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Artificial Intelligence

Tokens, Embeddings, and Vector Representations

A High School & College Primer on How AI Models Turn Words Into Math

You've heard that AI models like ChatGPT "read" text — but computers don't actually understand words. So what's really happening under the hood? If you've tried to dig into natural language processing and hit a wall of jargon, this guide cuts through it.

**TLDR: Tokens, Embeddings, and Vector Representations** explains, step by step, how a language model takes raw text and turns it into something a neural network can actually compute with. You'll learn why models chop sentences into tokens instead of letters or whole words, how Byte-Pair Encoding decides where those cuts happen, and what an embedding layer is doing when it converts a token ID into a list of hundreds of numbers. If you've ever wondered how AI language models process text before generating a single output, this is the answer.

From there, the guide covers the geometry that makes it all work: why similar words land near each other in vector space, how cosine similarity measures meaning, and why the famous "king − man + woman = queen" analogy actually holds mathematically. The final sections bridge theory to practice — covering contextual embeddings from transformers, semantic search, and retrieval-augmented generation (RAG).

This book is written for high school and early college students, developers who are curious about the ML stack beneath their tools, and anyone who wants tokenization and embeddings explained simply without wading through academic papers. It's short on purpose: 15 focused pages, no filler.

Grab it, read it in an afternoon, and walk into your next AI course or project with the foundation everyone assumes you already have.

What you'll learn
  • Explain what a token is and how tokenizers like BPE split text into subword units.
  • Describe what an embedding vector is and why high-dimensional space can encode meaning.
  • Interpret cosine similarity and vector arithmetic (king - man + woman ≈ queen).
  • Distinguish static embeddings (word2vec, GloVe) from contextual embeddings (BERT, GPT).
  • Connect tokens and embeddings to real applications like search, RAG, and LLM inputs.
What's inside
  1. 1. From Text to Numbers: Why Models Need Tokens
    Why neural networks can't read text directly and the basic idea of breaking language into discrete units a model can index.
  2. 2. How Tokenizers Actually Work: BPE and Subwords
    A walkthrough of Byte-Pair Encoding and subword tokenization, with concrete examples of how words split and why.
  3. 3. Embeddings: Turning Token IDs Into Meaning
    How an embedding layer maps a token ID to a dense vector and why those vectors place similar words near each other.
  4. 4. The Geometry of Meaning: Similarity and Vector Math
    Cosine similarity, distance, and the famous analogy arithmetic that made embeddings famous.
  5. 5. Static vs. Contextual Embeddings
    Why 'bank' needs two different vectors depending on context, and how transformers produce embeddings that change with the sentence.
  6. 6. Where This Shows Up: Search, RAG, and LLM Inputs
    How tokens and embeddings power semantic search, retrieval-augmented generation, and the input pipeline of every modern LLM.
Published by Solid State Press
Tokens, Embeddings, and Vector Representations cover
TLDR STUDY GUIDES

Tokens, Embeddings, and Vector Representations

A High School & College Primer on How AI Models Turn Words Into Math
Solid State Press

Tokens, Embeddings, and Vector Representations

A High School & College Primer on How AI Models Turn Words Into Math

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you're taking an intro computer science or data science course, preparing for a class unit on artificial intelligence, or simply curious about how ChatGPT turns words into numbers, this book was written for you. It's equally useful for a self-taught programmer who keeps hitting terms like "embeddings" and "tokens" without a clear explanation, and for a parent or tutor helping a student navigate NLP concepts for the first time.

This is a machine learning text representation primer that covers, in plain language, how AI language models process text — from splitting a sentence into tokens, to mapping those tokens onto word vectors. Tokenization and embeddings explained simply, without buried prerequisites. You'll also see how transformers use these vectors as inputs and how cosine similarity captures meaning geometrically. About 15 pages, no padding.

Read straight through the first time — each section builds on the last. Work through the worked examples in place, then test yourself with the problem set at the end.

Contents

  1. 1 From Text to Numbers: Why Models Need Tokens
  2. 2 How Tokenizers Actually Work: BPE and Subwords
  3. 3 Embeddings: Turning Token IDs Into Meaning
  4. 4 The Geometry of Meaning: Similarity and Vector Math
  5. 5 Static vs. Contextual Embeddings
  6. 6 Where This Shows Up: Search, RAG, and LLM Inputs
Chapter 1

From Text to Numbers: Why Models Need Tokens

Every neural network, at its core, is a machine that multiplies numbers. It adds them, scales them, passes them through functions — but everything it does reduces to arithmetic on numerical arrays. Hand it a sentence like "The cat sat on the mat" and it has no idea what to do, for the same reason a calculator has no idea what to do when you type a letter: the hardware simply isn't built for it. Before a model can process language, that language has to become numbers.

The question is how. That turns out to matter enormously.

Tokens are the answer — discrete units of text that a model works with one at a time. Think of a token as the atomic unit of language for a given model: not necessarily a word, not necessarily a letter, but whatever chunk the model has been built to recognize and process. A vocabulary is the complete list of all tokens a model knows. Every token in the vocabulary gets assigned a unique integer ID, so the sentence "The cat sat" might become the list [464, 3797, 6096]. Those integers are something a neural network can actually use.

Why not just feed in raw characters?

The most obvious approach is to treat each character as a unit. There are only about 100 printable ASCII characters, so the vocabulary stays tiny. But character-level models have a serious problem: they have to learn everything about language from scratch, one character at a time. The word "Saturday" is eight characters, and the model has to figure out, across thousands of examples, that those eight characters together mean a day of the week. Relationships between distant characters in a long document become extremely hard to track. In practice, character-level models struggle to learn grammar, meaning, and long-range context as well as models that work with larger chunks.

Why not just use whole words?

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon