Algebra
Algebra is often referred to as a generalization of arithmetic. As such, it is a collection of rules: rules for translating words into the symbolic notation of mathematics, rules for formulating mathematical statements using symbolic notation, and rules for rewriting mathematical statements in a manner that leaves their truth unchanged.
The power of elementary algebra, which grew out of a desire to solve problems in arithmetic, stems from its use of variables to represent numbers. This allows the generalization of rules to whole sets of numbers. For example, the solution to a problem may be the variable x or a rule such as ab=ba can be stated for all numbers represented by the variables a and b.
Elementary algebra is concerned with expressing problems in terms of mathematical symbols and establishing general rules for the combination and manipulation of those symbols. There is another type of algebra, however, called abstract algebra, which is a further generalization of elementary algebra, and often bears little resemblance to arithmetic. Abstract algebra begins with a few basic assumptions about sets whose elements can be combined under one or more binary operations, and derives theorems that apply to all sets, satisfying the initial assumptions.
Elementary 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 16th 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 (1), multiplication (x), subtraction (2), and division (4) 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. 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.
Applications
Applications of algebra are found everywhere. The principles of algebra are applied in all branches of mathematics, for instance, calculus, geometry, topology. They are applied every day by men and women working in all types of business. As a typical example of applying algebraic methods, consider the following problem. A painter is given the job of whitewashing three billboards along the highway. The owner of the billboards has told the painter that each is a rectangle, and all three are the same size, but he does not remember their exact dimensions. He does have two old drawings, one indicating the height of each billboard is two feet less than half its width, and the other indicating each has a perimeter of 68 feet. The painter is interested in determining how many gallons of paint he will need to complete the job, if a gallon of paint covers 400 square feet. To solve this problem three basic steps must be completed. First, carefully list the available information, identifying any unknown quantities. Second, translate the information into symbolic notation by assigning variables to unknown quantities and writing equations. Third, solve the equation, or equations, for the unknown quantities.
Step one, list available information: (a) three billboards of equal size and shape, (b) shape is rectangular, (c) height is 2 feet less than 1/2 the width, (d) perimeter equals 2 times sum of height plus width equals 68 feet, (e) total area, equals height times width times 3, is unknown, (f) height and width are unknown, (g) paint covers 400 sq.ft. per gallon, (h) total area divided by 400 equals required gallons of paint.
Step two, translate. Assign variables and write equations.
Let: A = area; h = height; w = width; g = number of gallons of paint needed.
Then:(1) h = 1/2w - 2 (from [c] in step 1) (2) 2(h+w) = 68 (from [d] in step 1) (3) A = 3hw (from [e] in step 1) (4) g = A/400 (from [h] in step 1) Step three, solve the equations. The right hand side of equation (1) can be substituted into equation (2) for h giving 2(1/2w-2+w) = 68. By the commutative property, the quantity in parentheses is equal to (1/2w+w-2), which is equal to (3/2w-2). Thus, the equation 2(3/2w-2)=68 is exactly equivalent to the original. Applying the distributive property to the left hand side of this new equation results in another equivalent expression, 3w-4 = 68. To isolate w on one side of the equation, add 4 to both sides giving 3w-4+4 = 68+4 or 3w = 72. Finally, divide the expressions on each side of this last expression by 3 to isolate w. The result is w = 24 ft. Next, put the value 24 into equation (1) wherever w appears, h = (1/2(24)-2), and do the arithmetic to find h = (12-2) = 10ft. Then, put the values of h and w into equation (3) to find the area, A = 3x10x24 = 720 sq. ft. Finally, substitute the value of A into equation (4) to find g = 720/400 = 1.8 gallons of paint.
Graphing algebraic equations
The methods of algebra are extended to geometry, and vice versa, by graphing. The value of graphing is two-fold. It can be used to describe geometric figures using the language of algebra, and it can be used to depict geometrically the algebraic relationship between two variables. For example, suppose that Fred is twice the age of his sister Irma. Since Irma's age is unknown, Fred's age is also unknown. The relationship between their ages can be expressed algebraically, though, by letting y represent Fred's age and x represent Irma's age. The result is y = 2x. Then, a graph, or picture, of the relationship can be drawn by indicating the points (x,y) in the Cartesian coordinate system for which the relationship y = 2x is always true. This is a straight line, and every point on it represents a possible combination of ages for Fred and Irma (of course negative ages have no meaning so x and y can only take on positive values). If a second relationship between their ages is given, for instance, Fred is three years older than Irma, then a second equation can be written, y = x+3, and a second graph can be drawn consisting of the ordered pairs (x,y) such that the relationship y = x+3 is always true. This second graph is also a straight line, and the point at which it intersects the line y = 2x is the point corresponding to the actual ages of Irma and Fred. For this example, the point is (3,6), meaning that Irma is three years old and Fred is six years old.
