respect-to-x-mean-when-integrating.htmlWhen dealing with derivatives, "with respect to $x$" means we are observing how a small change in $x$ (the input) affects a change in $y$ (the output). I found this conceptualization very ...Sat, 29 Aug 2026 03:16:45 -0400https://liberty.io/how-to-find-a-normal-vector-on-a-surface-from-a-point-out-of-the-surfa.htmlhttps://liberty.io/how-to-find-a-normal-vector-on-a-surface-from-a-point-out-of-the-surfa.htmlFor example, if we consider a shape in 2D like a circle at center $(x_0,y_0)$ with radius R so the equation would be: $$f(x,y)=(x-x_0)^2+(y-y_0)^2-R^2$$ now the normal on the surface is $${\bf \h......Sat, 29 Aug 2026 03:15:35 -0400https://liberty.io/find-the-least-nonnegative-residue.htmlhttps://liberty.io/find-the-least-nonnegative-residue.htmlFind the least nonnegative residue of $5^{18} \mod 11$. To do this I took $5^2 \equiv 3 \mod 11$. Then I did $(5^2)^5 \equiv 3^5 \mod 11$. And $3^5 \equiv 1 \mod 11$. So now I have $5^{10} \equi......Sat, 29 Aug 2026 02:59:21 -0400https://liberty.io/determine-whether-the-following-statements-are-true-or-false-and-then-.htmlhttps://liberty.io/determine-whether-the-following-statements-are-true-or-false-and-then-.htmlDetermine whether statement is true or false, and then prove answer. For all integers $a,b,c$: a) if $a|bc$ and gcd$(a,b) = 1$ then $a|c$ b) if $a|c$ and $a|b$ and gcd$(a,b) = 1$ then $ab|c$. I've ...Sat, 29 Aug 2026 02:56:33 -0400https://liberty.io/finding-the-equation-for-a-line-tangent-to-a-parametric-curve.htmlhttps://liberty.io/finding-the-equation-for-a-line-tangent-to-a-parametric-curve.htmlI have the parametric equation $x = 2t - 1$ $y = 3t + 5$ $t = -1$ (defined as $t_0$) I am trying to find the line tangent to it. My book says if $x'(t_0) \not = 0$ then you can use the equa......Sat, 29 Aug 2026 02:40:03 -0400https://liberty.io/difference-between-horizontal-and-vertical-line-tests.htmlhttps://liberty.io/difference-between-horizontal-and-vertical-line-tests.htmlTrying to understand what the differnce is between a vertical and horizontal line test. If an equation fails the vertical line test, what does that tell you about the graph? If an equation fails the ...Sat, 29 Aug 2026 02:37:53 -0400https://liberty.io/integrating-sec-2-x-from-first-principles.htmlhttps://liberty.io/integrating-sec-2-x-from-first-principles.htmlIs it possible to solve $\int \sec^2 x ~dx$ without knowing that $\frac{d}{dx}\tan x = \sec^2 x$?...Sat, 29 Aug 2026 02:24:59 -0400https://liberty.io/preparations-for-reading-algebraic-number-theory-by-serge-lang.htmlhttps://liberty.io/preparations-for-reading-algebraic-number-theory-by-serge-lang.htmlI am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...Sat, 29 Aug 2026 02:24:02 -0400https://liberty.io/how-to-prove-the-limit-of-a-sequence-using-epsilon-n.htmlhttps://liberty.io/how-to-prove-the-limit-of-a-sequence-using-epsilon-n.htmlI think I have a proper understanding of the general procedure, but I'm having difficulty manipulating my inequality so that I can isolate $n$ by itself. Sadly I wasn't given many examples to model......Sat, 29 Aug 2026 02:17:56 -0400https://liberty.io/wolfram-alpha-how-to-show-functions-with-abs-and-sgn-as-a-piecewise-fu.htmlhttps://liberty.io/wolfram-alpha-how-to-show-functions-with-abs-and-sgn-as-a-piecewise-fu.htmlI was trying to obtain this result: integrate e^|x| dx The result is $$ \int e^{|x|} dx = \frac12 e^{-x} \left( (e^x-1)^2 \operatorname{sgn}(x) -2e^x +e^{2x} - 1 \right) \color{gray}{\,+ \text{...Sat, 29 Aug 2026 02:05:48 -0400https://liberty.io/a-nice-number-is-an-integer-ending-in-3-or-7-when-written-out-in-decim.htmlhttps://liberty.io/a-nice-number-is-an-integer-ending-in-3-or-7-when-written-out-in-decim.htmlMy teacher just asked me a question like this but i do not know how to start and work it out at all. Can someone help me out with that?...Sat, 29 Aug 2026 01:59:47 -0400https://liberty.io/what-is-meant-by-maximal-proper-factors-of-a-integer.htmlhttps://liberty.io/what-is-meant-by-maximal-proper-factors-of-a-integer.htmlI understand what is meant by proper factors, e.g. the proper factors of 36 are 2, 3, 4, 6, 9, 12, & 18. However I've just seen the phrase "maximal proper factors" used in the context of determ......Sat, 29 Aug 2026 01:49:15 -0400https://liberty.io/richardson-039-s-extrapolation-of-composite-trapezoidal-rule.htmlhttps://liberty.io/richardson-039-s-extrapolation-of-composite-trapezoidal-rule.htmlI have applied Richardson's formula to the composite trapezoidal rule, $I_h(f)=\frac{h}{2}(f(a)+\sum_{k=1}^{n-1}f(a+kh)+f(b))$, in an attempt to better approximate the integral $I(f)=\int_0^1 e^{-x......Sat, 29 Aug 2026 01:47:28 -0400https://liberty.io/the-decimal-expansion-of-the-quotient-of-two-integers.htmlhttps://liberty.io/the-decimal-expansion-of-the-quotient-of-two-integers.htmlIt is an exercise in a book on discrete mathematics.How to prove that in the decimal expansion of the quotient of two integers, eventually some block of digits repeats. For example: $\frac { 1 }{ 6......Sat, 29 Aug 2026 01:47:17 -0400https://liberty.io/how-to-know-spot-a-repeated-root-in-a-graph.htmlhttps://liberty.io/how-to-know-spot-a-repeated-root-in-a-graph.htmlThe diagram shows a cubic curve passing through (–1, 0), (2, 0) and (0, –8). What is the equation of the curve. So I would have said: $-8 = k(x + 1)(x-2)$ $k = 4$ $y = 4(x+1)(x-2)$ but the root at ......Sat, 29 Aug 2026 01:46:56 -0400https://liberty.io/clean-proof-of-baker-campbell-hausdorff-formula.htmlhttps://liberty.io/clean-proof-of-baker-campbell-hausdorff-formula.htmlI am thinking of the cleanest way to prove the BCH formula and I have come up with this. First, work out $e^{\lambda A}Be^{-\lambda A}$ by expanding the exponentials (sums go from $0$ to $\infty$):......Sat, 29 Aug 2026 01:46:40 -0400https://liberty.io/is-this-proof-of-0-a-0-correct.htmlhttps://liberty.io/is-this-proof-of-0-a-0-correct.htmlI need to prove that $0*a = a*0 = 0$ using only axiomatic properties for real numbers(It's a exercise from Apostol's book. My try: We can write $0$ as $(a - a)$. Hence we can rewrite the equation as ...Sat, 29 Aug 2026 01:46:04 -0400https://liberty.io/finding-radius-of-circle.htmlhttps://liberty.io/finding-radius-of-circle.htmlTaking the angle at center as $x$, I got $r \sin x =2$. But after that how to proceed?...Sat, 29 Aug 2026 01:42:04 -0400https://liberty.io/length-of-cross-product-vector.htmlhttps://liberty.io/length-of-cross-product-vector.htmlI have two vectors $u$, and $v$, I know that $\mid u \mid$ = 3 and $\mid v \mid$ = 5, and that $u\cdot v = -12$. I need to calculate the length of the vector $(3u+2v) \times (3v-u)$. Because I kno......Sat, 29 Aug 2026 01:34:09 -0400https://liberty.io/why-is-1-mod-3-2-closed.htmlhttps://liberty.io/why-is-1-mod-3-2-closed.htmlTitle says it all. Google doesn't have anything. Trying to get it to let me submit by typing more words. There is nothing really to be added but it wont let me submit without this....Sat, 29 Aug 2026 01:25:43 -0400https://liberty.io/what-is-an-intuitive-meaning-of-e-overline-x-and-var-overline-x.htmlhttps://liberty.io/what-is-an-intuitive-meaning-of-e-overline-x-and-var-overline-x.htmlLet $X$ be a random variable distributed over, for example say, the Binomial Distribution. Then $P(X)$ is the probability of getting $x$ successful trials in $n$ total trials. So I saw a notation......Sat, 29 Aug 2026 01:19:13 -0400https://liberty.io/solution-vector-equation-involving-cross-product.htmlhttps://liberty.io/solution-vector-equation-involving-cross-product.htmlI am starting to study linear algebra, and a problem appeared at the beginning of the textbook, the problem is the following: What 3-vector u satisfies $(1,1,0)\times u=(0,1,1)$. My immediate respo......Sat, 29 Aug 2026 01:12:45 -0400https://liberty.io/tail-of-a-divergent-p-series-with-p-1.htmlhttps://liberty.io/tail-of-a-divergent-p-series-with-p-1.htmlConsider a divergent $p$-series with $p<1$, for instance $\sum_{j=0}^{\infty} \frac{1}{\sqrt{j}}$. Let $R_{n}=\sum_{j=n}^{\infty} \frac{1}{\sqrt{j}}$ be the tail of this series. Obviously $R_n$ ...Sat, 29 Aug 2026 01:09:06 -0400https://liberty.io/riemann-sum-with-subintervals-of-unequal-width.htmlhttps://liberty.io/riemann-sum-with-subintervals-of-unequal-width.htmlAt the opening of a chapter on Riemann Sums and definite integrals, my book gives the following example problem: Consider the region bounded by the graph of $$f(x)=\sqrt{x}$$ and the x-axis for ......Sat, 29 Aug 2026 01:07:23 -0400https://liberty.io/average-value-using-triple-integrals.htmlhttps://liberty.io/average-value-using-triple-integrals.htmlFind the average value of the function $f(x,y,z)=xyz$ over the tetrahedron with vertices, $(0,0,0),(1,0,0), (1,1,0), \text{and }(1,1,1)$...Sat, 29 Aug 2026 01:02:24 -0400https://liberty.io/when-to-begin-a-new-paragraph-closed.htmlhttps://liberty.io/when-to-begin-a-new-paragraph-closed.htmlWhen writing a mathematical article, when should someone begin a new paragraph? Is there some specific rule or convention? And more generally, what rules are there about articles-writing?...Sat, 29 Aug 2026 00:59:35 -0400https://liberty.io/why-there-is-no-formula-log-a-log-b-something.htmlhttps://liberty.io/why-there-is-no-formula-log-a-log-b-something.htmlWhen I was studying in high school, my teacher taught me about log(a) + log(b) = log(ab) log(a) - log(b) = log(a/b) log(a) / log(b) = log$_b$(a) Then, I asked my teacher "Why there is no formul......Sat, 29 Aug 2026 00:53:36 -0400https://liberty.io/a-function-that-is-bounded-and-measurable-but-not-lebesgue-integrable.htmlhttps://liberty.io/a-function-that-is-bounded-and-measurable-but-not-lebesgue-integrable.htmlCould you give me concrete examples about "a function that is bounded and measurable but not Lebesgue integrable". Royden's textbook "Real analysis" says a bounded measurable function is said to be ...Sat, 29 Aug 2026 00:51:08 -0400https://liberty.io/method-for-finding-marginal-cdf.htmlhttps://liberty.io/method-for-finding-marginal-cdf.htmlI wish to calculate the marginal CDF of a joint probability distribution function. However, I am unsure of the bounds I am supposed to use, and wish to verify it. Suppose I have the expression: $$f......Sat, 29 Aug 2026 00:44:31 -0400https://liberty.io/are-matrices-rank-2-tensors.htmlhttps://liberty.io/are-matrices-rank-2-tensors.htmlI know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular......Sat, 29 Aug 2026 00:28:47 -0400https://liberty.io/how-do-you-simplify-cos-4-x-by-using-power-reducing-formulae-closed.htmlhttps://liberty.io/how-do-you-simplify-cos-4-x-by-using-power-reducing-formulae-closed.htmlHow do you simplify $\cos^4(x)$ using the power reducing formula?...Sat, 29 