Algebra

Algebra was popularized in the early ninth century by al-Khowarizmi, an Arab mathematician, and the author of the first algebra book, Al-jabr wa'l Muqabalah, from which the English word algebra is derived. An influential book in its day, it remained the standard text in algebra for a long time. The title translates roughly to "restoring and balancing," referring to the primary algebraic method of performing an operation on one side of an equation and restoring the balance, or equality, by doing the same thing to the other side. In his book, al-Khowarizmi did not use variables as we recognize them today, but concentrated on procedures and specific rules, presenting methods for solving numerous types of problems in arithmetic. Variables based on letters of the alphabet were first used in the late sixteenth century by the French mathematician François Viète. The idea is simply that a letter, usually from the English or Greek alphabet, stands for an element of a specific set. For example, x, y, and z are often used to represent a real number, z to represent a complex number, and n to stand for an integer. Variables are often used in mathematical statements to represent unknown quantities.

The rules of elementary algebra deal with the four familiar operations of addition (+), multiplication (×), subtraction (−), and division (÷) of real numbers. Each operation is a rule for combining the real numbers, two at a time, in a way that gives a third real number. A combination of variables and numbers that are multiplied together, such as 64x², 7yt, s/2, 32xyz, is called a monomial. The sum or difference of two monomials is referred to as a binomial, examples include 64x²+7yt, 13t+6x, and 12y−3ab/4. The combination of three monomials is a trinomial (6xy+3z−2), and the combination of more than three is a polynomial. All are referred to as algebraic expressions.

One primary objective in algebra is to determine what conditions make a statement true. Statements are usually made in the form of comparisons. One expression is greater than (>), less than (<), or equal to (=) another expression, such as 6x+3 > 5, 7x²−4 < 2, or 5x²+6x = 3y+4. The application of algebraic methods then proceeds in the following way. A problem to be solved is stated in mathematical terms using symbolic notation. This results in an equation (or inequality). The equation contains a variable; the value of the variable that makes the equation true represents the solution to the equation, and hence the solution to the problem. Finding that solution requires manipulation of the equation in a way that leaves it essentially unchanged, that is, the two sides must remain equal at all times. The object is to select operations that will isolate the variable on one side of the equation, so that the other side will represent the solution. Thus, the most fundamental rule of algebra is the principle of al-Khowarizmi: whenever an operation is performed on one side of an equation, an equivalent operation must be performed on the other side bringing it back into balance. In this way, both sides of an equation remain equal.