MagicInternetMath
Magic Internet Math

https://mathacademy-cyan.vercel.app

Daily Insights from Magic Internet Math courses. Learn at https://mathacademy-cyan.vercel.app

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Last Notes

2026-04-04 08:24:07 UTC

📖 Field
A field $F$ is a set which, under addition, forms an additive abelian group, and for which the nonzero elements form a multiplicative group. The two operations are connected by the distributive law $a(b+c) = ab + ac$ and $(b+c)a = ba + ca$.
From: gal-artin
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2026-04-04 06:40:18 UTC

📖 Solvable Group (Preview)
A group $G$ is solvable if there exists a chain of subgroups $\\{e\\} = G_0 \\triangleleft G_1 \\triangleleft G_2 \\triangleleft \\cdots \\triangleleft G_k = G$ where each $G_i$ is normal in $G_{i+1}$ and each quotient $G_{i+1}/G_i$ is cyclic of prime order. $S_3$ and $S_4$ are solvable, but $S_5$ is not.
From: gal-edwards
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2026-04-04 05:13:00 UTC

📐 Division Algorithm
For any two polynomials $f(x)$ and $g(x)$ in $F$ with $g \\neq 0$: $f(x) = q(x) \\cdot g(x) + r(x)$ where $q(x)$ and $r(x)$ are unique and $\\deg(r) < \\deg(g)$.
Proof: Subtract suitable multiples of $g(x)$ from $f(x)$ to reduce the degree. Since the degree decreases at each step, the process terminates with $\\deg(r) < \\deg(g)$. Uniqueness: if $q_1 g + r_1 = q_2 g + r_2$, then $(q_1 - q_2)g = r_2 - r_1$, and the degree constraint forces $q_1 = q_2$, $r_1 = r_2$.
From: gal-artin
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2026-04-03 06:50:48 UTC

📖 Norm and Trace
For a finite extension $K/F$ and $\\alpha \\in K$, the **norm** $N_{K/F}(\\alpha) = \\det(L_\\alpha)$ and the **trace** $T_{K/F}(\\alpha) = \\operatorname{tr}(L_\\alpha)$, where $L_\\alpha: K \\to K$ is left multiplication by $\\alpha$.
From: gal-morandi
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2026-04-02 10:38:01 UTC

📖 Separable Polynomial
A polynomial $f \\in F[X]$ is separable if it has no repeated roots in any extension of $F$. An algebraic element is separable if its minimal polynomial is separable.
From: gal-weintraub
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2026-03-02 03:20:11 UTC

💡 The Significance of Aura Colors
Each color in the aura corresponds to a specific quality of soul life. Blue tones indicate devotion and religious feeling; red expresses passion and anger; yellow radiates from intellectual activity; green relates to adaptability and empathy; violet indicates spiritual aspiration. The clarity, brightness, and stability of the colors reveal the degree to which these qualities are purified and de...
From: steiner-GA90a
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2026-03-01 23:56:56 UTC

💡 Proposition 6.1.7 (Uniqueness of Limits)
Let $(a_n)$ be a real sequence. If $(a_n)$ converges to $L$ and also to $L\
Proof: Suppose $(a_n)$ converges to both $L$ and $L' \\neq L$.
Let $\\varepsilon := |L - L'|/3 > 0$.
There exists $N$ with $|a_n - L| \\leq \\varepsilon$ for $n \\geq N$, and $M$ with $|a_n - L'| \\leq \\varepsilon$ for $n \\geq M$.
For $n := \\max(N, M)$:
$|L - L'| \\leq |L - a_n| + |a_n - L'| \\leq...
From: tao-analysis-1
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2026-03-01 22:58:06 UTC

📖 Definition: Pivot
A pivot is the first nonzero entry in a row of a matrix in echelon form. The pivot positions determine the structure of the solution space.
From: Linear Algebra
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2026-03-01 21:54:50 UTC

