Algebra history of the word

Algebra (from Arabic ‏الجبر‎ (al-jabr) ‘reunion of broken parts,^[1] bonesetting’^[2]) is the study of variables and the rules for manipulating these variables in formulas;^[3] it is a unifying thread of almost all of mathematics.^[4]

Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having «algebra» in their name, such as commutative algebra, and some not, such as Galois theory.

The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.

Etymology

The word algebra comes from the Arabic: الجبر, romanized: al-jabr, lit. ‘reunion of broken parts,^[1] bonesetting^[2]‘ from the title of the early 9th century book ʿIlm al-jabr wa l-muqābala «The Science of Restoring and Balancing» by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala «balancing» referred to adding equal terms to both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.^[6]

Different meanings of «algebra»

The word «algebra» has several related meanings in mathematics, as a single word or with qualifiers.

• As a single word without an article, «algebra» names a broad part of mathematics.
• As a single word with an article or in the plural, «an algebra» or «algebras» denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term «algebra», they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word «algebra» refers to a generalization of the above concept, which allows for n-ary operations.
• With a qualifier, there is the same distinction:
  • Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).
  • With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra.
  • Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.^[7] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

can be any numbers whatsoever (except that cannot be ), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity which satisfy the equation. That is to say, to find all the solutions of the equation.

Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as «does an equation have a solution?», «how many solutions does an equation have?», «what can be said about the nature of the solutions?» are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors, matrices, and polynomials. The structural properties of these non-numerical objects were then formalized into algebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown considerably and includes many branches of mathematics, as can be seen in the Mathematics Subject Classification^[8]
where none of the first level areas (two digit entries) are called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

History

The use of the word «algebra» for denoting a part of mathematics dates probably from the 16th century.^{[citation needed]} The word is derived from the Arabic word al-jabr that appears in the title of the treatise Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing), written circa 820 by Al-Kwarizmi.

Al-jabr referred to a method for transforming equations by subtracting like terms from both sides, or passing one term from one side to the other, after changing its sign.

Therefore, algebra referred originally to the manipulation of equations, and, by extension, to the theory of equations. This is still what historians of mathematics generally mean by algebra.^{[citation needed]}

In mathematics, the meaning of algebra has evolved after the introduction by François Viète of symbols (variables) for denoting unknown or incompletely specified numbers, and the resulting use of the mathematical notation for equations and formulas. So, algebra became essentially the study of the action of operations on expressions involving variables. This includes but is not limited to the theory of equations.

At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so-called mathematical structures such as groups, fields and vector spaces. This new algebra was called modern algebra by van der Waerden in his eponymous treatise, whose name has been changed to Algebra in later editions.

Early history

The roots of algebra can be traced to the ancient Babylonians,^[9] who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid’s Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.^[10]

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.^[7] Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations,^[11] and have led, in number theory, to the modern notion of Diophantine equation.

Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.^[12]

The Hellenistic mathematicians Hero of Alexandria and Diophantus^[13] as well as Indian mathematicians such as Brahmagupta, continued the traditions of Egypt and Babylon, though Diophantus’ Arithmetica and Brahmagupta’s Brāhmasphuṭasiddhānta are on a higher level.^{[14][better source needed]} For example, the first complete arithmetic solution written in words instead of symbols,^[15] including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD.^[16] Later, Persian and Arab mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi’s contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.^[17]

In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the «father of algebra» and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as «the father of algebra».^{[18][19][20][21][22][23][24]} It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as «the father of algebra». Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.^[25] Those who support Al-Khwarizmi point to the fact that he introduced the methods of «reduction» and «balancing» (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) which the term al-jabr originally referred to,^[26] and that he gave an exhaustive explanation of solving quadratic equations,^[27] supported by geometric proofs while treating algebra as an independent discipline in its own right.^[22] His algebra was also no longer concerned «with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study». He also studied an equation for its own sake and «in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems».^[28]

Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe.^[29] Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.^[30] He also developed the concept of a function.^[31] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,^[32] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took «the first steps toward the introduction of algebraic symbolism». He also computed Σn², Σn³ and used the method of successive approximation to determine square roots.^[33]

Modern history

François Viète’s work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper «Réflexions sur la résolution algébrique des équations« devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.

Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues.^[34] George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).^[35]

Areas of mathematics with the word algebra in their name

Some subareas of algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word «algebra» in the name.

• Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.
• Abstract algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
• Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied.
• Boolean algebra, a branch of algebra abstracting the computation with the truth values false and true.
• Commutative algebra, the study of commutative rings.
• Computer algebra, the implementation of algebraic methods as algorithms and computer programs.
• Homological algebra, the study of algebraic structures that are fundamental to study topological spaces.
• Universal algebra, in which properties common to all algebraic structures are studied.
• Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view.
• Algebraic geometry, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.
• Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.
• Relational algebra: a set of finitary relations that is closed under certain operators.

