data/knowledge_base/mathematics.md

Mathematics Knowledge Base

Algebra

Linear Algebra

• Vectors: Mathematical objects with magnitude and direction
• Matrices: Rectangular arrays of numbers used for linear transformations
• Eigenvalues: Special numbers associated with square matrices
• Linear Transformations: Functions that preserve vector addition and scalar multiplication

Polynomials

• Quadratic Equations: ax² + bx + c = 0, solved using quadratic formula
• Factoring: Breaking down polynomials into simpler factors
• Synthetic Division: Method for dividing polynomials

Calculus

Differential Calculus

• Derivatives: Rate of change of a function
• Chain Rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)
• Product Rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
• Quotient Rule: d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/g²(x)

Integral Calculus

• Antiderivatives: Functions whose derivatives equal the original function
• Definite Integrals: Area under a curve between two points
• Integration by Parts: ∫udv = uv - ∫vdu
• Substitution: Method for solving complex integrals

Geometry

Euclidean Geometry

• Triangles: Three-sided polygons with various properties
• Circles: Set of points equidistant from a center
• Pythagorean Theorem: a² + b² = c² for right triangles

Trigonometry

• Sine, Cosine, Tangent: Basic trigonometric functions
• Unit Circle: Circle with radius 1 centered at origin
• Trigonometric Identities: Mathematical relationships between trig functions

Statistics

Descriptive Statistics

• Mean: Average of a set of numbers
• Median: Middle value when data is ordered
• Mode: Most frequently occurring value
• Standard Deviation: Measure of data spread

Probability

• Probability Rules: Addition and multiplication rules
• Conditional Probability: P(A|B) = P(A∩B)/P(B)
• Bayes' Theorem: P(A|B) = P(B|A)P(A)/P(B)

Number Theory

Prime Numbers

• Prime Factorization: Breaking numbers into prime factors
• Euclidean Algorithm: Finding greatest common divisor
• Modular Arithmetic: Arithmetic with remainders

Sequences and Series

• Arithmetic Sequences: Sequences with constant difference
• Geometric Sequences: Sequences with constant ratio
• Fibonacci Sequence: Each number is sum of two preceding ones