World Of Mathematics: October 2013

Algebra

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"Algebraist" redirects here. For the novel by Iain M. Banks, see The Algebraist.

For beginner's introduction to algebra, see Wikibooks: Algebra.

Three-dimensional right conoid surface plot, described by the elementary algebraic trigonometrical equations
$x=v \times \cos(u)$ , $y=v \times \sin(u)$ , $z=2 \times \sin(u)$

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis.
For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

As a single word without article, "algebra" names a broad part of mathematics (see below).
As a single word with article or in plural, "algebra" denote a specific mathematical structure. See algebra (ring theory) and algebra over a field.
With a qualifier, there is the same distinction:
- Without article, it means a part of algebra, like 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 abstract structure, like a Lie algebra or an associative algebra.
- Frequently both meanings exist for the same qualifier, like in the sentence: Commutative algebra is the study of commutative rings, that all are commutative algebras over the integers.
Sometimes "algebra" is also used to denote the operations and methods related to algebra in the study of a structure that does not belong to algebra. For example algebra of infinite series may denotes the methods for computing with series without using the notions of infinite summation, limits and convergence.

The adjective "algebraic" usually means relation to algebra, as in "algebraic structure". For historical reasons, it may also mean relation with the roots of polynomial equations, like in algebraic number, algebraic extension or algebraic expression.

Algebra as a branch of mathematics

Algebra can essentially be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects.^[1] Initially, these objects represented either numbers that were not yet known (unknowns) or unspecified numbers (indeterminate or parameter), allowing one to state and prove properties that are true no matter which numbers are involved. For example, in the quadratic equation

$ax^2+bx+c=0,$

$a, b, c$ are indeterminates and $x$ is the unknown. Solving this equation amounts to computing with the variables to express the unknowns in terms of the indeterminates. Then, substituting any numbers for the indeterminates, gives the solution of a particular equation after a simple arithmetic computation.
As it developed, algebra was extended to other non-numerical objects, like vectors, matrices or polynomials. Then, the structural properties of these non-numerical objects were abstracted to define algebraic structures like groups, rings, fields and algebras.
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 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, some of them included in algebra, either totally or partially.
It follows that algebra, instead of being a true branch of mathematics, appears nowadays, to be a collection of branches sharing common methods. This is clearly seen in the Mathematics Subject Classification^[2] where none of the first level areas (two digit entries) is called algebra. In fact, algebra is, roughly speaking, the union of sections 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. Some other first level areas may be considered to belong partially to algebra, like 11-Number theory (mainly for algebraic number theory) and 14-Algebraic geometry.
Elementary algebra is the part of algebra that is usually taught in elementary courses of mathematics.
Abstract algebra is a name usually given to the study of the algebraic structures themselves.

History

Main articles: History of algebra and Timeline of algebra

The start of algebra as an area of mathematics may be dated to the end of 16th century, with François Viète's work. Nevertheless some earlier works may be considered as algebra and constitute the prehistory of algebra.

Prehistory of algebra

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,^[3] 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.^[4]
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.^[1] Diophantus (3rd century AD), sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations.^[5]
The word algebra comes from the Arabic language (الجبر al-jabr "restoration") and much of its methods from Arabic/Islamic mathematics. Earlier traditions discussed above had a direct influence on Muhammad 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.^[6]
The Hellenistic mathematicians Hero of Alexandria and Diophantus ^[7] as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level.^[8] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim 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 contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he has to distinguish several types of equations.^[9]
The Greek mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.^[10] 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.^[11] 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,^[12] and that he gave an exhaustive explanation of solving quadratic equations,^[13] supported by geometric proofs, while treating algebra as an independent discipline in its own right.^[14] 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".^[15]
The Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.^[16] He also developed the concept of a function.^[17] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,^[18] 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. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.

History of algebra

In 1545, the Italian mathematician Girolamo Cardano published Ars magna -The great art, a 40-chapter masterpiece in which he gave for the first time a method for solving the general cubic and quartic equations.

François Viète's work at the close of the 16th century marks the start of the classical discipline of 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 Kowa Seki 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.^[19] The "modern algebra" has deep nineteenth-century roots in the work, for example, of Richard Dedekind and Leopold Kronecker and profound interconnections with other branches of mathematics such as algebraic number theory and algebraic geometry.^[20] 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).^[21]

Topics containing the word "algebra"

Areas of mathematics:

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.
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 by solutions of polynomials equations.
Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.

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:
In measure theory,
- Sigma-algebra
- Algebra over a set
In category theory
- F-algebra and F-coalgebra
- T-algebra
In logic,
- Relational algebra, in which a set of finitary relations that is closed under certain operators.
- Boolean algebra, a structure abstracting the computation with the truth values false and true. See also Boolean algebra (structure).
- Heyting algebra

Elementary algebra

Main article: 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 denoted by symbols (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".)

Polynomials

The graph of a polynomial function of degree 3.

Main article: Polynomial

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.

Teaching algebra

Abstract algebra

Main articles: Abstract algebra and Algebraic structure

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, 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 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 a ∗ e = a and e ∗ a = a. 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 = 1 and a⁻¹ ∗ a = 1 .
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

Main article: Group (mathematics)

See also: Group theory and Examples of 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 nonzero 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.
Semigroups, quasigroups, 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 semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup 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 semigroups.





		Rings and fields

Main articles: ring (mathematics) and field (mathematics)

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, and specifies the order in which the operators should be applied, (called the precedence). 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.

World Of Mathematics

Wednesday, October 30, 2013

Algebra

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