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📐 Euclidean Algorithm $\\gcd(a,b) = \\gcd(b, a \\mod b)$. Repeated application computes $\\gcd(a,b)$ in $O((\\log b)^2)$ bit operations. Proof: If $a = bq + r$, then any common divisor of $a$ and $b$ also divides $r = a - bq$. Conversely, any common divisor of $b$ and $r$ also divides $a = bq + r$. Therefore $\\gcd(a,b) = \\gcd(b,r)$. The algorithm terminates because remainders strictly decrease. Each step halves the larger numbe... From: Algebraic Number Theory Learn more:

Number Theory and Cryptography | Math Academy

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📖 Injective Module An $R$-module $Q$ is \\textbf{injective} if for every injection $f: M \\to N$ and map $g: M \\to Q$, there exists an extension $\\tilde{g}: N \\to Q$ with $\\tilde{g} \\circ f = g$. From: df-course Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Absolute Convergence Implies Convergence If $\\sum |a_n|$ converges, then $\\sum a_n$ converges. From: Real Analysis Learn more:

Real Analysis | Math Academy

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

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📐 Pappus Let $A, B, C$ be collinear and $D, E, F$ be collinear on a parallel line. If $AB \\parallel ED$ and $FE \\parallel BC$, then $AF \\parallel CD$. Proof: By similar triangles formed by the parallel lines, we establish proportions that force $AF \\parallel CD$. 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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📐 Derived Demand Demand for factors is derived from demand for the consumers\ From: Man, Economy, and State Learn more:

Man, Economy, and State | Math Academy

Murray Rothbard

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📐 Theorem 2.36 (Nested Compact Sets) If $\\{K_\\alpha\\}$ is a collection of compact subsets of a metric space such that every finite subcollection has nonempty intersection, then $\\bigcap_\\alpha K_\\alpha \\neq \\emptyset$. From: rudin Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Vertical Angles Theorem When two lines intersect, vertically opposite angles are equal. Proof: Let the lines intersect at $O$, forming angles $\\alpha, \\beta, \\alpha', \\beta'$ in order. Then $\\alpha + \\beta = 180°$ (angles on a straight line). Also $\\beta + \\alpha' = 180°$. Therefore $\\alpha = \\alpha'$. Similarly, $\\beta = \\beta'$. From: numbers-geometry Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Cross-Ratio is Projectively Invariant The cross-ratio of four collinear points is preserved by any projectivity (and hence by any central projection). Proof: A Mobius transformation $f(x) = \\frac{ax + b}{cx + d}$ preserves cross-ratio: $(f(a), f(b); f(c), f(d)) = (a, b; c, d)$. Direct calculation shows the cross-ratio is unchanged after substituting the linear fractional expressions. From: Four Pillars of Geometry Learn more:

The Four Pillars of Geometry | Magic Internet Math

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

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💡 World Wisdom and Human Wisdom There exists a cosmic wisdom (world wisdom) that permeates and orders the entire universe, and a human wisdom that is the portion of this cosmic wisdom accessible to human consciousness at its present stage of development. The task of spiritual science is to progressively expand human wisdom until it encompasses ever greater portions of world wisdom. This expansion occurs not through abstract s... From: steiner-GA90a Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Function A function $f: A \\to B$ assigns to each element of $A$ exactly one element of $B$. From: intro-discrete Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.