Algebra

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The quadratic formula expresses the solution of the equation ax + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c. Quadratic formula.svg
The quadratic formula expresses the solution of the equation ax + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.

Algebra (from Arabic : الجبرal-jabr, meaning "reunion of broken parts" [1] and "bonesetting" [2] ) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; [3] it is a unifying thread of almost all of mathematics. [4] It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

Contents

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. [5] For example, in the letter is an unknown, but applying additive inverses can reveal its value: . In E = mc2 , the letters and are variables, and the letter is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.

A mathematician who does research in algebra is called an algebraist.

Etymology

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi. Muhammad ibn Musa al-Khwarizmi.png
The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

The word algebra comes from the Arabic الجبر (al-jabr lit. "the restoring of broken parts") from the title of the early 9th century book cIlm 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 fifteenth 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 sixteenth century. [7]

Different meanings of "algebra"

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

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers. [5] 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 abstracted 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 until it 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) is 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

Early history of algebra

A page from Al-Khwarizmi's al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala Image-Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala.jpg
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala

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. [5] 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 Arabic 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] A debate now exists whether who (in the general sense) is more entitled to be known 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 ∑n2, ∑n3 and used the method of successive approximation to determine square roots. [33]

Modern history of algebra

Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna. Gerolamo Cardano (colour).jpg
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna .

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 areas of mathematics that fall under the classification abstract 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.

Many mathematical structures are called algebras:

Elementary algebra

Algebraic expression notation:
1 - power (exponent)
2 - coefficient
3 - term
4 - operator
5 - constant term
x y c - variables/constants Algebraic equation notation.svg
Algebraic expression notation:
  1 – power (exponent)
  2 – coefficient
  3 – term
  4 – operator
  5 – constant term
 xyc – variables/constants

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:

Polynomials

The graph of a polynomial function of degree 3 Polynomialdeg3.svg
The graph of a polynomial function of degree 3

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 is started 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: a collection of all objects (called elements) selected by property specific for the set. 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 in the 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 abstracted to give a binary operation, ∗ say. 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, ab 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 abstracted 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 ae = a and ea = 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 aa−1 = e and a−1a = 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 (ab) ∗ c = a ∗ (bc). 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 ab = ba. 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:

If a group is also commutative – that is, for any two members a and b of S, ab is identical to ba – 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 ¼, which is not an integer.

The theory of groups is studied in group theory. A major result in 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 structure similar to groups, but more general. 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 (also real R and complex C numbers)Integers modulo 3: Z3 = {0, 1, 2}
Operation+× (w/o zero)+× (w/o zero)+× (w/o zero)÷ (w/o zero)+× (w/o zero)
ClosedYesYesYesYesYesYesYesYesYesYes
Identity01010N/A1N/A01
InverseN/AN/AaN/AaN/A1/aN/A0, 2, 1, respectivelyN/A, 1, 2, respectively
AssociativeYesYesYesYesYesNoYesNoYesYes
CommutativeYesYesYesYesYesNoYesNoYesYes
Structure monoid monoid abelian group monoid abelian group quasi-group abelian group quasi-group abelian group abelian group (Z2)

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

Related Research Articles

Arithmetic Elementary branch of mathematics

Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory.

Binary operation Mathematical operation that combines two elements for producing a third one

In mathematics, a binary operation or dyadic operation is a calculation that combines two elements to produce another element. More formally, a binary operation is an operation of arity two.

Field (mathematics) Algebraic structure with addition, multiplication and division

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

Group (mathematics) Algebraic structure with one binary operation

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namely associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

Monoid Algebraic structure with an associative operation and an identity element

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.

Modular arithmetic Computation modulo a fixed integer

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

Multiplication Arithmetical operation

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product.

Number theory Branch of mathematics

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.

Ring (mathematics) Algebraic structure with addition and multiplication

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Division (mathematics) Arithmetic operation

Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. The division sign ÷, a symbol consisting of a short horizontal line with a dot above and another dot below, is often used to indicate mathematical division. This usage, though widespread in anglophone countries, is neither universal nor recommended: the ISO 80000-2 standard for mathematical notation recommends only the solidus / or fraction bar for division, or the colon for ratios; it says that this symbol "should not be used" for division.

Galois theory Mathematical connection between field theory and group theory

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing to group theory certain problems in field theory; this makes them simpler in some sense, and allows a better understanding.

In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A of finite arity, and a finite set of identities, known as axioms, that these operations must satisfy.

Muhammad ibn Musa al-Khwarizmi 9th century Persian mathematician, astronomer and geographer

Muḥammad ibn Mūsā al-Khwārizmī, Arabized as al-Khwarizmi and formerly Latinized as Algorithmi, was a Persian polymath who produced vastly influential works in mathematics, astronomy, and geography. Around 820 CE he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.

Mathematics encompasses a growing variety and depth of subjects over history, and comprehension of it requires a system to categorize and organize these various subjects into a more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve.