📐 Simple Algebraic Extensions
Let $K$ be a given field and let $G(X)$ be an irreducible polynomial with coefficients in $K$. Then one can construct a field $K(t)$ such that: (1) $K(t)$ contains $K$, (2) $K(t)$ contains an element $t$ with $G(t) = 0$, and (3) every element of $K(t)$ can be expressed as a polynomial $b_0 + b_1 t + \\cdots + b_\\nu t^\\nu$ where $\\nu < \\deg G$. Moreover, any two such fields are naturally iso...
Proof: Let $R$ be the set of all polynomials in $X$ with coefficients in $K$. Two elements are congruent mod $G$ if their difference is divisible by $G(X)$. The quotient $L$ is a ring. The mapping $k \\mapsto$ [class of constant $k$] embeds $K$ into $L$. The class of $X$ is a root of $G$ in $L$. The Euc...
From: gal-edwards
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2026-03-01 20:58:14 UTC

💡 Proposition I.18
In any triangle the greater side subtends the greater angle.
From: Euclid's Elements
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2026-03-01 19:54:11 UTC

💡 Proposition III.23
On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
From: Euclid's Elements
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2026-03-01 19:02:44 UTC

📐 Lines in Homogeneous Coordinates
A line in $\\mathbb{RP}^2$ is the set of points $[x : y : z]$ satisfying $ax + by + cz = 0$ for some $(a, b, c) \\neq (0, 0, 0)$. The line can be represented by $[a : b : c]$.
From: Four Pillars of Geometry
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2026-03-01 17:56:16 UTC

📖 Carmichael Number
A Carmichael number is composite $n$ that is a Fermat pseudoprime to all bases $a$ with $\\gcd(a,n) = 1$.
From: Algebraic Number Theory
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2026-03-01 16:59:04 UTC

📐 Successive Derivatives of a Polynomial
For a polynomial of degree $n$, the $(n+1)$th derivative and all subsequent derivatives are zero.
From: Beginner Calculus
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2026-03-01 15:55:31 UTC

📐 Inverse Function Theorem
If f: R^n → R^n is C¹ and Df(a) is invertible, then f is a local diffeomorphism near a.
From: analysis-pugh
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2026-03-01 14:57:14 UTC

💡 The Threefold Soul
Between the ego and the three lower bodies lie three soul members: the Sentient Soul (the ego working in the astral body), the Intellectual Soul or Mind Soul (the ego working in the etheric body), and the Consciousness Soul (the ego working in the physical body). These three soul members represent stages of the ego\
From: steiner-GA90a
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2026-03-01 14:00:44 UTC

📐 Spectral Theorem for Symmetric Matrices
A real symmetric matrix $A$ has all real eigenvalues, and eigenvectors from different eigenspaces are orthogonal. $A$ can be diagonalized as $A = Q\\Lambda Q^T$ with orthogonal $Q$.
Proof: Real eigenvalues: For $A\\mathbf{x} = \\lambda\\mathbf{x}$ with $A = A^T$,
$\\bar{\\mathbf{x}}^TA\\mathbf{x} = \\lambda\\|\\mathbf{x}\\|^2$ and
$\\bar{\\mathbf{x}}^TA^T\\mathbf{x} = \\bar{\\mathbf{x}}^TA\\mathbf{x} = \\bar{\\lambda}\\|\\mathbf{x}\\|^2$.
Thus $\\lambda = \\bar{\\lambda}$, so $\...
From: Linear Algebra
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2026-03-01 13:28:25 UTC

📐 Archimedean Property
For every real number x, there exists a natural number n such that n > x. Equivalently, for any ε > 0, there exists n ∈ N with 1/n < ε.
From: analysis-pugh
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2026-03-01 11:53:29 UTC

📐 When is 2 a QR?
$(2/p) = 1$ if $p \\equiv \\pm 1 \\pmod{8}$; $(2/p) = -1$ if $p \\equiv \\pm 3 \\pmod{8}$.
From: Algebraic Number Theory
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2026-03-01 10:57:18 UTC

