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

\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]

\[ \nabla \cdot \mathbf{B} = 0 \]

\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]

\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]

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.