Understanding LLMs: How a Large Language Model Works — From the Mathematics to the Practice

Some of the answers are brilliant. Some are nonsense delivered with total conviction. Both come out of the same mathematics.Language models arrived at work faster than anyone could explain how they work. We use them for writing, for analysis, for code and for decisions — at work and at home — on the strength of intuition and the vendor's marketing copy.This book turns intuition into understanding.It starts with Bayes' theorem of 1763 and runs through the perceptron, backpropagation and the transformer to today's systems: APIs, RAG, agents and MCP. Along the way it explains tokenisation, embeddings and the attention mechanism — with concrete metaphors, with working Python code, with a bibliography of 56 sources and notes pointing to the original publications.By the end you will know:how the model knows what it knows — and why it sometimes invents things with a straight face;when to trust it and when not to — and how to check: golden sets, LLM-as-judge, monitoring;what can realistically be built — and what is smoke;what security looks like in practice — prompt injection, red-teaming, guardrails, OWASP;where the limits are — mathematical, not marketing. The mathematics is explained but never takes over: the formulas sit in "Math Under the Hood" boxes you can skip without losing the thread. The appendices hold a glossary of 87 terms, a mathematical cheat sheet and six checklists for practitioners.For everyday users of chat assistants, for product managers, lawyers, marketers and doctors deciding where to let a model into a process — and for anyone who simply likes to understand how things work.And when you have finished, try answering the last question in this book: did a human write these sentences?
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