# The 20 Most Important Mathematical Equations 

A journey through the most elegant and influential formulas in mathematics

## 1. Euler's Identity

$$
e^{i \pi}+1=0
$$

Connects five fundamental constants ( $\mathrm{e}, \mathrm{i}, \mathrm{n}, 1,0$ ), revealing the profound relationship between exponential functions and trigonometry.

## 2. Pythagorean Theorem

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

In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.

## 3. The Fundamental Theorem of Calculus

$$
\int_{a}^{b} f(x) d x=F(b)-F(a)
$$

Establishes that differentiation and integration are inverse operations. If $F$ is an antiderivative of $f$, the definite integral equals $F(b)-F(a)$. Revolutionized mathematical problem-solving.

## 4. Maxwell's Equations

$$
\begin{gathered}
\nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_{0}} \\
\nabla \cdot \mathbf{B}=0 \\
\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}
\end{gathered}
$$

Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.