Many mathematical structures are called algebras:

• Algebra over a field or more generally algebra over a ring.
  Many classes of algebras over a field or over a ring have a specific name:
  • Associative algebra
  • Non-associative algebra
  • Lie algebra
  • Composition algebra
  • Hopf algebra
  • C*-algebra
  • Symmetric algebra
  • Exterior algebra
  • Tensor algebra
• In measure theory,
  • Sigma-algebra
  • Algebra over a set
• In category theory
  • F-algebra and F-coalgebra
  • T-algebra
• In logic,
  • Relation algebra, a residuated Boolean algebra expanded with an involution called converse.
  • Boolean algebra, a complemented distributive lattice.
  • Heyting algebra

Elementary algebra

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). This is useful because:

• It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
• It allows the reference to «unknown» numbers, the formulation of equations and the study of how to solve these. (For instance, «Find a number x such that 3x + 1 = 10″ or going a bit further «Find a number x such that ax + b = c«. This step leads to the conclusion that it is not the nature of the specific numbers that allow us to solve it, but that of the operations involved.)
• It allows the formulation of functional relationships. (For instance, «If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied».)

Polynomials

A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, x² + 2x − 3 is a polynomial in the single variable x. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (x − 1)(x + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.

Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Education

It has been suggested that elementary algebra should be taught to students as young as eleven years old,^[36] though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States.^[37] However, in some US schools, algebra instruction starts in ninth grade.

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are the listed fundamental concepts in abstract algebra.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: collections of objects called elements. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors of a plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is generalized to the notion of binary operation (denoted here by ∗). The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a ∗ b is another element in the set; this condition is called closure. Addition (+), subtraction (−), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are generalized to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a, and is necessarily unique, if it exists. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, …) has no identity element for addition.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written −a, and for multiplication the inverse is written a⁻¹. A general two-sided inverse element a⁻¹ satisfies the property that a ∗ a⁻¹ = e and a⁻¹ ∗ a = e, where e is the identity element.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes a ∗ b = b ∗ a. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

Groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

• An identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a⁻¹ such that a ∗ a⁻¹ and a⁻¹ ∗ a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).

If a group is also commutative – that is, for any two members a and b of S, a ∗ b is identical to b ∗ a – then the group is said to be abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The non-zero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.

The theory of groups is studied in group theory. A major result of this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which separates the finite simple groups into roughly 30 basic types.

Semi-groups, quasi-groups, and monoids are algebraic structures similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an associative binary operation but might not have an identity element. A monoid is a semi-group which does have an identity but might not have an inverse for every element. A quasi-group satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however, the binary operation might not be associative.

All groups are monoids, and all monoids are semi-groups.

Examples

Set	Natural numbers N	Integers Z	Rational numbers Q Real numbers R Complex numbers C	Integers modulo 3 Z/3Z = {0, 1, 2}
Operation	+	×	+	×	+	−	×	÷	+	×
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes
Identity	0	1	0	1	0	N/A	1	N/A	0	1
Inverse	N/A	N/A	−a	N/A	−a	N/A	1/a (a ≠ 0)	N/A	0, 2, 1, respectively	N/A, 1, 2, respectively
Associative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Commutative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Structure	monoid	monoid	abelian group	monoid	abelian group	quasi-group	monoid	quasi-group	abelian group	monoid

Rings and fields

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings and fields.

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have an identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

Distributivity generalises the distributive law for numbers. For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a⁻¹.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

See also

• Algebra tile
• Outline of algebra
• Outline of linear algebra

References

Citations

Works cited

• Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons. ISBN 978-0-471-54397-8.
• Gandz, S. (January 1936). «The Sources of Al-Khowārizmī’s Algebra». Osiris. 1: 263–277. doi:10.1086/368426. JSTOR 301610. S2CID 60770737.
• Herstein, I. N. (1964). Topics in Algebra. Ginn and Company. ISBN 0-471-02371-X.

Further reading

• Allenby, R. B. J. T. (1991). Rings, Fields and Groups. ISBN 0-340-54440-6.
• Asimov, Isaac (1961). Realm of Algebra. Houghton Mifflin.
• Euler, Leonhard (November 2005). Elements of Algebra. ISBN 978-1-899618-73-6. Archived from the original on 2011-04-13.
• Herstein, I. N. (1975). Topics in Algebra. ISBN 0-471-02371-X.
• Hill, Donald R. (1994). Islamic Science and Engineering. Edinburgh University Press.
• Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books. ISBN 9780140277784.
• O’Connor, John J.; Robertson, Edmund F. (2005). «History Topics: Algebra Index». MacTutor History of Mathematics archive. University of St Andrews. Archived from the original on 2016-03-03. Retrieved 2011-12-10.
• Sardar, Ziauddin; Ravetz, Jerry; Loon, Borin Van (1999). Introducing Mathematics. Totem Books.