Aug 2026 00:20:02 -0400https://liberty.io/how-to-calculate-the-lower-and-upper-bound-of-the-error-in-estimating-.htmlhttps://liberty.io/how-to-calculate-the-lower-and-upper-bound-of-the-error-in-estimating-.htmlI have the following problem, I am trying to estimate the conversion rate of product sales online, my data is simple, I have the number of clicks and the number of sales for each product. Sales are ...Sat, 29 Aug 2026 00:06:20 -0400https://liberty.io/what-does-the-inverted-v-represent-in-math.htmlhttps://liberty.io/what-does-the-inverted-v-represent-in-math.htmlI know that A V B represents Logical disjunction which means A OR B and the result of it is false only when both A and B are false . But I still didn't understand what an inverted V means as shown ......Fri, 28 Aug 2026 23:47:44 -0400https://liberty.io/why-is-glide-symmetry-its-own-type.htmlhttps://liberty.io/why-is-glide-symmetry-its-own-type.htmlArtin's Algebra pages 155 & 156 list the types of symmetry of a plane figure as: Reflective Rotational Translational Glide He then goes on to say "Figures such as wallpaper patterns may have ......Fri, 28 Aug 2026 23:40:57 -0400https://liberty.io/what-are-collinear-lines.htmlhttps://liberty.io/what-are-collinear-lines.htmlI know about collinear points (points lying on a same straight line). I know about collinear vector (parallel vectors). But, coming to collinear lines, I am not even able to imagine them, and I can't ...Fri, 28 Aug 2026 23:40:08 -0400https://liberty.io/determine-whether-the-series-is-divergent-or-convergent-and-find-its-s.htmlhttps://liberty.io/determine-whether-the-series-is-divergent-or-convergent-and-find-its-s.htmlAny tips on how to start this would be great. I'm aware its not geometric but the answer key indicates it diverges, however I have no idea how to physically show that with this equation. $$\sum_{n=......Fri, 28 Aug 2026 23:38:16 -0400https://liberty.io/what-are-all-the-elements-of-the-group-of-symmetries-of-a-regular-tetr.htmlhttps://liberty.io/what-are-all-the-elements-of-the-group-of-symmetries-of-a-regular-tetr.htmlI can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on 'top' and we do to the other $3$ as we did in a r......Fri, 28 Aug 2026 23:37:32 -0400https://liberty.io/if-i-is-a-number-then-what-is-its-numerical-value.htmlhttps://liberty.io/if-i-is-a-number-then-what-is-its-numerical-value.html$ i $ is the unit imaginary part of complex number , but there is a question which it is mixed me probably i missed the definition of a number , wolfram alpha $ i $ is assumed to be a number , and ...Fri, 28 Aug 2026 23:29:49 -0400https://liberty.io/proofs-for-parallel-and-perpendicular-lines.htmlhttps://liberty.io/proofs-for-parallel-and-perpendicular-lines.htmlDoes anyone know a good proof to show that all lines parallel to a specific line AB have the same slope? Also, does anyone know a good proof to show that any line perpendicular to a specific line C......Fri, 28 Aug 2026 23:26:57 -0400https://liberty.io/find-the-relative-maximum-and-relative-minimum-of-the-function-f-x-49x.htmlhttps://liberty.io/find-the-relative-maximum-and-relative-minimum-of-the-function-f-x-49x.htmlFind the relative maximum and relative minimum of the function $f(x) = 49x + \frac{4}{x}=49x+4x^{-1}$ Solution: Step 1: Find the values of $x$ where $f'(x)=0$ and $f'(x)$ DNE. $f'(x)=49-4x^{-2}=......Fri, 28 Aug 2026 23:16:34 -0400https://liberty.io/numerically-solving-a-poisson-equation-with-neumann-boundary-condition.htmlhttps://liberty.io/numerically-solving-a-poisson-equation-with-neumann-boundary-condition.htmlThe