Learning Library
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Introductory Statistics
By Barbara Illowsky, Susan Dean
A comprehensive introductory statistics course from OpenStax, covering descriptive statistics, probability, discrete and continuous random variables, normal distributions, and more.
Linear Algebra
By Jim Hefferon
An open-source textbook on linear algebra, focusing on systems of linear equations, vector spaces, linear maps, determinants, and eigenvalues.
Algebra Primer
By MathMech Team
A specially curated native introductory guide covering core algebraic concepts.
Combinatorics I
By MathMech Team
An introduction to combinations, permutations, and discrete counting principles.
Counting Specific Poker Hands
By MathMech Team
A focused lesson on the Rank-Then-Suit strategy for counting exact poker hand configurations using combinations.
Burnside's Lemma
By MathMech Team
A step-by-step guide to Burnside's Lemma for counting distinct objects under rotational symmetry.
Probability Fundamentals
By MathMech Team
A broad introduction to probability: sample spaces, events, axioms, and the classical formula. Covers complementary counting, uniform probability, and worked real-world examples.
Binomial Distribution
By MathMech Team
The binomial model for repeated independent trials. Covers the binomial formula, mean, variance, and normal approximation with worked examples.
Conditional Probability & Bayes' Theorem
By MathMech Team
How new information updates probability estimates. Covers conditional probability, the multiplication rule, and Bayes' theorem with medical-test and dice examples.
Expected Value & Linearity
By MathMech Team
The weighted average of a random variable's outcomes. Covers the definition, linearity of expectation, and applications to dice, games, and the coupon-collector problem.
Geometric Distribution
By MathMech Team
Counts the number of trials until the first success. Covers the PMF, CDF, memoryless property, and expected value.
Poisson Distribution
By MathMech Team
Models rare events over a fixed interval. Covers the Poisson PMF, relationship to the binomial, and the Poisson approximation theorem.
Negative Binomial Distribution
By MathMech Team
Counts trials until the r-th success. Covers the PMF, relationship to the geometric distribution, and worked examples.
Hypergeometric Distribution
By MathMech Team
Sampling without replacement from a finite population. Covers the PMF, comparison with binomial, and quality-control applications.
Geometric Probability
By MathMech Team
Probability defined by areas and lengths in continuous spaces. Covers uniform distributions on intervals and regions, and dart-board style examples.
Independent Events
By MathMech Team
When the occurrence of one event has no effect on another. Covers the product rule, mutual independence, and common misconceptions.
The Monty Hall Problem
By MathMech Team
A deep dive into the famous three-door paradox. Covers the conditional probability argument, simulation intuition, and generalizations.
Martingales & the Gambler's Ruin
By MathMech Team
Fair-game stochastic processes and stopping times. Covers martingale definition, optional stopping theorem, and the Gambler's Ruin problem.
Pólya Urn & Exchangeability
By MathMech Team
A reinforcement model for probability. Covers the Pólya urn process, exchangeability, and the Beta-Binomial connection.
Probability Generating Functions
By MathMech Team
A transform technique for discrete distributions. Covers the PGF definition, recovering probabilities and moments, and convolution of distributions.
Random Walks
By MathMech Team
Sequential coin-flip paths on the integer line. Covers simple random walks, return-to-origin probability, the reflection principle, and streak probabilities.
The Birthday Problem
By MathMech Team
How many people does it take before two share a birthday? Covers the complementary counting approach, the birthday paradox formula, and generalizations.
Algebra Essentials
By MathMech Team
Core algebraic techniques: solving quadratics, factoring, sequences, logarithms, and polynomial theorems. Includes the quadratic formula, Vieta's formulas, and the Factor Theorem.
Number Theory
By MathMech Team
Divisibility, primes, modular arithmetic, and number-theoretic functions. Covers GCD, LCM, Euler's totient, Wilson's theorem, and the Chinese Remainder Theorem.
Derivatives of Trigonometric Functions
By MathMech Team
An introduction to the derivatives of sine, cosine, and tangent. Covers the geometric definition of trig derivatives, key formulas, and applications.
