Nostr notes by MagicInternetMath

📖 Field A field $F$ is a set which, under addition, forms an ...

2026-04-04T08:24:07Z

📖 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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📖 Solvable Group (Preview) A group $G$ is solvable if there ...

2026-04-04T06:40:18Z

📖 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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📐 Division Algorithm For any two polynomials $f(x)$ and $g(x)$ ...

2026-04-04T05:13:00Z

📐 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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📖 Norm and Trace For a finite extension $K/F$ and $\\alpha ...

2026-04-03T06:50:48Z

📖 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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📖 Separable Polynomial A polynomial $f \\in F[X]$ is separable ...

2026-04-02T10:38:01Z

📖 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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💡 The Significance of Aura Colors Each color in the aura ...

2026-03-02T03:20:11Z

💡 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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💡 Proposition 6.1.7 (Uniqueness of Limits) Let $(a_n)$ be a ...

2026-03-01T23:56:56Z

💡 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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📖 Definition: Pivot A pivot is the first nonzero entry in a ...

2026-03-01T22:58:06Z

📖 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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📐 Simple Algebraic Extensions Let $K$ be a given field and let ...

2026-03-01T21:54:50Z

📐 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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💡 Proposition I.18 In any triangle the greater side subtends ...

2026-03-01T20:58:14Z

💡 Proposition I.18

In any triangle the greater side subtends the greater angle.

From: Euclid's Elements
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💡 Proposition III.23 On the same straight line there cannot be ...

2026-03-01T19:54:11Z

💡 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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📐 Lines in Homogeneous Coordinates A line in $\\mathbb{RP}^2$ ...

2026-03-01T19:02:44Z

📐 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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📖 Carmichael Number A Carmichael number is composite $n$ that ...

2026-03-01T17:56:16Z

📖 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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📐 Successive Derivatives of a Polynomial For a polynomial of ...

2026-03-01T16:59:04Z

📐 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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📐 Inverse Function Theorem If f: R^n → R^n is C¹ and Df(a) ...

2026-03-01T15:55:31Z

📐 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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💡 The Threefold Soul Between the ego and the three lower ...

2026-03-01T14:57:14Z

💡 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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📐 Spectral Theorem for Symmetric Matrices A real symmetric ...

2026-03-01T14:00:44Z

📐 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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📐 Archimedean Property For every real number x, there exists a ...

2026-03-01T13:28:25Z

📐 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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📐 When is 2 a QR? $(2/p) = 1$ if $p \\equiv \\pm 1 \\pmod{8}$; ...

2026-03-01T11:53:29Z

📐 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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📖 OP_CHECKLOCKTIMEVERIFY (BIP-65) OP_CLTV compares the top ...

2026-03-01T10:57:18Z

📖 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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💡 Conservative Design I wanted to design a system that could ...

2026-03-01T09:57:41Z

💡 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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💡 Fermat Prime Condition If $2^n + 1$ is prime, then $n$ must ...

2026-03-01T09:01:36Z

💡 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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📐 Upper-Triangular Matrix Every operator on a ...

2026-03-01T07:57:09Z

📐 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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💡 Fee Market Transition In a few decades when the reward gets ...

2026-03-01T07:22:42Z

💡 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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📐 Theorem 8.3.SB (Schröder-Bernstein) If there exist ...

2026-03-01T06:28:29Z

📐 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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📐 Bézout For any integers $a$ and $b$, there exist integers ...

2026-03-01T05:10:43Z

📐 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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📐 Difference of Squares $a^2 - b^2 = (a + b)(a - b)$. ...

2026-03-01T03:34:35Z

📐 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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Robin Linus just had the first Binohash transaction mined on ...

2026-02-26T00:00:09Z

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

📐 Menger The maximum number of edge-disjoint $u$-$v$ paths ...

2026-02-10T15:47:59Z

📐 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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📐 Second Isomorphism Theorem If $H \\le G$ and $N ...

2026-02-10T14:08:44Z

📐 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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💡 New Metrics Required Each institution type needs ...

