PYTHAGORAS’ THEOREM a² + b² = c² _Pythagoras, 530 BC_ -------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------ --------- LOGARITHMS logxy = logx + logy _John Napier, 1610_ CALCULUS $ f}{ t} = {h}$ _Newton, 1668_ LAW OF GRAVITY $F = G {r^2}$ _Newton, 1687_ THE SQUARE ROOT OF MINUS ONE i² = −1 _Euler, 1750_ EULER’S FORMULA FOR POLYHEDRA V − E + F = 2 _Euler, 1751_ NORMAL DISTRIBUTION $\psi(x) = {} e^{2~\rho^2}$ _C. F. Gauss, 1810_ WAVE EQUATION ${\partial t^2} = c^2 {\partial x^2}$ _J. D‘Ambert, 1746_ FOURIER TRANSFORM f(ω)=∫−∞∞f(x)e−2 π i x ωdx _J. Fourier, 1822_ NAVIER-STOKES EQUATION $\rho \left ( }{\partial t} + \cdot \nabla \right ) = - \nabla p + \nabla \cdot T + f$ _C. Navier, G. Stokes, 1845_ MAXWELL’S EQUATIONS ∇ ⋅ E = 0 _J. C. Maxwell, 1865_ $\nabla \times E = - {e} {\partial t}$ ∇ ⋅ H = 0 $\nabla \times H = {e} {\partial t}$ SECOND LAW OF THERMODYNAMICS dS ≥ 0 _L. Boltzmann, 1874_ PAYWALL RELATIVITY E = mc² _Einstein, 1905_ PAYWALL SCHRÖDINGER’S EQUATION $ \hbar {\partial t} \psi = H \psi$ _E. Schrödinger, 1927_ PAYWALL INFORMATION THEORY H = −∑p(x)logp(x) _C. Shannon, 1949_ PAYWALL CHAOS THEORY xt + 1 = k xt(1 − xt) _Robert May, 1975_ PAYWALL BLACK-SCHOLES EQUATION ${2} \sigma^2 S^2 {\partial S^2} + r S {\partial S} + {\partial t} - r V = 0$ _F. Black, M. Scholes, 1990_ PAYWALL EULER’S TRANSFORMATION $^\infty (-1)^n a_n = ^\infty (-1)^n {2^{n+1}}$ _Euler, 1755_ PAYWALL RUSSELL’S PARADOX Let R = {x ∣ x ∉ x}, then R ∈ R ⇔ R ∉ R _Russell, 1902_ GÖDEL’S INCOMPLETENESS THEOREM G(x):=¬Prov(sub(x, x)) ⇒ PA ⊢ G(⌜G⌝) ↔ ¬Prov(⌜G(⌜G⌝)⌝) _Gödel, 1931_