# The 20 Most Important Mathematical Equations

* A journey through the most elegant and influential formulas in mathematics

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