Algebra
The word algebra comes from the Arabic term al-jabr, which originally referred to the surgical treatment of bonesetting. In the 9th century, Persian mathematician Muhammad ibn Musa al-Khwarizmi employed this term to name a method for transforming equations. He used it in the title of his treatise The Compendious Book on Calculation by Completion and Balancing. This work was translated into Latin as Liber Algebrae. The word entered the English language in the 16th century from Italian, Spanish, and medieval Latin. Initially, its meaning was restricted to the theory of equations. It described the art of manipulating polynomial equations in view of solving them. This changed in the 19th century when the scope of algebra broadened. The field began to cover diverse types of algebraic operations and structures together with their underlying axioms.
One of the earliest documents on algebraic problems is the Rhind Mathematical Papyrus from ancient Egypt. It was written around 1650 BCE and discusses solutions to linear equations. Babylonian clay tablets from around the same time explain methods to solve linear and quadratic polynomial equations. They include the method of completing the square. Many insights found their way to the ancient Greeks starting in the 6th century BCE. Their main interest was geometry rather than algebra. They employed algebraic methods to solve geometric problems. Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called Arithmetica in the 3rd century CE. He was the first to experiment with symbolic notation to express polynomials. In ancient China, The Nine Chapters on the Mathematical Art explored various techniques for solving algebraic equations. It was composed over the period spanning from the 10th century BCE to the 2nd century CE.
In the 16th and 17th centuries, French mathematicians François Viète and René Descartes introduced letters and symbols to denote variables and operations. Their predecessors had relied on verbal descriptions of problems and solutions. This development made it possible to express equations in an concise and abstract manner. Some historians see this as a key turning point in the history of algebra. They consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation. In 1545, Italian polymath Gerolamo Cardano published his book Ars Magna. It covered many topics in algebra and discussed imaginary numbers. It was the first to present general methods for solving cubic and quartic equations. At the end of the 18th century, German mathematician Carl Friedrich Gauss proved the fundamental theorem of algebra. It describes the existence of zeros of polynomials of any degree without providing a general solution.
Starting in the mid-19th century, interest in algebra shifted from the study of polynomials towards a more general inquiry into algebraic structures. This marked the emergence of abstract algebra. The approach explored the axiomatic basis of arbitrary algebraic operations. Influential early developments were made by German mathematicians David Hilbert, Ernst Steinitz, and Emmy Noether. Austrian mathematician Emil Artin also researched different forms of algebraic structures. They categorized them based on their underlying axioms into types like groups, rings, and fields. In 1898, English mathematician Alfred North Whitehead conceived the idea of universal algebra in his book A Treatise on Universal Algebra. Starting in the 1930s, American mathematician Garrett Birkhoff expanded these ideas. He developed many foundational concepts of this field. Mathematicians soon realized the relevance of group theory to other fields. They applied it to disciplines like geometry and number theory.
Elementary algebra is the oldest and most basic form of algebra. It relies on variables and examines how mathematical statements may be transformed. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values. Linear algebra starts with the study of systems of linear equations. An equation is linear if it can be expressed in the form ax + b = c. Matrices are rectangular arrays of values that have been originally introduced for having a compact notation for systems of linear equations. Abstract algebra studies algebraic structures. These consist of a set of mathematical objects together with one or several operations defined on that set. One of the most basic types is a group. It has one operation and requires that this operation is associative and has an identity element and inverse elements. The natural numbers with addition do not form a group since they contain only positive integers and therefore lack inverse elements.
Algebraic methods are commonly employed in other areas like the natural sciences. They are used to express scientific laws and solve equations in physics, chemistry, and biology. Similar applications are found in fields like economics, geography, engineering, and computer science. Linear algebra plays a central role in artificial intelligence and machine learning. It enables the efficient processing and analysis of large datasets. Physical sciences like crystallography and quantum mechanics make extensive use of group theory. Group theory is also employed to study puzzles such as Sudoku and Rubik's Cubes. Both coding theory and cryptology rely on abstract algebra to solve problems associated with data transmission. Topological algebra arose in the early 20th century studying algebraic structures such as topological groups and Lie groups. In the 1940s and 50s, homological algebra emerged employing algebraic techniques to study homology.
Common questions
What is the origin of the word algebra?
The word algebra comes from the Arabic term al-jabr, which originally referred to the surgical treatment of bonesetting. It entered the English language in the 16th century from Italian, Spanish, and medieval Latin.
When did Muhammad ibn Musa al-Khwarizmi write his treatise on algebra?
Muhammad ibn Musa al-Khwarizmi wrote The Compendious Book on Calculation by Completion and Balancing in the 9th century. This work was translated into Latin as Liber Algebrae and established the method for transforming equations.
Who introduced symbolic notation to express polynomials in algebra?
Diophantus provided a detailed treatment of how to solve algebraic equations in Arithmetica during the 3rd century CE. He was the first to experiment with symbolic notation to express polynomials.
What year did Gerolamo Cardano publish Ars Magna?
Italian polymath Gerolamo Cardano published his book Ars Magna in 1545. It covered many topics in algebra and discussed imaginary numbers while presenting general methods for solving cubic and quartic equations.
How does linear algebra define an equation?
An equation is linear if it can be expressed in the form ax + b = c. Linear algebra starts with the study of systems of linear equations using matrices as rectangular arrays of values.