📖 OP_CHECKLOCKTIMEVERIFY (BIP-65)
OP_CLTV compares the top stack item to the transaction nLockTime. If nLockTime < stack value, the script fails. Enables absolute time locks: "cannot spend until block N or time T".
From: bips
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2026-03-01 09:57:41 UTC

💡 Conservative Design
I wanted to design a system that could do everything that might be needed in the future, without having to go back and change the fundamentals.
From: satoshi
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2026-03-01 09:01:36 UTC

💡 Fermat Prime Condition
If $2^n + 1$ is prime, then $n$ must be a power of $2$.
Proof: Suppose $n$ has an odd divisor $k > 1$, so $n = km$ for some $m$.
Using the factorization for odd $k$: $x^k + 1 = (x+1)(x^{k-1} - x^{k-2} + \\cdots + 1)$:
$2^n + 1 = 2^{km} + 1 = (2^m)^k + 1 = (2^m + 1)(\\cdots)$.
Since $2^m + 1 > 1$, this shows $2^n + 1$ is composite.
Therefore $n$ can h...
From: numbers-geometry
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2026-03-01 07:57:09 UTC

📐 Upper-Triangular Matrix
Every operator on a finite-dimensional nonzero complex vector space has an upper-triangular matrix with respect to some basis.
From: linalg-axler
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2026-03-01 07:22:42 UTC

💡 Fee Market Transition
In a few decades when the reward gets too small, the transaction fee will become the main compensation for nodes. When there are more transactions, fees can remain low.
From: satoshi
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2026-03-01 06:28:29 UTC

📐 Theorem 8.3.SB (Schröder-Bernstein)
If there exist injections $f : A \\to B$ and $g : B \\to A$, then there exists a bijection between $A$ and $B$.
From: tao-analysis-1
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2026-03-01 05:10:43 UTC

📐 Bézout
For any integers $a$ and $b$, there exist integers $s$ and $t$ such that $\\gcd(a, b) = sa + tb$.
Proof: Run the Euclidean algorithm: $a = q_1 b + r_1$, $b = q_2 r_1 + r_2$, $\\ldots$
Back-substitute from the last nonzero remainder to express $\\gcd(a,b)$ as a linear combination of $a$ and $b$.
This is the extended Euclidean algorithm.
From: num-theor-stilwell
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2026-03-01 03:34:35 UTC

📐 Difference of Squares
$a^2 - b^2 = (a + b)(a - b)$. Geometrically: the gnomon (L-shaped region) in a square can be rearranged into a rectangle.
Proof: Consider a square of side $a$. Remove a square of side $b$ from one corner, leaving an L-shaped gnomon.
The gnomon can be cut and rearranged into a rectangle of dimensions $(a + b) \\times (a - b)$.
Therefore $a^2 - b^2 = (a + b)(a - b)$.
From: Four Pillars of Geometry
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2026-02-26 00:00:09 UTC

Robin Linus just had the first Binohash transaction mined on Bitcoin mainnet.
This means covenants — transaction introspection — without a soft fork. Using FindAndDelete, a piece of Satoshi's original code that sat dormant for 15 years.
MARA's Slipstream mined the non-standard TX.
The paper requires understanding ECDSA signature puzzles, SIGHASH modes, the birthday paradox, binomial coefficients, HORS subset signatures, and more.
Magic Internet Math Academy has a complete course on Binohash — and courses on every prerequisite to understand it.
The math is the sovereignty.
magicinternetmath.com
#binohash #bitvm #bitcoin #covenants #math

2026-02-10 15:47:59 UTC

📐 Menger
The maximum number of edge-disjoint $u$-$v$ paths equals the minimum size of an edge cut separating $u$ and $v$.
Proof: Reduce to vertex version: construct $G\
From: Introduction to Graph Theory
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2026-02-10 14:08:44 UTC

