The idea that language can be meaningfully represented as numbers initially seems completely absurd because words inherently carry meaning, subtle context, and emotional resonance that numbers can’t possibly capture. Yet this seemingly impossible transformation from rich human text to cold numerical representations is precisely what allows LLMs to work their magic. Understanding how this conversion actually happens is absolutely key to understanding how modern AI systems process and generate language.
Table of contents
Open Table of contents
Challenge & Solution
The fundamental problem we need to solve: Computers can only process numbers while language fundamentally carries meaning, so we need some way to bridge this gap. Simple encoding schemes like assigning A=1, B=2, C=3 completely fail because they capture absolutely no semantic meaning or linguistic structure. Manual feature engineering where we define features like has_fur=1 for animals also can’t capture the subtle nuances and complex patterns that make language work.
How LLMs approach this challenge: They learn mathematical representations where similar words automatically map to similar numerical values, related concepts naturally cluster nearby in high-dimensional space, and context dynamically changes these numerical representations based on surrounding words. This learned approach genuinely works where hard-coded approaches completely failed.
The Process
Step 1 - Tokenization breaks text into manageable pieces: The system breaks raw text into subword pieces called tokens, so a word like “unhappiness” gets split into [“un”, “happiness”] which helps the model understand word components. Each unique token gets assigned a unique numerical ID from the vocabulary, so “Hello, world!” might become the sequence [15496, 11, 1917, 0] where each number represents a specific token. Modern systems use algorithms like BPE (Byte Pair Encoding) which automatically learns which character pairs and subword combinations appear most frequently in the training data and should be treated as single tokens.
Step 2 - Embedding converts IDs into meaningful vectors: Each token ID gets transformed into a high-dimensional vector containing 768 to 12,288 individual numbers depending on the model size, so token 15496 might become [0.23, -0.45, 0.12, …] extending across hundreds or thousands of dimensions. These vectors are carefully arranged so that similar tokens end up nearby in this high-dimensional space, creating mathematical relationships that mirror linguistic relationships. The famous example is that “king” minus “man” plus “woman” approximately equals “queen” in this vector space, demonstrating that vector math operations genuinely capture linguistic meaning and relationships.
Step 3 - Contextual representations adapt to meaning: Static embeddings give each word exactly one vector representation regardless of context, like “bank” having a single fixed vector. Contextual embeddings are far more sophisticated where “bank” gets completely different vector representations in “river bank” versus “money bank” based on surrounding context. The surrounding context dynamically modifies the representation so the same word with different meanings receives appropriately different numerical vectors that reflect those different meanings.
Step 4 - Attention mechanisms let words influence each other: Words in a sentence mutually influence each other’s representations through attention, so in “The cat sat on the mat” the representation for “cat” actively attends to both “sat” and “mat” to understand its role in the sentence. Multi-head attention means the model simultaneously considers multiple different perspectives on these relationships, capturing different aspects of how words relate to each other. The attention weights that the model computes reveal exactly what information each word is focusing on from the surrounding context.
Step 5 - Deep layers progressively refine understanding: Deep neural networks with 12 to 96 layers progressively transform these representations through increasingly sophisticated processing at each level. Early layers near the input primarily capture syntactic patterns like grammatical structure and word order, middle layers encode rich semantic meaning about what concepts actually mean, and late layers near the output specialize in task-specific transformations. Each layer successively refines the representation to extract progressively more abstract and task-relevant information from the text.
Step 6 - Generation converts numbers back to language: The final step transforms these numerical representations back into actual words by computing a probability distribution over all possible tokens in the vocabulary. The model samples from this probability distribution with temperature control determining randomness, where temperature=0 produces completely deterministic output and temperature=2 produces highly creative but risky generations. This sampling process repeats iteratively where each generated token influences the next token’s probabilities until the model produces a complete coherent sentence.
Why This Matters
Similarity becomes mathematically computable: We can use cosine similarity and other distance metrics to precisely measure how close two words are in meaning, where “happy” and “joyful” end up very close together in vector space while “happy” and “sad” are positioned far apart, allowing the model to quantify semantic relationships that were previously just intuitive.
