"The topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes."
Study of numbers, quantities, and shapes, and the relationships between them.
Sets: This topic teaches about collections of objects and their properties, such as intersection and union.
Functions: This teaches about how elements of one set are related to elements of another set.
Algebra: This includes equations, inequalities, and polynomials, among others.
Calculus: This teaches about rates of change, integration and differentiation, and limits, among others.
Probability and Statistics: This teaches about likelihood, randomness, distribution, and data analysis, among others.
Linear Algebra: This teaches about linear equations and their solutions, matrix multiplication, and determinants, among others.
Geometry: This teaches about the properties of shapes, such as lines, angles, and curves.
Trigonometry: This teaches about the relationships between angles and sides of triangles.
Differential Equations: This teaches about equations that describe how variables change over time.
Graph Theory: This teaches about graphs, which are comprised of vertices and edges, and their applications in various fields.
"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."