Problem Suppose I have an equation of the form $\nabla^2 \phi(x) = f(x)$ on the interval $A \le x \le B$, where $f(x)$ is known and $\phi(x)$ is unknown. I have Neumann-type boundary condition......Fri, 28 Aug 2026 23:13:52 -0400https://liberty.io/understanding-how-to-prove-when-a-subset-is-a-subgroup.htmlhttps://liberty.io/understanding-how-to-prove-when-a-subset-is-a-subgroup.htmlLemma 3.4. Let $(G ,*)$ be a group. A nonempty subset $H$ of $G$ is a subgroup of $(G,*)$, iff, for every $a, b\in H$, $a*b^{-1}\in H$. Proof. First, suppose $H$ is a subgroup. If $b\in H$, then $b......Fri, 28 Aug 2026 23:04:53 -0400https://liberty.io/graph-isomorphisms-on-6-vertices-with-degree-3.htmlhttps://liberty.io/graph-isomorphisms-on-6-vertices-with-degree-3.htmlI want to find another graph that has 6 vertices and each has degree $3$ that is not isomorphic to these two graphs below. I know that these two graphs are isomorphic. They will all have the same d......Fri, 28 Aug 2026 22:54:33 -0400https://liberty.io/necessary-to-prove-the-inverse-is-invertible.htmlhttps://liberty.io/necessary-to-prove-the-inverse-is-invertible.htmlI am just starting out on linear algebra and I have come to a section the book that confuses me somewhat. The authour defines an invertible matrix A as: "A square matrix A is said to be invertible......Fri, 28 Aug 2026 22:53:13 -0400https://liberty.io/what-039-s-the-proof-that-the-euler-totient-function-is-multiplicative.htmlhttps://liberty.io/what-039-s-the-proof-that-the-euler-totient-function-is-multiplicative.htmlThat is, why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel ......Fri, 28 Aug 2026 22:47:34 -0400https://liberty.io/maximum-modulus-principle-and-open-mapping-theorem.htmlhttps://liberty.io/maximum-modulus-principle-and-open-mapping-theorem.htmlThe Maximum Modulus Principal states: let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_{0}$ is a point in $D$ s......Fri, 28 Aug 2026 22:31:18 -0400https://liberty.io/pursuit-curves-solution.htmlhttps://liberty.io/pursuit-curves-solution.htmlFor our math class we have to do some calculations with respect to pursuit curves. The chased object starts at point $(p,0)$. Chaser starts at $(0,0)$.(x,y) Speed of the chased object is $u$. Speed ...Fri, 28 Aug 2026 22:27:27 -0400https://liberty.io/find-the-limit-as-x-approaches-infinity-of-e-x-lnx-and-sin2x-x.htmlhttps://liberty.io/find-the-limit-as-x-approaches-infinity-of-e-x-lnx-and-sin2x-x.htmlI'm not sure how to evaluate the limit as $x \rightarrow \infty $ of the following: $$ \bullet \lim_{x\to\infty}e^{-x}\cdot \log x $$ $$\bullet\lim_{x\to\infty} \frac{\sin(2x)}{x}$$...Fri, 28 Aug 2026 22:22:40 -0400https://liberty.io/what-are-the-ways-of-proving-that-the-cantor-set-is-uncountable-apart-.htmlhttps://liberty.io/what-are-the-ways-of-proving-that-the-cantor-set-is-uncountable-apart-.htmlWhat are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization? Are there any based on dynamical systems?...Fri, 28 Aug 2026 22:19:54 -0400https://liberty.io/limit-of-sin-x-as-x-approaches-zero-from-the-left.htmlhttps://liberty.io/limit-of-sin-x-as-x-approaches-zero-from-the-left.htmlI'm trying to find a proof that: $$ \lim_{x\to0^-}\sin(x)=0$$ I'd like to be able to do the proof without reference to advanced theorems (mean value theorem, series, etc). I have a geometric approa......Fri, 28 Aug 2026 22:14:14 -0400https://liberty.io/what-is-the-difference-between-square-of-sum-and-sum-of-square.htmlhttps://liberty.io/what-is-the-difference-between-square-of-sum-and-sum-of-square.htmlWhat is difference