https://liberty.ioCurated star highlights with fresh energy.en-usSun, 16 Aug 2026 19:20:49 -0400https://liberty.io/is-it-possible-to-draw-this-picture-without-lifting-the-pen.htmlhttps://liberty.io/is-it-possible-to-draw-this-picture-without-lifting-the-pen.htmlSome days ago, our math teacher said that he would give a good grade to the first one that will manage to draw this: To draw this without lifting the pen and without tracing the same line more tha......Sun, 16 Aug 2026 23:20:02 -0400https://liberty.io/what-is-a-regular-submanifold.htmlhttps://liberty.io/what-is-a-regular-submanifold.htmlDefinition: Let $M$ be a non-empty subset of $\mathbb{R}^n$ and consider $m\in\mathbb{N}$ such that $1\leqslant m\leqslant n$. M is named a regular sub manifold of $\mathbb{R}^n$ of dimension $m$ ......Sun, 16 Aug 2026 23:17:14 -0400https://liberty.io/calculating-a-limit-with-floor-function.htmlhttps://liberty.io/calculating-a-limit-with-floor-function.htmlCan someone help me understand how to calculate the limit: $\lim_{n \to \infty} n [\sqrt{n+4}-\sqrt{n} ] $ ? How can I det rid the floor function ?! (Multiplying by $ \sqrt{n+4}+\sqrt{n} $ gives me ...Sun, 16 Aug 2026 23:16:26 -0400https://liberty.io/method-of-least-squares-why-is-it-preferred-duplicate.htmlhttps://liberty.io/method-of-least-squares-why-is-it-preferred-duplicate.htmlPossible Duplicate: Why do we use a Least Squares fit? To find the normal equations for derivation of the regression line we use the method of least squares, We want to make the error smallest......Sun, 16 Aug 2026 23:15:38 -0400https://liberty.io/why-multiplication-of-zero-by-any-number-is-zero-closed.htmlhttps://liberty.io/why-multiplication-of-zero-by-any-number-is-zero-closed.htmlIf I multiply $a$ by $0$, it means $a$ is added $0$ times, but I am not able to visualize $0$ times!...Sun, 16 Aug 2026 23:13:33 -0400https://liberty.io/plotting-f-x-sin-x-cos-x-by-converting-it-to-another-form.htmlhttps://liberty.io/plotting-f-x-sin-x-cos-x-by-converting-it-to-another-form.htmlAre there any trigonometric identity that can make $f(x) = \sin(x)+\cos(x)$ easier to plot? I have no idea how it becomes a sin graph shape in the end....Sun, 16 Aug 2026 23:06:14 -0400https://liberty.io/questions-tagged-multivariable-calculus.htmlhttps://liberty.io/questions-tagged-multivariable-calculus.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 16 Aug 2026 23:02:51 -0400https://liberty.io/tangent-the-trig-ratio-vs-tangent-the-geometric-concept.htmlhttps://liberty.io/tangent-the-trig-ratio-vs-tangent-the-geometric-concept.htmlDo the terms "tangent" (the trigonometric ratio), which is defined as "opposite-side / adjacent-side", and "tangent" (of the curve), which is a line that touches a curve at a point, have the SAME m......Sun, 16 Aug 2026 23:02:35 -0400https://liberty.io/ratio-test-vs-cauchy-hadamard.htmlhttps://liberty.io/ratio-test-vs-cauchy-hadamard.htmlI'm sorry if this had been asked before, but there are many other questions that make it difficult to find an answer to my own. I know that the radius of convergence of a power series can be deter......Sun, 16 Aug 2026 22:56:22 -0400https://liberty.io/how-to-explain-if-a-quantity-is-a-pivot-quantity.htmlhttps://liberty.io/how-to-explain-if-a-quantity-is-a-pivot-quantity.htmlfrom what i have gathered . I know that pivots are functions that give approximate or exact assumption of CI . I'm not entirely sure what is the quantity the question is referring to. does the quan......Sun, 16 Aug 2026 22:56:05 -0400https://liberty.io/classificating-the-isometries-of-the-hyperbolic-plane.htmlhttps://liberty.io/classificating-the-isometries-of-the-hyperbolic-plane.htmlI just did the classification of the isometries of the Euclidean Plane using some results of linear algebra and euclidean geometry, guiding myself through this article: http://repositorio.ul.pt/...Sun, 16 Aug 2026 22:47:47 -0400https://liberty.io/find-the-equation-of-the-cone-z-sqrt-x-2-y-2-in-spherical-coordinates.htmlhttps://liberty.io/find-the-equation-of-the-cone-z-sqrt-x-2-y-2-in-spherical-coordinates.htmlI have the following... $$z=\sqrt{x^2+y^2}$$ I need to write this as an equation in spherical coordinates. I know that $p^2 = x^2+y^2+z^2$ and that... $$x = p\sin\phi \cos\theta$$ $$y=p\sin\phi......Sun, 16 Aug 2026 22:46:36 -0400https://liberty.io/equivalence-between-middle-excluded-law-and-double-negation-eliminatio.htmlhttps://liberty.io/equivalence-between-middle-excluded-law-and-double-negation-eliminatio.htmlIt's