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| # 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 | |