Mathematics learning guides

Build intuition for the logic, probability, and algebra that power computer science.

Articles in Mathematics

Sets, Functions, and Relations: The Language of CS — A gentle introduction to describing collections, mappings, and relationships precisely. 5 min read · 2026-10-28
Proof by Induction: Climb an Infinite Ladder — Prove a statement for every natural number by checking one starting step and showing how each step follows. 5 min read · 2026-10-27
Modular Arithmetic and the Euclidean Algorithm — Work with remainders like clock numbers, then use repeated division to find the greatest common divisor. 6 min read · 2026-10-26
Probability for Computer Science Beginners — Reason about random outcomes with sample spaces, events, and conditional probability. 6 min read · 2026-10-25
Vectors and Matrices: Transformations in Disguise — Represent data as vectors and see how matrix multiplication can rotate, scale, and combine it. 6 min read · 2026-10-24
Fourier Series: Building Signals from Simple Waves — See how sine and cosine waves combine to approximate complex periodic signals, then experiment with an interactive Fourier Lab. 8 min read · 2026-10-28
Propositional Logic and Truth Tables — Propositional logic represents statements as true or false and combines them with operators such as AND, OR, and NOT. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Boolean Algebra and Logic Simplification — Boolean algebra applies identities to simplify expressions while preserving their truth values. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Set Operations: Union, Intersection, and Difference — Set operations describe which elements belong to combined collections and underpin database and query reasoning. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Combinatorics and the Product Rule — The product rule counts sequences of independent choices by multiplying the number of options at each step. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Permutations and Combinations — Permutations count ordered selections, while combinations count selections where order does not matter. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
The Binomial Theorem and Pascal Structure — The binomial theorem expands powers of two-term sums using coefficients that count how many ways terms are selected. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Conditional Probability and the Sample Space — Conditional probability restricts attention to outcomes where a condition is true, then measures the event inside that smaller space. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Bayes’ Theorem and Updating Beliefs — Bayes’ theorem reverses a conditional relationship by combining prior probability with how likely the evidence is under each case. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Expected Value for Randomized Algorithms — Expected value is the probability-weighted average outcome and helps describe randomized algorithm behavior. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Variance and Measuring Spread — Variance measures the average squared distance from a distribution's mean, emphasizing outcomes far from the center. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Markov Chains and State Transitions — A Markov chain models transitions among states where the next-state distribution depends on the current state. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Modular Inverses and Solving Congruences — A modular inverse of a modulo m is a value x for which ax leaves remainder one when divided by m. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
The Sieve of Eratosthenes for Prime Tables — The Sieve of Eratosthenes marks multiples of each discovered prime to list all primes up to a bound. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Complex Numbers on the Plane — A complex number a+bi corresponds to point (a,b), where multiplication combines scaling and rotation. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Vector Magnitude and Normalization — A vector's magnitude is its Euclidean length, and normalization scales a nonzero vector to length one. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
The Dot Product and Angle Between Vectors — The dot product combines corresponding vector components and relates direction alignment to the cosine of their angle. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Matrix Inverses and Reversible Transformations — A square matrix has an inverse when it represents a reversible linear transformation with nonzero determinant. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Eigenvectors and Stable Directions — An eigenvector keeps its direction under a matrix transformation and is scaled by its associated eigenvalue. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Derivatives as Local Rates of Change — A derivative measures the limiting rate at which a function changes near an input value. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Limits and Continuity at a Point — A limit describes the value a function approaches as its input gets close to a point, even when the function value differs or is undefined there. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Logarithms as Inverse Exponents — A logarithm answers which exponent produces a value, turning multiplicative growth into additive reasoning. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Linear Recurrences and Characteristic Roots — A linear recurrence defines each sequence value as a weighted combination of earlier values and can often be solved using characteristic roots. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Entropy and Information Content — Entropy measures uncertainty in a probability distribution and is largest when outcomes are evenly likely. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Floating-Point Precision and Rounding — Floating-point formats approximate many real numbers with finite binary precision, so some decimal fractions cannot be represented exactly. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29
Coordinate Transforms and Change of Basis — A coordinate transform changes how a vector is represented, while the underlying geometric vector can remain the same. This guide works through a concrete example and the trade-offs to remember. 2 min read · 2026-10-29