Encyclopedia
Algebra is a branch of
mathematics concerning the study of structure, relation and quantity. 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 variable, 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.
The name
Algebra is derived from the treatise written by the
Persian mathematician titled
Al-Kitab al-Jabr wa-l-Muqabala , 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.
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 ;
- 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 ;
- 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 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
- Main article: 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 occur, in algebra, numbers are often denoted by symbols . This is useful because:
- It allows the general formulation of arithmetical laws , 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 .
- It allows the formulation of functional relationships .
Abstract algebra
- Main article: Abstract algebra; see also algebraic structure.
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 , 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, .
Addition ,
subtraction , multiplication , and division are all binary operations as in 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 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. 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: +4=2+. In general, this becomes *
c =
a * . 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
- Main article: group; see also group theory, examples of 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 binary operation
- An identity element exists,
- Every element has an inverse,
- The operation is associative.
If commutativity is included as well, then we get 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, because for any integers
a,
b and
c, +
c =
a + .
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 | Integers mod 3: |
|---|
| operation | + | × | + | × | + | - | × | ÷ | + | × |
|---|
| Closed | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
|---|
| identity | 0 | 1 | 0 | 1 | 0 | 0 | 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 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 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
- Main articles: ring , field ; see also ring theory, glossary of ring theory, field theory, glossary of field theory.
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, . For the integers × c =
a×
c+
b×
c and
c × =
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 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, real number and complex numbers are all examples of fields.
Algebras
The word
algebra is also used for various algebraic structures:
History
The origins of algebra can be traced to the ancient Babylonians, 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
Moscow and Rhind Mathematical Papyri,
Sulba Sutras,
Euclid's Elements, and
The Nine Chapters on the Mathematical Art is a Chinese mathematics [i] book, probably composed by th ...
. 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.
Indian mathematicians proceeded to write treatises on algebraic means of solving equations from the end of the first millennium BC, followed by
Hellenistic mathematicians from the early first millennium AD. Important algebraic works from this general era include the
Bakhshali Manuscript, the works of
Hero of Alexandria, the
Arithmetica, an ancient Greek [i] text on mathematics [i] written by the Hellenistic ...
of
Diophantus, the
Aryabhatiya of Aryabhata, and the
Brahma Sputa Siddhanta of Brahmagupta.
The word "algebra" is named after the
Arabic word "
al-jabr" from the title of the book
, meaning
The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the
Persian Muslim mathematician in 820. The word
al-jabr means
"reunion". Al-Khwarizmi is often considered the "father of algebra" , as much of his works on reduction are still in use today. 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
...
.
A timeline of key algebraic developments are as follows:
- Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.
- Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.
- Circa 800 BC: Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax2 = c and ax2 + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.
- Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.
- Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.
- Circa 300 BC: A geometric construction for the solution of the cubic is sought . It is now well known that the general cubic has no such solution using Euclidean tools
...
.
- Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu is a Chinese mathematics [i] book, probably composed by th ...
, which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations. - Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.
- 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 the "father of algebra", writes his famous Arithmetica, an ancient Greek [i] text on mathematics [i] written by the Hellenistic ...
, a work featuring solutions of algebraic equations and on the theory of numbers. - 499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a differential equation.
- Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.
- 628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell's equation, and gives rules for solving linear and quadratic equations. He discovers that quadratic equations have two roots, including both negative as well as irrational roots.
- 820: The word algebra is derived from operations described in the treatise written by the Persian mathematician titled Al-Kitab al-Jabr wa-l-Muqabala on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.
- Circa 850: Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.
- Circa 850: Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations.
- Circa 990: Persian Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and gives rules for the products of any two of these.
- Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations.
- 1072: Persian mathematician Omar Khayyam develops algebraic geometry and, in the Treatise on Demonstration of Problems of Algebra, gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.
- 1114: Indian mathematician Bhaskara, in his Bijaganita , recognizes that a positive number has both a positive and negative square root, and solves quadratic equations with more than one unknown, various cubic, quartic and higher-order polynomial equations, Pell's equation, the general indeterminate quadratic equation, as well as indeterminate cubic, quartic and higher-order equations.
- 1150: Bhaskara, in his Siddhanta Shiromani, solves differential equations.
- 1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.
- Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.
- Circa 1400: Indian mathematician Madhava of Sangamagramma finds the solution of transcendental equations by iteration, iterative methods for the solution of non-linear equations, and solutions of differential equations.
- 1515: Scipione del Ferro solves a cubic such that the quadratic term is missing.
- 1535: Nicolo Fontana Tartaglia solves a cubic such that the linear term is missing.
- 1545: Girolamo Cardano publishes Ars magna -The great art which gives solutions for a variety of cubics as well as Ludovico Ferrari's solution of a special quartic equation.
- 1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.
- 1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.
- 1631: Thomas Harriot in a posthumus publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than".
- 1682: Gottfried Wilhelm Leibniz
...
develops his notion of symbolic manipulation with formal rules which he calls
characteristica generalis.
- 1683: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, discriminant, and Bernoulli numbers.
- 1685: Kowa Seki solves the general cubic equation, as well as some quartic and quintic equations.
- 1693: Leibniz solves systems of simultaneous linear equations using matrices and determinants.
- 1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.
- 1830: Galois theory is developed by Évariste Galois in his work on abstract algebra.
References
- Donald R. Hill, Islamic Science and Engineering .
- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics .
- George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics .
- John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive .
See also
- Fundamental theorem of algebra
- Computer algebra system
References
- An online algebra text by James W. Brennan.
- An online algebra problem solver.
- Online algebra tutorials.
-
-
- I.N. Herstein: Topics in Algebra. ISBN 0-471-02371-X
- R.B.J.T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
External links
-
- Example problems and solutions from and algebra.
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