between square of sum $(\sum_{i=1}^{n}x_i)^2$ and sum of square $\sum_{i=1}^{n}x_i^2$? I think square of sum is bigger than sum of square but i can not find a relation between t......Fri, 28 Aug 2026 22:13:48 -0400https://liberty.io/on-laplace-transform-of-periodic-functions.htmlhttps://liberty.io/on-laplace-transform-of-periodic-functions.htmlI recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ......Fri, 28 Aug 2026 22:08:09 -0400https://liberty.io/least-squares-estimate-b1-formula.htmlhttps://liberty.io/least-squares-estimate-b1-formula.htmlMy regression textbook textbook says that $\sum_{i=1}^{n} (x_i - \overline{x}) = 0$. I know that: $\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})$ $= \sum_{i=1}^{n} (x_i - \overline......Fri, 28 Aug 2026 21:58:20 -0400https://liberty.io/linear-ode-roots-of-characteristic-equation-having-multiplicity-1.htmlhttps://liberty.io/linear-ode-roots-of-characteristic-equation-having-multiplicity-1.htmlI know the method. For a linear homogeneous ordinary differential equation with a root of the characteristic polynomial in $\alpha$ has multiplicity $k$, then $y=x^me^{\alpha x}$ with $m=0,1,\cdot......Fri, 28 Aug 2026 21:55:12 -0400https://liberty.io/explicit-formula-for-floor-x.htmlhttps://liberty.io/explicit-formula-for-floor-x.htmlIn number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. (second Chebyshev ...Fri, 28 Aug 2026 21:51:07 -0400https://liberty.io/how-to-rotate-a-polar-equation.htmlhttps://liberty.io/how-to-rotate-a-polar-equation.htmlTake a simple polar equation like r = θ/2 that graphs out to: But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is there a way to easily shift the plot?...Fri, 28 Aug 2026 21:47:34 -0400https://liberty.io/finding-eigenpairs-of-matrix-a.htmlhttps://liberty.io/finding-eigenpairs-of-matrix-a.htmlHow do you find the eigenpairs of this? Given two matrices $A=\begin{bmatrix} a& b\\b&-a\end{bmatrix}$ and $B=\begin{bmatrix} a& -b\\b&a\end{bmatrix}$ where $b \neq 0$, find all ...Fri, 28 Aug 2026 21:44:19 -0400https://liberty.io/if-a-is-not-relatively-prime-to-n-prove-modulo-property.htmlhttps://liberty.io/if-a-is-not-relatively-prime-to-n-prove-modulo-property.htmlIf $n>1$ is integer and $1\le a \le n$ is integer such that $(a,n)\neq 1$ then prove there exist integer $1 \le b <n $ such that $ab \equiv 0(mod \; n)$ I have tried everything from going to ...Fri, 28 Aug 2026 21:42:37 -0400https://liberty.io/expression-for-the-current-as-a-function-of-time.htmlhttps://liberty.io/expression-for-the-current-as-a-function-of-time.htmlWhen the electromotive force (emf) is removed from a circuit containing inductance and resistance but no capacitors, the rate of decrease of current is proportional to the current. If the initial ...Fri, 28 Aug 2026 21:33:31 -0400https://liberty.io/evaluate-double-integral-of-triangular-region.htmlhttps://liberty.io/evaluate-double-integral-of-triangular-region.htmlEvaluate $$ \int\int_D e^{y^2} dA $$ where D is the triangular region with vertices (0,0), (0,1) and (2,1) My attempts: $$ \int^{2}_0 \int^{1}_\frac{x}{2} e^{y^2} dy dx $$ or $$ \int^{0}_1\int^{......Fri, 28 Aug 2026 21:32:22 -0400https://liberty.io/for-which-values-of-b-c-is-the-matrix-invertible.htmlhttps://liberty.io/for-which-values-of-b-c-is-the-matrix-invertible.html$A =\left(\begin{array}{ccc} 0 & 1 & b \\ -1 & 0 & c \\ -b & -c & 0 \end{array}\right)$ I tried to come up with the RREF($A$), but in the final step I have the matrix: $A ......Fri, 28 Aug 2026 21:27:37 -0400https://liberty.io/is-there-any-deep-reason-that-23456789-is-prime-closed.htmlhttps://liberty.io/is-there-any-deep-reason-that-23456789-is-prime-closed.htmlI