The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics

1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.

2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
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
\[ \int_a^b f(x) \, dx = F(b) - F(a) \]
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.

4. Maxwell's Equations
\[
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} = 0 \\
\n\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\
\n\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
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.