📐 Solvability of Linear Diophantine Equations The equation $ax + by = c$ has integer solutions if and only if $\\gcd(a,b)$ divides $c$. Proof: (⟹) If $x, y$ are solutions, then $\\gcd(a,b)$ divides $ax + by = c$. (⟸) Suppose $d = \\gcd(a,b)$ divides $c$. Write $c = dk$. By the Linear Representation Theorem, $d = ax_0 + by_0$ for some integers. Then $c = dk = a(kx_0) + b(ky_0)$. So $x = kx_0$ and $y = ky_0$ is a solution. From: numbers-geometry Learn more:

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💡 Events in the Second Half of Lemurian Development In the second half of the Lemurian epoch, three decisive events occurred: first, the Moon separated from the Earth, allowing the human form to solidify sufficiently for ego-incarnation. Second, the Luciferic beings penetrated the human astral body, bestowing the capacity for freedom and error simultaneously. Third, the Sons of Manas descended and implanted the spark of individual mind (Manas) i... From: steiner-GA90a Learn more:

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📖 The Directly Given (das unmittelbar Gegebene) The world-picture as it presents itself before any cognitive activity has been applied to it. It is an undifferentiated totality containing no distinctions of substance and attribute, cause and effect, matter and spirit, body and soul. These categories are all products of thinking. The directly given is what remains when every concept is stripped away. From: steiner-ga3 Learn more:

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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: orwell-1984 Learn more:

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💡 Proposition VII.38 If a number have any part whatever, it will be measured by a number called by the same name as the part. 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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💡 Who is John Galt? An expression of despair and the central mystery of the novel, eventually revealing the identity of the man organizing the strike of the mind. From: Atlas Shrugged Learn more:

Atlas Shrugged | Math Academy

An interactive course exploring Atlas Shrugged by Ayn Rand

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📖 Definition V.8 (Proportion) A proportion in three terms is the least possible. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

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📐 Polygon Area Any polygon can be decomposed into triangles, so its area is the sum of triangle areas. From: numbers-geometry Learn more:

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📖 Vector Space A vector space $V$ over a field $F$ is a nonempty set with operations of addition and scalar multiplication satisfying the axioms: closure, associativity, commutativity, identity, inverses, and distributivity. From: adv_linalg Learn more:

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📖 Minimum A minimum is a point where a function reaches a local lowest value. At a minimum, the derivative is zero and the second derivative is positive. From: Beginner Calculus Learn more:

Introductory Calculus | Math Academy

Calculus Made Easy - An interactive introduction to calculus

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📐 Theorem 1.21 (Existence of nth Roots) For every real $x > 0$ and every positive integer $n$, there is one and only one positive real $y$ such that $y^n = x$. From: rudin Learn more:

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📐 Cyclic Decomposition Theorem Every finitely generated module $M$ over a PID $R$ is isomorphic to $R^r \\oplus R/(a_1) \\oplus \\cdots \\oplus R/(a_k)$ where $a_1 | a_2 | \\cdots | a_k$. Proof: Apply the primary decomposition first, then decompose each primary component into cyclic modules. The divisibility chain follows from the uniqueness of elementary divisors. From: adv_linalg Learn more:

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🎮 Interactive: Constructible Polygon Demo See which regular polygons can be constructed with compass and straightedge. Gauss proved the 17-gon is constructible at age 19! From: Disquisitiones Arithmeticae Try it:

Disquisitiones Arithmeticae | Math Academy

Gauss

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📖 Point of Inflection A point of inflection is where a curve changes from concave up to concave down (or vice versa). At such points, $\\frac{d^2y}{dx^2} = 0$. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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📐 The Action Axiom Human beings engage in purposeful behavior toward chosen goals. This is self-evident and cannot be denied without self-refutation—the very act of denying it is itself purposeful action. Proof: The action axiom is apodictically certain—true by the nature of human thought itself. To argue against it, one must: 1. Have a goal (refuting the axiom) 2. Choose means (constructing arguments) 3. Believe the argument can achieve the goal But this is precisely purposeful action. The denial... From: Human Action Learn more:

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Ludwig von Mises

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📐 Area of Circle The area of a circle is proportional to the square of its diameter: $A \\propto d^2$ Proof: Proved using the Method of Exhaustion: inscribe and circumscribe polygons with increasing numbers of sides. As the number of sides approaches infinity, both polygon areas approach the circle's area. Since polygon areas scale with the square of linear dimensions, so does the circle's area. From: Men of Mathematics Learn more:

Men of Mathematics | Math Academy

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

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📖 Ring A \\textbf{ring} $R$ is a set with two binary operations $+$ and $\\cdot$ such that: (1) $(R, +)$ is an abelian group with identity $0$; (2) Multiplication is associative; (3) Distributive laws hold: $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$. From: df-course Learn more:

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📖 Perfect Special Honest Verifier Zero-Knowledge PSHVZK: A PPT simulator $\\mathcal{S}$ exists such that real and simulated transcript distributions are identical, given the verifier\ From: lcn Learn more:

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💡 The Salvation of Humanity Through the Christ Impulse Without the intervention of the Christ Being, humanity would have descended ever deeper into material existence without the possibility of return to the spirit. The Christ impulse, entering Earth evolution at the Baptism in the Jordan and culminating at Golgotha, planted a new spiritual force into the Earth\ From: steiner-GA90a Learn more:

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💡 Proposition 2.3.6 (Distributive Law) For any natural numbers $a$, $b$, $c$, we have $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$. From: tao-analysis-1 Learn more:

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