External links

• Khan Academy: Conceptual videos and worked examples
• Khan Academy: Origins of Algebra, free online micro lectures
• Algebrarules.com: An open source resource for learning the fundamentals of Algebra
• 4000 Years of Algebra, lecture by Robin Wilson, at Gresham College, October 17, 2007 (available for MP3 and MP4 download, as well as a text file).
• Pratt, Vaughan. «Algebra». In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.

Что такое алгебра?

Алгебра — это раздел математики, занимающийся символами и правилами обращения с этими символами. В элементарной алгебре эти символы (сегодня пишутся латинскими и греческими буквами) представляют величины без фиксированных значений, известные как переменные. Точно так же, как предложения описывают отношения между конкретными словами, в алгебре уравнения описывают отношения между переменными.

Algebra (Arabic: al-jebr‎, from الجبر al-jabr, meaning «reunion of broken parts»)^[1] is a branch of mathematics concerning the study of structure, relation, and quantity. Elementary algebra is the branch that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Many historians agree that the earliest mathematical research was done by the priest classes of ancient civilizations, most notably the Babylonians, to go along with religious rituals.^[2] The origins of algebra can thus be traced back to ancient Babylonian mathematicians roughly four thousand years ago. After further development among Hellenistic and Indian mathematicians, it was eventually the work of Islamic mathematicians that established algebra as an independent discipline in its own right.

Etymology

The word Algebra is derived from the Arabic word Al-Jabr, and this comes from the treatise written in 820 by the Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, entitled, in Arabic, كتاب الجبر والمقابلة or Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, which 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. Although the exact meaning of the word al-jabr is still unknown, most historians agree that the word meant something like «restoration», «completion»,^[3] «reuniter of broken bones» or «bonesetter.» 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.^[3]

Stages of algebra

See also: Timeline of algebra

Algebraic expression

Algebra did not always make use of the symbolism that is now ubiquitous in mathematics, rather, it went through three distinct stages. The stages in the development of symbolic algebra are roughly as follows:^[4]

• Rhetorical algebra, where equations are written in full sentences. For example, the rhetorical form of 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, where some symbolism is used but which does not contain all of the characteristic 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, where 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, though fully symbolic algebra sees its culmination in the work of Rene Descartes.

As important as the symbolism, or lack thereof, that was used in algebra was the degree of the equations that were used. 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 p and q are positive.
This trichotomy comes about because quadratic equations of the form , with p and q positive, have no positive roots.^[5]

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 A.

Conceptual stages

In addition to the three stages of expressing algebraic ideas, there were four conceptual stages in the development of algebra which occurred alongside the changes in expression. These four stages were as follows:^[6]

• 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 geometric algebra dates back to Diophantus and Brahmagupta, but algebra didn’t decisively move to the static equation-solving stage until Al-Khwarizmi’s Al-Jabr.
• Dynamic function stage, where motion is an underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra didn’t 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.

Early developments

Ancient Babylonian mathematics

The origins of algebra can be traced to the ancient Babylonians,^[7] who developed a positional number system which greatly aided them in solving their rhetorical algebraic equations. The Babylonians were not interested in exact solutions but approximations, and so they would commonly use linear interpolation to approximate intermediate values.^[8] One of the most famous tablets is the Plimpton 322 tablet, created around 1900 — 1600 BCE, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek mathematics.^[9]

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.^[8] 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.^[8] They were familiar with many simple forms of factoring,^[8] three-term quadratic equations with positive roots,^[10] and many cubic equations^[11] although it is not known if they were able to reduce the general cubic equation.^[11]

Ancient Egyptian mathematics

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.^[8]

The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written circa 1650 BCE by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 BCE.^[12] It is the most extensive ancient Egyptian mathematical document known to historians.^[13] The Rhind Papyrus contains problems where linear equations of the form and are solved, where a, b, and c are known and x, which is referred to as «aha» or heap, is the unknown.^[14] 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.^[14]

Ancient Indian mathematics

See also: Indian mathematics

The method known as «Modus Indorum» or the method of the Indians have become our algebra today. This algebra came along with the Hindu Number system to Arabia and then migrated to Europe. The earliest known Indian mathematical documents are dated to around the middle of the first millennium B.C.E (around the sixth century B.C.E.).^[15]

The recurring themes in Indian mathematics are, among others, determinate and indeterminate linear and quadratic equations, simple mensuration, and Pythagorean triples.^[16]

Ancient Chinese mathematics

See also: Chinese mathematics

The earliest known magic squares appeared in China,^[17] as early as 650 BCE.^[18]

