History of algebra


can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.
This article describes the history of the theory of equations, referred to in this article as "algebra", from the origins to the emergence of algebra as a separate area of mathematics.

Etymology

The word "algebra" is derived from the Arabic word الجبر, and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Al-Khwārizmī, whose Arabic title, Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, can be translated as The Compendious Book on Calculation by Completion and Balancing. The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "t is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the previous translation. The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation. Arabic influence in Spain long after the time of al-Khwarizmi is found in Don Quixote, where the word 'algebrista' is used for a bone-setter, that is, a 'restorer'." The term is used by al-Khwarizmi to describe the operations that he introduced, "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.

Stages of algebra

Algebraic expression

Algebra did not always make use of the symbolism that is now ubiquitous in mathematics; instead, it went through three distinct stages. The stages in the development of symbolic algebra are approximately as follows:
  • Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form of is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century.
  • Syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression first appeared in Diophantus' Arithmetica, followed by Brahmagupta's Brahma Sphuta Siddhanta.
  • Symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna and al-Qalasadi, although fully symbolic algebra was developed by François Viète. Later, René Descartes introduced the modern notation and showed that the problems occurring in geometry can be expressed and solved in terms of algebra.
As important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.
where and are positive.
This trichotomy comes about because quadratic equations of the form, with and positive, have no positive roots.
In between the rhetorical and syncopated stages of symbolic algebra, a geometric constructive algebra was developed by classical Greek and Vedic Indian mathematicians in which algebraic equations were solved through geometry. For instance, an equation of the form was solved by finding the side of a square of area

Conceptual stages

In addition to the three stages of expressing algebraic ideas, some authors recognized four conceptual stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows:
  • Geometric stage, where the concepts of algebra are largely geometric. This dates back to the Babylonians and continued with the Greeks, and was later revived by Omar Khayyám.
  • Static equation-solving stage, where the objective is to find numbers satisfying certain relationships. The move away from the geometric stage dates back to Diophantus and Brahmagupta, but algebra did not decisively move to the static equation-solving stage until Al-Khwarizmi introduced generalized algorithmic processes for solving algebraic problems.
  • Dynamic function stage, where motion is an underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra did not decisively move to the dynamic function stage until Gottfried Leibniz.
  • Abstract stage, where mathematical structure plays a central role. Abstract algebra is largely a product of the 19th and 20th centuries.

    Babylon

The origins of algebra can be traced to the ancient Babylonians, who developed a positional number system that greatly aided them in solving their rhetorical algebraic equations. The Babylonians were not interested in exact solutions, but rather approximations, and so they would commonly use linear interpolation to approximate intermediate values. One of the most famous tablets is the Plimpton 322 tablet, created around 1900–1600 BC, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek mathematics.
Babylonian algebra was much more advanced than the Egyptian algebra of the time; whereas the Egyptians were mainly concerned with linear equations the Babylonians were more concerned with quadratic and cubic equations. The Babylonians had developed flexible algebraic operations with which they were able to add equals to equals and multiply both sides of an equation by like quantities so as to eliminate fractions and factors. They were familiar with many simple forms of factoring, three-term quadratic equations with positive roots, and many cubic equations, although it is not known if they were able to reduce the general cubic equation.

Ancient Egypt

Ancient Egyptian algebra dealt mainly with linear equations while the Babylonians found these equations too elementary, and developed mathematics to a higher level than the Egyptians.
The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written c. 1650 BC by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 BC. It is the most extensive ancient Egyptian mathematical document known to historians. The Rhind Papyrus contains problems where linear equations of the form and are solved, where and are known and which is referred to as "aha" or heap, is the unknown. The solutions were possibly, but not likely, arrived at by using the "method of false position", or regula falsi, where first a specific value is substituted into the left hand side of the equation, then the required arithmetic calculations are done, thirdly the result is compared to the right hand side of the equation, and finally the correct answer is found through the use of proportions. In some of the problems the author "checks" his solution, thereby writing one of the earliest known simple proofs.

Greek mathematics

It is sometimes alleged that the Greeks had no algebra, but this is disputed. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them, and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas". "The application of areas" is only a part of geometric algebra and it is thoroughly covered in Euclid's Elements.
An example of geometric algebra would be solving the linear equation The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios and The Greeks would construct a rectangle with sides of length and then extend a side of the rectangle to length and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution.

Bloom of Thymaridas

in Introductio arithmatica says that Thymaridas worked with simultaneous linear equations. In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that:
If the sum of quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to of the difference between the sums of these pairs and the first given sum.

or using modern notation, the solution of the following system of linear equations in unknowns,











is,

Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.

Euclid of Alexandria

was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I. Neither the year nor place of his birth have been established, nor the circumstances of his death.
Euclid is regarded as the "father of geometry". His Elements is the most successful textbook in the history of mathematics. Although he is one of the most famous mathematicians in history there are no new discoveries attributed to him; rather he is remembered for his great explanatory skills. The Elements is not, as is sometimes thought, a collection of all Greek mathematical knowledge to its date; rather, it is an elementary introduction to it.