The Unit Circle
By MathMech Team
Exact values of sine, cosine, and tangent at the fundamental angles using the unit circle. Covers the construction of the unit circle and all quadrant signs.
Trigonometric Ratios
By MathMech Team
SOH-CAH-TOA and the definitions of the six trig functions via right-triangle ratios. Covers the 30-60-90 and 45-45-90 special triangles.
Reciprocal Trig Functions
By MathMech Team
Definitions and values of cosecant, secant, and cotangent. Covers their relationship to sin, cos, tan, and exact values at standard angles.
Pythagorean Identities
By MathMech Team
The three fundamental Pythagorean identities and how to derive and apply them. Covers proofs from the unit circle and uses in simplifying expressions.
Evaluating Trigonometric Expressions
By MathMech Team
A step-by-step guide to evaluating expressions involving sec, sin, and tan at special angles. Covers reciprocal trig functions, Pythagorean identities, and the 30-60-90 triangle.
Evaluating Trig Expressions (Practice 6)
By MathMech Team
A detailed lesson on evaluating secant, tangent, and cosecant expressions at special angles, using Pythagorean identities and special right triangles.
Evaluating Trig Expressions (Practice 1)
By MathMech Team
A detailed lesson on evaluating sine, cosine, and tangent expressions using the sine-cosine Pythagorean identity.
Evaluating Trig Expressions (Practice 2)
By MathMech Team
A detailed lesson on evaluating sine, cosine, and tangent expressions using the fundamental Pythagorean identity and special right triangles.
Evaluating Trig Expressions (Practice 3)
By MathMech Team
A detailed lesson on evaluating cosecant and cotangent expressions using the cosecant-cotangent Pythagorean identity.
Complementary Sine and Cosine Values
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Tangent and Cotangent Reciprocal Sum
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Double-Angle Sine at 45 Degrees
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Double-Angle Cosine at 30 Degrees
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Secant–Tangent Pythagorean Identity
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Cosecant–Cotangent Pythagorean Identity
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Sine–Cosine Pythagorean Identity
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Tangent Squared at 60 Degrees
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Cosecant Minus Cotangent at 30 Degrees
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Sine Double Angle at 30 Degrees
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Cosine of a Sum of Angles
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Sine with a 150-Degree Reference Angle
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Cosine in Quadrant II at 135 Degrees
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Tangent as a Sine–Cosine Ratio
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Secant as the Reciprocal of Cosine
By MathMech Team
A focused trigonometry lesson with an exact-value method and a worked evaluation for its linked problem.
Matrix Addition & Scalar Multiplication
By MathMech Team
Element-wise matrix operations. Covers matrix addition, subtraction, scalar multiplication, and the zero matrix.
The Matrix Trace
By MathMech Team
The sum of the diagonal entries of a square matrix. Covers the definition, linearity, cyclic property, and relationship to eigenvalues.
The Determinant (2×2)
By MathMech Team
Computing and interpreting the determinant of a 2×2 matrix. Covers the ad-bc formula, geometric meaning as area scaling, and conditions for invertibility.
Determinants (General)
By MathMech Team
Determinants of n×n matrices via cofactor expansion. Covers properties, the identity matrix determinant, and the effect of row operations.
Matrix Properties & Powers
By MathMech Team
Key algebraic properties of matrices including the determinant of a matrix power, transpose rules, and invertibility conditions.
The Dot Product
By MathMech Team
The inner product of two vectors. Covers the algebraic and geometric definitions, projection, angle between vectors, and the Cauchy-Schwarz inequality.
Orthogonal Vectors
By MathMech Team
Vectors that meet at a right angle. Covers the dot-product test for orthogonality, orthogonal complements, and the Gram-Schmidt process.
Singular Matrices & Linear Dependence
By MathMech Team
When a matrix has no inverse. Covers linear dependence of columns, zero determinant, and the connection to null spaces and inconsistent systems.
Visual Regression Tests
By MathMech Team
Quality assurance test book for verifying layout, styles, and KaTeX math equation rendering.