Mathematics

A comprehensive guide to mathematical concepts, from foundational principles to advanced theories, with structured learning paths and reference materials.

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Practice & Exercises

Master mathematical concepts through our comprehensive collection of interactive exercises with solutions.

References & Tools

Access our comprehensive collection of formulas, tables, calculators, and mathematical glossary.

Structured Learning PathsFor Every Mathematics Level

Follow our carefully designed learning roadmaps to master mathematical concepts in the optimal order, from beginner to advanced level.

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Explore Mathematics Topics

Browse our comprehensive library of mathematical topics, organized by field and complexity level

Mathematical Foundations

The fundamental concepts, principles, and methods that form the essential groundwork for mathematics, including logic, set theory, methods of proof, and number systems.

Mathematical Logic

The formal study of logical reasoning, symbolic representation of statements, and principles of valid inference in mathematics.

Set Theory

The study of sets as collections of objects, their properties, operations, and relations, forming a fundamental framework for mathematics.

Number Systems

The development, structure, and properties of various number systems, from natural numbers to complex numbers.

Mathematical Proofs

The principles, methods, and techniques for constructing rigorous mathematical proofs.

Foundations of Analysis

The fundamental concepts that underpin mathematical analysis, including limits, continuity, sequences, and series.

Foundations of Geometry

The fundamental concepts, axioms, and methods that form the basis of geometric reasoning and study.

Basic Mathematical Structures

Introduction to fundamental algebraic structures that appear throughout mathematics.

Historical and Philosophical Perspectives

Historical development and philosophical aspects of mathematical foundations.

Number Theory

The branch of mathematics concerned with the properties and relationships of integers, divisibility, primes, congruences, and their applications in cryptography and other fields.

Elementary Number Theory

Fundamental concepts of number theory including divisibility, prime numbers, greatest common divisors, and modular arithmetic.

Quadratic Residues

The study of squares in modular arithmetic systems, focusing on which integers have square roots when working modulo a given number.

Continued Fractions

The representation of real numbers as a sequence of integers using recursive fractions, with applications in approximation and number theory.

Advanced Diophantine Equations

The study of polynomial equations for which integer or rational solutions are sought, including quadratic forms and higher-degree equations.

Algebraic Number Theory

The study of algebraic structures related to number theory, including algebraic integers, number fields, and their properties.

Analytic Number Theory

The application of methods from mathematical analysis to problems in number theory, particularly those involving prime numbers.

Computational Number Theory

The study of algorithms and computational methods for solving number-theoretic problems efficiently.

Cryptographic Applications

Applications of number theory in modern cryptography, including public-key systems and digital signatures.

Advanced Topics in Number Theory

Specialized and advanced areas of number theory, including connections to other mathematical disciplines and recent developments.

Historical Development

The evolution of number theory from ancient times to the modern era, including key figures, breakthroughs, and historical context.

Algebra

Comprehensive study of algebraic concepts, from elementary operations and equations to advanced abstract structures and applications.

Elementary Algebra

Foundational algebraic concepts including number systems, operations, expressions, equations, and inequalities.

Intermediate Algebra

Advanced algebraic concepts building upon elementary algebra, including polynomials, rational expressions, radicals, quadratics, systems of equations, and sequences and series.

Advanced Algebra

Higher-level algebraic concepts including advanced functions, polynomial theory, matrix algebra, and introductory abstract structures that bridge toward modern algebra.

Linear Algebra

Comprehensive study of vector spaces, linear transformations, matrices, systems of linear equations, eigenvalues, and their applications in mathematics and other disciplines.

History of Algebra

The historical development of algebraic concepts, notation, and theory from ancient civilizations to modern abstract algebra, highlighting key figures and pivotal advancements.

Geometry

The branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional spaces, as well as their measures and transformations.

Euclidean Geometry

The study of geometry based on Euclid's axioms, primarily concerned with properties of figures in two and three dimensions that remain invariant under rigid motions.

Analytic Geometry

The study of geometry using coordinate systems and algebraic equations, bridging geometry and algebra to solve geometric problems analytically.

Non-Euclidean Geometry

Geometric systems that differ from Euclidean geometry by modifying or replacing Euclid's parallel postulate, leading to hyperbolic, elliptic, and other geometries.

Differential Geometry

The study of geometric properties and structures using differential calculus, including curves, surfaces, manifolds, and their intrinsic properties.

Projective Geometry

The study of geometric properties that are invariant under projective transformations, focusing on incidence relations and projective spaces.

Algebraic Geometry

The study of geometric objects defined by polynomial equations, combining abstract algebra with geometric intuition.

Computational Geometry

The study of algorithms for solving geometric problems, with applications in computer graphics, geographic information systems, and robotics.

Fractal Geometry

The study of geometric patterns that exhibit self-similarity at different scales, often possessing non-integer dimensions.

Introduction to Topology

The study of properties of space that are preserved under continuous deformations, such as stretching, twisting, and bending, but not tearing or gluing.

Applications of Geometry

Practical applications of geometric principles and theories in various fields including physics, engineering, computer science, art, and architecture.

Historical Development of Geometry

The evolution of geometric thought and methods from ancient civilizations to modern times, highlighting key figures, discoveries, and paradigm shifts.

Trigonometry

The study of relationships between angles and sides of triangles, and the trigonometric functions that describe these relationships, with applications in various fields of mathematics and science.

Basic Concepts of Trigonometry

Fundamental trigonometric concepts including angle measurement, the unit circle, and basic definitions of trigonometric functions.

Trigonometric Functions

Comprehensive study of the six trigonometric functions, their definitions, properties, domains, ranges, and graphical representations.

Trigonometric Identities

Fundamental equations that relate trigonometric functions, essential for simplification, problem-solving, and advanced applications.

Inverse Trigonometric Functions

Functions that determine angles from trigonometric ratios, including arcsine, arccosine, arctangent, and their properties and applications.

Trigonometric Equations

Methods for solving equations involving trigonometric functions, including simple, multiple angle, and complex equations.

Trigonometry of Triangles

Applications of trigonometry to solve problems involving triangles, including area, the law of sines, the law of cosines, and various triangle properties.

Applications of Trigonometry

Real-world applications of trigonometric principles in various fields, including physics, engineering, navigation, and other sciences.

Hyperbolic Functions

Functions analogous to trigonometric functions but based on the unit hyperbola rather than the unit circle, with applications in engineering and calculus.

Complex Trigonometry

The extension of trigonometric functions to complex numbers, including Euler's formula and relationships with exponential functions.

Historical Development of Trigonometry

The evolution of trigonometric concepts and methods through history, from ancient civilizations to modern mathematics.

Advanced Topics in Trigonometry

Specialized and advanced trigonometric concepts and techniques beyond the fundamental level, connecting to various mathematical disciplines.

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By Federico Airoldi· 2025-02-28· 5 min read· all-levels