17 Equations that Changed the World

From the wonderful book by Ian Stewart, here are the equations themselves; read the book to know more about them.

1. Pythagoras Theorem

$$ a^2 + b^2 = c^2 $$

Pythagoras, 530 BC (Wikipedia)

2. Logarithms

$$ \log xy = \log x + \log y $$

John Napier, 1610 (Wikipedia)

3. Calculus

$$ \frac{df}{dt} = \lim\limits_{h \to 0} \frac{f(t+h) - f(t)}{h} $$

Newton, 1668 (Wikipedia)

4. Law of Gravity

$$ F = G \frac{m_1 m_2}{d^2} $$

Newton, 1687 (Wikipedia)

5. The Square Root of Minus One

$$ i^2 = -1 $$

Euler, 1750 (Wikipedia)

6. Euler’s Formula for Polyhedra

$$ F - E + V = 2 $$

Euler, 1751 (Wikipedia)

7. Normal Distribution

$$ \Phi(x) = \frac{1}{\sigma\sqrt{2 \pi}}e^{- \frac{(x - \mu)^2}{2 \sigma^2}} $$

C.F. Gauss, 1810 (Wikipedia)

8. Wave Equation

$$ \frac{\partial ^2 u}{\partial t^2} = c^2 \frac{\partial ^2 u}{\partial x^2} $$

J. d’Alembert, 1746 (Wikipedia)

9. Fourier Transform

$$ f(\omega) = \int_{-\infty}^\infty f(x)\,e^{-2 \pi i x \omega} \,dx $$

J. Fourier, 1822 (Wikipedia)

10. Navier-Stokes Equation

$$ \rho(\frac{\partial v}{\partial t} + v \cdot \nabla{v}) = - \nabla \rho + \nabla \cdot T + f $$

C. Navier, G. Stokes, 1845 (Wikipedia)

11. Maxwell’s Equations

$$ \nabla \cdot E = 0 $$ $$ \nabla \times E = - \frac{1}{c} \frac{\partial H}{\partial t} $$ $$ \nabla \cdot H = 0 $$ $$ \nabla \times H = \frac{1}{c} \frac{\partial E}{\partial t} $$

J. C. Maxwell, 1865 (Wikipedia)

12. Second Law of Thermodynamics

$$ dS \geq 0 $$

J. Boltzmann, 1874 (Wikipedia)

13. Relativity

$$ E = mc^2 $$

Einstein, 1905 (Wikipedia)

14. Schrödinger’s Equation

$$ ih \frac{\partial}{\partial t}\psi = H \psi $$

E. Schrödinger, 1927 (Wikipedia)

15. Information Theory

$$ H = - \sum p(x) \log p(x) $$

C. Shannon, 1949 (Wikipedia)

16. Chaos Theory

$$ x_{t + 1} = k x_t (1 - x_t) $$

Robert May, 1975 (Wikipedia)

17. Black-Scholes Equation

$$ \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} - rV = 0 $$

F. Black, M. Scholes, 1990 (Wikipedia)

PS: reference for $\LaTeX$ symbols and syntax, used to write the symbols in the Markdown source file of this entry.