📖 Leibniz Notation for Integrals The integral is written as $\\int y\\,dx$, where the elongated S stands for "summa" (sum) and $dx$ represents an infinitesimally thin width. From: Calculus: A Liberal Art Learn more:

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📐 Cooperation from Self-Interest People cooperate because it makes them better off through productivity gains from division of labor—not from altruism or compulsion. From: Human Action Learn more:

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Ludwig von Mises

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📐 Planetary Motion Equation The motion of a planet under gravitational attraction is governed by the differential equation: $\\frac{d^2\\vec{r}}{dt^2} = -\\frac{GM}{r^2}\\hat{r}$ From: Calculus: A Liberal Art Learn more:

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📐 Theorem 1.19 (Existence of Real Numbers) There exists an ordered field $\\mathbb{R}$ which has the least upper bound property. Moreover, $\\mathbb{Q}$ is a subfield of $\\mathbb{R}$. From: rudin-course Learn more:

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💡 Bitcoin Treasury Optionality A company holding Bitcoin treasury has optionality advantage over competitors. They can hold for appreciation or deploy to lower operating costs, providing flexibility not available to cash holders. From: bfi Learn more:

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📐 Least Squares Solution The least squares solution to $A\\mathbf{x} = \\mathbf{b}$ minimizes $\\|A\\mathbf{x} - \\mathbf{b}\\|^2$ and satisfies the normal equations $A^TA\\hat{\\mathbf{x}} = A^T\\mathbf{b}$. Proof: The error $\\mathbf{e} = \\mathbf{b} - A\\hat{\\mathbf{x}}$ is minimized when $\\mathbf{e}$ is perpendicular to $C(A)$. This means $A^T\\mathbf{e} = \\mathbf{0}$, so $A^T(\\mathbf{b} - A\\hat{\\mathbf{x}}) = \\mathbf{0}$. Rearranging: $A^TA\\hat{\\mathbf{x}} = A^T\\mathbf{b}$. From: Linear Algebra Learn more:

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: hoppe Learn more:

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📖 Field A \\textbf{field} is a commutative ring with unity $1 \\neq 0$ in which every nonzero element is a unit. From: df-course Learn more:

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📐 Subgroups of Cyclic Groups Every subgroup of a cyclic group is cyclic. If $|G| = n$, subgroups correspond to divisors of $n$. From: intro-discrete Learn more:

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📐 Theorem 6.6.8 (Bolzano-Weierstrass) Every bounded sequence has a convergent subsequence. Proof: Let $(a_n)$ be bounded. Then $L := \\limsup a_n$ is finite. By definition of limsup, for every $\\varepsilon > 0$ and every $N$, there exists $n \\geq N$ with $a_n$ within $\\varepsilon$ of $L$. Construct subsequence: Choose $n_1$ with $|a_{n_1} - L| < 1$. Given $n_k$, choose $n_{k+1} > n_k$ wi... From: tao-analysis-1 Learn more:

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💡 Proposition III.2 If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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🎯 Converse of Thales Theorem If $\\angle ACB = 90°$ and $A$, $B$ are endpoints of segment $AB$, then $C$ lies on the circle with diameter $AB$. Proof: Let $M$ be the midpoint of $AB$. We show $|MC| = |MA| = |MB|$. By the Pythagorean theorem in $\\triangle ACB$: $|AC|^2 + |CB|^2 = |AB|^2$. The median to the hypotenuse of a right triangle equals half the hypotenuse. Therefore $|MC| = \\frac{1}{2}|AB| = |MA| = |MB|$. So $C$ lies on the circle with... From: Four Pillars of Geometry Learn more:

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Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📖 Linear Combination Let $S$ be a nonempty subset of a linear space $V$. An element $x$ in $V$ of the form $x = \\sum_{i=1}^k c_i x_i$, where $x_1, \\ldots, x_k$ are all in $S$ and $c_1, \\ldots, c_k$ are scalars, is called a \\textbf{finite linear combination} of elements of $S$. From: calc2-course Learn more:

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📖 Cyclic Group A group $G$ is cyclic if $G = \\langle g \\rangle = \\{g^n : n \\in \\mathbb{Z}\\}$ for some generator $g$. From: intro-discrete Learn more:

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📖 R-squared (Coefficient of Determination) $R^2 = \\frac{\\text{TSS} - \\text{RSS}}{\\text{TSS}} = 1 - \\frac{\\text{RSS}}{\\text{TSS}}$ where $\\text{TSS} = \\sum(y_i - \\bar{y})^2$. It measures the proportion of variance explained by the model. From: Intro to Statistical Learning Learn more:

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📐 Unique Factorization for Polynomials Every non-constant polynomial factors uniquely (up to order and constants) into irreducible polynomials. From: intro-discrete Learn more:

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📐 Space of Linear Transformations The set $\\mathcal{L}(V, W)$ of all linear transformations from $V$ to $W$ is itself a linear space with operations: $(S + T)(x) = S(x) + T(x)$ and $(cT)(x) = cT(x)$. From: calc2-course Learn more:

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🎯 Euler $e^{i\\pi} + 1 = 0$ Proof: From Euler's formula with $\\theta = \\pi$: $e^{i\\pi} = \\cos\\pi + i\\sin\\pi = -1 + 0i = -1$ Therefore $e^{i\\pi} + 1 = 0$ This equation connects five fundamental constants: $e$, $i$, $\\pi$, 1, and 0. From: Men of Mathematics Learn more:

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An interactive journey through 2500 years of mathematical history with E.T. Bell

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📖 Limit of a Series If the partial sums of a series get closer and closer to a number $k$, then $k$ is called the limit of the series. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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📐 Properties of Characteristic Polynomial If $A$ is an $n \\times n$ matrix, then $f(\\lambda) = \\det(\\lambda I - A)$ is a polynomial of degree $n$. The leading term is $\\lambda^n$ and the constant term is $f(0) = (-1)^n \\det A$. From: calc2-course Learn more:

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