"The topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes."
Mathematics is the study of patterns, structures, and relationships involving numbers, quantities, shapes, and symbols.
Arithmetic: The basics of arithmetic including addition, subtraction, multiplication, and division of natural numbers, integers, fractions, and decimals.
Algebra: Algebra is a branch of mathematics that deals with the manipulation of mathematical symbols and the study of equations and their properties. It includes topics like polynomials, equations, groups, rings, and fields.
Geometry: The study of the properties, measurements, and relationships of points, lines, shapes, and solids.
Trigonometry: The study of angles and triangles, including trigonometric functions, identities, and applications.
Calculus: The study of limits, derivatives, integrals, and their applications to functions and equations.
Number theory: The study of the properties and relationships of positive integers and related concepts like divisibility, prime numbers, GCD, LCM, etc.
Statistics: The discipline of collecting, analyzing, and interpreting numerical data to make informed decisions.
Probability: The study of random events and their likelihood, including the rules of probability and statistical inference.
Set Theory: A branch of mathematics that deals with the study of sets, which are collections of distinct objects.
Combinatorics: The study of counting, permutations, combinations, and other discrete mathematical concepts used in solving problems.
Graph theory: The study of graphs and networks, including connectivity, paths, cycles, and algorithms.
Topology: The study of spatial relationships, including continuity, connectedness, and convergence.
Series and sequences (mathematics): The study of infinite series and sequences, including convergence and divergence.
Linear algebra: The study of matrices, vectors, and linear equations, including linear transformations and matrix algebra.
Differential equations: The study of equations involving derivatives, including techniques for solving and modeling dynamic systems.
Number systems: The study of number systems beyond the natural numbers such as complex numbers, hyperreal numbers, and p-adic numbers.
Analytic geometry: The branch of Mathematics that deals with the relationship between algebraic equations and geometric shapes.
Kinematics: The study of motion of bodies without considering the forces that cause the motion.
Dynamics (mathematics): The study of motion of bodies considering the forces that cause the motion.
Cryptography: The study of encoding and decoding secret messages based on mathematical algorithms.
Mathematical Logic: Logic is the study of reasoning and argumentation. It includes topics like propositional logic, predicate logic, set theory, and proof theory.
Applied Mathematics: Applied mathematics is the use of mathematical theory and techniques in various fields, including physics, engineering, biology, and finance.
Mathematical Physics: Mathematical physics is the study of the mathematical models used to describe physical phenomena. It includes topics like quantum mechanics, relativity, and statistical mechanics.
Game Theory: Game theory is the branch of mathematics that deals with the study of strategic decision-making, particularly in situations where the outcome depends on the actions of multiple agents.
"Number theory, algebra, geometry, and analysis."
"No general consensus among mathematicians about a common definition."
"The discovery of properties of abstract objects and the use of pure reason to prove them."
"A proof consists of a succession of applications of deductive rules to already established results."
"Independent from any scientific experimentation."
"The natural sciences, engineering, medicine, finance, computer science, and the social sciences."
"Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications."
"Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications."
"The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks."
"Historically, the concept of a proof and its associated mathematical rigor first appeared in Greek mathematics, most notably in Euclid's Elements."
"Mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions)."
"The 16th and 17th centuries."
"The interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both."
"The foundational crisis of mathematics at the end of the 19th century."
"It heralded a dramatic increase in the number of mathematical areas and their fields of application."
"More than 60 first-level areas of mathematics."
"Mathematics is extensively used for modeling phenomena."
"Certain properties called axioms."
"Some basic properties that are considered true starting points of the theory under consideration."