<i>The Compendious Book on Calculation by Completion and Balancing</i>

The Compendious Book on Calculation by Completion and Balancing, also known as Al-Jabr (ٱلْجَبْر), is an Arabic mathematical treatise on algebra written by the Polymath Muḥammad ibn Mūsā al-Khwārizmī around 820 CE while he was in the Abbasid capital of Baghdad, modern-day Iraq. Al-Jabr was a landmark work in the history of mathematics, establishing algebra as an independent discipline, and with the term "algebra" itself derived from Al-Jabr.

The history of mathematical notation includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

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.

Abstract algebra Mathematical study of algebraic structures

In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belong to a given field, commonly the real or the complex numbers. This article is devoted to the same problems where "field" is replaced by "commutative ring", or, typically "Noetherian integral domain".

References

Citations

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  3. See Herstein 1964, page 1: "An algebraic system can be described as a set of objects together with some operations for combining them".
  4. See Herstein 1964, page 1: "...it also serves as the unifying thread which interlaces almost all of mathematics".
  5. 1 2 3 See Boyer 1991, Europe in the Middle Ages, p. 258: "In the arithmetical theorems in Euclid's Elements VII–IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."
  6. Esposito, John L. (2000-04-06). The Oxford History of Islam. Oxford University Press. p. 188. ISBN   978-0-19-988041-6.
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  8. "2010 Mathematics Subject Classification" . Retrieved 2014-10-05.
  9. Struik, Dirk J. (1987). A Concise History of Mathematics . New York: Dover Publications. ISBN   978-0-486-60255-4.
  10. See Boyer 1991.
  11. Cajori, Florian (2010). A History of Elementary Mathematics – With Hints on Methods of Teaching. p. 34. ISBN   978-1-4460-2221-4.
  12. Roshdi Rashed (November 2009). Al Khwarizmi: The Beginnings of Algebra. Saqi Books. ISBN   978-0-86356-430-7.
  13. "Diophantus, Father of Algebra". Archived from the original on 2013-07-27. Retrieved 2014-10-05.
  14. "History of Algebra" . Retrieved 2014-10-05.
  15. Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).
  16. Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).
  17. Meri, Josef W. (2004). Medieval Islamic Civilization. Psychology Press. p. 31. ISBN   978-0-415-96690-0 . Retrieved 2012-11-25.
  18. Corona, Brezina (February 8, 2006). Al-Khwarizmi: The Inventor Of Algebra. New York, United States: Rosen Pub Group. ISBN   978-1404205130.
  19. See Boyer 1991, page 181: "If we think primarily of the matter of notations, Diophantus has good claim to be known as the 'father of algebra', but in terms of motivation and concept, the claim is less appropriate. The Arithmetica is not a systematic exposition of the algebraic operations, or of algebraic functions or of the solution of algebraic equations".
  20. See Boyer 1991, page 230: "The six cases of equations given above exhaust all possibilities for linear and quadratic equations...In this sense, then, al-Khwarizmi is entitled to be known as 'the father of algebra'".
  21. See Boyer 1991, page 228: "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi".
  22. 1 2 See Gandz 1936, page 263–277: "In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
  23. Christianidis, Jean (August 2007). "The way of Diophantus: Some clarifications on Diophantus' method of solution". Historia Mathematica . 34 (3): 289–305. doi:10.1016/j.hm.2006.10.003. It is true that if one starts from a conception of algebra that emphasizes the solution of equations, as was generally the case with the Arab mathematicians from al-Khwārizmī onward as well as with the Italian algebraists of the Renaissance, then the work of Diophantus appears indeed very different from the works of those algebraists
  24. Cifoletti, G. C. (1995). "La question de l'algèbre: Mathématiques et rhétorique des homes de droit dans la France du 16e siècle". Annales de l'École des Hautes Études en Sciences Sociales, 50 (6): 1385–1416. Le travail des Arabes et de leurs successeurs a privilégié la solution des problèmes.Arithmetica de Diophantine ont privilégié la théorie des equations
  25. See Boyer 1991, page 228.
  26. See Boyer 1991, The Arabic Hegemony, p. 229: "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. 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".
  27. See Boyer 1991, The Arabic Hegemony, p. 230: "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions".
  28. Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN   978-0-7923-2565-9. OCLC   29181926.
  29. Mathematical Masterpieces: Further Chronicles by the Explorers. p. 92.
  30. O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive , University of St Andrews .
  31. Victor J. Katz, Bill Barton; Barton, Bill (October 2007). "Stages in the History of Algebra with Implications for Teaching". Educational Studies in Mathematics. 66 (2): 185–201 [192]. doi:10.1007/s10649-006-9023-7. S2CID   120363574.
  32. See Boyer 1991, The Arabic Hegemony, p. 239: "Abu'l Wefa was a capable algebraist as well as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax2n + bxn = c (only equations with positive roots were considered),"
  33. "Al-Qalasadi biography". www-history.mcs.st-andrews.ac.uk. Retrieved 2017-10-17.
  34. "The Origins of Abstract Algebra". University of Hawaii Mathematics Department.
  35. "The Collected Mathematical Papers". Cambridge University Press.
  36. "Hull's Algebra" (PDF). New York Times. July 16, 1904. Retrieved 2012-09-21.
  37. Quaid, Libby (2008-09-22). "Kids misplaced in algebra" (Report). Associated Press . Retrieved 2012-09-23.

Works cited

Further reading