The Mathematics Behind Large Language Models

Since OpenAI launched ChatGPT in November 2022, the pace of innovation in Large Language Models (LLMs) has accelerated, with nearly every major technology company releasing its own generative-AI models.

But have you ever stopped and asked yourself 'How do these models actually work?' How do they produce such human-like, fluent responses? At its core, the answer lies in the mathematics powering them.

In this article, we’ll explore the mathematical principles enabling models to “understand” language and why these mathematical insights make AI feel so human.

The Paper That Changed AI: Attention Is All You Need

In 2017, deep learning was already revolutionising language processing and audio analysis through Recurrent Neural Networks (RNNs). But these models were slow and forgetful, processing sentences one word at a time and struggling to maintain context over long sequences.

Then came a groundbreaking insight. Researchers at Google Brain proposed a radical alternative: what if we completely replaced recurrent processing with a mechanism that allowed the model to “pay attention” to all parts of a sentence simultaneously?

This marked the birth of the Transformer architecture, introducing self-attention, a mathematical mechanism allowing each word to weigh its relationship to every other word, in parallel.

The concept was shared in the research paper humbly titled “Attention Is All You Need” and this breakthrough delivered unprecedented improvements in performance, scalability, and interpretability which ignited the era of Large Language Models.

So how does this architecture actually work?

Step One: Turning Words Into Numbers

Before a language model can “understand” text, it must translate these words into numerical representations.

Each word (or word fragment) is called a token and is mapped to a numeric vector. These vectors encode both the meaning of the word and its relationships to other words. For instance:

• “Cat” and “Kitten” share similar vector representations.
• “Run” and “Jog” point toward similar directions within the mathematical space.

This conversion, known as embedding, is foundational. During training, the model learns these vectors by adjusting them based on context, so that words appearing together frequently (like cat and kitten) end up close in the mathematical space. When you type a prompt, the model looks up each token’s vector from this learned space to provide the numerical representation for the next step.

Step Two: Attention Through Dot Products

With words represented numerically, the model begins to assess relationships within a sentence. The self-attention mechanism enables each word to decide the relevance of every other word in a sentence:

1. The model calculates a dot product, a basic linear algebra operation between two vectors. This produces a score indicating how relevant two tokens are to each other in this sentence.
2. These raw similarity scores are then passed through the softmax function, which normalises them into probabilities that sum to 1. Softmax guarantees that the model distributes its attention in a stable, interpretable way: highly relevant words receive large weights, while less relevant ones receive small weights.

For example, in the sentence: “The cat sat on the mat because it was tired.”

• The word “it” gives strong attention to “cat” (the likely referent).
• The word “mat” gets little attention from “it”, since it doesn’t make sense to be “tired.”

By converting dot-product scores into softmax-normalised weights, LLMs determine precisely how much attention one word should pay to another, capturing context, relationships, and nuance.

Step Three: Parallelism via Matrix Multiplication

Transformers perform these calculations simultaneously across every word in a sentence. They leverage matrix multiplication, a parallel-friendly mathematical operation highly optimised on modern GPUs to compute all attention scores at once. In practice, the model arranges each sentence into three learned tables (often called queries, keys, and values) and multiplies them in one big sweep, sometimes through multiple attention heads that each look for a different kind of pattern. This all-at-once approach explains why GPUs from companies like NVIDIA have become so sought-after in recent years and why Transformers can process even long paragraphs with impressive speed.

Step Four: Encoding Word Order

Self-attention alone doesn’t track word order. Transformers solve this by adding positional encodings. These numeric tags are generated from smooth sine-cosine waves. By simply adding these tags to each word’s vector, the model knows whether a token came first or last, letting it distinguish between “the dog chased the cat” and “the cat chased the dog.” Because the tags extend naturally to longer sequences, this subtle mathematical tweak preserves meaning while allowing the model to read anything from short tweets to multi-page documents.

Final Thoughts

The success of Large Language Models stems from surprisingly basic mathematical concepts that work together to create a whole far greater than the sum of its parts.

The embeddings, dot products, softmax normalisation, matrix operations and positional encodings:

Convert words to numbers → Measure their relationships → Prioritise relevant information → Combine mathematically.

The result is an AI that reads and writes with remarkable contextual understanding. As enterprises navigate the future of AI, understanding some of these mathematical foundations provides input into architectural decisions, reveals performance trade-offs, and unlocks the potential of emerging generative technologies.