📐 Second Isomorphism Theorem
If $H \\le G$ and $N \\trianglelefteq G$, then $HN \\le G$, $H \\cap N \\trianglelefteq H$, and $HN/N \\cong H/(H \\cap N)$.
From: df-course
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2026-02-10 12:38:01 UTC

💡 New Metrics Required
Each institution type needs Bitcoin-specific metrics: Pensions need funded status/termination timeline, corporates need future value focus, credit needs subordination redefinition.
From: bfi
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2026-02-10 11:37:12 UTC

💡 The Stages of Cosmic Formation
The cosmos came into manifestation through a sequence of progressive densification: First, the spiritual archetypes existed in the formless (arupa) region of the highest spiritual world. Then these archetypes were given form (rupa) in the lower spiritual world. Then they were expressed as astral images in the astral world. Finally they condensed into physical-etheric matter. This sequence — aru...
From: steiner-GA90a
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2026-02-10 10:39:15 UTC

📐 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
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2026-02-10 09:43:12 UTC

💡 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
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2026-02-10 08:29:32 UTC

📖 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
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2026-02-10 06:47:03 UTC

📖 Sample Definition
A function $f: A \\to B$ is a mapping from set $A$ to set $B$.
From: orwell-1984
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2026-02-10 05:22:15 UTC

💡 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
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2026-02-10 03:43:05 UTC

💡 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
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2026-02-10 00:06:48 UTC

📖 Definition V.8 (Proportion)
A proportion in three terms is the least possible.
From: Euclid's Elements
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2026-02-09 23:22:07 UTC

📐 Polygon Area
Any polygon can be decomposed into triangles, so its area is the sum of triangle areas.
From: numbers-geometry
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2026-02-09 22:20:27 UTC

📖 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
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2026-02-09 21:24:20 UTC

📖 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
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2026-02-09 19:53:06 UTC

📐 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
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2026-02-09 18:37:15 UTC

📐 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
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2026-02-09 17:36:48 UTC

🎮 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
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2026-02-09 16:43:06 UTC

📖 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
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2026-02-09 15:38:02 UTC

📐 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
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2026-02-09 14:01:34 UTC

📐 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
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2026-02-09 12:33:55 UTC

📖 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
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2026-02-09 11:32:46 UTC

📖 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
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2026-02-09 10:42:22 UTC

💡 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
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2026-02-09 09:43:06 UTC

💡 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
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2026-02-09 08:30:24 UTC

📐 Whitney
A graph $G$ with at least 3 vertices is 2-connected if and only if every pair of edges lies on a common cycle.
Proof: If 2-connected, any two edges $e = xy$ and $f = uv$ can be put on a cycle: by Menger, there are 2 internally disjoint $x$-$u$ paths. Together with edges $e$ and $f$ and paths from $y$ and $v$, we can form a cycle through both edges. Conversely, if every pair of edges lies on a cycle, there\
From: Introduction to Graph Theory
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2026-02-09 06:47:34 UTC

📐 Schur
(1) Any $G$-homomorphism between irreducible representations is either zero or an isomorphism. (2) If $k$ is algebraically closed, any $G$-endomorphism of an irreducible representation is a scalar multiple of the identity.
From: df-course
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2026-02-09 05:21:22 UTC

📐 Division Algorithm
For any integers $a$ and $b$ with $b > 0$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $a = bq + r$ and $0 \\le r < b$.
Proof: \\textbf{Existence:} Let $S = \\{a - bk : k \\in \\mathbb{Z}, a - bk \\geq 0\\}$. Since $S$ is non-empty (choose $k$ sufficiently negative) and bounded below by 0, by the Well-Ordering Principle, $S$ has a minimum element $r = a - bq$ for some $q$. If $r \\geq b$, then $a - b(q+1) = r - b \\geq 0...
From: df-course
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2026-02-09 03:35:30 UTC

💡 Proposition I.20 (Triangle Inequality)
In any triangle two sides taken together in any manner are greater than the remaining one.
From: Euclid's Elements
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2026-02-08 23:59:34 UTC

