Mathematician Ian Stewart compiled this list of seventeen equations in his book *In Pursuit of the Unknown: 17 Equations That Changed the World*. Stewart highlights how each equation, from Pythagoras’s theorem to the Black-Scholes equation, has significantly impacted human history and modern science. His selection showcases the power of mathematics to explain natural phenomena, shape technological innovations, and influence fields as diverse as physics, finance, and information theory. Stewart’s work emphasizes the profound and far-reaching implications of these equations in shaping our understanding of the world.

In the field of machine learning and AI, many of these equations play a crucial role in developing algorithms and optimizing models. Calculus, particularly derivatives and integrals, forms the backbone of optimization techniques used in training models, such as gradient descent. The normal distribution is fundamental in statistics, which is deeply embedded in machine learning for tasks like probabilistic modeling, hypothesis testing, and generating predictions. Fourier transforms are applied in signal processing, which aids in feature extraction from audio or image data. Information theory guides the efficient transmission and processing of data, essential for training neural networks. Chaos theory helps in understanding the sensitivity of algorithms to initial conditions, which can lead to overfitting or instability in AI models. Additionally, principles from the Black-Scholes equation have parallels in reinforcement learning, where similar techniques are used for decision-making under uncertainty. These equations provide the mathematical framework that powers advancements in machine learning and AI, enabling intelligent systems to learn, predict, and adapt effectively.

**Pythagoras’s Theorem**

In Euclidean geometry, the Pythagorean theorem, or Pythagoras’s theorem, defines the relationship between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

**Logarithms**

A logarithm is the power to which a fixed number (the base) must be raised to yield a particular number.

**Calculus**

Calculus is the mathematical study of change, focused on finding derivatives and integrals. It originally evolved from methods based on summing infinitesimal differences. The two main branches are differential calculus and integral calculus.

**Law of Gravity**

Newton’s law of universal gravitation explains that every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

**Square Root of Minus One**

The unit imaginary number, denoted as ( i ), is defined as the square root of -1. In electronics, the symbol ( j ) is used instead of ( i ) to avoid confusion with electric current.

**Euler’s Polyhedral Formula**

Euler’s formula for polyhedra is a mathematical relation that states for any convex 3D polyhedron, the number of vertices ( V ), faces ( F ), and edges ( E ) satisfy the equation ( V + F – E = 2 ).

**Normal Distribution**

The normal distribution, or Gaussian distribution, is a continuous probability distribution commonly used in statistics and the natural and social sciences to model real-valued random variables with unknown distributions.

**Wave Equation**

The wave equation is a second-order linear partial differential equation that describes the propagation of waves, such as sound, light, and water waves, in physics. It is applied in fields like acoustics, electromagnetics, and fluid dynamics.

**Fourier Transform**

A Fourier transform is a mathematical operation that decomposes a function into a sum of sinusoidal functions, each representing a frequency component of the original function.

**Navier-Stokes Equations**

The Navier-Stokes equations describe the motion of viscous fluids, named after Claude-Louis Navier and George Gabriel Stokes, and are fundamental in fluid dynamics.

**Maxwell’s Equations**

Maxwell’s equations are a set of partial differential equations that, along with the Lorentz force law, describe classical electromagnetism, optics, and electrical circuits.

**Second Law of Thermodynamics**

The second law of thermodynamics describes the relationships between heat and other forms of energy, emphasizing that in any energy transfer, entropy (disorder) tends to increase.

**Relativity**

Relativity is the theory that explores how physical phenomena vary with the relative motion of observers and objects, particularly with respect to light, space, time, and gravity.

**Schrodinger’s Equation**

Schrodinger’s equation describes the allowed energy levels of quantum mechanical systems, like atoms, and is used to calculate the probability distribution (wavefunction) of particles at various positions.

**Information Theory**

Information theory is the mathematical study of how information is encoded, transmitted, and processed, often applied in computer circuits and telecommunications.

**Chaos Theory**

Chaos theory deals with dynamical systems that exhibit extreme sensitivity to initial conditions, leading to seemingly random, unpredictable behavior.

**Black-Scholes Equation**

In mathematical finance, the Black-Scholes equation is a partial differential equation that governs the pricing of European call and put options under the Black-Scholes model, and is broadly applicable to various financial derivatives.