Linear algebra
Linear algebra involves the extension of techniques from elementary algebra to the solution of systems of linear equations. A linear equation is one in which no two variables are multiplied together, so that terms like xy, yz, x2, y2, and so on, do not appear. A system of equations is a set of two or more equations containing the same variables. Systems of equations arise in situations where two or more unknown quantities are present. In order for a unique solution to exist there must be as many independent conditions given as there unknowns, so that the number of equations that can be formulated equals the number of variables. Thus, we speak of two equations in two unknowns, three equations in three unknowns, and so forth. Consider the example of finding two numbers such that the first is six times larger than the second, and the second is 10 less that the first. This problem has two unknowns, and contains two independent conditions. In order to determine the two numbers, let x represent the first number and y represent the second number. Using the information provided, two equations can be formulated, x = 6y, from the first condition, and x-10 = y, from the second condition. To solve for y, replace x in the second equation with 6y from the first equation, giving 6y-10=y. Then, subtract y from both sides to obtain 5y-10=0, add 10 to both sides giving 5y=10, and divide both sides by 5 to find y=2. Finally, substitute y=2 into the first equation to obtain x=12. The first number, 12, is six times larger than the second, 2, and the second is 10 less than the first, as required. This simple example demonstrates the method of substitution. More general methods of solution involve the use of matrix algebra.
Matrix algebra
A matrix is a rectangular array of numbers, and matrix algebra involves the formulation of rules for manipulating matrices. The elements of a matrix are contained in square brackets and named by row and then column. For example the matrix has two rows and two columns, with the element (-6) located in row one column two. In general, a matrix can have i rows and j columns, so that an element of a matrix is denoted in double subscript notation by aij. The four elements in A are a11 = 1, a12 = -6, a21 = 3, a22 = 2. A matrix having m rows and n columns is called an "m by n" or (m x n) matrix. When the number of rows equals the number of columns the matrix is said to be square. In matrix algebra, the operations of addition and multiplication are extended to matrices and the fundamental principles for combining three or more matrices are developed. For example, two matrices are added by adding their corresponding elements. Thus, two matrices must each have the same number of rows and columns in order to be compatible for addition. When two matrices are compatible for addition, both the associative and commutative principles of elementary algebra continue to hold. One of the many applications of matrix algebra is the solution of systems of linear equations. The coefficients of a set of simultaneous equations are written in the form of a matrix, and a formula (known as Cramer's rule) is applied which provides the solution to n equations in n unknowns. The method is very powerful, especially when there are hundreds of unknowns, and a computer is available.
Abstract algebra
Abstract algebra represents a further generalization of elementary algebra. By defining such constructs as groups, based on a set of initial assumptions, called axioms, provides theorems that apply to all sets satisfying the abstract algebra axioms. A group is a set of elements together with a binary operation that satisfies three axioms. Because the binary operation in question may be any of a number of conceivable operations, including the familiar operations of addition, subtraction, multiplication, and division of real numbers, an asterisk or open circle is often used to indicate the operation. The three axioms that a set and its operation must satisfy in order to qualify as a group, are: (1) members of the set obey the associative principle [a x (b x c) = (a x b) x c]; (2) the set has an identity element, I, associated with the operation x, such that a x I = a; (3) the set contains the inverse of each of its elements, that is, for each a in the set there is an inverse, a', such that a x a' = I. A well known group is the set of integers, together with the operation of addition. If it happens that the commutative principle also holds, then the group is called a commutative group. The group formed by the integers together with the operation of addition is a commutative group, but the set of integers together with the operation of subtraction is not a group, because subtraction of integers is not associative. The set of integers together with the operation of multiplication is a commutative group, but division is not strictly an operation on the integers because it does not always result in another integer, so the integers together with division do not form a group. The set of rational numbers, however, together with the operation of division is a group. The power of abstract algebra derives from its generality. The properties of groups, for instance, apply to any set and operation that satisfy the axioms of a group. It does not matter whether that set contains real numbers, complex numbers, vectors, matrices, functions, or probabilities, to name a few possibilities.
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