well-know that in intuitionistic logic, middle excluded law and double negation elimination are equivalent. For example, in Johnstone - Topo theory, I read that, in a Heyting algebra, $p\vee\......Sun, 16 Aug 2026 22:46:01 -0400https://liberty.io/show-v-x-0-is-the-only-solution-for-y-039-039-cy-0-v-0-0-v-039-0-0-c-g.htmlhttps://liberty.io/show-v-x-0-is-the-only-solution-for-y-039-039-cy-0-v-0-0-v-039-0-0-c-g.htmlShow $v(x) = 0$ is the only solution for $y'' + cy=0, v(0) = 0, v'(0)=0, c\gt 0$ I have this question as the only practice problem I can't do for my exam in thirty minutes. I can see that it is t......Sun, 16 Aug 2026 22:17:30 -0400https://liberty.io/the-pattern-in-mathematical-induction-proofs.htmlhttps://liberty.io/the-pattern-in-mathematical-induction-proofs.htmlWhen given a statement to be proven by mathmatical induction the statement tends to look like this $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ so going about the proof. 1) Prove the base case $\......Sun, 16 Aug 2026 22:12:24 -0400https://liberty.io/for-compass-and-straightedge-problems-are-you-allowed-to-use-the-compa.htmlhttps://liberty.io/for-compass-and-straightedge-problems-are-you-allowed-to-use-the-compa.htmlFor compass and straightedge problems, you could have a line between two points A and B, and want to make a line the same size between C and line DE. If you placed the two points of the compass be......Sun, 16 Aug 2026 22:09:36 -0400https://liberty.io/principal-value-of-complex-number.htmlhttps://liberty.io/principal-value-of-complex-number.htmlLet $z=-1-i$, find the principal value ? Here $x=-1,y=-1$ therefore $\arg(z)=\tan \alpha=|\frac{y}{x}|=|\frac{-1}{-1}|$ Therefore, $\alpha =\tan^{-1}=\frac{\pi}{4}$ which lies between $0$ and $\f......Sun, 16 Aug 2026 22:06:15 -0400https://liberty.io/regular-and-irregular-points.htmlhttps://liberty.io/regular-and-irregular-points.htmlIn the following 2 examples, I am trying to find all the singular points of the given equations and determine whether each one is regular or irregular I can determine the singular points of the ...Sun, 16 Aug 2026 22:04:55 -0400https://liberty.io/do-carmo-differential-geometry-exercise-4-4-17.htmlhttps://liberty.io/do-carmo-differential-geometry-exercise-4-4-17.htmlLet $\alpha:I\to\Bbb R^3$ be a curve parametrized by arc length $s$, with nonzero curvature and torsion. Consider the parametrized surface $$\textbf{x}(s,v) = \alpha(s)+vb(s),\ s\in I,-\epsilon<......Sun, 16 Aug 2026 22:03:26 -0400https://liberty.io/why-do-we-need-the-commutative-axiom-of-multiplication.htmlhttps://liberty.io/why-do-we-need-the-commutative-axiom-of-multiplication.htmlI think I can prove that $ab = ba$ by other axioms. Am I wrong? Why? Edit: proof updated with notes showing where I actually used the commutative property without noticing. Proof. For any numb......Sun, 16 Aug 2026 22:01:34 -0400https://liberty.io/how-many-balls-does-he-put-into-each-green-bag.htmlhttps://liberty.io/how-many-balls-does-he-put-into-each-green-bag.htmlThis is a homework problem my son brought home -- 5th grade. Raul has 56 bouncy balls. He puts three times as many balls into red gift bags as he puts into green gift bags. If he puts the same ...Sun, 16 Aug 2026 21:54:06 -0400https://liberty.io/find-all-the-primitive-roots-of-13.htmlhttps://liberty.io/find-all-the-primitive-roots-of-13.htmlFind all the primitive roots of $13$ My attempt: Since that $13$ is a prime I need to look for $g$ such that $g^{13-1}\equiv 1\pmod{13}$ There are $\phi(12)=4$ classes modulo $12$ how can I fin......Sun, 16 Aug 2026 21:48:41 -0400https://liberty.io/integration-shortcut-to-avoid-tedious-integration-by-parts-closed.htmlhttps://liberty.io/integration-shortcut-to-avoid-tedious-integration-by-parts-closed.htmlIn addition to the common integration shortcut of $\int x \cdot e^{ax}=\frac{xe^{ax}}{a}-\frac {e^{ax}}{a^2}+c$ Is there supplemental shortcut that can help me to avoid tedious repetitive integr......Sun, 16 Aug 2026 21:36:34 -0400https://liberty.io/why-do-graphing-calculators-plot-this-function-wrong-wolfram-mathemati.htmlhttps://liberty.io/why-do-graphing-calculators-plot-this-function-wrong-wolfram-mathemati.htmlWrong graph of the function f(x)=-1^x The function in question is $f(x)=-1^x$ and the plot example is from Wolfram Mathematica. The graph is supposed to show a zig-zag otherwise known as the trian......Sun, 16 Aug 2026 21:24:34 -0400https://liberty.io/good-function-039-s.htmlhttps://liberty.io/good-function-039-s.htmlI'm trying to solve the following question: Let $(X,d)$ be a metric space. We call a continuous function $f:X\to \mathbb R$ "good function" if for every continuous function $g:X\to \mathbb R$,......Sun, 16 Aug 2026 21:19:06 -0400https://liberty.io/what-is-the-growth-rate-and-continuous-growth-rate.htmlhttps://liberty.io/what-is-the-growth-rate-and-continuous-growth-rate.htmlThe problem: $\;f(t) = 4 \cdot 2^{\,t/5}$ I know continuous is e^k, but this problem doesn't seem to work for that. Is the initial value, 4, able to be used to find the growth rates?...Sun, 16 Aug 2026 21:11:29 -0400https://liberty.io/using-logarithmic-differentiation-with-trig-functions.htmlhttps://liberty.io/using-logarithmic-differentiation-with-trig-functions.htmlMy first intuition to derive the following function - $f(x) = \sec(x^x)$ - was to take the natural log of both sides and simplify to: $\ln(f(x)) = x(\ln(\sec(x))$. However, this does not lead me to......Sun, 16 Aug 2026 20:47:36 -0400https://liberty.io/how-to-get-all-the-factors-of-a-number-using-its-prime-factorization.htmlhttps://liberty.io/how-to-get-all-the-factors-of-a-number-using-its-prime-factorization.htmlFor example, I have the number $420$. This can be broken down into its prime factorization of $$2^2 \times3^1\times5^1\times7^1 = 420 $$ Using $$\prod_{i=1}^r (a_r + 1)$$ where $a$ is the magnitud......Sun, 16 Aug 2026 20:36:49 -0400https://liberty.io/should-this-definition-of-set-equality-be-an-axiom.htmlhttps://liberty.io/should-this-definition-of-set-equality-be-an-axiom.htmlThis question is about set equality, as it is presented in Chapter 3, Set Theory, Section 1, Fundamentals, of Terence Tao's textbook, Analysis I. However, I think it is prudent to begin earlier (...Sun, 16 Aug 2026 20:23:11 -0400https://liberty.io/does-the-series-sum-n-1-infty-1-n-ln-n-converge-or-diverge.htmlhttps://liberty.io/does-the-series-sum-n-1-infty-1-n-ln-n-converge-or-diverge.htmlSuppose you have the series: $$\sum_{n=1}^\infty (-1)^n\ln(n)$$ Does it converge or diverge? You cannot apply the alternating series test since $b_n$ is not decreasing. Similarly, you cannot apply ......Sun, 16 Aug 2026 20:18:44 -0400https://liberty.io/existence-and-uniqueness-theorems-for-first-order-ode-039-s.htmlhttps://liberty.io/existence-and-uniqueness-theorems-for-first-order-ode-039-s.htmlThere is a theorem regarding the existence and uniqueness of solutions to first-order ODE's: Thm.: Consider the initial value problem $y'=f(t,y)$, $y(t_0)=y_0$. Suppose $f$ and $\frac{∂f}{∂y}......Sun, 16 Aug 2026 20:04:46 -0400https://liberty.io/what-is-the-formula-for-finding-the-summation-of-an-exponential-functi.htmlhttps://liberty.io/what-is-the-formula-for-finding-the-summation-of-an-exponential-functi.htmlI am having a hard time researching how to handle summations of functions with exponential growth or decay. I know that simple summations can be calculated as follows: $$\sum_{i=1}^{n} i = \frac{n(......Sun, 16 Aug 2026 20:03:50 -0400https://liberty.io/a-pattern-appearing-in-the-powers-of-phi.htmlhttps://liberty.io/a-pattern-appearing-in-the-powers-of-phi.html\begin{align} \phi^5 &= 11,\underline{0}901699\cdots\\ \phi^6 &= 17,\underline{9}44271\cdots\\ \phi^7 &= 29,\underline{6}34441\cdots\\ \phi^8 &= 46,\underline{9}7871\cdots\\ \phi^9 ......Sun, 16 Aug 2026 20:00:37 -0400https://liberty.io/surface-integral-first-octant.htmlhttps://liberty.io/surface-integral-first-octant.htmlI have to evaluate $$\iint_TxdS$$ Where $T$ is the part of the sphere $x^2+y^2+z^2=a^2$ which lies in the first octant $x,y,z\geq0$. So I used polar coordinates with $dS=rdrd\theta$ where $0\leq r......Sun, 16 Aug 2026 19:53:56 -0400https://liberty.io/the-autocovariance-function-of-arma-1-1.htmlhttps://liberty.io/the-autocovariance-function-of-arma-1-1.htmlSo I am reading Brockwell and Davis introduction to Time Series analysis on page 89 where he derives the ACVF of an $ARMA(1,1)$ given by: $X_t - \phi X_{t-1}=Z_t+\theta Z_{t-1}$ with ${Z_t}$ is $W......Sun, 16 Aug 2026 19:50:43 -0400https://liberty.io/why-is-the-frobenius-norm-energy-of-a-rectangular-matrix-a-equal-to-th.htmlhttps://liberty.io/why-is-the-frobenius-norm-energy-of-a-rectangular-matrix-a-equal-to-th.htmlReading through a book, it is mentioned that $||{A}||^2_F=\text{Energy}(A)=\text{tr}(AA^T)=\text{tr}(A^TA)$. I understand that for square matrices the square