Mathematical operations gain linguistic meaning: Word analogies genuinely work through vector arithmetic, where “Paris” minus “France” plus “Italy” approximately equals “Rome” in the vector space, demonstrating that mathematical relationships between vectors accurately preserve and mirror the conceptual relationships between the words they represent.
Context gets dynamically captured in representations: The same word adapts its position in vector space based on context, so “Apple” positions near “fruit” and “orchard” in sentences about eating but shifts to position near “tech” and “iPhone” in sentences about technology, with these adjustments happening dynamically based on the surrounding words in each specific usage.
Learning reduces to mathematical optimization: Training an LLM fundamentally involves adjusting these numerical representations to minimize prediction error across billions of examples, which is pure mathematical optimization rather than programming symbolic rules, allowing the model to discover patterns that humans never explicitly specified.
Interpolation enables concept blending: You can meaningfully blend concepts by combining their vector representations, where 0.5 times “formal” plus 0.5 times “casual” produces a representation that captures semi-formal style, enabling smooth transitions between different linguistic registers and tones.
What Numbers Capture
Semantic meaning and word relationships: Words with similar meanings like “dog,” “puppy,” and “canine” end up positioned very close together in vector space, where synonyms naturally cluster together without anyone explicitly programming these relationships. Syntactic roles and grammatical patterns: Words that serve similar grammatical functions cluster together, where verbs naturally group with other verbs and nouns cluster with other nouns, capturing part-of-speech patterns purely from observing how words are used in context. Relational patterns across words: Consistent relationships like “bigger” minus “big” equaling “smaller” minus “small” emerge automatically, where the model discovers and encodes comparative patterns and other systematic linguistic transformations. Abstract conceptual meaning: Even abstract concepts like “justice,” “freedom,” and “love” receive meaningful numerical encodings that capture their semantic content, with these representations learned purely from observing how these words are used across millions of different contexts. Domain-specific specialized knowledge: Technical vocabulary from specialized domains automatically clusters together, where medical terminology groups with other medical terms and legal jargon clusters with related legal concepts, demonstrating that the model learns domain structure from specialized vocabulary usage patterns.
Limitations
These numerical representations capture statistical patterns in language but absolutely not consciousness or genuine understanding of the world. LLMs “know” that 2+2=4 in the sense that they can predict this pattern, but they don’t genuinely “understand” quantity the way humans do through grounded physical experience. Multimodal models that handle both text and images require fundamentally different embedding spaces for each modality, with complex bridging mechanisms needed to connect these different representation spaces. Floating-point arithmetic inevitably accumulates numerical errors through long sequences of calculations, which can occasionally produce nonsensical or inconsistent results. Individual dimensions aren’t interpretable in any meaningful way, so you can’t look at dimension 742 and determine what concept or feature it represents.
Conclusion
The complete pipeline flows from text to tokens to embeddings to progressive transformations through layers and finally to generation of new text. Language possesses inherent mathematical structure in how words combine and relate to each other, and this structure can be encoded numerically in high-dimensional vector spaces. Neural networks learn these numerical encodings automatically from massive text datasets without explicit programming, and the result is AI systems that genuinely “understand” language in a functional if not conscious sense.
The fundamental insights are that similar meanings automatically map to similar numerical representations, that linguistic relationships are preserved through vector arithmetic, that context dynamically modifies these representations, that mathematical operations over vectors carry genuine linguistic meaning, and that learning involves adjusting these numbers to better predict language patterns.
Language as numbers is the breakthrough that unlocked modern AI capabilities, where every word becomes a vector of numbers, those numbers somehow capture rich semantic meaning and relationships, LLMs process these numerical representations through deep neural networks, and genuine magic happens through pure mathematics.
The philosophical question “What if meaning could be precisely measured and computed?” seemed absurd until recently, but it turns out that meaning genuinely can be quantified and manipulated mathematically.