## 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 |
|----------------------------------------|------------------------|
| \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] | \[
\begin{align*}
\nabla \cdot \mathbf{E} &= \frac{Q}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\n\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\n\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{align*}
\] |
| 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. | 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. |