Algebra

Algebra is often referred to as a generalization of arithmetic: problems and operations are expressed in terms of variables as well as constants. A constant is some number that always has the same value, such as 3 or 14.89. A variable is a number that may have different values. In algebra, letters such as a, b, c, x, y, and z are often used to represent variables. In any given situation, a variable such as x may stand for one, two, or any number of values. For example, in the expression x + 5 = 7, the only value that x can have is 2. In the expression x² = 4, however, x can be either +2 or −2. And in the expression x + y = 9, x can have an unlimited number of values, depending on the value of y.

Origins of algebra

Algebra became popular as a way of expressing mathematical ideas in the early ninth century. Arab mathematician Al-Khwarizmi is credited with writing the first algebra book, Al-jabr waʾl Muqabalah, from which the English word algebra is derived. The title of the book translates as "restoring and balancing," which refers to the way in which equations are handled in algebra. Al-Khwarizmi's book was influential in its day and remained the most important text in algebra for many years.

Al-Khwarizmi did not use variables in the same way they are used today. He concentrated instead on developing procedures and rules for solving many types of problems in arithmetic. The use of letters to stand for variables was first suggested in the sixteenth century by French mathematician Françoise Vièta (1540–1603). Vièta appears to have been the first person to recognize that a single letter (such as x) can be used to represent a set of numbers.

Elementary algebra

The rules of elementary algebra deal with the four familiar operations of addition, subtraction, multiplication, and division of real numbers. A real number can be thought of as any number that can be expressed as a point on a line. Constants and variables can be combined in various ways to produce algebraic expressions. Numbers such as 64x², 7yt, s/2, and 32xyz are examples. Such numbers combined by multiplication and division only are monomials. The combination of two or more monomials is a polynomial. The expression a + 2b − 3c + 4d + 5e − 7x is a polynomial because it consists of six monomials added to and subtracted from each other. A polynomial containing only two parts (two terms) is a binomial, and one containing three parts (three terms) is a trinomial. Examples of a binomial and trinomial, respectively, are 3x² + 2y² and 4a + 2b² + 8c³.

One primary objective in algebra is to determine the conditions under which some statement is true. Such statements are usually made in the form of a comparison. One expression can be said to be greater than (>), less than (<), or equal to (=) a second expression. The purpose of an algebraic operation, then, is to find out precisely when such conditions are true.

For example, suppose the question is to find all values of x for which the expression x + 3 = 12 is true. Obviously, the only value of x for which this statement is true is x = 9. Suppose the problem, however, is to find all x for which x + 3 > 12. In this case, an unlimited possible number of answers exists. That is, x could be 10 (because 10 + 3 > 12), or 11 (because 11 + 3 > 12), or 12 (because 12 + 3 > 12), and so on. The answer to this problem is said to be indeterminate because no single value of x will satisfy the conditions of the algebraic statement.

In most instances, equations are the tool by which problems can be solved. One begins with some given equality, such as the fact that 2x + 3 = 15, and is then asked to find the value of the variable x. The rule for dealing with equations such as this one is that the same operation must always be performed on both sides of the equation. In this way, the equality between the two sides of the equation remains true.

In the above example, one could subtract the number 3 from both sides of the equation to give: 2x + 3 − 3 = 15 − 3, or 2x = 12. The condition given by the equation has not changed since the same operation (subtracting 3) was done to both sides. Next, both sides of the equation can be divided by the same number, 2, to give: 2x/2 = 12/2, or x = 6. Again, equality between the two sides is maintained by performing the same operation on both sides.

Applications. Algebra has applications at every level of human life, from the simplest day-to-day mathematical situations to the most complicated problems of space science. Suppose that you want to know the original price of a compact disc for which you paid $13.13, including a 5 percent sales tax. To solve this problem, you can let the letter x stand for the original price of the CD. Then you know that the price of the disc plus the 5 percent tax totaled $13.13.

That information can be expressed algebraically as x (the price of the CD) + 0.05x (the tax on the CD) = 13.13. In other words: x + 0.05x = 13.13. Next, it is possible to add both of the x terms on the left side of the equation: 1x + 0.05x = 1.05x. Then you can say that 1.05x = 13.13. Finally, to find the value of x, you can divide both sides of the equation by 1.05: 1.05x/1.05 = 13.13/1.05, or x = 12.50. The original price of the disc was $12.50.

Higher forms of algebra

Other forms of algebra have been developed to deal with more difficult and special kinds of problems. Matrix algebra, as an example, deals with sets of numbers that are arranged in rectangular boxes, known as matrices (the plural form of matrix). Two or more matrices can be added, subtracted, multiplied, or divided according to rules from matrix algebra. Abstract algebra is another form of algebra that constitutes a generalization of algebra, just as algebra itself is a generalization of arithmetic.