The Chou Pei Suan Ching from 300 BCE is generally considered to be one of the oldest Chinese mathematical documents.^[19]

Magic squares

See also: Lo Shu Square

The earliest known magic squares appeared in China,^[17] as early as 650 BCE.^[18]

In Nine Chapters on the Mathematical Art, 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.^[17]

Nine Chapters on the Mathematical Art

See also: The Nine Chapters on the Mathematical Art

Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 BCE, 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.^[19]

Greek geometric mathematics

See also: Greek mathematics

While the Greeks had no algebra in the modern sense, it would be inaccurate to say there was nothing like algebra.^[20] By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric equivalent of algebra, where terms were represented by sides of geometric objects,^[21] usually lines, that had letters associated with them,^[22] and with this form they were able to find solutions to equations by using a process that they invented which is known as «the application of areas».^[21] «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 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 a:b and c:x. The Greeks would construct a rectangle with sides of length b and c, then extend a side of the rectangle to length a, and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution.^[21]

Bloom of Thymaridas

Iamblichus in Introductio arithmatica tells us that Thymaridas (ca. 400 BCE — ca. 350 BCE) worked with simultaneous linear equations.^[23] 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 n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/ (n + 2) of the difference between the sums of these pairs and the first given sum.^[24]

or using modern notion, the solution of the following system of n linear equations in n unknowns,^[23]

x + x1 + x2 + … + xn-1 = s

x + x1 = m1

x + x2 = m2

.

.

.

x + xn-1 = mn-1

is,

Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.^[23]

Conic sections

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^[25] (ca. 380 BCE – ca, 320 BCE) 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 y² = lx holds, where 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.^[26] 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.^[26]

We are informed by Eutocius that the method he used to solve the cubic equation was due to Dionysodorus (250 BCE — 190 BCE). 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.

Hellenistic mathematics in Egypt

Euclidean geometric mathematics

Euclid (Greek: Εὐκλείδης) was a Hellenistic Egyptian mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC).^[27][28] Neither the year nor place of his birth^[27] 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.^[27] 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.^[29] 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.^[30]

Elements

See also: Euclid’s Elements

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.^[20] 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.^[20]

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, ; and in Books V and VII of the Elements the commutative and associative laws for multiplication are demonstrated.^[20]

Many basic equations were also proved geometrically. For instance, proposition 4 in Book II proves that ,^[31] and proposition 5 in Book II proves that .^[20]

Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation , and proposition 11 of Book II gives a solution to .^[32]

Data

See also: Data (Euclid)

Data is a work written by Euclid for use at the university 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.^[33] Some of these statements are geometric equivalents to solutions of quadratic equations.^[33] For instance, Data contains the solutions to the equations and the familiar Babylonian equation , x ± y = b.^[33]

Diophantine mathematics

Diophantus was a Hellenized Babylonian mathematician who lived in Alexandria, Egypt, circa 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.^[34] Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, but resembles Babylonian mathematics to a much greater extent, though it is also quite different from it in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.^[35]

In Arithmetica, Diophantus used symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;^[35] 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.^[36] So, for example, what we would write as

Diophantus would have written as

Κ^Υ α̅ς ι̅  ⫛ Δ^Υ β̅ Μ  α̅ ἴσ Μ  ε̅

where the symbols represent the following:^[37][38]

Symbol	Representation
α̅	represents 1
β̅	represents 2
ε̅	represents 5
ι̅	represents 10
ς	represents the unknown quantity (i.e. the variable)
ἴσ	(short for ἴσος) represents «equals»
⫛	represents the subtraction of everything that follows it up to ἴσ
Μ	represents the zeroth power of the variable (i.e. a constant term)
ΔΥ	represents the second power of the variable, from Greek δύναμις, meaning strength or power
ΚΥ	represents the third power of the variable, from Greek κύβος, meaning a cube
ΔΥΔ	represents the fourth power of the variable
ΔΚΥ	represents the fifth power of the variable
ΚΥΚ	represents the sixth power of the variable

Note that 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:^[37]

and, to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:^[37]

Arithmetica is a collection of some 150 solved problems with specific numbers and there is no postulational development nor is a general method explicitly explained, although generality of method may have been intended and there is no attempt to find all of the solutions to the equations.^[35] Arithmetica does contain solved problems involving several unknown quantities, which are solved, if possible, by expressing the unknown quantities in terms of only one of them.^[35] Arithmetica also makes use of the identities:^[39]

It seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to earlier Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”^[40]

According to mathematics historian Odile Kouteynikoff:

According to the fact that Al-Khwarizmi founded Algebra during the 9th century, it is not surprising that, when being translated into Arabic in the late 9th century by Lebanese Ibn Luqa whose native language was Greek, Diophante’s Arithmetics seemed to be considered as a treatise about Algebra since algebraic vocabulary and way of thinking were most widely shared. Only few people understood that it was actually an arithmetic treatise: Al-Khazin (900–971) did, and therefore he is one of those who laid the foundations for the integer Diophantine analysis.^[41]

