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Equations That Changed Civilization

The formulas that redirected human history — from geometry to relativity.

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  1. 1

    Pythagorean Theorem

    Beginner
    a2+b2=c2a^2 + b^2 = c^2

    In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

    Formal Sciences
    Mathematics
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  2. 2

    Newton's Second Law

    Beginner
    F=maF = ma

    Force equals mass times acceleration—the fundamental law of motion.

    Physical Sciences
    Classical Mechanics
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  3. 3

    Mass-Energy Equivalence

    Intermediate
    E=mc2E = mc^2

    Mass and energy are equivalent—a tiny amount of mass contains enormous energy.

    Physical Sciences
    Relativity
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  4. 4

    Maxwell's Equations

    Advanced
    ∇⋅E=ρε0,∇×E=−∂B∂t\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \quad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

    Four equations unifying electricity and magnetism—predicting electromagnetic waves and light.

    Physical Sciences
    Electromagnetism
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  5. 5

    Shannon Entropy

    Intermediate
    H=−∑ipilog⁡2piH = -\sum_{i} p_i \log_2 p_i

    Measures the average information (surprise) in a random variable—how many bits needed to encode it.

    Formal Sciences
    Information Theory
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