💡 Proposition VII.31
Any composite number is measured by some prime number.
From: Euclid's Elements
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2026-02-08 23:02:54 UTC

📐 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: tontines
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2026-02-08 21:59:26 UTC

📖 Legendre Symbol
For odd prime $p$: $(a/p) = 0$ if $p|a$; $(a/p) = 1$ if $a$ is QR mod $p$; $(a/p) = -1$ if $a$ is QNR mod $p$.
From: Algebraic Number Theory
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2026-02-08 21:01:08 UTC

📐 Rationality of Action
Action is necessarily always rational in the sense that it involves selecting means believed suitable for attaining ends. Irrational action does not exist—people can be mistaken about which means achieve their ends, but error is not irrationality.
From: Human Action
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2026-02-08 19:55:45 UTC

📐 Theorem 2.47 (Connected Subsets of ℝ)
A subset $E$ of $\\mathbb{R}$ is connected if and only if: whenever $x, y \\in E$ and $x < z < y$, we have $z \\in E$. That is, the connected subsets of $\\mathbb{R}$ are precisely the intervals.
From: rudin
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2026-02-08 19:07:32 UTC

📐 Theorem 10.5.1 (L
Let $f, g : (a, b) \\to \\mathbb{R}$ be differentiable with $g\
From: tao-analysis-1
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2026-02-08 17:59:06 UTC

📖 Euler
$\\phi(n)$ is the number of integers from 1 to $n$ that are relatively prime to $n$.
From: intro-discrete
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2026-02-08 17:03:32 UTC

📖 Definition 5.5.1 (Upper Bound)
Let $E$ be a subset of $\\mathbb{R}$, and let $M$ be a real number. We say $M$ is an **upper bound** for $E$ iff $x \\leq M$ for every $x \\in E$.
From: tao-analysis-1
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2026-02-08 15:59:31 UTC

📐 Distance Formula
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Proof: By the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical displacements:
$d^2 = (x_2-x_1)^2 + (y_2-y_1)^2$
Therefore $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
From: Men of Mathematics
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2026-02-08 15:01:13 UTC

📖 Definition 4.3.4 (ε-Closeness for Rationals)
Let $\\varepsilon > 0$ be a rational. We say rationals $x$ and $y$ are **$\\varepsilon$-close** iff $|x - y| \\leq \\varepsilon$.
From: tao-analysis-1
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2026-02-08 14:06:01 UTC

📖 Exponential Distribution
A random variable $X$ has an \\textbf{exponential distribution} with parameter $\\lambda > 0$ if its distribution function is $F(t) = 1 - e^{-\\lambda t}$ for $t \\geq 0$, and $F(t) = 0$ for $t < 0$. The density function is $f(t) = \\lambda e^{-\\lambda t}$ for $t \\geq 0$.
From: calc2
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2026-02-08 13:32:57 UTC

💡 Proposition III.12
If two circles touch one another externally, the straight line joining their centres will pass through the point of contact.
From: Euclid's Elements
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2026-02-08 11:55:02 UTC

💡 The Principle of Correlation Between Planes
Every being, force, and form on the physical plane has its counterpart on the astral and devachanic planes. Physical matter is the condensed expression of astral forces, which are in turn the densified expression of devachanic archetypes. What appears as a physical object to the senses appears as a living, luminous form on the astral plane and as a sounding, creative thought-being on the devach...
From: steiner-GA90a
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2026-02-08 11:01:18 UTC

📐 Theorem 4.14 (Image of Compact)
If $f: X \\to Y$ is continuous and $K \\subset X$ is compact, then $f(K)$ is compact.
Proof: Let $\\{V_\\alpha\\}$ be an open cover of $f(K)$.
Since $f$ is continuous, each $f^{-1}(V_\\alpha)$ is open.
These sets cover $K$: $K \\subset \\bigcup_\\alpha f^{-1}(V_\\alpha)$.
Since $K$ is compact, finitely many suffice: $K \\subset f^{-1}(V_{\\alpha_1}) \\cup \\cdots \\cup f^{-1}(V_{\\alp...
From: rudin
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2026-02-08 09:59:11 UTC

