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)

## Comments

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TopNewestBinomial theorem was discovered in 11 century CE?!?! – Ashish Siva · 4 months, 4 weeks ago

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– Steven Zheng · 4 months, 4 weeks ago

Yes it was.Log in to reply

– Ashish Siva · 4 months, 3 weeks ago

Simply amazing.Log in to reply

i have a real problem with 15th. why we named it Lorentz force ? what is such a big deal about it , He just superimposed 2 forces , nothing more. Any high school student can do that . i also just superimposed gravitational and electrostatic force , should i name it SHUBHAM'S force ? – Shubham Dhull · 5 months, 1 week ago

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– Steven Zheng · 5 months ago

The discovery of simple scientific equations does not mean it was not important in building our modern world. Sure, it is the sum of two forces (electric and magnetic), our particle accelerators and other electromagnetic technologies would not be properly designed without Heaviside's or Lorentz' work.Log in to reply