[See also Arithmetic; Calculus; Complex numbers; Geometry; Topology ]

Algebra

Algebra is a branch of mathematics that uses variables to solve equations. When solving an algebraic problem, at least one variable will be unknown. Using the numbers and expressions that are given, the unknown variable(s) can be determined.

Early Algebra

The history of algebra began in ancient Egypt and Babylon. The Rhind Papyrus, which dates to 1650 b.c.e., provides insight into the types of problems being solved at that time.

The Babylonians are credited with solving the first quadratic equation . Clay tablets that date to between 1800 and 1600 b.c.e. have been found that show evidence of a procedure similar to the quadratic equation. The Babylonians were also the first people to solve indeterminate equations, in which more than one variable is unknown.

The Greek mathematician Diophantus continued the tradition of the ancient Egyptians and Babylonians into the common era. Diophantus is considered the "father of algebra," and he eventually furthered the discipline with his book Arithmetica. In the book he gives many solutions to very difficult indeterminate equations. It is important to note that, when solving equations, Diophantus was satisfied with any positive number whether it was a whole number or not.

By the ninth century, an Egyptian mathematician, Abu Kamil, had stated and proved the basic laws and identities of algebra. In addition, he had solved many problems that were very complicated for his time.

Medieval Algebra. During medieval times, Islamic mathematicians made great strides in algebra. They were able to discuss high powers of an unknown variable and could work out basic algebraic polynomials . All of this was done without using modern symbolism. In addition, Islamic mathematicians also demonstrated knowledge of the binomial theorem .

Modern Algebra

An important development in algebra was the introduction of symbols for the unknown in the sixteenth century. As a result of the introduction of symbols, Book III of La géometrie by René Descartes strongly resembles a modern algebra text.

Descartes's most significant contribution to algebra was his development of analytical algebra. Analytical algebra reduces the solution of geometric problems to a series of algebraic ones. In 1799, German mathematician Carl Friedrich Gauss was able to prove Descartes's theory that every polynomial equation has at least one root in the complex plane .

Following Gauss's discovery, the focus of algebra began to shift from polynomial equations to studying the structure of abstract mathematical systems. The study of the quaternion became extensive during this period.

The study of algebra went on to become more interdisciplinary as people realized that the fundamental principles could be applied to many different disciplines. Today, algebra continues to be a branch of mathematics that people apply to a wide range of topics.

Current Status of Algebra. Today, algebra is an important day-to-day tool; it is not something that is only used in a math course. Algebra can be applied to all types of real-world situations. For example, algebra can be used to figure out how many right answers a person must get on a test to achieve a certain grade. If it is known what percent each question is worth, and what grade is desired, then the unknown variable is how many right answers one must get to reach the desired grade.

Not only is algebra used by people all the time in routine activities, but many professions also use algebra just as often. When companies figure out budgets, algebra is used. When stores order products, they use algebra. These are just two examples, but there are countless others.

Just as algebra has progressed in the past, it will continue to do so in the future. As it is applied to more disciplines, it will continue to evolve to better suit peoples' needs. Although algebra may be something not everyone enjoys, it is one branch of mathematics that is impossible to ignore.

see also Descartes and His Coordinate System; Mathematics, Very Old.

Brook E. Hall

Bibliography

Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.

Internet Resources

History of Algebra. Algebra.com. <http://www.algebra.com/algebra/about/history/>.

algebra branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements. Thus, a + a =2 a and a + b = b + a no matter what numbers a and b represent.

Principles of Classical Algebra

In elementary algebra letters are used to stand for numbers. For example, in the equation ax² + bx +c=0, the letters a, b, and c stand for various known constant numbers called coefficients and the letter x is an unknown variable number whose value depends on the values of a, b, and c and may be determined by solving the equation. Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots , or values of the unknowns, that upon substitution into the original equation will make it a numerical identity. For example, x =-2 is a root of x² -2 x -8=0 because (-2) ² -2(-2)-8=4+4-8=0; substitution will verify that x =4 is also a root of this equation.

The equations of elementary algebra usually involve polynomial functions of one or more variables (see function ). The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic ). One method of finding the zeros of the polynomial function f ( x ), i.e., the roots of the equation f ( x )=0, is to factor the polynomial, if possible. The polynomial x² -2 x -8 has factors ( x +2) and ( x -4), since ( x +2)( x -4)= x² -2 x -8, so that setting either of these factors equal to zero will make the polynomial zero. In general, if ( x - r ) is a factor of a polynomial f ( x ), then r is a zero of the polynomial and a root of the equation f ( x )=0. To determine if ( x - r ) is a factor, divide it into f ( x ); according to the Factor Theorem, if the remainder f ( r )—found by substituting r for x in the original polynomial—is zero, then ( x - r ) is a factor of f ( x ). Although a polynomial has real coefficients, its roots may not be real numbers; e.g., x² -9 separates into ( x +3)( x -3), which yields two zeros, x =-3 and x =+3, but the zeros of x² +9 are imaginary numbers.

