{ "api": "YAML JSON TOON Database", "version": "1.0.0", "format": "json", "category": { "id": 15, "slug": "mathematics", "name": "Mathematics", "description": "Core mathematical concepts, theorems, and notation.", "icon": "box", "created_at": 1783090350, "topic": { "overview": "Mathematics reference data collects the facts that are looked up rather than derived: counting rules, constants, geometric formulas, set notation, descriptive statistics, trigonometric identities and the primes. These are exact by definition, which makes them the cleanest kind of reference data for an agent: a lookup either matches a known value or it does not. The usual trouble is not accuracy but context, such as which formula applies to which shape, what a symbol means in a given convention, and how many digits of a constant are meaningful.", "key_ideas": [ { "term": "Exact vs approximate", "meaning": "Constants such as pi and e are irrational: any decimal value is a rounded approximation. Keep symbolic form where exactness matters." }, { "term": "Permutation vs combination", "meaning": "A permutation counts ordered selections, a combination unordered ones. Choosing seats is a permutation; choosing a committee is a combination." }, { "term": "Set operation", "meaning": "Union, intersection, difference and complement combine sets by membership. De Morgan's laws relate them." }, { "term": "Descriptive statistic", "meaning": "A number that summarizes a dataset, such as the mean, median or standard deviation. Different summaries respond differently to outliers." }, { "term": "Identity", "meaning": "An equation true for every value of its variables, such as sin squared plus cos squared equals 1, as opposed to an equation true only for some values." }, { "term": "Prime number", "meaning": "An integer greater than 1 whose only positive divisors are 1 and itself. The prime factorization of any integer is unique." }, { "term": "Radian", "meaning": "The angle measure in which a full turn is 2 pi, used by calculus and most programming-language trig functions." } ], "pitfalls": [ "Using degrees where a library expects radians, or the reverse.", "Quoting a decimal expansion of an irrational constant as exact; floating-point arithmetic keeps only about 15 to 17 significant digits.", "Mixing population and sample formulas for variance and standard deviation (dividing by n versus n minus 1).", "Reading 1 as prime. By the usual definition it is not, which keeps unique factorization true." ], "agent_tips": [ "Show the formula and the meaning of each variable before substituting numbers, so the user can check the setup.", "For computed results, state the unit and whether angles are in degrees or radians.", "Prefer a tool or code execution for arithmetic with many steps instead of mental arithmetic, and use these datasets for definitions and constants." ], "start_with": [ "common-mathematical-constants", "statistics-basics" ] } }, "datasets": [ { "id": 1, "slug": "combinatorics-fundamentals", "title": "Combinatorics Fundamentals", "description": "Core counting principles and combinatorial formulas.", "tags": "math,combinatorics,counting,permutations,combinations", "view_count": 0, "created_at": 1783090350, "entry_count": 16, "summary": "The counting principles used to enumerate possibilities: the sum and product rules, permutations and combinations (with and without repetition), the pigeonhole principle and the binomial theorem, plus inclusion-exclusion, derangements, stars and bars, Catalan and Stirling numbers. Each entry has the principle, a one-line description and the formula." }, { "id": 2, "slug": "common-mathematical-constants", "title": "Common Mathematical Constants", "description": "Widely used mathematical constants and their approximate values.", "tags": "math,constants,reference", "view_count": 0, "created_at": 1783090350, "entry_count": 17, "summary": "The mathematical constants that recur across fields, each with its symbol, a decimal value to ten places and what it measures: pi, e, the golden ratio, square roots, the imaginary unit, and further constants such as tau, the Euler-Mascheroni constant, Catalan's constant, Apery's constant and the Feigenbaum constant." }, { "id": 1020, "slug": "experimental-design-pitfalls", "title": "Experimental Design and Inference Pitfalls", "description": "Common threats to experiment validity, with a practical design adjustment and reporting consequence.", "tags": "mathematics,experimental design pitfalls,reference,engineering", "view_count": 0, "created_at": 1791799200, "entry_count": 10, "summary": "Common threats to experiment validity, with a practical design adjustment and reporting consequence. Each row includes a practical application and a failure to check before using it." }, { "id": 3, "slug": "geometric-formulas", "title": "Geometric Formulas", "description": "Common formulas for area, perimeter, surface area, and volume.", "tags": "math,geometry,formulas", "view_count": 0, "created_at": 1783090350, "entry_count": 20, "summary": "Area, surface area and volume formulas for the common plane and solid shapes, with the Pythagorean theorem, plus trapezoids, ellipses, cones, pyramids, regular polygons, Heron's formula, the distance formula and the law of cosines. Each row names the variables so the formula can be applied directly." }, { "id": 4, "slug": "prime-numbers-list", "title": "Prime Numbers List", "description": "The first 100 prime numbers with their position in the sequence.", "tags": "math,primes,numbers", "view_count": 0, "created_at": 1783090350, "entry_count": 100, "summary": "The first one hundred prime numbers with their position in the sequence (the 1st prime is 2, the 100th is 541). It is a ready reference for factorization practice, hashing table sizes, test data and checking a primality claim." }, { "id": 1019, "slug": "probability-distribution-reference", "title": "Probability Distribution Selection Reference", "description": "Common probability distributions with support, mean, and practical modeling boundaries.", "tags": "mathematics,probability distribution reference,reference,engineering", "view_count": 0, "created_at": 1791799200, "entry_count": 10, "summary": "Common probability distributions with support, mean, and practical modeling boundaries. Each row includes a practical application and a failure to check before using it." }, { "id": 5, "slug": "set-theory-basics", "title": "Set Theory Basics", "description": "Fundamental set operations and relations.", "tags": "math,set-theory,logic", "view_count": 0, "created_at": 1783090350, "entry_count": 14, "summary": "The core vocabulary and notation of set theory: union, intersection, complement, subsets, power sets and Cartesian products, with set difference, symmetric difference, cardinality, membership, disjoint sets and De Morgan's laws." }, { "id": 6, "slug": "statistics-basics", "title": "Statistics Basics", "description": "Fundamental statistical measures and concepts.", "tags": "math,statistics,probability,data-science", "view_count": 0, "created_at": 1783090350, "entry_count": 15, "summary": "The basic measures of central tendency, spread and association: mean, median, mode, variance, standard deviation, correlation and the z-score, plus range, interquartile range, covariance, skewness, standard error, coefficient of variation, confidence intervals and the p-value, each with its formula." }, { "id": 7, "slug": "trigonometric-identities", "title": "Trigonometric Identities", "description": "Fundamental identities used in trigonometry.", "tags": "math,trigonometry,identities", "view_count": 0, "created_at": 1783090350, "entry_count": 17, "summary": "The trigonometric identities most used in calculation and proof: the Pythagorean identities and double-angle formulas, with angle-sum and angle-difference formulas, half-angle formulas, parity and periodicity, product-to-sum rules and Euler's formula." } ] }