History of algebra

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 (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property).

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.

The word "algebra" is derived from the Arabic word الجبر al-jabr, 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, "[i]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'."^[1] 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.^[2]

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:^[3]

• Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form of ${\displaystyle x+1=2}$ 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 (3rd century AD), followed by Brahmagupta's Brahma Sphuta Siddhanta (7th century).
• 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 (13th–14th centuries) and al-Qalasadi (15th century), although fully symbolic algebra was developed by François Viète (16th century). Later, René Descartes (17th century) introduced the modern notation (for example, the use of x—see below) and showed that the problems occurring in geometry can be expressed and solved in terms of algebra (Cartesian geometry).

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.

• ${\displaystyle x^{2}+px=q}$
• ${\displaystyle x^{2}=px+q}$
• ${\displaystyle x^{2}+q=px}$

where ${\displaystyle p}$ and ${\displaystyle q}$ are positive. This trichotomy comes about because quadratic equations of the form ${\displaystyle x^{2}+px+q=0}$, with ${\displaystyle p}$ and ${\displaystyle q}$ positive, have no positive roots.^[4]

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 ${\displaystyle x^{2}=A}$ was solved by finding the side of a square of area ${\displaystyle A.}$

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:^[5]

• 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.

The origins of algebra can be traced to the ancient Babylonians,^[6] 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.^[7] 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.^[8]

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.^[7] 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.^[7] They were familiar with many simple forms of factoring,^[7] three-term quadratic equations with positive roots,^[9] and many cubic equations,^[10] although it is not known if they were able to reduce the general cubic equation.^[10]

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.^[7]

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.^[11] It is the most extensive ancient Egyptian mathematical document known to historians.^[12] The Rhind Papyrus contains problems where linear equations of the form ${\displaystyle x+ax=b}$ and ${\displaystyle x+ax+bx=c}$ are solved, where ${\displaystyle a,b,}$ and ${\displaystyle c}$ are known and ${\displaystyle x,}$ which is referred to as "aha" or heap, is the unknown.^[13] 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.^[13]

It is sometimes alleged that the Greeks had no algebra, but this is disputed.^[15] 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,^[16] usually lines, that had letters associated with them,^[17] 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".^[16] "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 ${\displaystyle ax=bc.}$ The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios ${\displaystyle a:b}$ and ${\displaystyle c:x.}$ The Greeks would construct a rectangle with sides of length ${\displaystyle b}$ and ${\displaystyle c,}$ then extend a side of the rectangle to length ${\displaystyle a,}$ and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution.^[16]

Iamblichus in Introductio arithmatica says that Thymaridas (c. 400 BC – c. 350 BC) worked with simultaneous linear equations.^[18] 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 ${\displaystyle n}$ quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to ${\displaystyle 1/(n-2)}$ of the difference between the sums of these pairs and the first given sum.^[19]

or using modern notation, the solution of the following system of ${\displaystyle n}$ linear equations in ${\displaystyle n}$ unknowns,^[18]

${\displaystyle x+x_{1}+x_{2}+\cdots +x_{n-1}=s}$
${\displaystyle x+x_{1}=m_{1}}$
${\displaystyle x+x_{2}=m_{2}}$
${\displaystyle \vdots }$
${\displaystyle x+x_{n-1}=m_{n-1}}$

is,

${\displaystyle x={\cfrac {(m_{1}+m_{2}+...+m_{n-1})-s}{n-2}}={\cfrac {(\sum _{i=1}^{n-1}m_{i})-s}{n-2}}.}$

Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.^[18]

Euclid (Greek: Εὐκλείδης) was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC).^[20][21] Neither the year nor place of his birth^[20] 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.^[20] 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.^[22] 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.^[23]

The geometric work of the Greeks, typified in Euclid's Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.

Book II of the Elements contains fourteen propositions, which in Euclid's time were extremely significant for doing geometric algebra. These propositions and their results are the geometric equivalents of our modern symbolic algebra and trigonometry.^[15] Today, using modern symbolic algebra, we let symbols represent known and unknown magnitudes (i.e. numbers) and then apply algebraic operations on them, while in Euclid's time magnitudes were viewed as line segments and then results were deduced using the axioms or theorems of geometry.^[15]

Many basic laws of addition and multiplication are included or proved geometrically in the Elements. For instance, proposition 1 of Book II states:

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

But this is nothing more than the geometric version of the (left) distributive law, ${\displaystyle a(b+c+d)=ab+ac+ad}$; and in Books V and VII of the Elements the commutative and associative laws for multiplication are demonstrated.^[15]

Many basic equations were also proved geometrically. For instance, proposition 5 in Book II proves that ${\displaystyle a^{2}-b^{2}=(a+b)(a-b),}$^[24] and proposition 4 in Book II proves that ${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}.}$^[15]

Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation ${\displaystyle ax+x^{2}=b^{2},}$ and proposition 11 of Book II gives a solution to ${\displaystyle ax+x^{2}=a^{2}.}$^[25]

Data is a work written by Euclid for use at the schools of Alexandria and it was meant to be used as a companion volume to the first six books of the Elements. The book contains some fifteen definitions and ninety-five statements, of which there are about two dozen statements that serve as algebraic rules or formulas.^[26] Some of these statements are geometric equivalents to solutions of quadratic equations.^[26] For instance, Data contains the solutions to the equations ${\displaystyle dx^{2}-adx+b^{2}c=0}$ and the familiar Babylonian equation ${\displaystyle xy=a^{2},x\pm y=b.}$^[26]

A conic section is a curve that results from the intersection of a cone with a plane. There are three primary types of conic sections: ellipses (including circles), parabolas, and hyperbolas. The conic sections are reputed to have been discovered by Menaechmus^[27] (c. 380 BC – c. 320 BC) and since dealing with conic sections is equivalent to dealing with their respective equations, they played geometric roles equivalent to cubic equations and other higher order equations.