European mathematics in Dark Ages

Just as the death of Hypatia signals the close of the Library of Alexandria as a mathematical center, so does the death of Boethius signal the end of mathematics in the Western Roman Empire. Although there was some work being done at Athens, it came to a close when in 529 the Byzantine emperor Justinian closed the pagan philosophical schools. The year 529 is now taken to be the beginning of the medieval period. Scholars fled the West towards the more hospitable East, particularly towards Persia, where they found haven under King Chosroes and established what might be termed an «Athenian Academy in Exile».^[42] Under a treaty with Justinian, Chosroes would eventually return the scholars to the Eastern Empire. During the Dark Ages, European mathematics was at its nadir with mathematical research consisting mainly of commentaries on ancient treatises; and most of this research was centered in the Byzantine Empire. The end of the medieval period is set as the fall of Constantinople to the Turks in 1453.

Classical Indian mathematics

See also: Indian mathematics

Aryabhata and Aryabhatiya

Aryabhata (476–550 A.D.) was an Indian mathematician who authored Aryabhatiya. In it he gave the rules,^[43]

and

Brahmagupta and Brahma Sphuta Siddhanta

Brahmagupta (fl. 628) was an Indian mathematician who authored Brahma Sphuta Siddhanta. In his work, Brahmagupta solves the general quadratic equation for both positive and negative roots.^[44] In indeterminate analysis Brahmagupta gives the Pythagorean triads , , , but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with.^[45] He was the first to give a general solution to the linear Diophantine equation ax + by = c, where a, b, and c are integers. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave all integer solutions; but that Brahmagupta used some similar examples as Diophantus has led some historians to consider the possibility of influence on Brahmagupta’s work, or at least a common Babylonian source.^[46]

Like the work of Diophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.^[46] The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.^[46]

Classical Islamic algebra

The first century of the Islamic Arab Empire saw very little mathematical achievements, since the Arabs, with their newly conquered empire, had not yet gained the intellectual drive, while research in other parts of the world had faded. In the eighth century, Islam had a cultural awakening, and research in mathematics and the sciences increased.^[47] The Muslim Abbasid caliph Al-Mamun (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence Al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy’s Almagest and Euclid’s Elements. Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace.^[47] Many of these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.^[48]

There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Hellenistic-Egyptian influence. Many scholars believe that it is the result of a combination of all three sources.^[49]

The Arabs initially used a fully rhetorical algebra, where often even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs did not initially develop a syncopated or symbolic algebra,^[48] until the work of Ibn al-Banna in the 13th century and Abū al-Hasan ibn Alī al-Qalasādī in the 15th century.

Al-Khwarizmi

The Muslim^[50] Persian mathematician Muhammad ibn Mūsā al-Khwārizmī was a faculty member of the «House of Wisdom» (Bait al-Hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 CE, wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind.^[47]

Al-jabr wa’l muqabalah

One of al-Khwarizmi’s most famous books is entitled Al-jabr wa’l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree.^[51] The book also introduced the fundamental concept of «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. This is the operation which Al-Khwarizmi originally described as al-jabr.^[3]

R. Rashed and Angela Armstrong write:

«Al-Khwarizmi’s text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus’ Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.»^[52]

Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax² = bx), the second chapter deals with squares equal to number (ax² = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax² + bx = c), the fifth chapter deals with squares and number equal roots (ax² + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax²).^[53]

In Al-Jabr, al-Khwarizmi uses geometric proofs,^[22] he does not recognize the root x = 0,^[53] and he only deals with positive roots.^[54] He also recognizes that the discriminant must be positive and described the method of completing the square, though he does not justify the procedure.^[55] The Greek influence is shown by Al-Jabr’s geometric foundations^[49][56] and by one problem taken from Heron.^[57] He makes use of lettered diagrams but all of the coefficients in all of his equations are specific numbers since he had no way of expressing with parameters what he could express geometrically; although generality of method is intended.^[22]

Al-Khwarizmi most likely did not know of Diophantus’s Arithmetica,^[58] which became known to the Arabs sometime before the tenth century.^[59] And even though al-Khwarizmi most likely knew of Brahmagupta’s work, Al-Jabr is fully rhetorical with the numbers even being spelled out in words.^[58] So, for example, what we would write as

Diophantus would have written as^[60]

Δ^Υα̅ ςι̅ ‘ίσ Μ λ̅θ̅

And al-Khwarizmi would have written as^[60]

One square and ten roots of the same amount to thirty-nine dirhems; that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?

