# Algebra

# Algebra

Algebra is a generalization of arithmetic. In practice, it is a collection of 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 while increasing their meaningfulness for human beings.

Elementary algebra grew out of a desire to solve problems in arithmetic. Its power stems from its substitution of variables—non-numeric symbols standing for no specific value—for numbers. This allows the generalization of rules to whole sets or ranges 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 sixteenth century by the French mathematician François Vie`te. 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* and *y* are often used to represent real numbers, *z* to represent a complex number, and *n* to represent 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 of real numbers: addition (+), multiplication (×), subtraction (–), and division (÷). Each operation is a rule for combining 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 64 *x* ^{2}, 7*yt*, *s* /2, or 32*xyz*, is called a monomial. The sum or difference of two monomials is referred to as a binomial, examples include 64 *x* ^{2}+7*yt*, 13 *t* +6 *x*, and 12 *y* –3*ab* /4. The combination of three monomials is a trinomial (6*xy* +3 *z* –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 6 *x* +3 >5, 7 *x* ^{2}–4 < 2, or 5 *x* ^{2}+6 *x* = 3 *y* +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, and 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, 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* = 1/2*w* –2 (from [c] in step 1); (2) 2(*h* + *w* ) = 68 (from [d] in step 1); (3) A = 3*hw* (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(1/2 *w* –2+*w* ) = 68. By the commutative property, the quantity in parentheses is equal to (1/2*w* + *w* –2), which is equal to (3/2 *w* –2). Thus, the equation 2(3/2*w* –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, 3*w* –4 = 68. To isolate *w* on one side of the equation, add 4 to both sides, giving 3 *w* –4 + 4 = 68+4 or 3 *w* = 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). Doing the arithmetic, we find *h* = (12–2) = 10 ft. Then put the values of *h* and *w* into equation (3) to find the area, *A* = 3×10×24 = 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 = 2 *x*. 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* = 2 *x* 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* = 2 *x* 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*, *x* ^{2}, *y* ^{2}, 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* = 6 *y*, from the first condition, and *x* –10 = *y*, from the second condition. To solve for *y*, replace *x* in the second equation with 6 *y* from the first equation, giving 6y–10=y. Then, subtract y from both sides to obtain 5 *y* –10 = 0, add 10 to both sides giving 5 *y* =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 *a _{ij}*. The four elements in

*A*are

*a*

_{11}= 1,

*a*

_{12}=–6,

*a*

_{21}= 3, and

*a*

_{22}= 2. A matrix having m rows and n columns is called an “

*m*by

*n*” or (

*m*×

*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* × (*b* × *c* ) = (*a* × *b* ) × *c* ]; (2) the set has an identity element, *I*, associated with the operation, such that *a* × *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* × *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.

*See also* Associative property; Solution of equation.

## Resources

### BOOKS

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications*. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Blitzer, Robert. *Algebra and Trigonometry*. 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2003.

Larson, Ron, et al. *Algebra I (Equations, Applications, and Graphs)*. Evanston, IL: McDougal Littell/Houghton Mifflin, 2004.

Martin-Gay, K. Elayn. *Intermediate Algebra*. Upper Saddle River, NJ: Prentice Hall, 2006.

Stedall, Jacqueline and Timothy Edward Ward. *The Greate Invention of Algebra: Thomas Harriot’s Treatise on Equations*. Oxford: Oxford University Press, 2003.

### OTHER

*Algebra Blaster 3 CD-ROM for Windows*. Torrance, CA: Davidson and Associates Inc., 1994.

J. R. Maddocks

# Algebra

# 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 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.

## Applications

Applications of algebra are found everywhere. The principles of algebra are applied in all branches of mathematics, for instance, **calculus** , **geometry** , and **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, 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 = 3×10×24 = 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 × 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 × (b × c) = (a × b) × c]; (2) the set has an **identity element** , I, associated with the operation ×, such that a × 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 × 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.

See also Associative property; Solution of equation.

## Resources

### books

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Blitzer, Robert. *Algebra and Trigonometry.* 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2003.

Gelfond, A.O. *Transcendental and Algebraic Numbers.* Dover Publications, 2003.

Immergut, Brita and Jean Burr Smith. *Arithmetic and Algebra**Again.* New York: McGraw-Hill, 1994.

Stedall, Jacqueline and Timothy Edward Ward. *The Greate Invention of Algebra: Thomas Harriot's Treatise on Equations.* Oxford: Oxford University Press, 2003.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* New York: CRC Press, 1998.

### Other

*Algebra Blaster 3 CD-ROM for Windows.* Torrance, CA: Davidson and Associates Inc., 1994.

J.R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Graph**—A picture of an equation showing the points of a plane that satisfy the equation.

**Operation**—A method of combining the members of a set, two at a time, so the result is also a member of the set. Addition, subtraction, multiplication, and division of real numbers are familiar examples of operations.

# Algebra

# 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 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.

## 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 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.

## 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

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

**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.

# algebra

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-__K__wārizmī (see algorithm).

# algebra

**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

**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.

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