# The 20 Most Important Mathematical Equations

A journey through the most elegant and influential formulas in mathematics

| 1. Euler's Identity | 3. The Fundamental Theorem of Calculus |
|---------------------|---------------------------------------|
| $e^{i\pi} + 1 = 0$  | $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ |

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

| 2. Pythagorean Theorem | 4. Maxwell's Equations |
|-------------------------|------------------------|
| $a^2 + b^2 = c^2$       | $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ |

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.

|                 |                                 |
|-----------------|---------------------------------|
|                 | $\nabla \cdot \mathbf{B} = 0$  |
|                 | $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ |
|                 | $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ |

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.