Frobenius norm would be the the squared......Sun, 16 Aug 2026 19:46:43 -0400https://liberty.io/proving-that-a-galois-group-gal-e-q-is-isomorphic-to-mathbb-f-p-times.htmlhttps://liberty.io/proving-that-a-galois-group-gal-e-q-is-isomorphic-to-mathbb-f-p-times.htmlI have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group $Gal(E:Q)\s......Sun, 16 Aug 2026 19:42:52 -0400https://liberty.io/singular-value-decomposition-of-product-of-matrices.htmlhttps://liberty.io/singular-value-decomposition-of-product-of-matrices.htmlGiven SVD(A) and SVD(B) and B is a diagonal matrix, is there a way or method to construct SVD(AB) ?...Sun, 16 Aug 2026 19:28:29 -0400https://liberty.io/find-the-volume-of-the-solid-above-the-cone-z-sqrt-x-2-y-2-and-below-t.htmlhttps://liberty.io/find-the-volume-of-the-solid-above-the-cone-z-sqrt-x-2-y-2-and-below-t.htmlI actually have found the solution using double integral in polar coordinate. However, I am curious about whether I could find the same exact solution using triple integral in spherical coordinates......Sun, 16 Aug 2026 19:27:23 -0400https://liberty.io/how-to-raise-a-fraction-to-a-fractional-exponent.htmlhttps://liberty.io/how-to-raise-a-fraction-to-a-fractional-exponent.htmlI am trying to simplify this but I'm not sure how to approach it....Sun, 16 Aug 2026 19:19:13 -0400https://liberty.io/distance-point-on-ellipse-to-centre.htmlhttps://liberty.io/distance-point-on-ellipse-to-centre.htmlI'm trying to calculate the distance of a certain point of an ellipse to the centre of that ellipse: The blue things are known: The lengths of the horizontal major radius and vertical minor radius......Sun, 16 Aug 2026 19:17:11 -0400https://liberty.io/confusing-qualifiers-in-logic.htmlhttps://liberty.io/confusing-qualifiers-in-logic.htmlSuppose we know that there exists widgets. For any widget $X$, we have that the statements $A(X)$ and $B(X)$. Suppose it's true that for any $X$, $A(X) \not \Rightarrow B(X)$. How do I interpret th......Sun, 16 Aug 2026 19:13:32 -0400https://liberty.io/given-the-determinant-of-a-2-times-2-matrix-calculate-the-determinant-.htmlhttps://liberty.io/given-the-determinant-of-a-2-times-2-matrix-calculate-the-determinant-.htmlIf $$\det \underbrace{\begin{bmatrix} a&b\\ c&d\end{bmatrix}}_{=: A} = -3$$ calculate the determinant $$\det \underbrace{\begin{bmatrix} 2&-2&0\\ c+1&-1&2a\\ d-2&2&2......Sun, 16 Aug 2026 19:12:17 -0400https://liberty.io/what-point-on-the-line-y-2x-4-is-closest-to-the-origin.htmlhttps://liberty.io/what-point-on-the-line-y-2x-4-is-closest-to-the-origin.htmlWhat point on the line $y=2x+4$ is closest to the origin in $\mathbb{R}^2$? What is the distance from this point to the origin? I know the point closest to the origin is perpendicular to the origin, ...Sun, 16 Aug 2026 19:06:30 -0400https://liberty.io/level-curve-of-function-that-passes-through-given-point.htmlhttps://liberty.io/level-curve-of-function-that-passes-through-given-point.htmlI have a function $f(x,y)= 16-x^2-y^2$ that passes through the point $(2\sqrt(2),\sqrt(2))$ . How do I find out the equation of level curve? I dont have any idea . Pls provide hints. What I am do......Sun, 16 Aug 2026 19:05:17 -0400https://liberty.io/finding-the-matrix-associated-with-a-linear-map.htmlhttps://liberty.io/finding-the-matrix-associated-with-a-linear-map.htmlFind the matrix associated with the linear map $f:R^2 \rightarrow R^2$ defined by $f(x,y)=(3x-y,y-x)$ with respect to the ordered basis ${(1,0),(1,1)}$ Let the matrix be $A$ and let $f(x)=AX$ wher......Sun, 16 Aug 2026 19:01:43 -0400https://liberty.io/add-1100-0110-in-binary.htmlhttps://liberty.io/add-1100-0110-in-binary.htmlI'm studying encryption in class and we're doing "One-Time-Pad" encryption and I've come across this situation. I want to code $1100$, where my key is $0110$). In my book it says that the cipher is......Sun, 16 Aug 2026 18:55:49 -0400https://liberty.io/derivative-of-definite-integral.htmlhttps://liberty.io/derivative-of-definite-integral.htmlI am learning Calculus online and this problem stumped me. The solution doesn't really make sense either. I'm starting to think that the user made a typo. Here is the question to simplify: $$\f......Sun, 16 Aug 2026 18:53:10 -0400https://liberty.io/how-to-determine-the-values-of-the-coefficient-c-in-the-objective-func.htmlhttps://liberty.io/how-to-determine-the-values-of-the-coefficient-c-in-the-objective-func.htmlLet's say we have the following optimization problem : Max