allenai--olmocr/olmocr/bench/sample_data/olmocr_pipeline/mathfuncs_pg1_repeat1.md

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) )	( \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} )

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.