📖 Ends and Means
Every action aims at an end (goal) and employs means to achieve it. Means are always scarce relative to ends; otherwise no action would be needed.
From: Man, Economy, and State
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2026-02-08 09:03:47 UTC

💡 Hidden Purpose and Sacrifice
The apparent abandonment of great potential by Francisco and other strikers is actually purposeful sacrifice in service of a larger cause.
From: Atlas Shrugged
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2026-02-08 08:01:47 UTC

📖 Invertible Matrix
An $n \\times n$ matrix $A$ is \\textbf{invertible} (or \\textbf{nonsingular}) if there exists an $n \\times n$ matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I_n$. A matrix that is not invertible is called \\textbf{singular}.
From: calc2
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2026-02-08 07:28:02 UTC

📐 Theorem 11.41 (Riesz-Fischer)
$L^2(\\mu)$ is complete with respect to the norm $\\|f\\|_2 = \\left(\\int |f|^2\\right)^{1/2}$.
From: rudin
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2026-02-08 05:51:24 UTC

📖 Linear Transformation
A linear transformation $T: \\mathbb{R}^n \\to \\mathbb{R}^m$ satisfies $T(\\mathbf{u} + \\mathbf{v}) = T(\\mathbf{u}) + T(\\mathbf{v})$ and $T(c\\mathbf{v}) = cT(\\mathbf{v})$.
From: Four Pillars of Geometry
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2026-02-08 03:46:55 UTC

💡 Equilateral Triangle Construction (Euclid I.1)
Given a line segment $AB$, construct circles centered at $A$ and $B$ with radius $|AB|$. If they intersect at point $C$, then $\\triangle ABC$ is equilateral.
Proof: Since $C$ lies on the circle centered at $A$, we have $|AC| = |AB|$ by definition of circle.
Since $C$ lies on the circle centered at $B$, we have $|BC| = |AB|$ by definition of circle.
Therefore $|AB| = |AC| = |BC|$, making $\\triangle ABC$ equilateral.
From: Four Pillars of Geometry
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2026-02-08 00:03:02 UTC

📐 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
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2026-02-07 23:01:44 UTC

📖 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: https://mathacademy-cyan.vercel.app/#/section/84
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2026-02-07 21:58:50 UTC

📐 Absolute Convergence Implies Convergence
If $\\sum |a_n|$ converges, then $\\sum a_n$ converges.
From: Real Analysis
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2026-02-07 21:00:19 UTC

📐 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
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2026-02-07 19:54:48 UTC

📐 Derived Demand
Demand for factors is derived from demand for the consumers\
From: Man, Economy, and State
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2026-02-07 19:05:04 UTC

📐 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
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2026-02-07 17:58:06 UTC

📐 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
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2026-02-07 17:03:19 UTC

📐 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
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2026-02-07 15:58:19 UTC

💡 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
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2026-02-07 14:59:19 UTC

📖 Function
A function $f: A \\to B$ assigns to each element of $A$ exactly one element of $B$.
From: intro-discrete
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2026-02-07 14:04:14 UTC

💡 Proposition III.14
In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.
From: Euclid's Elements
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2026-02-07 13:32:25 UTC

📖 Entrepreneurial Loss
Arises from incorrect forecasts; result of misallocating resources. Penalty for poor judgment about future consumer wants.
From: Man, Economy, and State
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2026-02-07 11:55:04 UTC

📖 Pure Rate of Interest
The premium on present goods over future goods, determined by time preference. Interest is not the "price of money" but the price of time.
From: Man, Economy, and State
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2026-02-07 11:00:38 UTC

