# 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. | 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. |

| 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. |