📖 Pythagorean Triples A Pythagorean triple $(a, b, c)$ is a set of positive integers satisfying $a^2 + b^2 = c^2$. Examples: $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$. 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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💡 Monetary Sovereignty The ability to sell all assets and put value on a seed phrase enabling movement anywhere in the world represents a level of sovereignty previously unavailable to humans. From: bfi Learn more:

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📐 Equation of a Line Every line in $\\mathbb{R}^2$ can be described by a linear equation $ax + by = c$ where $(a, b) \\neq (0, 0)$. Conversely, every such equation describes a line. 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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📖 Definition: Orthogonal Complement The orthogonal complement $V^\\perp$ of subspace $V$ is the set of all vectors orthogonal to every vector in $V$: $V^\\perp = \\{\\mathbf{w} : \\mathbf{w} \\cdot \\mathbf{v} = 0 \\text{ for all } \\mathbf{v} \\in V\\}$. From: Linear Algebra Learn more:

Introduction to Linear Algebra | Math Academy

Interactive course based on Gilbert Strang

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📐 Arithmetic of Congruences If $a \\equiv b \\pmod{m}$ and $c \\equiv d \\pmod{m}$, then $a + c \\equiv b + d \\pmod{m}$ and $ac \\equiv bd \\pmod{m}$. Proof: Since $m \\mid (a-b)$ and $m \\mid (c-d)$, we have $m \\mid [(a-b) + (c-d)] = (a+c) - (b+d)$, so $a+c \\equiv b+d \\pmod{m}$. For multiplication: $ac - bd = ac - bc + bc - bd = c(a-b) + b(c-d)$. Since $m \\mid (a-b)$ and $m \\mid (c-d)$, we have $m \\mid [c(a-b) + b(c-d)] = ac - bd$. From: Disquisitiones Arithmeticae Learn more:

Disquisitiones Arithmeticae | Math Academy

Gauss

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📐 Diagonal Matrix Representation A linear transformation $T: V \\to V$ where $\\dim V = n$ has a diagonal matrix representation if and only if there exist independent elements $u_1, \\ldots, u_n$ in $V$ and scalars $\\lambda_1, \\ldots, \\lambda_n$ such that $T(u_k) = \\lambda_k u_k$ for $k = 1, \\ldots, n$. From: calc2-course Learn more:

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📖 Even and Odd Permutations A permutation is \\textbf{even} if it can be written as a product of an even number of transpositions, and \\textbf{odd} otherwise. The parity of a permutation is well-defined. From: df-course Learn more:

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📖 Definition 3.3.1 (Function) Let $X$, $Y$ be sets, and let $P(x, y)$ be a property pertaining to an object $x \\in X$ and an object $y \\in Y$, such that for every $x \\in X$, there is exactly one $y \\in Y$ for which $P(x, y)$ is true. Then we define the function $f : X \\to Y$ defined by $P$ on the domain $X$ and codomain $Y$ to be the object which assigns to each $x \\in X$ a single output $f(x) \\in Y$. From: tao-analysis-1 Learn more:

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📖 Projection onto a Subspace Let $S$ be a finite-dimensional subspace of a Euclidean space $V$, with orthonormal basis $\\{e_1, \\ldots, e_n\\}$. For $x \\in V$, the element $s = \\sum_{i=1}^n (x, e_i)e_i$ is called the \\textbf{projection of $x$ on the subspace $S$}. From: calc2-course Learn more:

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📐 Theorem 11.5.1 (Continuous Functions are Integrable) Let $f : [a, b] \\to \\mathbb{R}$ be continuous. Then $f$ is Riemann integrable. From: tao-analysis-1 Learn more:

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📐 Chain Rule If $y = f(u)$ and $u = g(x)$, then $\\frac{dy}{dx} = \\frac{dy}{du} \\times \\frac{du}{dx}$. This is the chain rule for composite functions. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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💡 Proposition I.38 Triangles which are on equal bases and in the same parallels are equal to 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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📖 Reflection A reflection in line $L$ maps each point $P$ to the point $P'$ such that $L$ is the perpendicular bisector of $PP'$. From: Four Pillars of Geometry Learn more:

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💡 Proposition 9.4.7 (Algebra of Continuous Functions) Let $X \\subseteq \\mathbb{R}$ and $f, g : X \\to \\mathbb{R}$ be continuous at $x_0$. Then $f + g$, $f - g$, $fg$, $\\max(f,g)$, $\\min(f,g)$ are continuous at $x_0$. If $g(x_0) \\neq 0$, then $f/g$ is continuous at $x_0$. From: tao-analysis-1 Learn more:

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📐 Transformation Theorem Given additive shares $a_i$ where $\\sum_i a_i = s$, computing $\\hat{s}_i = a_i \\cdot \\lambda_i$ gives values that satisfy $\\sum_i \\hat{s}_i = s$ with Lagrange reconstruction. Proof: Since $\\sum_i \\lambda_i = 1$ (Lagrange coefficients sum to 1): $\\sum_i \\hat{s}_i = \\sum_i a_i \\cdot \\lambda_i$ For the special case where all $a_i = s$ are equal: $\\sum_i s \\cdot \\lambda_i = s \\cdot \\sum_i \\lambda_i = s \\cdot 1 = s$ From: frost Learn more:

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📐 Fundamental Theorem of Galois Theory For a Galois extension $K/F$ with $G = \\text{Gal}(K/F)$: (1) There is a bijection between intermediate fields $F \\subseteq E \\subseteq K$ and subgroups $H \\le G$ via $E \\mapsto \\text{Gal}(K/E)$ and $H \\mapsto K^H$. (2) $[K:E] = |\\text{Gal}(K/E)|$ and $[E:F] = [G : \\text{Gal}(K/E)]$. (3) $E/F$ is Galois iff $\\text{Gal}(K/E) \\trianglelefteq G$, and then $\\text{Gal}(E/F) \\cong G/\\tex... From: df-course Learn more:

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📐 Zero Product Property In any field $F$, for all $a \\in F$: $a \\cdot 0 = 0$. Proof: For any $a \\in F$: $$a \\cdot 0 = a \\cdot (0 + 0) = a \\cdot 0 + a \\cdot 0$$ Adding $-(a \\cdot 0)$ to both sides: $$0 = a \\cdot 0$$ From: Advanced Linear Algebra Learn more:

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📐 Second Fundamental Theorem of Calculus If $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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📐 Triangles on Equal Base (Euclid I.37) Triangles on the same base and between the same parallels are equal in area. Proof: Let $\\triangle ABC$ and $\\triangle DBC$ have base $BC$ and vertices $A, D$ on a line parallel to $BC$. Construct parallelograms $ABCE$ and $DBCF$ by drawing parallels. By Euclid I.35, these parallelograms are equal. Each triangle is half its parallelogram (Euclid I.34). Therefore $\\triangle AB... From: Four Pillars of Geometry Learn more:

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💡 100% Bitcoin Allocation for Pensions Given medium-term problem horizon and ability to survive two Bitcoin halving cycles (eight years), 100% allocation is justified for pensions. Limited bailout availability argues for maximum volatility bet. From: bfi Learn more:

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