Algebra

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Algebra ( Arabic: الجبر, al-jabr) is a branch of mathematics which studies structure and quantity. Elementary algebra is often taught in high school and gives an introduction into the basic ideas of algebra: studying what happens when one adds and multiplies numbers and how one can make polynomials and find their roots.

Algebra is much broader than arithmetic and can be generalized. Rather than working on numbers, one can work over symbols or elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures called groups, rings and fields.

Together with geometry and analysis, algebra is one of the three main branches of mathematics.

Classification

Algebra may be roughly divided into the following categories:

elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
linear algebra, in which the specific properties of vector spaces are studied (including matrices);
universal algebra, in which properties common to all algebraic structures are studied.

In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:

Normed linear spaces
Banach spaces
Hilbert spaces
Banach algebras
Normed algebras
Topological algebras
Topological groups

Elementary algebra

Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as a, x, y) to denote numbers. 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 $3 x + 1 = 10$ ).
It allows the formulation of functional relationships (such as "if you sell x tickets, then your profit will be $3 x - 10$ dollars").

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of objects called elements. All 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 group 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. For two elements a and b in a set S a*b gives another element in the set, (technically this condition is called closure). Addition (+), subtraction (-), multiplication (×), and division (÷) are all binary operations as in addition and multiplication of matrices, vectors, and polynomials.

Identity elements: Zero and one are abstracted to give the notion of an identity element. 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, 1×a=a. However, if we take the positive natural numbers and addition, there is no identity element.

Inverse elements: The negative numbers gives rise to the concept of an inverse elements. For addition the inverse of a is -a, and for multiplication the inverse is ¹/_a. A general inverse element a^-1 must satisfy the property that a*a^-1=e and a^-1*a=e.

Associativity: The integers with addition have a property called associativity: (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.

Commutativity: The integers with addition also have a property called commutativity: 2+3=3+2. In general this becomes a+b=b+a. Only some binary operations have this property, it holds for the integers with addition and multiplication, but it does not hold for matrix multiplication.

Groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group consists of:

a set S of elements,
a(closed) binary operation (*)
an identity element exists,
every element has an inverse,
the operation is associative.

If commutativity is included as well then we obtain an Abelian group.

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 since 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 $1 \cdot a = a \cdot 1 = a$ for any any rational number a. The inverse of a is $\frac{1}{a}$ , since $a \cdot {1 \over a}=1$ .

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 in this theory is the Classification of finite simple groups a vast body of work which classified all the is a vast body of work, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups.

Examples of groups
Set:	Natural numbers $\mathbb{N}$		Integers $\mathbb{Z}$		Rational numbers $\mathbb{Q}$ (also real $\mathbb{R}$ and complex $\mathbb{C}$ numbers)				Integers mod 3: {0,1,2}
operation	+ (including zero)	× (excluding zero)	+	× (excluding zero)	+	−	× (excluding zero)	÷ (excluding zero)	+
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
identity	0	1	0	1	0	NA	1	NA	0
inverse	NA	NA	-1	NA	-1	NA	¹/_a	NA	0,2,1 receptively
Associative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes
Commutative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes
Structure	semigroup	quasigroup	Abelian group	Monoid	Abelian group	quasigroup	Abelian group	quasigroup	Abelian group

Many other types of algebraic structures exist. Among the most common are rings, fields, and monoids. These different structures can be used to model different types of mathematical objects. Different algebraic structures are often related. For example, a group is a specific kind of monoid, and rings and fields are similar to groups, but with more operations.

Algebras

The word algebra is also used for various algebraic structures:

algebra over a field
algebra over a set
Boolean algebra
sigma-algebra
F-algebra and F-coalgebra in category theory

History

Hellenistic mathematician Euclid details geometrical algebra in Elements.

The origins of algebra can be traced to the ancient Egyptians and Babylonians, who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.

Circa 800 BC: Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, finds geometric solutions of linear equations and quadratic equations of the forms ax² = c and ax² + bx = c.
Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation.
Circa 300 BC: Hellenistic mathematician Euclid, who lived in Alexandria, Egypt, addresses quadratic equations in Book 2 of his Elements, although in a strictly geometrical fashion.
Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu ( The Nine Chapters on the Mathematical Art).
Circa 150 AD: Hellenized Egyptian mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.
Circa 200: Hellenized Babylonian mathematician Diophantus, who lived in Egypt and is often considered as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
Circa 200: The Bakhshali Manuscript in ancient India contains solutions of linear equations with as many as five unknowns, the general algebraic formula for the quadratic equation, quadratic indeterminate equations, and simultaneous equations.
499: Indian mathematician Aryabhata, obtains whole number solutions to linear equations by a method equivalent to the modern one.
628: Indian mathematician Brahmagupta, invents the method of solving indeterminate equations of the second degree, gives rules for solving linear and quadratic equations, and discovers negative solutions for the quadratic equation. Indian mathematicians at the time recognized that quadratic equations have two roots, and included negative as well as irrational roots.
820: The word algebra is derived from operations described in the treatise first written by Persian mathematician Al-Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion". Al-Khwarizmi is often considered as the "father of modern algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.
1114: Indian mathematician Bhaskara II, who wrote the text Bijaganita (Algebra), recognizes that a positive number has two square roots, and also solves quadratic indeterminate equations and quadratic equations with more than one unknown.
1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .