Blog
Occasional Random thoughts from us.
Rational Functions: One Six-Step Routine That Covers AP Precalculus 1.7 to 1.10
End behavior, zeros, vertical asymptotes and holes are taught as four separate sections. They are really one procedure with four different questions asked of it — and the whole thing fits on six lines of scratch paper.
Bayes' Theorem and the Mathematics of AI
A 99%-accurate test for a rare condition is wrong most of the times it says "positive." Bayes' theorem explains why, and the same formula helped break Enigma, find a lost airliner and train AI systems.
Probability: The Mathematics Behind Language Models
A language model is a probability machine: it estimates how likely each next word is. Here is how that works, from counting word pairs to perplexity, and why always picking the most likely word produces terrible text.
The Mathematics Behind Large Language Models
A large language model turns text into numbers, runs them through layers of matrix math, and outputs a probability for every possible next token. Here is the full pipeline, step by step, with the key equations.
What Does "Attention" Mean in AI? The Mathematics Explained
Attention is a weighted average where the weights come from dot products. Here is the famous formula softmax(QKᵀ/√d)V worked through with real numbers, including why that square root is there.
Cosine Similarity: The Math Behind Semantic Search
Semantic search finds "affordable laptops" when you type "cheap computers" by measuring the angle between vectors. Here is the cosine similarity formula, a worked example, and how search scales to millions of documents.
What Is a Vector? The Math Behind AI Embeddings
In 2013, researchers found that "king" − "man" + "woman" lands near "queen" in a space of numbers. Here is what a vector is, how AI turns words into vectors, and why distance in that space captures meaning.
The Mathematics Behind Neural Network Training
Training a neural network is a loop: predict, measure error, compute gradients, update weights. Here is the math of each step, plus the tricks (initialization, Adam, dropout) that make deep networks trainable at all.
The Math Behind Neural Networks — Without the Hype
Strip away the brain metaphors and a neural network is weighted sums, a simple bend, and the chain rule, repeated at scale. Here is the math, step by step.
Gradient Descent Explained With Simple Mathematics
Gradient descent is walking downhill in fog: feel the slope, take a step, repeat. Here it is worked by hand on a simple function, including what happens when the step size is too small, just right, or too big.
Why Derivatives Matter in Machine Learning
A derivative tells a model which way to adjust each of its billions of weights. Here is what derivatives measure, why estimating them numerically is hopeless at scale, and how automatic differentiation makes training possible.
The Mathematics Behind Machine Learning
Machine learning is mostly curve fitting: pick a model, measure its error, and nudge its numbers downhill. Here are the four areas of math that make it work.