Father of algebra

The Islamic Persian mathematician Al-Khwarizmi is widely considered the father of algebra,^[61] though some have also given that title to the Hellenistic Babylonian mathematician Diophantus.^[61][62] Many agree that Al-Khwarizmi deserves this title most.^[61]

Those who support Diophantus point to the algebra found in Al-Jabr being more elementary than Arithmetica, and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.^[61] However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to earlier Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”^[40]

Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^[63] and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers.^[64] Al-Khwarizmi also introduced the fundamental concept of «reduction» and «balancing» (which he originally used the term al-jabr to refer to), 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.^[3] Other supporters of Al-Khwarizmi point to his algebra no longer being concerned «with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study.» They also point to his treatment of an equation for its own sake and «in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.»^[52] Al-Khwarizmi’s work established algebra as a mathematical discipline that is independent of geometry and arithmetic.^[65] In addition, R. Rashed and Angela Armstrong write:

«Al-Khwarizmi’s text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus’ Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.»^[66]

Ibn Turk and Logical Necessities in Mixed Equations

‘Abd al-Hamīd ibn Turk authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi’s Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr.^[59] The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution.^[59] The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and ‘Abd al-Hamid.^[59]

Abu Kamil: Irrational numbers and non-linear simultaneous equations

Arabic mathematicians treated irrational numbers as algebraic objects.^[67] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850-930) was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.^[68] He was also the first to solve three non-linear simultaneous equations with three unknown variables.^[69]

Al-Karaji: Pure algebra and algebraic calculus

Al-Karkhi (953-1029), also known as Al-Karaji, was the successor of Abū al-Wafā’ al-Būzjānī (940-998) and he was the first to discovered the numerical solution to equations of the form ax²ⁿ + bxⁿ = c.^[70] Al-Karkhi only considered positive roots.^[70] Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in Extrait du Fakhri, traité d’Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853), praised Al-Karaji for being «the first who introduced the theory of algebraic calculus». Stemming from this, Al-Karaji investigated binomial coefficients and Pascal’s triangle.^[71]

Brethren of Purity and magic squares

Magic squares (an early form of matrix) were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian culture, and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Brethren of Purity‘s Rasa’il Ikhwan al-Safa (Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab mathematicians.^[18]

Islamic mathematicians also solved more complex examples of magic squares. They developed two basic methods to solve odd-order magic squares: the «diamond» technique, and a more sophisticated magic torus method understood in terms of a virtual torus.^[72]

Omar Khayyám: Geometric algebra and algebraic geometry

Omar Khayyám (ca. 1050 — 1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.^[73] Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible.^[73] His method of solving cubic equations by using intersecting conics had been used by Ibn al-Haytham (Alhazen), but Omar Khayyám generalized the method to cover all cubic equations with positive roots.^[73] He only considered positive roots and he did not go past the third degree.^[73] He also saw a strong relationship between Geometry and Algebra.^[73]

Geometric solution of cubic equation

As shown in this graph, to solve the third-degree equation where Omar Khayyám constructed the parabola the circle with diameter having its center on the positive x-axis and intersecting the origin, and a vertical line through the point above the x-axis where the circle and parabola intersect. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.

Sharaf al-Dīn: Numerical analysis and dynamic functional algebra

In the 12th century, Sharaf al-Dīn al-Tūsī (1135–1213) wrote the Al-Mu’adalat (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the «Ruffini—Horner method» to numerically approximate the root of a cubic equation. He also developed the concepts of the maxima and minima of curves in order to solve cubic equations which may not have positive solutions.^[74] He understood the importance of the discriminant of the cubic equation and used an early version of Cardano‘s formula^[75] to find algebraic solutions to certain types of cubic equations. Some scholars such as Roshdi Rashed have pointed out that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance.^[76]

Sharaf al-Din also developed the concept of a function. In his analysis of
the equation for example, he begins by changing the equation’s form to . He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value . To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when , which gives the functional value . Sharaf al-Din then states that if this value is less than , there are no positive solutions; if it is equal to , then there is one solution at ; and if it is greater than , then there are two solutions, one between and and one between and .^[77]

Al-Hassar and symbolic notation

Al-Hassār, a mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appeared soon after in the work of Fibonacci in the 13th century.^[78]

The symbol now commonly denote an unknown variable. Even though any letter can be used, is the most common choice. This usage can be traced back to the Arabic word šay’ شيء = “thing,” used in Arabic algebra texts such as the Al-Jabr, and was taken into Old Spanish with the pronunciation “šei,” which was written xei, and was soon habitually abbreviated to . (The Spanish pronunciation of “x” has changed since). Some sources say that this is an abbreviation of Latin causa, which was a translation of Arabic شيء. This started the habit of using letters to represent quantities in algebra. In mathematics, an “italicized x” () is often used to avoid potential confusion with the multiplication symbol.

