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World of Mathematics

The formula library

Formulas. The equations that shaped mathematics.

Ten classical equations, each a turning point in the history of ideas — and ten modern additions that extend the tradition into the 20th and 21st centuries.

Classical

The classical ten.

Ordered chronologically, each a foundation of modern mathematics.

algebra

The Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The universal solution to any quadratic equation ax² + bx + c = 0 — a closed-form recipe that always works.

Open → geometry

The Pythagorean Theorem

a2+b2=c2a^2 + b^2 = c^2

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The foundation stone of Euclidean geometry.

Open → algebra

The Binomial Theorem

(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

A closed-form expansion of (a + b)^n — the identity that underlies combinatorics, probability, and the algebra of polynomials.

Open → analysis

The Fundamental Theorem of Calculus

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\, dx = F(b) - F(a)

The deep connection between differentiation and integration — the central result that unifies the whole of calculus.

Open → analysis

Euler's Identity

eiπ+1=0e^{i\pi} + 1 = 0

The fundamental relation linking the five essential constants of mathematics through the exponential map on the complex plane — and a gateway into modern analysis.

Open → geometry

Euler's Polyhedron Formula

V−E+F=2V - E + F = 2

For any convex polyhedron, vertices minus edges plus faces equals two. A simple relationship that launched topology.

Open → probability

Bayes' Theorem

P(A∣B)=P(B∣A)⋅P(A)P(B)P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}

The formula for updating probabilities in light of new evidence — the mathematical heart of inference, learning, and decision-making under uncertainty.

Open → probability

The Gaussian Distribution

f(x)=1σ2π e−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} \, e^{-\frac{(x - \mu)^2}{2\sigma^2}}

The bell curve — the probability distribution that appears whenever many independent random effects add up. The cornerstone of statistics.

Open → analysis

The Fourier Transform

f^(ξ)=∫−∞∞f(x) e−2πiξx dx\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i \xi x}\, dx

The integral transform that decomposes a function into its frequency spectrum — foundation of harmonic analysis, quantum mechanics, and modern signal processing.

Open → number theory

The Riemann Zeta Function

ζ(s)=∑n=1∞1ns\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

An infinite sum that hides the distribution of the primes. The object at the heart of the most important unsolved problem in mathematics.

Open →

Modern

Modern formulas.

Iconic equations of the 20th and 21st centuries — from general relativity to contemporary applied mathematics.

geometry

Atiyah–Singer Index Theorem

ind⁡(D)=∫Mch⁡(σ(D))⋅Td⁡(TM⊗C)\operatorname{ind}(D) = \int_M \operatorname{ch}(\sigma(D)) \cdot \operatorname{Td}(TM \otimes \mathbb{C})

The crown jewel of 20th-century mathematics — a formula equating the analytic index of an elliptic operator with a topological invariant of the underlying manifold.

Open → analysis

Black–Scholes Equation

∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=0\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0

The partial differential equation governing the price of European options, and the mathematical foundation of quantitative finance.

Open → analysis

Einstein Field Equations

Rμν−12R gμν+Λgμν=8πGc4TμνR_{\mu\nu} - \tfrac{1}{2} R\, g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

The tensorial equations governing the curvature of spacetime — the mathematical heart of general relativity.

Open → probability

Itô's Lemma

df(Xt,t)=∂f∂tdt+∂f∂x dXt+12∂2f∂x2 (dXt)2df(X_t, t) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}\,dX_t + \tfrac{1}{2}\frac{\partial^2 f}{\partial x^2}\,(dX_t)^2

The stochastic chain rule — the calculus identity that makes rigorous analysis of Brownian motion and stochastic differential equations possible.

Open → analysis

The Mandelbrot Set

zn+1=zn2+c,z0=0z_{n+1} = z_n^2 + c, \quad z_0 = 0

The iconic fractal defined by the simplest possible iteration in the complex plane — a touchstone of nonlinear dynamics, complex analysis, and the mathematics of chaos.

Open → geometry

Ricci Flow Equation

∂gij∂t=−2Rij\frac{\partial g_{ij}}{\partial t} = -2 R_{ij}

The geometric heat equation that deforms a Riemannian metric by its Ricci curvature — the engine Perelman used to prove the Poincaré and Thurston geometrization conjectures.

Open → analysis

Schrödinger Equation

iℏ∂ψ∂t=H^ψi\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi

The partial differential equation that governs the evolution of a quantum system's wavefunction — the mathematical core of quantum mechanics.

Open → analysis

Schubart–NEG Master Equation

P(t)=η⋅∫VΦeff(r,t)⋅σeff(E) dVP(t) = \eta \cdot \int_V \Phi_{\text{eff}}(\mathbf{r}, t) \cdot \sigma_{\text{eff}}(E) \, dV

The Schubart–NEG Master Equation as a mathematical framework: volume integral, energy-dependent coupling, functional structure. Developed in 2024 by Holger Thorsten Schubart.

Open → applied

Shannon Entropy

H(X)=−∑i=1npilog⁡piH(X) = -\sum_{i=1}^{n} p_i \log p_i

The measure of information content of a random source. Founded information theory in 1948 and sits at the heart of every modern communication system.

Open → analysis

Yang–Mills Equations

DμFμνa=JνaD^\mu F_{\mu\nu}^a = J_\nu^a

The non-abelian gauge-theory equations that extend Maxwell's electromagnetism to arbitrary compact Lie groups — the mathematical foundation of the Standard Model.

Open →

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