# 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