Menaechmus knew that in a parabola, the equation ${\displaystyle y^{2}=lx}$ holds, where ${\displaystyle l}$ is a constant called the latus rectum, although he was not aware of the fact that any equation in two unknowns determines a curve.^[28] He apparently derived these properties of conic sections and others as well. Using this information it was now possible to find a solution to the problem of the duplication of the cube by solving for the points at which two parabolas intersect, a solution equivalent to solving a cubic equation.^[28]

We are informed by Eutocius that the method he used to solve the cubic equation was due to Dionysodorus (250 BC – 190 BC). Dionysodorus solved the cubic by means of the intersection of a rectangular hyperbola and a parabola. This was related to a problem in Archimedes' On the Sphere and Cylinder. Conic sections would be studied and used for thousands of years by Greek, and later Islamic and European, mathematicians. In particular Apollonius of Perga's famous Conics deals with conic sections, among other topics.

Chinese mathematics dates to at least 300 BC with the Zhoubi Suanjing, generally considered to be one of the oldest Chinese mathematical documents.^[29]

Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 BC, is one of the most influential of all Chinese math books and it is composed of some 246 problems. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns.^[29]

Ts'e-yuan hai-ching, or Sea-Mirror of the Circle Measurements, is a collection of some 170 problems written by Li Zhi (or Li Ye) (1192 – 1279 AD). He used fan fa, or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations.^[30]

Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch'in Chiu-shao (c. 1202 – c. 1261). With the introduction of a method for solving simultaneous congruences, now called the Chinese remainder theorem, it marks the high point in Chinese indeterminate analysis^{[clarification needed]}.^[30]

The earliest known magic squares appeared in China.^[31] In Nine Chapters the author solves a system of simultaneous linear equations by placing the coefficients and constant terms of the linear equations into a magic square (i.e. a matrix) and performing column reducing operations on the magic square.^[31] The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. c. 1261 – 1275), who worked with magic squares of order as high as ten.^[32]

Ssy-yüan yü-chien《四元玉鑒》, or Precious Mirror of the Four Elements, was written by Chu Shih-chieh in 1303 and it marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. The Ssy-yüan yü-chien deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations.^[33]

The Precious Mirror opens with a diagram of the arithmetic triangle (Pascal's triangle) using a round zero symbol, but Chu Shih-chieh denies credit for it. A similar triangle appears in Yang Hui's work, but without the zero symbol.^[34]

There are many summation equations given without proof in the Precious mirror. A few of the summations are:^[34]

${\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}}$

${\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}}$

Diophantus was a Hellenistic mathematician who lived c. 250 AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.^[35] Arithmetica is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent the method of algebra, which existed before him.^[36] Algebra was practiced and diffused orally by practitioners, with Diophantus picking up techniques to solve problems in arithmetic.^[37]

In modern algebra a polynomial is a linear combination of variable x that is built of exponentiation, scalar multiplication, addition, and subtraction. The algebra of Diophantus, similar to medieval arabic algebra is an aggregation of objects of different types with no operations present^[38]

For example, in Diophantus a polynomial "6 4′ inverse Powers, 25 Powers lacking 9 units", which in modern notation is ${\displaystyle 6{\tfrac {1}{4}}x^{-1}+25x^{2}-9}$ is a collection of ${\displaystyle 6{\tfrac {1}{4}}}$ object of one kind with 25 object of second kind which lack 9 objects of third kind with no operation present.^[39]

Similar to medieval Arabic algebra Diophantus uses three stages to solve a problem by Algebra:

1) An unknown is named and an equation is set up

2) An equation is simplified to a standard form( al-jabr and al-muqābala in arabic)

3) Simplified equation is solved^[40]

Diophantus does not give a classification of equations in six types like Al-Khwarizmi in extant parts of Arithmetica. He does say that he would give solution to three terms equations later, so this part of the work is possibly just lost^[37]

In Arithmetica, Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;^[41] thus he used what is now known as syncopated algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.^[42]

So, for example, what we would write as

${\displaystyle x^{3}-2x^{2}+10x-1=5,}$

which can be rewritten as

${\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5,}$

would be written in Diophantus's syncopated notation as

${\displaystyle \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}$ἴ${\displaystyle \sigma \;\,\mathrm {M} {\overline {\varepsilon }}}$

where the symbols represent the following:^[43][44]

Symbol	What it represents
${\displaystyle {\overline {\alpha }}}$	1
${\displaystyle {\overline {\beta }}}$	2
${\displaystyle {\overline {\varepsilon }}}$	5
${\displaystyle {\overline {\iota }}}$	10
ἴσ	"equals" (short for ἴσος)
${\displaystyle \pitchfork }$	represents the subtraction of everything that follows ${\displaystyle \pitchfork }$ up to ἴσ
${\displaystyle \mathrm {M} }$	the zeroth power (i.e. a constant term)
${\displaystyle \zeta }$	the unknown quantity (because a number ${\displaystyle x}$ raised to the first power is just ${\displaystyle x,}$ this may be thought of as "the first power")
${\displaystyle \Delta ^{\upsilon }}$	the second power, from Greek δύναμις, meaning strength or power
${\displaystyle \mathrm {K} ^{\upsilon }}$	the third power, from Greek κύβος, meaning a cube
${\displaystyle \Delta ^{\upsilon }\Delta }$	the fourth power
${\displaystyle \Delta \mathrm {K} ^{\upsilon }}$	the fifth power
${\displaystyle \mathrm {K} ^{\upsilon }\mathrm {K} }$	the sixth power

Unlike in modern notation, the coefficients come after the variables and that addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following:^[43]