The Fundamental Theorem of Algebra states that every polynomial f ( x )= a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ , with a_n ≠0 and n ≥1, has at least one complex root, from which it follows that the equation f ( x )=0 has exactly n roots, which may be real or complex and may not all be distinct. For example, the equation x⁴ +4 x³ +5 x² +4 x +4=0 has four roots, but two are identical and the other two are complex; the factors of the polynomial are ( x +2)( x +2)( x + i )( x - i ), as can be verified by multiplication.

Principles of Modern Algebra

Modern algebra is yet a further generalization of arithmetic than is classical algebra. It deals with operations that are not necessarily those of arithmetic and that apply to elements that are not necessarily numbers. The elements are members of a set and are classed as a group , a ring , or a field according to the axioms that are satisfied under the particular operations defined for the elements. Among the important concepts of modern algebra are those of a matrix and of a vector space.

Bibliography

See M. Artin, Algebra (1991).

algebra
1. The investigation of mathematical properties of data, such as numbers, and of operations on data, such as the addition and multiplication of numbers.

2. A collection of sets together with a collection of operations over those sets. Many examples involve only one set, such as the following: (a) the set N = {0,1,2,…} of natural numbers together with, for example, the operations of addition, subtraction, and multiplication;(b) the set B = {TRUE, FALSE} of Boolean truth values together with the operations AND, OR, and NOT (see also Boolean algebra);(c) the set of all finite strings over a set of symbols together with the operation of concatenation;(d) a set of sets together with the operations of union, intersection, and complement (see also set algebra).

In computer science, however, it is natural to consider algebras involving more than one set. These are called many-sorted algebras, in contrast to single-sorted algebras with only one set. For example, in programming languages there are different data types such as Boolean, integer, real, character, etc., as well as user-defined types. Operations on elements of these types can then be seen as giving rise to a many-sorted algebra. By stating axioms that define properties of these operations, an abstract data type can be specified. See also algebraic structure, signature.

al·ge·bra / ˈaljəbrə/ • n. the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations. ∎  a system of this based on given axioms. DERIVATIVES: al·ge·bra·ist / -ˌbrā-ist/ n. ORIGIN: late Middle English: from Italian, Spanish, and medieval Latin, from Arabic al-jabr ‘the reunion of broken parts,’ ‘bone setting,’ from jabara ‘reunite, restore.’ The original sense, ‘the surgical treatment of fractures,’ probably came via Spanish, in which it survives; the mathematical sense comes from the title of a book, עilm al-jabr wa'l-mụkābala ‘the science of restoring what is missing and equating like with like,’ by the mathematician al-Kwārizmī (see algorithm).

algebra (Arabic, al-jabr ‘to find the unknown’) Branch of mathematics dealing with the study of equations that are written using numbers and alphabetic symbols, which themselves represent quantities to be determined. An algebraic equation may be thought of as a constraint on the possible values of the alphabetic symbols. For example, y + x = 8 is an algebraic equation involving the variables x and y. Given any value of x the value of y may be determined, and vice-versa. See also Boole

algebra †bone-setting XIV; branch of mathematics XVI. — medL. algebra — Arab. al-jabr, i.e. AL-2, jabr reunion of broken parts, f. jabara reunite.
Hence algebraic XVII, algebraical XVI, algebrist XVII.

algebra •airer, bearer, carer, Clara, darer, declarer, Demerara, Éire, habanera, Halmahera, parer, Perak, primavera, repairer, Rivera, Riviera, Sarah, scarer, sharer, snarer, sparer, squarer, starer, swearer, tearer, wearer •cause célèbre • torch-bearer •swordbearer • pallbearer • wayfarer •seafarer • capoeira • Phaedra •sacra, simulacra •Libra, vers libre •ex cathedra •chypre, Ypres •palaestra (US palestra) • urethra •joie de vivre •mirror, sirrah •Coimbra • Middlesbrough • Indra •Sintra •aspidistra, sistra •algebra • orchestra • vertebra •Beira, Fujairah, Hegira, Lyra, Myra, naira, palmyra, spirogyra •Hydra • Lycra •begorra, Gomorrah, horror •double entendre • genre • amour propre • Le Nôtre • contra •Cosa Nostra, rostra