Late Medieval algebra

Indian algebra

See also: Indian mathematics

Bhāskara II: Lilavati and Vija-Ganita

Bhāskara II (1114-ca. 1185) was the leading Indian mathematician of the twelfth century. In Algebra, he gave the general solution of the Pell equation.^[46] He is the author of Lilavati and Vija-Ganita, which contain problems dealing with determinate and indeterminate linear and quadratic equations, and Pythagorean triples,^[16] though he fails to distinguish between exact and approximate statements.^[79] Many of the problems in Lilavati and Vija-Ganita are derived from other Hindu sources, and so Bhaskara is at his best in dealing with indeterminate analysis.^[79]

Bhaskara uses the initial symbols of the names for colors as the symbols of unknown variables. So, for example, what we would write today as

Bhaskara would have written as

. _ .

ya 1 ru 1

.

ya 2 ru 8

.

Sum ya 1 ru 9

where ya indicates the first syllable of the word for black, and ru is taken from the word species. The dots over the numbers indicate subtraction.

Citrabhanu and simultaneous equations

Citrabhanu (c. 1530) was a 16th-century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.

Chinese algebra

See also: Chinese mathematics

Li Zhi and Sea-Mirror of the Circle Measurements

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 — 1272 A.D.). 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.^[80]

Ch’in Chiu-shao and Mathematical Treatise in Nine Sections

Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch’in Chiu-shao (ca. 1202 — ca. 1261 A.D.) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.^[80]

Yang Hui and magic squares

See also: Lo Shu Square

The earliest known magic squares of order greater than six are attributed to Yang Hui (fl. ca. 1261 — 1275), who worked with magic squares of order as high as ten.^[81]

Chu Shih-chieh and Precious Mirror of the Four Elements

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.^[82]

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.^[83]

There are many summation series equations given without proof in the Precious mirror. A few of the summation series are:^[83]

Medieval European algebra

The twelfth century saw a flood of translations from Arabic into Latin and by the thirteenth century, European mathematics was beginning to rival the mathematics of other lands.

Fibonacci

In the thirteenth century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra in the 13th century.

Islamic algebra

Al-Kashi and numerical analysis

In the early 15th century, Jamshīd al-Kāshī developed an early form of Newton’s method to numerically solve the equation to find roots of .^[84] Al-Kāshī also developed decimal fractions and claimed to have discovered it himself. However, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu’l-Hasan al-Uqlidisi as early as the 10th century.^[69]

Ibn al-Banna and Al-Qalasadi: Algebraic symbolism

Abū al-Hasan ibn Alī al-Qalasādī (1412–1482) was the last major medieval Arab algebraist, who made the first attempt at creating an algebraic notation since Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since Diophantus and Brahmagupta in ancient times.^[85] The syncopated notations of his predecessors, however, lacked symbols for mathematical operations.^[36] Al-Qalasadi «took the first steps toward the introduction of algebraic symbolism by using letters in place of numbers»^[85] and by «using short Arabic words, or just their initial letters, as mathematical symbols.»^[85]

Modern algebra

As the Islamic world was declining after the fifteenth century, the European world was ascending. And it is here that Algebra was further developed.

A key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century.

Japanese algebra

Main article: Japanese mathematics

The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices.

Modern European algebra

Along with Gottfried Leibniz in the 17th century, Gabriel Cramer also did some work on matrices and determinants in the 18th century.

The symbol commonly denotes an unknown variable. Even though any letter can be used, is the most common choice. This usage can be traced back to the Arabic word šay’ شيء = “thing,” used in Arabic algebra texts such as the Al-Jabr, and was taken into Old Spanish with the pronunciation “šei,” which was written xei, and was soon habitually abbreviated to . (The Spanish pronunciation of “x” has changed since). Some sources say that this is an abbreviation of Latin causa, which was a translation of Arabic شيء. This started the habit of using letters to represent quantities in algebra. In mathematics, an “italicized x” () is often used to avoid potential confusion with the multiplication symbol.

Gottfried Leibniz

Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Gottfried Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.^[86] In the 18th century, «function» lost these geometrical associations.

Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination. Leibniz also discovered Boolean algebra and symbolic logic, also relevant to algebra.

Abstract algebra

Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.