cx Subject to $Ax=b$ with $x\geq0$ How can we find the interval of the values for the coefficient $c_{j}$ of the objective where the o......Sun, 16 Aug 2026 18:50:20 -0400https://liberty.io/how-many-positive-integers-less-than-1000-are-there-which-contain-at-l.htmlhttps://liberty.io/how-many-positive-integers-less-than-1000-are-there-which-contain-at-l.htmlHow many positive integers less than 1000 are there which contain at least one 4 or at least one 9 (or both)? Obviously theres 100 in 400's and 900's but short of counting every other number whats......Sun, 16 Aug 2026 18:42:52 -0400https://liberty.io/integration-and-differentiation-of-fourier-series.htmlhttps://liberty.io/integration-and-differentiation-of-fourier-series.htmlI am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose th......Sun, 16 Aug 2026 18:41:03 -0400https://liberty.io/what-is-the-modulus-of-a-number.htmlhttps://liberty.io/what-is-the-modulus-of-a-number.htmlWhat is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb C,|z|=OM=\sqrt{a^2......Sun, 16 Aug 2026 18:33:00 -0400https://liberty.io/number-of-possible-combinations-in-which-ordering-doesn-039-t-matter.htmlhttps://liberty.io/number-of-possible-combinations-in-which-ordering-doesn-039-t-matter.htmlI tried to figure this out the other day, but failed to arrive at a formula. It seems like a simple problem, though. Forgive me if my use of terminology sounds off, I have never had any formal ...Sun, 16 Aug 2026 18:23:17 -0400https://liberty.io/complex-number-how-to-find-the-angle-between-the-imaginary-axis-and-re.htmlhttps://liberty.io/complex-number-how-to-find-the-angle-between-the-imaginary-axis-and-re.htmlAssume I have complex number $z = a + ib$. $z$ can be represented by a polar representation as $r(\cos \theta+i\sin \theta)$, when $r$ is the absolute value of $z$, $\sqrt{a^2 + b^2}$. But how......Sun, 16 Aug 2026 18:14:51 -0400https://liberty.io/can-you-find-an-equation-parallel-or-perpendicular-to-a-line-when-it-i.htmlhttps://liberty.io/can-you-find-an-equation-parallel-or-perpendicular-to-a-line-when-it-i.htmlFor example, if I need to find the equation of the line parallel to $$2x-3y=4$$ which passes through the point $(1,-5)$ I know how to do this by putting it into slope-intercept form first to find the ...Sun, 16 Aug 2026 17:59:49 -0400https://liberty.io/sum-of-2-positive-irrational-numbers-is-irrational-duplicate.htmlhttps://liberty.io/sum-of-2-positive-irrational-numbers-is-irrational-duplicate.htmlSo, I know how the sum of 2 irractional numbers as a whole can be rational. For example, an irrational number $a$ and it's negative counterpart $-a$ have a sum of zero and so the sum is rational. But ...Sun, 16 Aug 2026 17:53:50 -0400https://liberty.io/what-is-the-definition-of-a-properly-immersed-arc.htmlhttps://liberty.io/what-is-the-definition-of-a-properly-immersed-arc.htmlI am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I hav......Sun, 16 Aug 2026 17:39:25 -0400https://liberty.io/is-73-37-the-only-pair-of-reversible-primes-p-q-s-t-p-2q-1.htmlhttps://liberty.io/is-73-37-the-only-pair-of-reversible-primes-p-q-s-t-p-2q-1.htmlIn addition to being probably the only Sheldon Cooper prime, $73$ is a reversible prime $p$ (or emirp), such that its reverse is $q=(p+1)/2$. It is not hard to see that all other reversible primes ......Sun, 16 Aug 2026 17:35:40 -0400https://liberty.io/how-to-find-the-equation-of-a-line-tangent-to-a-circle-that-passes-thr.htmlhttps://liberty.io/how-to-find-the-equation-of-a-line-tangent-to-a-circle-that-passes-thr.htmlI am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outs......Sun, 16 Aug 2026 17:17:40 -0400https://liberty.io/distributive-law-for-fraction-arithmetic.htmlhttps://liberty.io/distributive-law-for-fraction-arithmetic.htmlI'm in seventh grade and my teacher wasn't able to explain this to me. Why is $\ \frac{c}{a+b}\neq \frac ca +\frac cb,\,$ but $\ \frac{a+b}c = \frac{a}c + \frac{b}c$? I'm sorry if this is obvious.......Sun, 16 Aug 2026 17:15:57 -0400https://liberty.io/prove-that-the-equation-x-10000-x-100-1-0-has-a-solution-with-0-x-1.htmlhttps://liberty.io/prove-that-the-equation-x-10000-x-100-1-0-has-a-solution-with-0-x-1.htmlProve that the equation $x^{10000} + x^{100} - 1 = 0$ has a solution with $0 < x < 1$. This is a homework question. I know I could probably find a solution that would complete the proof, but......Sun, 16 Aug 2026 17:13:56 -0400https://liberty.io/what-are-the-two-039-sides-039-of-a-decimal-number-called-duplicate.htmlhttps://liberty.io/what-are-the-two-039-sides-039-of-a-decimal-number-called-duplicate.htmlIs there a fancy name for the "left side" and "right side" of a decimal number? (That is, the pre-decimal part and the post-decimal part.)...Sun, 16 Aug 2026 16:52:49 -0400https://liberty.io/internally-tangent-circles-and-circumcircle.htmlhttps://liberty.io/internally-tangent-circles-and-circumcircle.htmlThe circle w is internally tangent with the circumcircle of $\triangle ABC$ at point A. $W \cap AB = K$ and $w$ intersects $BC$. The line $CL$ is tangent to $w$ at point L and $KL \cap BC = T$. Prove ...Sun, 16 Aug 2026 16:48:52 -0400https://liberty.io/what-does-a-dot-after-a-number-mean.htmlhttps://liberty.io/what-does-a-dot-after-a-number-mean.htmlSo I'm making some calculations for numerical analysis and the output I get in Wolfram or Mathematica for input like: -0.666667 (x-1) (x-0.5) (x+0.5)+0.166666 (x-1) (x+1) (x+0.5)+0.499999 (x-1) (x......Sun, 16 Aug 2026 16:48:17 -0400https://liberty.io/comparing-log-2-3-to-log-3-5.htmlhttps://liberty.io/comparing-log-2-3-to-log-3-5.htmlHaving been asked to compare $\log_2(3)$ and $\log_3(5)$, this is my proof: $$\log_2(3)>\log_3(5)$$ Then one uses the rule $\log_a(b)=\frac{1}{\log_b(a)}$, so $$\log_3(2)>\frac{1}{\log_3(5)}$$ ...Sun, 16 Aug 2026 16:42:58 -0400https://liberty.io/composition-of-a-rotation-and-a-traslation-is-a-rotation.htmlhttps://liberty.io/composition-of-a-rotation-and-a-traslation-is-a-rotation.htmlAssuming that I know that: 1) A plane isometry is either a rotation, a traslation, a reflection, or a glide reflection 2) Every isometry can be expressed as a product of reflections, 3) Rotatio......Sun, 16 Aug 2026 16:37:47 -0400https://liberty.io/requirements-for-a-linear-transformation.htmlhttps://liberty.io/requirements-for-a-linear-transformation.htmlLet's say we have a transformation: $$T: \mathbb{R}^n \rightarrow \mathbb{R}^m.$$ This is a linear transformation iff: for all $ \vec{a} , \vec{b} \in \mathbb{R}^n$ and $c \in \Bbb{R}$, $T(\vec{......Sun, 16 Aug 2026 16:35:20 -0400https://liberty.io/how-do-i-prove-that-arccos-x-arccos-x-pi-when-x-in-1-1-closed.htmlhttps://liberty.io/how-do-i-prove-that-arccos-x-arccos-x-pi-when-x-in-1-1-closed.htmlProve that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$. How do I prove this? Where should I begin and what should I consider?...Sun, 16 Aug 2026 16:31:52 -0400https://liberty.io/chain-rule-for-partial-derivatives.htmlhttps://liberty.io/chain-rule-for-partial-derivatives.htmlMinton and Smith, in "Calculus" define the chain rule for full derivatives $\frac {dz} {dt}$ as it follows: Vretblad, however, in "Fourier Analysis and its Applications", mentions an "easy exercis......Sun, 16 Aug 2026 16:31:49 -0400https://liberty.io/solve-first-order-matrix-differential-equation.htmlhttps://liberty.io/solve-first-order-matrix-differential-equation.htmlIf I have a differential equation $ y'(t)=A y(t)$ where A is a constant square matrix that is not diagonalizable(although it is surely possible to calculate the eigenvalues) and no initial conditio......Sun, 16 Aug 2026 16:27:47 -0400https://liberty.io/slope-between-two-points-vectors-closed.htmlhttps://liberty.io/slope-between-two-points-vectors-closed.htmlI have two vector points, $\langle 1,3,7 \rangle$ and $\langle 23,-5,10 \rangle$. Point $1$ is at $t=13$ and Point $2$ is at $t=81$. I know how to find the slope between two vector points when $t=0......Sun, 16 Aug 2026 16:25:21 -0400https://liberty.io/equilateral-and-equiangular-polygon.htmlhttps://liberty.io/equilateral-and-equiangular-polygon.htmlCan we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-......Sun, 16 Aug 2026 16:14:12 -0400https://liberty.io/why-is-a-symmetric-group-a-group.htmlhttps://liberty.io/why-is-a-symmetric-group-a-group.htmlIf the identity axiom for groups is the following $ae = ea = a$ ($e$ the identity element) and the operation for a symmetric group is function composition then how does a symmetric group pass this ......Sun, 16 Aug 2026 16:03:36 -0400https://liberty.io/circle-and-a-quadrilateral.htmlhttps://liberty.io/circle-and-a-quadrilateral.htmlQ: The polygon circumscribes the circle. Find the perimeter....Sun, 16 Aug 2026 15:54:06 -0400https://liberty.io/properties-of-triangles-in-non-euclidean-geometries.htmlhttps://liberty.io/properties-of-triangles-in-non-euclidean-geometries.htmlAs we all know, the angles in all triangles