From Śhūnya to Super intelligence: 1,500 Years of Equations That Built Intelligence
- Zero (Śhūnya) - Āryabhata & Brahmagupta | 5th–7th Century CE
- Pythagoras’ Theorem - c. 530 BCE, Greece
- Logarithms — John Napier | 1614
- Calculus —Newton & Leibniz | 1665–1693
- Law of Gravity —Newton | 1687
- Wave Equation — Jean d’Alembert | 1747
- Imaginary Numbers (i² = –1) - Euler | 1572 1748
- Polyhedra F–E+V = 2 - Euler | 1758
- Normal Distribution – Moivre, Gauss | 1810
- Fourier Transform - Jean Fourier | 1822
- Navier–Stokes Equations | 1822–1845
- Maxwell’s Equations | 1861–1865
- Entropy - Clausius, Kelvin & Boltzmann | 1874
- Relativity - Albert Einstein | 1905
- Schrödinger’s Equation | 1926
- Information Theory - Claude Shannon | 1948
- Chaos Theory - P.–F. Verhulst / Robert May | 1838
- Black–Scholes Equation | 1973
In the age of Artificial Intelligence, we often think of algorithms as modern marvels, creations of technology. But the truth is far older and far more human.
Behind every AI model – from ChatGPT to Tesla Autopilot, from Google Maps to DeepMind Weather – lies a story of timeless mathematics and human curiosity.
A story of that began with a circle drawn on palm leaves – and continues today in silicon wafers.
This is not just a story of equations. It is a story of endurance…how ideas from monks, mathematicians, and dreamers became the invisible scaffolding for today’s Intelligent Machines.
Zero (Śhūnya): The Nothing That Created Everything
Origin: Āryabhata (5th c. CE), Brahmagupta (7th c. CE, India)
When India gave the world Zero, it did not just add a new number – it unlocked the concept of nothingness as a mathematical entity.
That leap made positional notation, algebra, and later calculus possible.
AI Application Explained:
Every digital system today – whether it is training a neural network or processing a chatbot query – runs on binary
(0s and 1s). Zero is not emptiness, it’s the ultimate reality. It allows machines to compute logic, store memory, and model intelligence.
Without Śhūnya, there would be no computation, no data, and certainly no AI.
Pythagoras’ Theorem: Geometry That Gave Vision to Machines
Formula: 𝑎2 + 𝑏2 = 𝑐2
Origin: Ancient Greece (popularized 530 BCE; Babylonian roots)
This simple equation of distance birthed the geometry of the visible world.
It allows AI to interpret space, shape, and form – from mapping roads to reconstructing human faces in 3D.
AI Application Explained:
Computer vision, LiDAR, and AR engines such as Google Maps or Apple ARKit calculate depth using the Pythagorean relationship.
It helps self-driving cars gauge distances and robots navigate with precision – guiding modern mobility.
Logarithms: Napier’s Gift of Simplicity in Complexity
Formula: log𝑏(𝑥𝑦) = log𝑏 𝑥 + log𝑏 𝑦
Origin: John Napier, 1614 (Scotland)
Napier transformed complexity into clarity – replacing laborious multiplications with elegant additions.
AI Application Explained:
In AI, logarithms stabilize model training. They appear in cross-entropy, log-loss, that keep neural networks numerically balanced. The core equation for the attention mechanism in today’s GenAI LLMs that solved human language’s nuances of context is the Attention(Q, K, V) SoftMax function. Every time your AI model avoids exploding gradients, it’s Napier bringing stability into the chaos of computation.
Calculus: The Pulse of Learning
Formula: 𝑑𝑦/𝑑𝑥 = limℎ→0 𝑓(𝑥+ℎ)−𝑓(𝑥)/ℎ
Origin: Isaac Newton & Gottfried Leibniz, 17th Century
Calculus quantifies change – and learning is nothing, but change measured wisely.
AI Application Explained:
The heart of machine learning – gradient descent – is calculus in motion.
When a neural network corrects itself during training, it computes derivatives to minimize prediction error. Without calculus, there would be no backpropagation, and AI would never learn from experience.
Law of Gravity: The Mathematics of Motion
Formula: 𝐹 = 𝐺 (𝑚1𝑚2/𝑟2)
Origin: Isaac Newton, 1687
Gravity unified the heavens and the Earth – a cosmic order written in mathematics.
AI Application Explained:
Today, physics-informed AI models use Newton’s principles to simulate orbits, planetary motion, and even fluid mechanics.
From NASA simulators to autonomous drones, AI learns to predict motion using the same formula that made an apple fall.
Wave Equation: The Sound of AI
Formula: ∂2𝑢/∂𝑡2 = 𝑐2(∂2𝑢/∂𝑥2)
Origin: Jean d’Alembert, 1747
D’Alembert’s equation explains how waves move through space – from sound to light.
AI Application Explained:
Voice synthesis models like WaveNet and Descript AI are built on this idea.
They generate speech by simulating how sound waves propagate – turning text into human-like voice. Every digital assistant you talk to owes its tone to this 18th-century equation.
Imaginary Numbers: Complex Signals, Real Magic
Formula: 𝑖2 = −1
Origin: Bombelli (1572); Euler notation (1748)
When mathematicians embraced the “imaginary,” they expanded reality.
AI Application Explained:
Imaginary numbers underpin Fourier transforms, the backbone of modern signal and image processing. Whether it’s an MRI scan, an audio equalizer, or computer vision filters – complex numbers make AI capable of “hearing” and “seeing” beyond the ordinary.
