Algebra
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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī titled (in Arabic كتاب الجبر والمقابلة )Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.
Together with geometry, analysis, and number theory, algebra is one of the several main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.
Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.
Contents
Classification
Algebra may be divided roughly 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 such as 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 a number of areas of functional analysis:
- 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. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. Although in arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, numbers are often denoted by symbols (such as a, x, y). 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").
- It allows the formulation of functional relationships (such as "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 the function is performed on.").
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 (ax^{2} + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups that 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 is 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. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operation * 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. However, if we take the positive natural numbers and addition, there is no identity element.
Inverse elements: The negative numbers give rise to the concept of 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: 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.
Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 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 is a combination of a set S and a binary operation '*' with the following properties:
- The operation is closed: if a and b are members of S, then so is a * b.
- In fact, it is redundant to mention this property, for every binary operation must be closed. So, the statement "a group is a combination of a set S and a binary operation '*'" is already saying that the operation is closed. However, closure is frequently emphasized repeating it as a group property.
- 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 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, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.
Examples | ||||||||||
Set: | Natural numbers | Integers | Rational numbers (also real and complex numbers) | Integers mod 3: {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 | NA | 1 | NA | 0 | 1 |
Inverse | NA | NA | -a | NA | -a | a | a | 0,2,1, respectively | NA, 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 | quasigroup | Abelian group | quasigroup | Abelian group | Abelian group () |
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 that 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 a unique pre- or post-operation; however the binary operation might not be associative.
All groups are monoids, and all monoids are semigroups.
Rings and fields—structures with two binary operations
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.
Distributivity generalised 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 +.
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 allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as -a.
The integers are an example of a ring. The integers have additional properties that 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, real number and complex numbers are all examples of fields.
Algebras
The word algebra is also used for various algebraic structures:
- Algebra over a field
- Algebra over a set
- Boolean algebra
- F-algebra and F-coalgebra in category theory
- Sigma-algebra
History
The origins of algebra can be traced to the ancient Babylonians,^{[1]} who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion. With the use of this system they were able to apply formulas and calculate solutions for unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Sulba Sutras, 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.
The word "algebra" is named after the Arabic word "al-jabr" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Persian Muslim mathematician Muhammad ibn Mūsā al-khwārizmī in 820. The word Al-Jabr means "reunion". The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi should take that title from Diophantus.^{[2]} Those who support Al-Khwarizmi point to the fact that much of his work on reduction is still in use today and that he gave an exhaustive explanation of solving quadratic equations. Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.^{[3]} Another Persian mathematician, Omar Khayyam, developed algebraic geometry and found the general geometric solution of the cubic equation. The Indian mathematicians Mahavira and Bhaskara, and the Chinese mathematician Zhu Shijie, solved various cubic, quartic, quintic and higher-order polynomial equations.
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 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. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.
The stages of the development of symbolic algebra are roughly as follows:
- Rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century;
- Geometric constructive algebra, which was emphasised by the Vedic Indian and classical Greek mathematicians;
- Syncopated algebra, as developed by Diophantus and in the Bakhshali Manuscript; and
- Symbolic algebra, which sees its culmination in the work of Leibniz.
Timeline
A timeline of key algebraic developments can be found on the Timelines subpage.