2026-02-10T12:38:01Z

💡 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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💡 The Stages of Cosmic Formation The cosmos came into ...

2026-02-10T11:37:12Z

💡 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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📐 Solvability of Linear Diophantine Equations The equation $ax ...

2026-02-10T10:39:15Z

📐 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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💡 Events in the Second Half of Lemurian Development In the ...

2026-02-10T09:43:12Z

💡 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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📖 The Directly Given (das unmittelbar Gegebene) The ...

2026-02-10T08:29:32Z

📖 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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📖 Sample Definition A function $f: A \\to B$ is a mapping from ...

2026-02-10T06:47:03Z

📖 Sample Definition

A function $f: A \\to B$ is a mapping from set $A$ to set $B$.

From: orwell-1984
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💡 Proposition VII.38 If a number have any part whatever, it ...

2026-02-10T05:22:15Z

💡 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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💡 Who is John Galt? An expression of despair and the central ...

2026-02-10T03:43:05Z

💡 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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📖 Definition V.8 (Proportion) A proportion in three terms is ...

2026-02-10T00:06:48Z

📖 Definition V.8 (Proportion)

A proportion in three terms is the least possible.

From: Euclid's Elements
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📐 Polygon Area Any polygon can be decomposed into triangles, ...

2026-02-09T23:22:07Z

📐 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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📖 Vector Space A vector space $V$ over a field $F$ is a ...

2026-02-09T22:20:27Z

📖 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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📖 Minimum A minimum is a point where a function reaches a ...

2026-02-09T21:24:20Z

📖 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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📐 Theorem 1.21 (Existence of nth Roots) For every real $x > ...

2026-02-09T19:53:06Z

📐 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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📐 Cyclic Decomposition Theorem Every finitely generated module ...

2026-02-09T18:37:15Z

📐 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: https://mathacademy-cyan.vercel.app/#/section/33

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🎮 Interactive: Constructible Polygon Demo See which regular ...

2026-02-09T17:36:48Z

🎮 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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📖 Point of Inflection A point of inflection is where a curve ...

2026-02-09T16:43:06Z

📖 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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📐 The Action Axiom Human beings engage in purposeful behavior ...

2026-02-09T15:38:02Z

📐 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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📐 Area of Circle The area of a circle is proportional to the ...

2026-02-09T14:01:34Z

📐 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: https://mathacademy-cyan.vercel.app/mom-deploy/#/section/2

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📖 Ring A \\textbf{ring} $R$ is a set with two binary ...

2026-02-09T12:33:55Z

📖 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: https://mathacademy-cyan.vercel.app/#/section/34

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📖 Perfect Special Honest Verifier Zero-Knowledge PSHVZK: A PPT ...

2026-02-09T11:32:46Z

📖 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: https://mathacademy-cyan.vercel.app/#/section/4

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💡 The Salvation of Humanity Through the Christ Impulse Without ...

2026-02-09T10:42:22Z

💡 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: https://mathacademy-cyan.vercel.app/#/section/23

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💡 Proposition 2.3.6 (Distributive Law) For any natural numbers ...

2026-02-09T09:43:06Z

💡 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: https://mathacademy-cyan.vercel.app/#/section/4

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📐 Whitney A graph $G$ with at least 3 vertices is 2-connected ...

2026-02-09T08:30:24Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/west-graphs-deploy/#/section/12

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📐 Schur (1) Any $G$-homomorphism between irreducible ...

2026-02-09T06:47:34Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/91

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📐 Division Algorithm For any integers $a$ and $b$ with $b > ...

2026-02-09T05:21:22Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/1

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💡 Proposition I.20 (Triangle Inequality) In any triangle two ...

2026-02-09T03:35:30Z

💡 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
Learn more: https://euclid-deploy.vercel.app/#/section/20

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💡 Proposition VII.31 Any composite number is measured by some ...

2026-02-08T23:59:34Z

💡 Proposition VII.31

Any composite number is measured by some prime number.

