📖 Binomial Coefficient For $0 \\leq k \\leq n$, $\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$ where $n! = n(n-1)\\cdots 1$ and $0! = 1$. From: intro-discrete Learn more:

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📐 Rank-Nullity Theorem (Dimension Theorem) If $V$ is finite-dimensional, then $T(V)$ is also finite-dimensional, and: $\\dim N(T) + \\dim T(V) = \\dim V$. In other words, the nullity plus the rank of a linear transformation equals the dimension of its domain. Proof: Let $n = \\dim V$ and let $e_1, \\ldots, e_k$ be a basis for $N(T)$ where $k = \\dim N(T)$. By Theorem 1.7, these are part of some basis for $V$: $e_1, \\ldots, e_k, e_{k+1}, \\ldots, e_n$ where $k + r = n$. We show that $T(e_{k+1}), \\ldots, T(e_n)$ form a basis for $T(V)$, proving $\\dim T(V) =... From: calc2 Learn more:

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📖 Rent The factor price of land services, determined by the discounted marginal value product of land in its most valuable use. From: Man, Economy, and State Learn more:

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Murray Rothbard

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📐 Fundamental Theorem of Calculus (Part 1) If $f$ is continuous on $[a, b]$ and $F(x) = \\int_a^x f(t)\\,dt$, then $F$ is differentiable and $F\ Proof: The derivative of an accumulation function is the original function. If you're accumulating something at rate $f(x)$, then the rate of change of your total is exactly $f(x)$. Formally: $F'(x) = \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h} = \\lim_{h \\to 0} \\frac{1}{h}\\int_x^{x+h} f(t)\\,dt = ... From: Calculus: A Liberal Art Learn more:

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📐 Cayley-Hamilton Theorem Every linear operator satisfies its characteristic polynomial: if $p(t) = \\det(tI - A)$, then $p(A) = 0$. Proof: The minimal polynomial divides the characteristic polynomial by structure theory. Since both are monic of the same degree when considering the module $F[t]/(m_\\tau)$, they are equal up to units. From: adv_linalg Learn more:

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📐 Uncertainty Action occurs in time, and the future is always uncertain. If the future were known with certainty, there would be no need to choose. From: Man, Economy, and State Learn more:

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Murray Rothbard

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📖 Leibniz Notation for Derivatives The derivative is written as $\\frac{dy}{dx}$, representing the ratio of infinitesimal changes. This notation suggests correctly that derivatives behave like fractions. From: Calculus: A Liberal Art Learn more:

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📐 Theorem 6.8 (Continuous ⟹ Integrable) If $f$ is continuous on $[a, b]$, then $f \\in \\mathcal{R}$ on $[a, b]$. From: rudin 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: foundation Learn more:

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📐 Fundamental Theorem of Arithmetic Every positive integer greater than $1$ has a unique factorization into prime numbers (up to order). From: Thales to Euclid Learn more:

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Journey through 2,500 years of mathematical history, from ancient Egypt through modern foundations

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📐 Newton Force equals mass times acceleration: $F = ma = m\\frac{d^2x}{dt^2}$. This is fundamentally a differential equation. From: Calculus: A Liberal Art Learn more:

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📖 Field A field $F$ is a set with addition and multiplication where $(F, +)$ is an abelian group, $(F \\setminus \\{0\\}, \\cdot)$ is an abelian group, and distributivity holds. From: Algebraic Number Theory Learn more:

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📐 Hyperbolic Isometries The isometries of the Poincare disk are the Mobius transformations that preserve the unit disk. These have the form $z \\mapsto e^{i\\theta}\\frac{z - a}{1 - \\bar{a}z}$ for $|a| < 1$. 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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📐 Theorem 7.12 (Uniform Limit of Continuous) If $f_n \\to f$ uniformly on $E$ and each $f_n$ is continuous on $E$, then $f$ is continuous on $E$. Proof: Fix $x \\in E$ and $\\varepsilon > 0$. By uniform convergence, there exists $N$ with $|f_n(t) - f(t)| < \\varepsilon/3$ for all $t \\in E$ when $n \\geq N$. Since $f_N$ is continuous at $x$, there exists $\\delta > 0$ with $|f_N(t) - f_N(x)| < \\varepsilon/3$ when $|t - x| < \\delta$. Then for... From: rudin Learn more:

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📖 Riemann Integrable $f$ is Riemann integrable if $\\sup_P L(f,P) = \\inf_P U(f,P)$. The common value is $\\int_a^b f$. From: Real Analysis Learn more:

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A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

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💡 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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An interactive course exploring F.A. Hayek

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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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A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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