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 x 2 = 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.
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.
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 64x 2 , 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 3x 2 + 2y 2 and 4a + 2b 2 + 8c 3 .
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.
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 ]
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