From: Euclid's Elements
Learn more: https://euclid-deploy.vercel.app/#/section/210

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, ...

2026-02-08T23:02:54Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/2

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📖 Legendre Symbol For odd prime $p$: $(a/p) = 0$ if $p|a$; ...

2026-02-08T21:59:26Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/koblitz-course-deploy/#/section/5

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📐 Rationality of Action Action is necessarily always rational ...

2026-02-08T21:01:08Z

📐 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
Learn more: https://human-action-deploy.vercel.app/#/section/1

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📐 Theorem 2.47 (Connected Subsets of ℝ) A subset $E$ of ...

2026-02-08T19:55:45Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/12

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📐 Theorem 10.5.1 (L Let $f, g : (a, b) \\to \\mathbb{R}$ be ...

2026-02-08T19:07:32Z

📐 Theorem 10.5.1 (L

Let $f, g : (a, b) \\to \\mathbb{R}$ be differentiable with $g\

From: tao-analysis-1
Learn more: https://mathacademy-cyan.vercel.app/#/section/52

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📖 Euler $\\phi(n)$ is the number of integers from 1 to $n$ ...

2026-02-08T17:59:06Z

📖 Euler

$\\phi(n)$ is the number of integers from 1 to $n$ that are relatively prime to $n$.

From: intro-discrete
Learn more: https://mathacademy-cyan.vercel.app/#/section/5

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📖 Definition 5.5.1 (Upper Bound) Let $E$ be a subset of ...

2026-02-08T17:03:32Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/19

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📐 Distance Formula The distance between two points $(x_1, ...

2026-02-08T15:59:31Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/mom-deploy/#/section/4

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📖 Definition 4.3.4 (ε-Closeness for Rationals) Let ...

2026-02-08T15:01:13Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/13

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📖 Exponential Distribution A random variable $X$ has an ...

2026-02-08T14:06:01Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/170

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💡 Proposition III.12 If two circles touch one another ...

2026-02-08T13:32:57Z

💡 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
Learn more: https://euclid-deploy.vercel.app/#/section/76

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💡 The Principle of Correlation Between Planes Every being, ...

2026-02-08T11:55:02Z

💡 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/27

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📐 Theorem 4.14 (Image of Compact) If $f: X \\to Y$ is ...

2026-02-08T11:01:18Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/29

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📖 Ends and Means Every action aims at an end (goal) and ...

2026-02-08T09:59:11Z

📖 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
Learn more: https://man-econ-state-deploy.vercel.app/#/section/0

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💡 Hidden Purpose and Sacrifice The apparent abandonment of ...

2026-02-08T09:03:47Z

💡 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
Learn more: https://atlas-shrugged-deploy.vercel.app/#/section/15

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📖 Invertible Matrix An $n \\times n$ matrix $A$ is ...

2026-02-08T08:01:47Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/19

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📐 Theorem 11.41 (Riesz-Fischer) $L^2(\\mu)$ is complete with ...

2026-02-08T07:28:02Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/85

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📖 Linear Transformation A linear transformation $T: ...

2026-02-08T05:51:24Z

📖 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
Learn more: https://four-pillars-deploy.vercel.app/#/section/49

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💡 Equilateral Triangle Construction (Euclid I.1) Given a line ...

2026-02-08T03:46:55Z

💡 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
Learn more: https://four-pillars-deploy.vercel.app/#/section/1

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📐 Euclidean Algorithm $\\gcd(a,b) = \\gcd(b, a \\mod b)$. ...

2026-02-08T00:03:02Z

📐 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: https://mathacademy-cyan.vercel.app/koblitz-course-deploy/#/section/1

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📖 Injective Module An $R$-module $Q$ is \\textbf{injective} if ...

2026-02-07T23:01:44Z

📖 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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📐 Absolute Convergence Implies Convergence If $\\sum |a_n|$ ...

2026-02-07T21:58:50Z

📐 Absolute Convergence Implies Convergence

If $\\sum |a_n|$ converges, then $\\sum a_n$ converges.

