Algebra I INTRODUCTION Algebra, branch of mathematics in which symbols (usually letters) represent unknown numbers in mathematical equations. Algebra allows the basic operations of arithmetic, such as addition, subtraction, and multiplication, to be performed without using specific numbers. People use algebra constantly in everyday life, for everything from calculating how much flour they need to bake a certain number of cookies to figuring out how long it will take to travel by car at a certain speed to a destination that is a specific distance away. Arithmetic alone cannot deal with mathematical relations such as the Pythagorean theorem, which states that the sum of the squares of the lengths of the two shorter sides of any right triangle is equal to the square of the length of the longest side. Arithmetic can only express specific instances of these relations. A right triangle with sides of length 3, 4, and 5, for example, satisfies the conditions of the theorem: 32 + 42 = 52. (32 stands for 3 multiplied by itself and is termed "three squared.") Algebra is not limited to expressing specific instances; instead it can make a general statement that covers all possible values that fulfill certain conditions--in this case, the theorem: a2 + b 2 = c2. This article focuses on classical algebra, which is concerned with solving equations, uses symbols instead of specific numbers, and uses arithmetic operations to establish ways of handling symbols. The word algebra is also used, however, to describe various modern, more abstract mathematical topics that also use symbols but not necessarily to represent numbers. Mathematicians consider modern algebra a set of objects with rules for connecting or relating them. As such, in its most general form, algebra may fairly be described as the language of mathematics. II SYMBOLS AND SPECIAL TERMS Mathematical Symbols Mathematics employs many symbols to describe numerical operations and relationships. This activity defines some common symbols and gives examples of their uses. © Microsoft Corporation. All Rights Reserved. The symbols of algebra include numbers, letters, and signs that indicate various arithmetic operations. Numbers are constants (values that do not change), but letters can represent either unknown constants or variables (values that vary). Letters that are used to represent constants are taken from the beginning of the alphabet; those used to represent variables are taken from the end of the alphabet. See also Mathematical Symbols. A Operation Symbols The basic operational signs of algebra are familiar from arithmetic: addition (+), subtraction (-), multiplication (×), and division (÷). The multiplication symbol × is often omitted or replaced by a dot, as in a · b. A group of consecutive symbols, such as abc, indicates the product (the result of multiplication) of a, b, and c. Division is commonly indicated by a horizontal bar (also called a vinculum), as in: A virgule, or slash (/), may also be used to indicate division: a/c. A power is the product of a number multiplied by itself. The notation 42 (read "four squared"), for example, is used as an abbreviation for 4 · 4 (4 times 4); thus 42 = 16. The 4 in 42 is called the base, and the small raised number 2 is called the exponent. An exponent indicates how many times the number is multiplied by itself: x3 (read "x cubed") means x · x · x. More generally xn (read "x to the n th power" or "x to the n th" where n is any number) means the product of x multiplied by itself n times. Fractions can take exponents as well: (y)2 = ? . A number whose n th power is equal to x is an n th root of x. When n is 2 the term "square root" is used and when n is 3 the term "cube root" is used. For example, 3 and -3 are both square roots of 9 since 32 = 9 and (-3)2 = 9; 2 is a cube root of 8 since 23 = 8; -2 is a cube root of -8; y is a cube root of ? The square root of x is . denoted like this: The number of times the root is multiplied by itself is called the index. The index is usually omitted for square roots, but appears as a small raised number just before the root symbol for higher roots: The two possible values of square roots, one positive and one negative, are often written using the plus or minus symbol: ±. The equation = 2 or -2, for instance, can be abbreviated = ±2. B Order of Operations and Grouping Algebra relies on an established sequence for performing arithmetic operations. This ensures that everyone who executes a string of operations arrives at the same answer. Multiplication is performed first, then division, followed by addition, then subtraction. For example: 1+2·3 equals 7 because 2 and 3 are multiplied first and then added to 1. Exponents and roots have even higher priority than multiplication: 3 · 22 = 3 · 4 = 12 Grouping symbols override the order of operations. All operations within a group are carried out first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and horizontal bars that are used most often for division and roots. Adding parentheses to a previous example: (1 + 2) · 3 indicates that 1 should be added to 2 first, and then the result multiplied by 3 for a total of 9 rather than 7. Brackets and braces are used in more complicated combinations that require multiple nested (one inside the other) groups. Operations within the innermost group are carried out