Algebra

It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.

It distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow, called axioms.

Algebra is relevant to many branches of mathematics, such as geometry, topology, number theory, and calculus, and other fields of inquiry, like logic and the empirical sciences.

[16] The word algebra comes from the Arabic term الجبر (al-jabr), which originally referred to the surgical treatment of bonesetting.

In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it in the title of a treatise on algebra, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah [The Compendious Book on Calculation by Completion and Balancing] which was translated into Latin as Liber Algebrae et Almucabola.

They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used.

[34] Factorization is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they evaluate to zero.

[41] Matrices are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations.

[65] The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements.

[67] The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups.

[93] The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities.

One of the earliest documents on algebraic problems is the Rhind Mathematical Papyrus from ancient Egypt, which was written around 1650 BCE.

For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified in Pythagoras' formulation of the difference of two squares method and later in Euclid's Elements.

[96] In the 3rd century CE, Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called Arithmetica.

They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications.

[101] This changed with the Persian mathematician al-Khwarizmi,[s] who published his The Compendious Book on Calculation by Completion and Balancing in 825 CE.

It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides.

[107] The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books including his Liber Abaci.

[108] In 1545, the Italian polymath Gerolamo Cardano published his book Ars Magna, which covered many topics in algebra, discussed imaginary numbers, and was the first to present general methods for solving cubic and quartic equations.

[109] In the 16th and 17th centuries, the French mathematicians François Viète and René Descartes introduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner.

[37] At the end of the 18th century, the German mathematician Carl Friedrich Gauss proved the fundamental theorem of algebra, which describes the existence of zeros of polynomials of any degree without providing a general solution.

They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields.

Starting in the 1930s, the American mathematician Garrett Birkhoff expanded these ideas and developed many of the foundational concepts of this field.

It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other.

[135] Similar applications are found in fields like economics, geography, engineering (including electronics and robotics), and computer science to express relationships, solve problems, and model systems.

[136] Linear algebra plays a central role in artificial intelligence and machine learning, for instance, by enabling the efficient processing and analysis of large datasets.

For example, physical sciences like crystallography and quantum mechanics make extensive use of group theory,[138] which is also employed to study puzzles such as Sudoku and Rubik's cubes,[139] and origami.

It is usually not introduced until secondary education since it requires mastery of the fundamentals of arithmetic while posing new cognitive challenges associated with abstract reasoning and generalization.

An additional difficulty for students lies in the fact that, unlike arithmetic calculations, algebraic expressions are often difficult to solve directly.

One method uses balance scales as a pictorial approach to help students grasp basic problems of algebra.