Algebra

Algebra is one of the broad parts of mathematics, together with number theory and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols, it includes everything from elementary equation solving to the study of abstractions such as groups and fields. The more basic parts of algebra are called elementary algebra. Elementary algebra is considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x = 3. In E = mc2, the letters E and m are variables, the letter c is a constant, the speed of light in a vacuum.

Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", the word is used, for example, in the phrases linear algebra and algebraic topology. A mathematician who does research in algebra is called an algebraist; the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wa'l-muḳābala by the Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the fifteenth century, from either Spanish, Italian, or Medieval Latin, it referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century; the word "algebra" has several related meanings as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics.

As a single word with an article or in plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the author. The structure has an addition, a scalar multiplication; when some authors use the term "algebra", they make a subset of the following additional assumptions: associative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations. With a qualifier, there is the same distinction: Without an article, it means a part of algebra, such as linear algebra, elementary algebra, or abstract algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers. Algebra began with letters standing for numbers.

This allowed proofs of properties. For example, in the quadratic equation a x 2 + b x + c = 0, a, b, c can be any numbers whatsoever, the quadratic formula can be used to and find the values of the unknown quantity x which satisfy the equation; that is to say. And in current teaching, the study of algebra starts with the solving of equations such as the quadratic equation above. More general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors and polynomials; the structural properties of these non-numerical objects were abstracted into algebraic structures such as groups and fields. Before the 16th century, mathematics was divided into only two subfields and geometry. Though some methods, 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 the 16th or 17th century.

From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, all of which used algebra. Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification where none of the first level areas is called algebra. Today algebra in