in Euclidean geometry must add up to $180^\circ$. As some of us may know, this is not true in non-Euclidean geometries; for example, on the surface of a ......Sun, 16 Aug 2026 15:53:03 -0400https://liberty.io/finding-the-basis-of-a-null-space.htmlhttps://liberty.io/finding-the-basis-of-a-null-space.htmlI am trying to understand why the method used in my linear algebra textbook to find the basis of the null space works. The textbook is 'Elementary Linear Algebra' by Anton. According to the textbo......Sun, 16 Aug 2026 15:50:34 -0400https://liberty.io/verification-of-my-solution-of-xy-039-2y.htmlhttps://liberty.io/verification-of-my-solution-of-xy-039-2y.htmlI am given: \begin{equation}xy'=2y\end{equation} Therefore, \begin{equation}y'=(2y)/x\end{equation} Solving for y, I have: $y=Cx^2$ (Originally, was $e^cx^2$, but C is arbitrary, so I simplifi......Sun, 16 Aug 2026 15:31:55 -0400https://liberty.io/solving-differential-equation-y-039-039-1-y-039-2.htmlhttps://liberty.io/solving-differential-equation-y-039-039-1-y-039-2.htmlFrom the Question $y''=1+(y')^2$ I used Reduction of Order by taking $y'=z$ to get, $$z\left(\frac{dz}{dy}\right)=1+z^2$$ Solve the equation by Separation of Variable, to get $$\frac{1}{2} \ln(1+z^......Sun, 16 Aug 2026 15:29:17 -0400https://liberty.io/can-any-right-triangle-be-inscribed-in-a-circle.htmlhttps://liberty.io/can-any-right-triangle-be-inscribed-in-a-circle.htmlCan every right triangle be inscribed in a circle? If so, we could infer: ": in any right triangle, the segment which cuts the hypotenuse in two and joins the opposite angle is equal to half the ...Sun, 16 Aug 2026 15:28:08 -0400https://liberty.io/how-to-construct-an-equilateral-triangle-on-2-concentric-circles.htmlhttps://liberty.io/how-to-construct-an-equilateral-triangle-on-2-concentric-circles.htmlConstruct an equilateral triangle with the given vertex so that the other vertices lie on the concentric circles respectively. I constructed the triangle, but I don't know how it works. How does......Sun, 16 Aug 2026 15:21:34 -0400https://liberty.io/divergence-of-laplacian.htmlhttps://liberty.io/divergence-of-laplacian.htmlIt seems to me that, in some derivations on fluid dynamic books I am reading, the identity $$\nabla \cdot (\nabla^2 u) = 0 $$ where $u$ is a vector field, is used. Does this identity exist? Is it ......Sun, 16 Aug 2026 15:10:05 -0400https://liberty.io/what-are-the-conditions-for-a-polygon-to-be-tessellated.htmlhttps://liberty.io/what-are-the-conditions-for-a-polygon-to-be-tessellated.htmlUpon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate ...Sun, 16 Aug 2026 15:04:37 -0400https://liberty.io/log-of-softmax-function-derivative.htmlhttps://liberty.io/log-of-softmax-function-derivative.htmlCould someone explain how that derivative was arrived at. According to me, the derivative of $\log(\text{softmax})$ is $$ \nabla\log(\text{softmax}) = \begin{cases} 1-\text{softmax}, & \text{......Sun, 16 Aug 2026 15:03:06 -0400https://liberty.io/prove-eigenfunctions-corresponding-to-different-eigenvalues-are-orthog.htmlhttps://liberty.io/prove-eigenfunctions-corresponding-to-different-eigenvalues-are-orthog.htmlI'm considering the eigenvalue problem $L(\phi) = -\lambda \sigma(x) \phi$, subject to a given set of homogeneous boundary conditions. Assume $\sigma(x) \gt 0$ on an interval $[a,b]$. Suppose tha......Sun, 16 Aug 2026 15:02:45 -0400https://liberty.io/polynomial-inequality-for-imaginary-roots.htmlhttps://liberty.io/polynomial-inequality-for-imaginary-roots.htmlI've to solve the following polynomial inequality $$x^2 - 6x + 11 > 0$$ By using quadratic formula, I got the value of $x$ as below $$ \frac{6\pm \sqrt{-8}}{2}$$ These are imaginary roots and......Sun, 16 Aug 2026 15:01:12 -0400https://liberty.io/infinity-times-zero-closed.htmlhttps://liberty.io/infinity-times-zero-closed.htmlAssuming the multiplication property of limits I can do the following: $\lim \limits_{x \to ∞}f(a)f(b)=\lim \limits_{x \to ∞}f(a)\lim \limits_{x \to ∞}f(b)$ Why cannot do this? The second one is ...Sun, 16 Aug 2026 14:49:17 -0400https://liberty.io/pivot-columns-and-basic-variables.htmlhttps://liberty.io/pivot-columns-and-basic-variables.htmlSuppose we are solving a system of linear equation and arrive at the following (augmented) matrix: $ \begin{bmatrix} 1& 6 & 0 & 3 & 0 & 0\\ 0& 0 & 1 & -4 &......Sun, 16 Aug 2026 14:48:49 -0400https://liberty.io/what-is-the-difference-between-topological-and-metric-spaces.htmlhttps://liberty.io/what-is-the-difference-between-topological-and-metric-spaces.htmlWhat is the difference between a topological and a metric space?...Sun, 16 Aug 2026 14:45:36 -0400https://liberty.io/how-do-i-calculate-a-dihedral-angle-given-cartesian-coordinates.htmlhttps://liberty.io/how-do-i-calculate-a-dihedral-angle-given-cartesian-coordinates.htmlI have the $(x, y, z)$ coordinates for four points in space (atoms). How do I calculate the dihedral angle ($\phi$ in the below pictures from WP)? I'm somewhat hazy on my math, and am trying to ...Sun, 16 Aug 2026 14:44:15 -0400https://liberty.io/what-is-the-j-invariant.htmlhttps://liberty.io/what-is-the-j-invariant.htmlI know that it has something to do with Elliptic Curves in the Complex Plane but I don't have an intuitive sense as to what its there for. I understand it is defined in terms of multiple Einstein ......Sun, 16 Aug 2026 14:35:17 -0400https://liberty.io/decompose-projection-matrix-into-a-matrix-and-its-pseudoinverse.htmlhttps://liberty.io/decompose-projection-matrix-into-a-matrix-and-its-pseudoinverse.htmlWhile researching regression, I encountered the situation where I am trying to reconstruct a data matrix $X \in \mathbb{R}^{n\times p}$ from the $n \times n$ hat matrix $H = X(X'X)^{-1}X' = XX^{+}$ (...Sun, 16 Aug 2026 14:31:30 -0400https://liberty.io/expected-value-of-a-binomial-distribution.htmlhttps://liberty.io/expected-value-of-a-binomial-distribution.htmlIf $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom nkp^k(1-p)^{......Sun, 16 Aug 2026 14:26:43 -0400https://liberty.io/finding-precise-solution-for-x-in-complicated-single-variable-equation.htmlhttps://liberty.io/finding-precise-solution-for-x-in-complicated-single-variable-equation.htmlI need an exact answer for $x$ in the following algebraic expression: $$ x^{4/3} - 2\sqrt{2}x^2 + 2\sqrt{2} = 0 $$ I don't need or want an approximate answer, I already know that $x≈1.20569$. So any ...Sun, 16 Aug 2026 14:23:51 -0400https://liberty.io/online-tools-for-doing-symbolic-mathematics.htmlhttps://liberty.io/online-tools-for-doing-symbolic-mathematics.htmlThis question is similar to this one, but specifically relates to resources available strictly as on-line web apps. Examples include: Wolfram Alpha Sage Notebook hicalc not sure what this is; th......Sun, 16 Aug 2026 14:18:50 -0400https://liberty.io/how-to-find-top-and-bottom-function-for-finding-area-between-two-curve.htmlhttps://liberty.io/how-to-find-top-and-bottom-function-for-finding-area-between-two-curve.htmlWhen I am trying to find the area between two curves below the $\mathbf{x}$-axis, should I always have the function furthest away from the x-axis as the function you are subtracting from, or should......Sun, 16 Aug 2026 14:18:16 -0400https://liberty.io/cofinite-topology.htmlhttps://liberty.io/cofinite-topology.htmlIf we have topological space $\mathbb{R}$ equipped with the co finite topology. If we have finite subsets in consideration they are definitely closed because open sets are of the form complement of ...Sun, 16 Aug 2026 14:10:25 -0400https://liberty.io/please-help-how-to-work-the-inequation-x-1-x-5-0.htmlhttps://liberty.io/please-help-how-to-work-the-inequation-x-1-x-5-0.htmlI am studying for college finals and I cannot solve the inequation (x-1)/(x-5)<0 using the method the professor taught us, which I have to use in the exam. This is another inequation using said ...Sun, 16 Aug 2026 14:06:06 -0400https://liberty.io/trace-of-a-matrix-when-to-use-what-is-trace-trick.htmlhttps://liberty.io/trace-of-a-matrix-when-to-use-what-is-trace-trick.htmlOn calculating log-likelihood function for some multivariate distributions, such as multivariate Normal, I see some examples where the matrices are suddenly changed to trace, even when the matrix i......Sun, 16 Aug 2026 14:00:56 -0400https://liberty.io/relation-between-the-roots-of-a-cubic-equation-and-the-coefficients.htmlhttps://liberty.io/relation-between-the-roots-of-a-cubic-equation-and-the-coefficients.html$ax^3 +bx^2 + cx + d= 0$ If the roots are $\alpha$ $\beta$ and $\gamma$, Is there any relationship between the sum of the squares of the roots and the coefficients of the quadratic equation.. In ...Sun, 16 Aug 2026 14:00:17 -0400https://liberty.io/generators-of-the-symmetric-group-s-3.htmlhttps://liberty.io/generators-of-the-symmetric-group-s-3.htmlI am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not foun......Sun, 16 Aug 2026 13:59:25 -0400