first: {2[5 + 3(1 + 4)]} = {2[5 + 3 · 5]} = {2[5 + 15]} = {2 · 20} = 40 When a slash is used to indicate division, care must be taken to group the terms appropriately. For example, cannot be written ax + b/c - dy. The second notation indicates that b should be divided by c before b is added to ax. Grouping symbols can be used to correctly represent the fraction when using a slash: (ax + b)/(c - dy). C Special Definitions Any statement that contains the equality relation (=), such as 3x = 9, is called an equation. An equation is called an identity if the equality is true for all values of its variables; if the equation is true for some values of its variables and false for others, the equation is conditional. The equation x + 0 = x, for example, is an identity while 3x = 9 is conditional because it is only true when x = 3. A term is any algebraic expression consisting only of products of constants and variables; 2x, -a, and ? 4x s are all examples of terms. The numerical part of a term is called its coefficient. The coefficients of each term above are, respectively, 2, -1, and ? . An expression containing one term, such as 2x3, is called a monomial. An expression involving the addition or subtraction of two terms, as in 2x2 + 3x, is called a binomial, while an expression with three terms, such as 4x5 - x4 + 7x, is known as a trinomial. Polynomial is the general name for expressions in which any number of terms are added or subtracted. The degree of a polynomial refers to the largest exponent of the variables in the polynomial. For example, if the largest exponent of a variable is 3, as in ax3 + bx2, the polynomial is said to be of degree 3. Similarly, the expression xn + xn-1 + xn-2 is of degree n. A linear equation with one variable is a polynomial equation of degree one--that is, of the form ax + b = 0. These are called linear equations because graphing these equations results in straight lines. A quadratic equation in one variable is a polynomial equation of degree two--that is, of the form ax2 + bx + c = 0. An indeterminate equation, such as x2 + y2 = z2, involves multiple unknowns. A prime number is any integer (the counting numbers: 1, 2, 3, ...; their negatives; and zero) that can be evenly divided only by itself and by the number 1 or the number -1. Thus, 2, 3, 5, 7, 11, 13, 17, and 19 are all prime numbers. A factor of a number is any integer by which the number can be divided evenly, with no remainder. The factors of 6, for example, are 1, 2, 3, and 6, because 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, and 6 ÷ 6 = 1. The prime factors of any number are those factors to which it can be reduced such that the number is expressed only as the product of primes and their powers. For example, the prime factors of 6 are 2 and 3. Similarly, because 60 = 22 × 3 × 5, the prime factors of 60 are 2, 3, and 5. III PUTTING POLYNOMIAL EQUATIONS INTO SOLVABLE FORMS Solving a polynomial equation usually requires altering its form. The most common tools used to manipulate equations into solvable form are the laws of arithmetic, factoring, and the finding of least common multiples. A Laws of Polynomial Arithmetic In manipulating polynomials, the usual laws of the arithmetic of numbers hold. This section lays out those laws. Normal arithmetic is limited to the set of rational numbers (positive and negative whole numbers and infinitely repeating decimals). Algebra and geometry can include irrational numbers (infinite decimals that do not repeat), such as pi (symbol p) and the square root of 2 (abbreviated ?2). The set of all rational and irrational numbers taken together constitutes the set of real numbers. See also Number (mathematics). A1 Laws of Addition 1. The sum of any two real numbers a and b is also a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots; this means that applying any of these operations to real numbers yields a quantity that also is a real number. 2. No matter how terms are grouped in carrying out additions, the sum will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition. 3. Given any real number a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a. 4. Given any real number a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0. 5. No matter in what order addition is carried out, the sum will always be the same: a + b = b + a. This is called the commutative law of addition. Any set of numbers obeying laws 1 through 4 is said to form a group. If the set also obeys law 5, it is said to be an Abelian, or commutative, group. Integers and real numbers are both Abelian groups. Since subtraction can be treated as the addition of negative numbers (3 - 4 is the same as 3 + -4), these laws also apply to subtraction. A2 Laws of Multiplication Laws similar to those for addition also apply to multiplication. Since powers are a special case of multiplication, these laws cover powers as well. 1. The product of any two real numbers a and b is also a real number, denoted a·b or ab. 2. No matter how terms are grouped in carrying out multiplications, the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication. 3. Given any real number a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a. 4. Given any nonzero real number a, there is a number (a-1), or (1/a), called the multiplicative inverse, such that a(a-1) = (a-1)a = 1. 