# The 20 Most Important Mathematical Equations

A journey through the most elegant and influential formulas in mathematics

|   |   |
|---|---|
| **1. Euler’s Identity** | **2. Pythagorean Theorem** |
| \[ e^{i\pi} + 1 = 0 \] | \[ a^2 + b^2 = c^2 \] |
| Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | 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** | **4. Maxwell’s Equations** |
| \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] | \[ \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} \]
| 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. | \[ \nabla \cdot \mathbf{B} = 0 \]
| \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
| \[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] |
| 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. |