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The quadratic formula expresses the solution of the equation ax2 + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.


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.[4] For example, in the letter is 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.


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


The name of algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi[5]

The word algebra comes from the Arabic الجبر (al-jabr lit. "the reunion of broken parts") from the title of the book Ilm al-jabr wa'l-muḳābala by the Persian mathematician and astronomer al-Khwarizmi. The word 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 the sixteenth century.[6]

Different meanings of "algebra"


Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.[4] 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 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[7] 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.


Early history of algebra


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.[4] 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,[10] 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.[11]

The Hellenistic mathematicians Hero of Alexandria and Diophantus[12] 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.[13][better source needed] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta.[citation needed] 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.[14]

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 context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".[15][16][17][18][19][20] A debate now exists whether who (in 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.[21] 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,[22] and that he gave an exhaustive explanation of solving quadratic equations,[23] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[18] 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".[24]

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.[25] Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.[26] He also developed the concept of a function.[27] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[28] 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.[29]

Modern history of algebra

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.[30] 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).[31]

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

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 allows 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".)


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 can not 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.


It has been suggested that elementary algebra should be taught to students as young as eleven years old,[32] 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.[33] 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 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.


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

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 are structures 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.

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)
Closed Yes Yes Yes Yes Yes Yes Yes Yes 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 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 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 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



  1. ^ "algebra". Oxford English Dictionary. Oxford University Press.
  2. ^ See Herstein 1964, page 1: "An algebraic system can be described as a set of objects together with some operations for combining them".
  3. ^ See Herstein 1964, page 1: "...it also serves as the unifying thread which interlaces almost all of mathematics".
  4. ^ a b c 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."
  5. ^ Esposito, John L. (2000-04-06). The Oxford History of Islam. Oxford University Press. p. 188. ISBN 978-0-19-988041-6.
  6. ^ T. F. Hoad, ed. (2003). "Algebra". The Concise Oxford Dictionary of English Etymology. Oxford: Oxford University Press. doi:10.1093/acref/9780192830982.001.0001. ISBN 978-0-19-283098-2.
  7. ^ "2010 Mathematics Subject Classification". Retrieved 2014-10-05.
  8. ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. ISBN 978-0-486-60255-4.
  9. ^ See Boyer 1991.
  10. ^ Cajori, Florian (2010). A History of Elementary Mathematics – With Hints on Methods of Teaching. p. 34. ISBN 978-1-4460-2221-4.
  11. ^ Roshdi Rashed (November 2009). Al Khwarizmi: The Beginnings of Algebra. Saqi Books. ISBN 978-0-86356-430-7.
  12. ^ "Diophantus, Father of Algebra". Archived from the original on 2013-07-27. Retrieved 2014-10-05.
  13. ^ "History of Algebra". Retrieved 2014-10-05.
  14. ^ Meri, Josef W. (2004). Medieval Islamic Civilization. Psychology Press. p. 31. ISBN 978-0-415-96690-0. Retrieved 2012-11-25.
  15. ^ See Boyer 1991, page 181: "If we think primarily of 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".
  16. ^ See Boyer 1991, page 230: "The six cases of equations given above exhaust all possiblities for linear and quadratic equations...In this sense, then, al-Khwarizmi is entitled to be known as 'the father of algebra'".
  17. ^ See Boyer 1991, page 228: "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi".
  19. ^ 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
  20. ^ 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
  21. ^ See Boyer 1991, page 228.
  23. ^ 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".
  24. ^ Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 978-0-7923-2565-9. OCLC 29181926.
  25. ^ Mathematical Masterpieces: Further Chronicles by the Explorers. p. 92.
  27. ^ 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.
  28. ^ 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),"
  29. ^ "Al-Qalasadi biography". www-history.mcs.st-andrews.ac.uk. Retrieved 2017-10-17.
  30. ^ "The Origins of Abstract Algebra". University of Hawaii Mathematics Department.
  31. ^ "The Collected Mathematical Papers". Cambridge University Press.
  32. ^ "Hull's Algebra" (pdf). New York Times. July 16, 1904. Retrieved 2012-09-21.
  33. ^ Quaid, Libby (2008-09-22). "Kids misplaced in algebra" (Report). Associated Press. Retrieved 2012-09-23.

Works cited

Further reading

External links