20 Equations That Built Our World

Here's a list of what I think are the 20 equations that built the modern world

1. Pythagorean Theorem ${ a }^{ 2 } + { b }^{ 2 } = { c }^{ 2 }$ Discovered: c. 1800 BCE (Babylonians)

2. Chinese Remainder Theorem $N=\sum_{i=1}^{n}{{r}_{i} M_{i} s_{i} (mod M)}$ Discovered: 3rd Century CE (Master Sun)

3. Binomial Theorem ${(a+b)}^{n}=\sum_{k=0}^{n}{\left( \begin{matrix} n \\ k \end{matrix} \right) {a}^{k} {b}^{n-k}}$ Discovered: 11th Century CE (Al-Karaji, Jia Xian, and Omar Khayyam)

4. Fundamental Theorem of Calculus $\int _{ a }^{ b }{ f(x)dx } =F(b)-F(a)$ Discovered: 1667 CE (James Gregory)

5. Compound Interest $P(t)={P}_{0} {\left(1+\frac{r}{n}\right)}^{nt}$ Discovered: 1683 CE (Jacob Bernoulli)

6. Newton’s Second Law of Motion $F=\frac{dp}{dt}$ Discovered: 1687 CE (Isaac Newton)

7. Taylor’s Theorem $f(x)=f(a)+\sum_{n=1}^{\infty}{\frac{{f}^{n} (a)}{n!} {(x-a)}^{n}}$ Discovered: 1712 CE (Brook Taylor)

8. Wave Equation $\frac{{\partial}^{2}u}{\partial {t}^{2}}={c}^{2} \frac{{\partial}^{2} u}{\partial {x}^{2} }$ Discovered: 1746 CE (Jean le Rond d’Alembert)

9. Euler’s Formula ${e}^{ix}=cos⁡(x)+isin(x)$ Discovered: 1748 CE (Leonhard Euler)

10. Euler-Lagrange Equation $\frac{d}{dt} \frac{\partial L(x,\dot x )}{\partial \dot x }- \frac{\partial L(x,\dot x )}{\partial x}=0$ Discovered: 1750s CE (Leonhard Euler and Joseph-Louis Lagrange)

11. Normal Distribution $\phi (x)=\frac{1}{\sqrt{2π{\sigma}^{2}} } {e}^{-\frac{{(x-μ)}^{2}}{2{\sigma}^{2}} }$ Discovered: 1809 CE (Carl Friedrich Gauss)

12. Fourier Transform $F(k)=\int_{-\infty}^{\infty}f(x) {e}^{-2πikx}dx$ Discovered: 1822 CE (Jean-Baptiste Fourier)

13. Maxwell’s Equations $\nabla \times E=-\frac{\partial B}{\partial t}$ $\nabla \times B={\mu}_{0} \left(J+{\epsilon}_{0} \frac{∂E}{∂t}\right)$ $\nabla \bullet E=\frac{\rho}{{\epsilon}_{0}}$ $\nabla \bullet B=0$ Discovered: 1861 CE (James Clerk Maxwell)

14. Boltzmann Entropy $S={k}_{B} \ln {⁡\Omega}$ Discovered: 1875 CE (Ludwig Boltzmann)

15. Lorentz Force $F=q(E+v \times B)$ Discovered: 1889 CE (Oliver Heaviside), 1892 CE (Hendrik Lorentz)

16. Lorentz Transformation ${x}^{\prime}={\gamma}(x-vt)$ ${t}^{\prime}=\gamma(t-vx/{c}^{2})$ Discovered: 1890s CE (Hendrik Lorentz and Joseph Larmor)

17. Einstein Mass-Energy Relation ${E}^{2}={m}^{2} {c}^{4}+{p}^{2} {c}^{2}$ Discovered: 1905 CE (Albert Einstein)

18. Schrödinger Equation $i\hbar \frac{\partial \Psi}{\partial t}=H\Psi$ Discovered: 1925 CE (Erwin Schrödinger)

19. Cobb-Douglas Production Function $Y=A{K}^{\alpha} {L}^{\beta}$ Discovered: 1927 CE (Charles Cobbs and Paul Douglas)

20. Black-Scholes Equation $\frac{\partial V}{\partial t}+\frac{1}{2} {\sigma}^{2} {S}^{2}=rV-rS \frac{\partial V}{\partial S}$ Discovered: 1973 CE (Fischer Black and Myron Scholes)