💡 Tocqueville\ "Democracy extends the sphere of individual freedom, socialism restricts it. Democracy attaches all possible value to each man; socialism makes each man a mere agent. Democracy and socialism have nothing in common but one word: equality. But notice the difference: while democracy seeks equality in liberty, socialism seeks equality in restraint and servitude." From: The Road to Serfdom Learn more:

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📐 Finite Implies Algebraic Every finite extension is algebraic. Proof: If $[K:F] = n$ and $\\alpha \\in K$, then $1, \\alpha, \\alpha^2, \\ldots, \\alpha^n$ are $n+1$ vectors in an $n$-dimensional space, hence linearly dependent. A dependency relation gives a polynomial with $\\alpha$ as a root. From: df-course Learn more:

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📐 Centroid of a Triangle The centroid of a triangle with vertices $\\mathbf{a}$, $\\mathbf{b}$, $\\mathbf{c}$ is $\\frac{1}{3}(\\mathbf{a} + \\mathbf{b} + \\mathbf{c})$. Proof: The centroid is the intersection of medians. A median from $\\mathbf{a}$ to midpoint $\\frac{1}{2}(\\mathbf{b} + \\mathbf{c})$ can be parameterized as $\\mathbf{a} + t(\\frac{1}{2}(\\mathbf{b} + \\mathbf{c}) - \\mathbf{a})$. The medians intersect at $t = \\frac{2}{3}$, giving $\\frac{1}{3}\\mathb... 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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📐 Galois Group of Cyclotomic Extensions $\\text{Gal}(\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}) \\cong (\\mathbb{Z}/n\\mathbb{Z})^\\times$. From: df-course Learn more:

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📖 Span The span of a set $S \\subseteq V$ is the set of all linear combinations of vectors in $S$: $\\text{span}(S) = \\{c_1 v_1 + \\cdots + c_n v_n : c_i \\in F, v_i \\in S\\}$. From: Advanced Linear Algebra Learn more:

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📐 Theorem 4.2 (Sequential Characterization) $\\lim_{x \\to p} f(x) = q$ if and only if for every sequence $\\{p_n\\}$ in $E$ with $p_n \\neq p$ and $p_n \\to p$, we have $f(p_n) \\to q$. From: rudin Learn more:

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📐 Principle of Duality In projective geometry, any theorem remains valid when "point" and "line" are interchanged, and "lies on" and "passes through" are interchanged. From: Four Pillars of Geometry Learn more:

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📐 Archimedes: Surface Area of Sphere The surface area of a sphere equals four times the area of a great circle: $S = 4\\pi r^2$. From: Thales to Euclid Learn more:

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📐 Arc Length Differential For a small piece of a curve, the arc length element is $ds = \\sqrt{(dx)^2 + (dy)^2} = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2} \\, dx$. Integrating these infinitesimal pieces gives the total arc length. From: Beginner Calculus Learn more:

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📐 Kuratowski A graph is planar if and only if it contains no subdivision of $K_5$ or $K_{3,3}$. Proof: Subdivisions of $K_5$ and $K_{3,3}$ are not planar (edge bound: $K_5$ has 10 edges but $3(5)-6=9$; $K_{3,3}$ is triangle-free with 9 edges but $2(6)-4=8$). Conversely, suppose $G$ is non-planar and edge-minimal. Show $G$ is 3-connected. Contract an edge to get a smaller non-planar graph with a Ku... From: Introduction to Graph Theory Learn more:

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💡 Proposition I.14 If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. 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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📐 Power Rule for Differentiation For any power $n$, if $y = x^n$, then $\\frac{dy}{dx} = nx^{n-1}$. From: Beginner Calculus Learn more:

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📐 Wagner A graph is planar if and only if it has no $K_5$ minor and no $K_{3,3}$ minor. From: Introduction to Graph Theory Learn more:

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📐 Theorem 2.41 (Heine-Borel Theorem) A subset of $\\mathbb{R}^k$ is compact if and only if it is closed and bounded. Proof: ($\\Rightarrow$) Compact sets are closed (Theorem 2.34) and bounded (covered by finitely many bounded balls). ($\\Leftarrow$) If $E$ is closed and bounded, then $E \\subset I$ for some $k$-cell $I$. $I$ is compact by Theorem 2.38. Since $E$ is closed, $E$ is a closed subset of compact $I$. By ... From: rudin Learn more:

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📖 Rectangle Area Area of a rectangle = length $\\times$ width. From: numbers-geometry Learn more:

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📐 Three Prerequisites of Action Action requires: (1) Uneasiness—felt dissatisfaction with present state; (2) Image of improvement—conception of a better state; (3) Expectation of efficacy—belief that action can help achieve improvement. From: Human Action Learn more:

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Ludwig von Mises

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📐 Polar Decomposition Every linear operator $T$ on an inner product space can be written as $T = UP$ where $U$ is unitary and $P = \\sqrt{T^*T}$ is positive. Proof: Define $P = \\sqrt{T^*T}$ using the spectral theorem. On $\\mathrm{im}\\, P$, define $U$ by $U(Pv) = Tv$. This is well-defined and isometric. Extend to an orthonormal basis. From: adv_linalg Learn more:

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📖 Interior Point Let $S \\subseteq \\mathbb{R}^n$ and $a \\in S$. Then $a$ is an \\textbf{interior point} of $S$ if there exists an open n-ball $B(a; r)$ such that $B(a; r) \\subseteq S$. The set of all interior points of $S$ is called the \\textbf{interior} of $S$, denoted $\\text{int } S$. From: calc2 Learn more:

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📖 Euclid A finite straight line can be extended continuously in a straight line. From: Thales to Euclid Learn more:

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💡 Operating Profitability Enhancement Companies can systematically sell Bitcoin gains and include them in operating revenue, improving operating profitability. This creates genuine cost reduction because Bitcoin outperforms USD by design. From: bfi Learn more:

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