From: Real Analysis
Learn more: https://ra-deploy-murex.vercel.app/#/section/7

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📐 Pappus Let $A, B, C$ be collinear and $D, E, F$ be collinear ...

2026-02-07T21:00:19Z

📐 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: https://four-pillars-deploy.vercel.app/#/section/4

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📐 Derived Demand Demand for factors is derived from demand for ...

2026-02-07T19:54:48Z

📐 Derived Demand

Demand for factors is derived from demand for the consumers\

From: Man, Economy, and State
Learn more: https://man-econ-state-deploy.vercel.app/#/section/4

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📐 Theorem 2.36 (Nested Compact Sets) If $\\{K_\\alpha\\}$ is a ...

2026-02-07T19:05:04Z

📐 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: https://mathacademy-cyan.vercel.app/#/section/10

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📐 Vertical Angles Theorem When two lines intersect, vertically ...

2026-02-07T17:58:06Z

📐 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: https://mathacademy-cyan.vercel.app/#/section/12

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📐 Cross-Ratio is Projectively Invariant The cross-ratio of ...

2026-02-07T17:03:19Z

📐 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: https://four-pillars-deploy.vercel.app/#/section/37

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💡 World Wisdom and Human Wisdom There exists a cosmic wisdom ...

2026-02-07T15:58:19Z

💡 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: https://mathacademy-cyan.vercel.app/#/section/0

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📖 Function A function $f: A \\to B$ assigns to each element of ...

2026-02-07T14:59:19Z

📖 Function

A function $f: A \\to B$ assigns to each element of $A$ exactly one element of $B$.

From: intro-discrete
Learn more: https://mathacademy-cyan.vercel.app/#/section/7

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💡 Proposition III.14 In a circle equal straight lines are ...

2026-02-07T14:04:14Z

💡 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
Learn more: https://euclid-deploy.vercel.app/#/section/78

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📖 Entrepreneurial Loss Arises from incorrect forecasts; result ...

2026-02-07T13:32:25Z

📖 Entrepreneurial Loss

Arises from incorrect forecasts; result of misallocating resources. Penalty for poor judgment about future consumer wants.

From: Man, Economy, and State
Learn more: https://man-econ-state-deploy.vercel.app/#/section/7

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📖 Pure Rate of Interest The premium on present goods over ...

2026-02-07T11:55:04Z

📖 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
Learn more: https://man-econ-state-deploy.vercel.app/#/section/5

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💡 The Esoteric Interpretation of Genesis The seven ...

2026-02-07T11:00:38Z

💡 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/20

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📐 Unique Solution Characterization For an $n \\times n$ square ...

2026-02-07T09:58:13Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/17

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📐 Total Differential If both $x$ and $y$ change simultaneously ...

2026-02-07T09:02:59Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/calceasy-deploy/#/section/18

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📖 The Double-Spend Problem The problem is the payee cannot ...

2026-02-07T07:58:46Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/2

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📖 Tangent Line to a Curve The tangent line to a curve at a ...

2026-02-07T07:21:06Z

📖 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
Learn more: https://mathacademy-cyan.vercel.app/calceasy-deploy/#/section/12

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📐 Riesz Representation Theorem Every continuous linear ...

2026-02-07T06:24:07Z

📐 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
Learn more: https://mathacademy-cyan.vercel.app/#/section/51

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📐 Intermediate Value Theorem If $f$ is continuous on $[a,b]$ ...

2026-02-07T04:58:15Z

📐 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
Learn more: https://ra-deploy-murex.vercel.app/#/section/10

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📖 Definition 2.45 (Connected Set) A set $E$ is connected ...

2026-02-07T03:13:43Z

📖 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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📖 Composition of Transformations Let $U, V, W$ be sets. Let ...

2026-02-06T23:58:04Z

📖 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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📐 Baby-Step Giant-Step Algorithm Compute discrete log in ...

2026-02-06T22:58:56Z

📐 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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📐 Theorem 3.2.1 (Russell There is no "set of all ...

2026-02-06T21:57:44Z

📐 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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