Algebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics.
Elementary algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving.
Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalised and their precise definitions lead to structures such as groups, rings and fields.
History
Main article: History of algebraSee also: Timeline of algebraWhile the word algebra comes from the Arabic language (al-jabr, الجبر literally, restoration ), its origins can be traced to ancient Indian mathematics. Muhammad ibn Musa al-Khwarizmi learned the technique of Indian mathematics and introduced it to the world through his famous book on arithmetic text, the Kitab al-jam’wal tafriq bi hisab al-Hindi ("Book on Addition and Subtraction after the Method of the Indians"), around 780-850 A.D. The first use of algebraic transformation was descibed by Brahmagupta in his book Brahmasphutasiddhanta , where first proposed solution of Linear Algebra and Quadratic Equation.
Algebra is also linked to the ancient Babylonians, 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 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 , 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, though this would not be realized until the medieval Muslim mathematicians.
The Hellenistic mathematicians Hero of Alexandria and Diophantus 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. Later, Arab 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-Khowarazmi was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.
The word "algebra" is named after the Arabic word " al-jabr , الجبر " from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala , الكتاب المختصر في حساب الجبر والمقابلة , meaning The book of Summary Concerning Calculating by Transposition and Reduction , a book written by the Islamic Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī (considered the "father of algebra"), in 820. The word Al-Jabr means "reunion" . The Hellenistic 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. 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. 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, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs, while treating algebra as an independent discipline in its own right. 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."
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. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In 1637 Rene 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 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.
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. This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). University-level courses in group theory may also be called elementary 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.
- Algebraic number theory , in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
- Algebraic geometry in its algebraic aspect.
- Algebraic combinatorics , in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or 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
Main article: Elementary algebraElementary 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 , x , or 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, "Fi
Data Entry Home Work - Associated Content - associatedcontent.com
Data entry home work requires typing skills and computer with Internet. With data entry home work, there is no need to go an office to earn money.
Make fast easy data entry jobs smart cash free money making online ...
data entry jobs, Make money at home by online data entry jobs to work from home easy earn fast money making millionaire getting paid for taking surveys by earning cash from ...
Work From Home. Data entry jobs Other Free UK Classified Ads
No special experience is necessary for work at home data entry. Earn money with simple internet Home work by online data entry jobs for 100`s international companies directly on ...
Finding Legitimate Home Based Work At Home Data Entry Jobs Online ...
If you have been trying to find a work at home data entry jobs, you will realized that these data entry home work are definitely not easy to find.
From Home Data Entry
Data entry from home. Work from home free with data entry. Free data entry from home. work at home processing data. data processing at home.
Online Data Entry Jobs at http://www.workathomeinus.com/148.html A ...
... entry jobs, at home data entry jobs, data entry at home jobs, data entry clerk, data entry from home jobs, data entry home job, data entry home jobs, data entry home work, data ...
Moneyfromhome.com - DATA ENTRY, Homework
Lists and lists of REAL scam-free work at home jobs as featured in FORTUNE Magazine and LIFETIME Television! Over 1000 legitimate home based work at home jobs including data entry ...
Work At Home Opportunities - Best Offers | BREAK THE MATRIX
data entry work from home work at home websites work at home institute jobs to work at home work at home scams work from home moms working from home data entry
Work From Home Data Entry - Online Data Entry Jobs
Data entry home work; Data entry jobs wiki; Earn money working at home; Data entry jobs home ... Principal Data Entry Machine Operator (Career Examination Series C-2866)
DATA ENTRY FROM HOME
data entry from home - work at home and make money online