Understanding LLMs – A Mathematical Approach to the Engine Behind AI

Published on: 2025-09-01 Last updated on: 2025-11-17 Version: 4

Introduction

Large Language Models (LLMs) have rapidly become one of the most transformative technologies of the 21st century. They write, analyze, summarize, translate, reason, and increasingly participate in the workflows of researchers, engineers, and creators. Yet, beneath all of these capabilities lies a simple and enduring truth: LLMs are mathematical machines.

They are built from equations, trained by optimization, and guided by statistical principles that have been studied for decades. The impressive “intelligence” we see emerges from a small set of mathematical ideas interacting at scale. This book is about those ideas.

Why I Am Writing This Book

Today, many people can use LLMs. But far fewer can explain why they work. The gap between capability and understanding is widening. Engineers, researchers, and business leaders often understand what LLMs can do, but not the mathematical principles that govern their behavior.

This book aims to fill that gap. It is written for readers who want durable knowledge— not a list of model names or APIs that will soon become outdated, but the stable mathematical bedrock that has defined modern AI for decades. I am writing this book because:

  • Mathematics is the part of AI that does not become obsolete. Architectures change, API shapes change, model names change. But probability, information theory, optimization, and linear algebra remain constant.
  • The major breakthroughs in LLMs were discovered through mathematical insight. Attention, scaling laws, embeddings, and training dynamics are all fundamentally mathematical discoveries.
  • Engineers deserve clear explanations that do not assume a PhD. Existing materials are often either overly simplified or unnecessarily technical. This book stays balanced—rigorous yet readable.
  • Understanding the math transforms how you think about AI. LLMs stop being mysterious and begin to look like logical, structured systems with clearly defined strengths and limitations.

Why These Fundamentals Will Not Change

It may seem like AI changes every few months, but the deeper the concept, the longer it lasts. The mathematics that define LLMs—probability distributions, vector spaces, entropy, gradient-based optimization—are not temporary engineering choices. They are structural necessities for any system that learns from data and represents meaning in a high-dimensional space.

Even if a radically new architecture replaces Transformers someday, such a model will still need to:

  • represent information in structured, geometric form,
  • estimate probabilities,
  • update beliefs based on data,
  • optimize a loss function,
  • operate efficiently in high-dimensional spaces.

These requirements are mathematical, not architectural. They will remain true unless we discover a fundamentally different paradigm of computation. That is why this book focuses on ideas that will stay relevant for years—or decades—to come.

What This Book Covers

The book is divided into four major parts:

Part I — Mathematical Foundations for Understanding LLMs

We build intuition for probability, information theory, embeddings, and the historical context of modern NLP.

Part II — The Mathematics of Transformers

We explore attention, positional encodings, Transformer blocks, and the geometry of representation learning.

Part III — Optimization and Large-Scale Training

We examine how LLMs learn: loss functions, gradient descent, implicit bias, scaling laws, and the mathematics of training at scale.

Part IV — Applications, Limitations, and the Road Ahead

We connect mathematical foundations to real-world tasks, societal considerations, engineering practices, and the future directions of the field.

Who This Book Is For

This book is written for:

  • engineers building AI applications,
  • researchers transitioning into LLMs,
  • students seeking mathematical clarity,
  • professionals who want to understand how AI reasons,
  • anyone who prefers intuition supported by rigorous explanation.

You do not need to be a mathematician. You only need curiosity and the willingness to peer beneath the surface.

How to Read This Book

The chapters do not need to be read strictly in order. Each is self-contained, but together they form a comprehensive mental model. Focus on intuition first; equations support understanding, not replace it.

By the end of the book, my goal is that when you watch an LLM produce an output, you can mentally trace the full process:

representation → attention → transformation → probability → output

You will understand not only what LLMs do, but why they think in the way they do.

A Final Thought Before We Begin

We are living through a defining moment in technology. Capabilities will grow, models will evolve, and new discoveries will reshape the field. But the mathematical foundations explored here—those are the roots.

And if this book has a single guiding belief, it is this:

To understand the future of AI, we must first understand the mathematics that built it.

Let us begin.

Table of Contents

  • Introduction
  • Part I — Mathematical Foundations for Understanding LLMs
  • Part II — The Mathematics of Transformers
    • Chapter 4 — Attention: The Heart of Modern LLMs
      • 4.1 Self-Attention: Equation, Geometry, Intuition
      • 4.2 Multi-Head Attention, Norms, Residuals
      • 4.3 Softmax: Stability, Temperature, and Kernel Interpretation
      • 4.4 Attention as a Kernel Method
    • Chapter 5 — Positional Encodings and Sequence Structure
      • 5.1 Why Position Matters
      • 5.2 Sinusoidal Encodings and Periodicity
      • 5.3 Relative Position Encodings
      • 5.4 RoPE: Rotation Groups and Complex Geometry
      • 5.5 Positional Encodings and Fourier Analysis
    • Chapter 6 — Transformer Blocks and Representation Power
      • 6.1 FFN Structure
      • 6.2 Activation Functions
      • 6.3 Why “Attention + FFN” Is Powerful
      • 6.4 Expressivity of Transformers
      • 6.5 Universal Approximation & Depth/Width
    • Chapter 7 — Efficiency and Variants of Transformers
      • 7.1 Computational Complexity of Attention
      • 7.2 GPU Memory Math
      • 7.3 FlashAttention
      • 7.4 Multi-Query Attention, Gated Layers
      • 7.5 Low-Rank Approximations of Attention
  • Part III — Optimization and Large-Scale Training
    • Chapter 8 — How Models Learn: Loss and Optimization
      • 8.1 Generalization in Overparameterized Models
      • 8.2 Implicit Bias of Gradient Descent
      • 8.3 Scaling Laws (Kaplan, Chinchilla, Hoffman)
      • 8.4 Open Mathematical Questions in LLM Theory
    • Chapter 9 — Training at Scale
      • 9.1 Data Preprocessing and Its Mathematical Effects
      • 9.2 Mini-Batch Learning, Efficiency, and GPU Parallelism
  • Part IV — Applications, Limitations, and the Road Ahead
    • Chapter 10 — Real-World Applications of LLMs
      • 10.1 Text Generation and Summarization
      • 10.2 Question Answering, Translation, and Other Tasks
    • Chapter 11 — Challenges and Future Directions
      • 11.1 Model Size and Computational Cost
      • 11.2 Bias, Ethics, and Societal Impact
    • Chapter 12 — Practical Knowledge for Engineers
      • 12.1 Next Steps for Deepening Your Understanding
      • 12.2 Tools, Libraries, and Practical Resources
  • Appendix A — LLM Math Cheat Sheet
    • Vectors and Linear Algebra
    • Embeddings
    • Attention Mechanism
    • Softmax Function
    • Loss Functions
    • Optimization and Learning
    • Model Computation Estimate
    • KL divergence and cross-entropy equivalence
    • Maximum likelihood derivation
  • Appendix B — A Statistical Perspective on LLMs
    • Core Concepts in Statistics
    • Information Theory and Entropy
    • Bayesian Thinking: Learning from New Evidence
    • Statistical Inference and Model Evaluation
    • Why LLMs Are Fundamentally Statistical Models
  • Closing
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SHO
SHO
CTO of Receipt Roller Inc., he builds innovative AI solutions and writes to make large language models more understandable, sharing both practical uses and behind-the-scenes insights.