was recently coding a prime factorization function and wanted to test that it could factor large numbers reasonably well. Arbitrarily, I slid my finger across the number pad, and got 23456789 as ......Fri, 28 Aug 2026 21:26:47 -0400https://liberty.io/recurrent-and-transient-states.htmlhttps://liberty.io/recurrent-and-transient-states.htmlI know which are the recurrent states {2,4,5} and It's clear to see it drawing a diagram but how can I prove that {2,4,5} are the recurrent states? In a formal way? Also the transient states are ......Fri, 28 Aug 2026 21:22:08 -0400https://liberty.io/ambiguity-in-rule-for-determining-an-outlier.htmlhttps://liberty.io/ambiguity-in-rule-for-determining-an-outlier.htmlI have the following set of numbers, in which I need to determine if there are any outliers: 11, 14, 21, 26, 29, 33, 61. I graphed them in Desmos (see above), and visually, it appears to me that 6......Fri, 28 Aug 2026 21:12:57 -0400https://liberty.io/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlhttps://liberty.io/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlUsing the method of characteristics, find a solution to Burgers' equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g......Fri, 28 Aug 2026 20:57:09 -0400https://liberty.io/computing-the-iterated-logarithm-log-star-by-hand.htmlhttps://liberty.io/computing-the-iterated-logarithm-log-star-by-hand.htmlI'm having trouble figuring out how to compute the iterated logarithm (log-star) by hand without using a calculator or programming language. I wrote out the following program in Java to check my an......Fri, 28 Aug 2026 20:54:31 -0400https://liberty.io/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlhttps://liberty.io/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlSometimes in exercises we are asked to calculate the fourier series of a function. But there are two ways to do that. If $f:\Bbb R\mapsto\Bbb R$ is $T$-periodic over $\Bbb R$ then what conditions......Fri, 28 Aug 2026 20:50:02 -0400https://liberty.io/why-does-two-terms-immediately-adjacent-mean-multiply.htmlhttps://liberty.io/why-does-two-terms-immediately-adjacent-mean-multiply.htmlI am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication? I understand the rule that two ...Fri, 28 Aug 2026 20:40:35 -0400https://liberty.io/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlhttps://liberty.io/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlThe fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are XOR......Fri, 28 Aug 2026 20:18:57 -0400https://liberty.io/remainder-when-2-108-is-divided-by-11-closed.htmlhttps://liberty.io/remainder-when-2-108-is-divided-by-11-closed.htmlWhat is the remainder obtained when $2^{108}$ is divided by $11$? I tried bringing in $11$ in the given no. such as in $(11-3)^{36}$ and then using binomial expansion...but its not helping. Any ...Fri, 28 Aug 2026 20:08:18 -0400https://liberty.io/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlhttps://liberty.io/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlI am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...Fri, 28 Aug 2026 20:04:37 -0400https://liberty.io/relationship-between-groups-and-permutations.htmlhttps://liberty.io/relationship-between-groups-and-permutations.htmlWhat is the relationship between groups and permutations? The way I understand it, a permutation is a set of numbers that you can do things with. A group is a set of elements equipped with an opera......Fri, 28 Aug 2026 20:03:35 -0400