💡 The Esoteric Interpretation of Genesis
The seven "days" of creation in Genesis do not describe the physical formation of the Earth in seven literal days but the seven great stages of cosmic evolution. The first "day" (separation of light from darkness) describes the Saturn stage. The second "day" (separation of waters above from waters below) describes the Sun stage. Each subsequent day corresponds to a further stage of densificatio...
From: steiner-GA90a
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2026-02-07 09:58:13 UTC

📐 Unique Solution Characterization
For an $n \\times n$ square matrix $A$, the following are equivalent: (a) $AX = B$ has a unique solution for every $B$, (b) $AX = 0$ has only the trivial solution, (c) The columns of $A$ are linearly independent, (d) $\\text{rank}(A) = n$, (e) $A$ is invertible.
From: calc2
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2026-02-07 09:02:59 UTC

📐 Total Differential
If both $x$ and $y$ change simultaneously in a function $f(x, y)$, the total change is approximately $df = \\frac{\\partial f}{\\partial x}dx + \\frac{\\partial f}{\\partial y}dy$.
From: Beginner Calculus
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2026-02-07 07:58:46 UTC

📖 The Double-Spend Problem
The problem is the payee cannot verify that one of the owners did not double-spend the coin. A common solution is to introduce a trusted central authority that checks every transaction for double spending, but this requires trust.
From: satoshi
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2026-02-07 07:21:06 UTC

📖 Tangent Line to a Curve
The tangent line to a curve at a point is the line that touches the curve at that point without crossing it locally. Its slope equals the derivative of the function at that point.
From: Beginner Calculus
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2026-02-07 06:24:07 UTC

📐 Riesz Representation Theorem
Every continuous linear functional $f$ on a Hilbert space $H$ has the form $f(v) = \\langle v, u \\rangle$ for a unique $u \\in H$.
Proof: If $f = 0$, take $u = 0$. Otherwise, $\\ker f$ is a closed hyperplane. Take $u_0 \\perp \\ker f$ with $f(u_0) = 1$. Set $u = \\overline{f(u_0)/\\|u_0\\|^2} \\cdot u_0$.
From: adv_linalg
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2026-02-07 04:58:15 UTC

📐 Intermediate Value Theorem
If $f$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then $f(c) = k$ for some $c \\in (a,b)$.
From: Real Analysis
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2026-02-07 03:13:43 UTC

📖 Definition 2.45 (Connected Set)
A set $E$ is **connected** if it cannot be written as the union of two nonempty separated sets.
From: rudin
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2026-02-06 23:58:04 UTC

📖 Composition of Transformations
Let $U, V, W$ be sets. Let $T: U \\to V$ and $S: V \\to W$. The \\textbf{composition} $ST$ is the function $ST: U \\to W$ defined by $(ST)(x) = S[T(x)]$ for every $x$ in $U$.
From: calc2
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2026-02-06 22:58:56 UTC

📐 Baby-Step Giant-Step Algorithm
Compute discrete log in $O(\\sqrt{n})$ time and space, where $n = |G|$.
Proof: Set $m = \\lceil\\sqrt{n}\\rceil$.
Baby steps: compute $g^0, g^1, \\ldots, g^{m-1}$ and store in table.
Giant steps: compute $hg^{-m}, hg^{-2m}, \\ldots$ until match found.
If $g^j = hg^{-im}$, then $x = im + j$.
At most $m$ steps of each type, so $O(\\sqrt{n})$ total.
From: Algebraic Number Theory
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2026-02-06 21:57:44 UTC

📐 Theorem 3.2.1 (Russell
There is no "set of all sets." More precisely, if we assume a universal set $\\Omega$ exists (containing all objects), we get a contradiction.
Proof: Suppose for sake of contradiction that a universal set $\\Omega$ exists.
Define $S := \\{x \\in \\Omega : x \\notin x\\}$. Since $S \\in \\Omega$ (by universality), we can ask: is $S \\in S$?
- If $S \\in S$, then by definition of $S$, we have $S \\notin S$. Contradiction!
- If $S \\notin S$, t...
From: tao-analysis-1
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