See also

• Algebra
• Timeline of algebra
• History of Mathematics

Footnotes and citations

References

• Bashmakova, I., and Smirnova, G. (2000) The Beginnings and Evolution of Algebra, Dolciani Mathematical Expositions 23. Translated by Abe Shenitzer. The Mathematical Association of America.
• Boyer, Carl B. (1991), A History of Mathematics (Second Edition ed.), John Wiley & Sons, Inc., ISBN 0471543977
• Cooke, Roger (1997), The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0471180823
• Derbyshire, John (2006), Unknown Quantity: A Real And Imaginary History of Algebra, Joseph Henry Press, ISBN 030909657X
• Stillwell, John (2004), Mathematics and its History (Second Edition ed.), Springer Science + Business Media Inc., ISBN 0387953361
• Burton, David M. (1997), The History of Mathematics: An Introduction (Third Edition ed.), The McGraw-Hill Companies, Inc., ISBN 0070094659
• Heath, Thomas Little (1981a), A History of Greek Mathematics, Volume I, Dover publications, ISBN 0486240738
• Heath, Thomas Little (1981b), A History of Greek Mathematics, Volume II, Dover publications, ISBN 0486240746
• Flegg, Graham (1983), Numbers: Their History and Meaning, Dover publications, ISBN 0486421651

External links

• History of Algebra Articles at Convergence

Developmental Stages Of Algebra

There are three critical developmental stages in «Symbolic Algebra» which is as follows :

1. Rhetorical Algebra

It was developed by ancient Babylonians where the equation was written in the form of words that remained up to the 16th century.

Example: x + 5 = 8, is written as » The thing plus five equal to eight»

2. Syncopated Algebra

Its expression first appeared in Diophantus Arithmetica (3rd century ) Brahmagupta‘s «Brahmagupta’s sputa Siddhanta» (9th century), where few symbols were used, and subtraction was used only once in the equation.

3. Symbolic Algebra

At this stage, all the symbols were used in Algebra. Many Islamic mathematicians like Ibn Al-Banna and Al Qalasadi wrote in their books about Symbolic Algebra. Francois Viete fully developed it in the 16th century. Rene Descartes introduced a modern notation that can solve geometrical problems in terms of algebra known as Cartesian Geometry.

[begin{align}&x^2 + px = q\[6pt]&x^2 = px + q\[6pt]&x^2 + q = px end{align}]

In early algebra, Quadratic equations played an important role, where it is said to belong one among the above three equations.

# The Greeks and Vedic Indian Mathematicians developed two more stages of algebra which lie between rhetorical and syncopated stages known as Geometric Constructive stages.

Contributions of different countries

I. Babylon

Ancient Babylonians developed a rhetorical stage of algebra where equations were written in the form of words. They used linear interpolation to approximate intermediate values as they were not interested more in exact solutions. Plimpton 322 tablet, one of the most famous tablets designed around 1900 — 1600 BC gives the tables of «Pythagorean Triplets».

Learn more about the .

Following are the significant contributions :

• They developed flexible algebraic operations to eliminate fractions and factors by adding equals to equals and multiplying like quantities on both sides of the equation.
• They also knew simple forms of factoring, three-term quadratic equations with the positive term, and cubic equations

II. Ancient Egypt

Ahmed, an Egyptian mathematician, wrote an Egyptian Papyrus in 1650 BC known as «The Rhind Papyrus» which is considered to be the most extensive ancient Egyptian Mathematical document in history. They mainly used linear equations.

The Rhind Papyrus contains problems of linear equation in the form of

x + xa = b and x + xa + bx = c where a, b and c are known terms and x is referred as » aha» or heap. The equations were solved by «method of false position» or «regular falsi» , in which a specific value is substituted to the left-hand side of the equation, and the obtained answer, after performing the required arithmetic operation is compared to the right-hand side of the equation.

III. China

• ZHOUBI SUANGJING is one of the oldest Chinese Mathematical documents.

The following are the five major books by Chinese mathematicians in algebra.

Nine Chapters on Mathematical Art	It is one of the most influential books that gives solutions for determining and indeterminate simultaneous linear equations using both positive and negative numbers. In one of the problems, it has a solution for five unknowns in four equations.
Sea-mirror of the Circle- measurement	LI ZHI wrote this book where he solved equations with the highest degree as six by using Horner’s method.
Mathematical Treatise in nine section	Ch’in Chiu-Shao, a wealthy governor and minister, invented the Chinese Remainder Theorem to solve simultaneous congruences.
Magic Square	In this book, the author Yang Hui formed a magic square or matrix by placing coefficients and constants to solve simultaneous linear equations. He worked with column reduction methods to get the solution.
Precious mirror of the four elements	Chu Shih-Chieh wrote this book in 1303 where unknown quantities in algebraic equations were represented as heaven, man, earth, and matter. Horner’s method is used to solve the simultaneous equation with the highest degree of fourteen.

Precious mirror of the four elements

IV Greece

The Greek mathematician represented the sides of geometric objects, lines, and letters associated with them, which is called Geometric Algebra.

They invented «The application of areas « to obtain the solutions for equations solved in geometric algebra.

Following are few Greek mathematicians whose contribution are the milestone in the history of Algebra :

1. Thymaridas (c 400 BCE — 350 BCE) created a famous rule called

 «Blooms Of Thymaridas» which states that

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