Euler’s Formula for Polyhedra: Geometry of 3D Intelligence
Formula: 𝐹 − 𝐸 +𝑉 = 2
Origin: Leonhard Euler, 1758
Euler discovered a symmetry that connects the faces, edges, and vertices of every solid object.
AI Application Explained:
In 3D reconstruction, computer graphics, and GAN based image generation, this rule ensures that AI models generate geometrically consistent worlds. Without Euler, 3D AI wouldn’t have been possible.
Normal Distribution: The Logic of Uncertainty
Formula: Φ(𝑥) = 1/√2𝜋𝜎 𝑒 ^ (𝑥−𝜇)2/2𝜎^2
Origin: de Moivre (1733), Gauss (1809)
The bell curve defines randomness and order together.
AI Application Explained:
In AI, Gaussian models measure uncertainty – predicting probabilities in everything from ad targeting to fraud detection.
Whenever an AI says, “I’m 80% confident,” that confidence comes from Gauss’s curve.
Fourier Transform: The Mathematics of Perception
Formula: ℱ(𝑓)(𝜉) = ∫𝑓(𝑥)𝑒^−2𝜋𝑖𝜉𝑥𝑑𝑥
Origin: Jean Fourier, 1822
Fourier decomposed complexity into rhythm.
AI Application Explained:
Music recognition on Spotify, speech analysis on YouTube, and vision in FNet architectures all rely on Fourier transforms to break signals into frequency patterns. Without Fourier, AI would be almost deaf.
Navier–Stokes: The Flow of Prediction
Formula: 𝜌(∂𝑣/∂𝑡 + 𝑣⋅∇𝑣) = −∇𝑝+𝜇∇^2𝑣
Origin: Navier (1822), Stokes (1845)
Navier–Stokes equations describe how fluids move.
AI Application Explained:
Today’s weather and climate AI systems (DeepMind, IBM Watson) use neural surrogates of these equations for turbulence and fluid flow – enabling real-time forecasting. Mathematics that once described rivers now can predict cyclones.
Maxwell’s Equations: The Pulse of Wireless Intelligence
Formula: ∇ ⋅ 𝐸 = 𝜌/𝜀0,∇×𝐵 = 𝜇0𝐽+𝜇0𝜀0(∂𝐸/∂𝑡)
Origin: James Clerk Maxwell, 1860s
Maxwell united electricity, magnetism, and light – and birthed the modern communication era.
AI Application Explained:
From Tesla’s sensors to 5G networks, electromagnetic simulations in AI-driven systems rely on these laws to interpret signals.
They power radar, wireless optimization, and even autonomous perception.
Second Law of Thermodynamics: Entropy as Exploration
Formula: 𝑑𝑆 ≥ 0
Origin: Clausius, Kelvin, Boltzmann (1850–1870s)
Nature tends toward disorder – but learning thrives in controlled chaos.
AI Application Explained:
Reinforcement learning uses entropy regularization to explore new actions rather than sticking to the safe ones.
It’s how AlphaGo learned creativity – through a mathematical dance with disorder.
Relativity: Energy as Computation
Formula: 𝐸 = 𝑚𝑐2
Origin: Albert Einstein, 1905
Einstein’s elegance connects mass and energy, showing equivalence across domains.
AI Application Explained:
Relativity inspires energy-efficient AI hardware design and physics-based simulation models, where conservation laws guide neural predictions.
Schrödinger’s Equation: The Quantum Inspiration
Formula: 𝑖ℏ(∂𝜓/∂𝑡) =𝐻̂𝜓
Origin: Erwin Schrödinger, 1926
Quantum theory teaches us that observation changes outcome.
AI Application Explained:
Quantum neural networks and probabilistic AI systems use similar superposition principles to represent many possibilities at once – bringing quantum logic closer to human intuition.
Information Theory: Shannon’s Language of Intelligence
Formula: 𝐻=−∑𝑝(𝑥)log𝑝(𝑥)
Origin: Claude Shannon, 1948
Shannon defined information and uncertainty mathematically – the foundation of all communication.
AI Application Explained:
Every large language model – from ChatGPT to Gemini – optimizes cross-entropy to minimize uncertainty. When an AI predicts the next word, it performs Shannon’s calculation billions of times per day.
Chaos Theory: Predicting the Unpredictable
Formula: 𝑥𝑛+1=𝑘𝑥𝑛(1−𝑥𝑛)
Origin: Verhulst (1838); Robert May (1976)
Chaos reveals patterns within unpredictability.
AI Application Explained:
In financial forecasting, biology, and climate prediction, AI models use chaos equations to understand nonlinear systems – learning when small causes lead to massive outcomes.
Black–Scholes Equation: Mathematics That Moves Money
Formula: (∂𝑉/∂𝑡) + (1/2 ) 𝜎2𝑆2(∂2𝑉/∂𝑆2) +𝑟𝑆(∂𝑉/∂𝑆) − 𝑟𝑉=0
Origin: Fischer Black, Myron Scholes, Robert Merton (1973)
This equation priced risk and birthed the modern financial world.
AI Application Explained:
Autonomous trading systems, portfolio optimization models, and risk simulators use Black–Scholes derivatives to quantify volatility.
Mathematics not only predicts motion – it predicts markets.
The Immortality of Ideas
Mathematics doesn’t just calculate; it endures.
Every equation here survived centuries because it captured something universal.
And in that endurance lies the story of every intelligent system ever built – human or artificial.
By Hrishikesh Sherlekar, Founder & CEO, S3K Technologies / MYAIGURU™
Master AI. Lead the Future.
hrishikesh@s3ktech.ai
