Algebra is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.
The Rhind Papyrus from ancient Egypt, dated around 1650 BCE, is one of the earliest documents discussing algebraic problems.
Al-Khwarizmi's The Compendious Book on Calculation by Completion and Balancing provided a general account of how linear and quadratic equations can be solved through the methods of "reducing" and "balancing".
Garrett Birkhoff developed many of the foundational concepts of universal algebra.
Image: Francois Viete
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.
On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC
A page from al-Khwārizmī's Algebra