5. No matter in what order multiplication is carried out, the product will always be the same: ab = ba. This is called the commutative law of multiplication. Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero (because division by zero is impossible), forms such a commutative group under multiplication. Law number 4 allows the laws of multiplication to be extended to division, since dividing by a number is the same as multiplying by its inverse. A3 Distributive Laws Another important property of the set of real numbers links addition and multiplication in two distributive laws as follows: 1. a(b + c) = ab + ac 2. (b + c)a = ba + ca Any set of elements with an equality relation and for which two operations (such as addition and multiplication) are defined, and which obeys all the laws for addition, the laws for multiplication, and the distributive laws, constitutes a field. A4 Exponent Laws The laws governing exponents are extensions of the addition, multiplication, and distributive laws. Since a2 = a · a and a3 = a · a · a, for example, a2 · a3 = a · a · a · a · a = a2+3 = a5. In general, therefore, for real numbers: 1. am·an = am+n 2. (am)n = am·n 3. am·bm = (a·b)m B Multiplying Polynomials To multiply polynomials, each term of each polynomial is multiplied in turn by all of the terms in all of the other polynomials. The results are then added together. The following is a simple example of the product of a binomial and a monomial: (x + 2)(4x2) = x(4x2) + 2(4x2) = 4x3 + 8x2 Each term of the one polynomial, (x + 2), is multiplied by the single term of the monomial, 4x2. This same principle is directly extended to polynomials of any number of terms. For example, the product of a binomial and a trinomial is carried out as follows: (5x + 2)(3x3 + x2 -4x) = [5x(3x3 + x2 -4x)] + [2(3x3 + x2 -4x)] = [5x(3x3) + 5x(x2) + 5x(-4x)] + [2(3x3) + 2(x2) + 2(-4x)] = [15x4 + 5x3 - 20x2] + [6x3 + 2x2 -8x] = After such operations have been performed, all terms of the same degree should be combined whenever possible to simplify the entire expression: 15x4 + 11x3 - 18x2 - 8x C Factoring Polynomials Given a complicated algebraic expression, it is often useful to factor it into the product of simpler terms. For example, 2x2 + 4xy can be factored as 2x(x + 2y) because (x + 2y) multiplied by 2x is equal to 2x(x) + 2x(2y), which simplifies to 2x2 + 4xy. Determining the factors of a given polynomial may be a simple matter of inspection or may require trial and error. Not all polynomials, however, can be factored using real-number coefficients, and these are called prime polynomials. Some common factorizations are given in the following examples. 1. Trinomials of the general form x2 + (a + b )x + ab can be factored (x + a)(x + b ). If a = 5 and b = 2, for example, x2 + 7x + 10 can be factored (x + 5)(x + 2). The problem can be worked backwards by multiplying out the factors to prove this factoring is correct: (x + 5)(x + 2) = x(x + 2) + 5(x + 2) = x2 + 2x + 5x + 10 = x2 + 7x + 10 2. Trinomials of the general form a2x2 + (2ab)(xy) + b 2y2 can be factored (ax + by)(ax + by) or simply (ax + by)2. In the simple example where a and b are both equal to 1, x2 + 2xy + y2 can be factored (x + y)2: (x + y)2 = (x + y)(x + y) = x2 + xy + xy + y2 = x2 +2xy + y2 Similarly, when a = 1 and b = -1, x2 - 2xy +y2 = (x - y)2. In a more complex example, where a = 5 and b = 2, 25x2 + 20xy + 4y2 can be factored (5x + 2y)2: (5x + 2y)2 = (5x + 2y) (5x + 2y) = 25x2 + 10xy + 10xy + 4y2 = 25x2 + 20xy + 2y2 3. The difference of squares of the form a2x2 - b 2y2 may be factored (ax + by)(ax - by). In the simplest case, when a and b equal 1, x2 - y2 = (x + y)(x - y) because the terms of lower degree add up to zero and thus cancel each other out: (x + y)(x - y) = x2 - xy + xy - y2 = x2 - y2 In the more complex case where a = 5 and b = 4, 25x2-16y2 = (5x + 4y)(5x - 4y): (5x + 4y)(5x - 4y) = 25x2 - 20xy + 20xy - 16y2 = 25x2 - 16y2 4. The sums and differences of cubes can also be factored according to the general formulas a3x3 + b 3y3 = (ax + by)(a2x2 - axby + b 2y2) and a3x3 - b 3y3 = (ax by)(a2x2 - axby + b 2y2). When a and b are both equal to 1, x3 + y3 = (x + y)(x2 - xy + y2): (x + y)(x2 - xy + y2) = x3 - x2y + xy2 + x2y - xy2 + y3 = x3 + y3 and x3 - y3 = (x - y)(x2 + xy + y2). Grouping may often be useful in factoring; terms that are similar are grouped wherever possible, as in the following example: 10x3 + 16x2y + 18xy2 -2x3 + 8x2y = 8x3 + 24x2y + 18xy2 = 2x(4x2 +12xy + 9y2) = 2x(2x+3y)2. D Identifying Highest Common Factors Given a polynomial, it is frequently important to isolate the greatest common factor from each term of the polynomial. For example, in the binomial 6x2 + 12x, the number 6 is a factor of both terms, as is x. After factoring, 6x(x + 2) is obtained, and 6x is the greatest common factor for all terms of the original binomial. Similarly, for the trinomial 6a2x3 + 9abx + 15cx2, the number 3 is the largest numerical factor common to 6, 9, and 15, and x is the largest variable factor common to all three terms. Thus, the greatest common factor of the trinomial is 3x and the trinomial can be factored 3x(2a2x2 + 3ab + 5cx). E Identifying Least Common Multiples Finding least common multiples is useful in combining algebraic fractions. The least common multiple (LCM) of a set of numbers is the smallest number into which each number in the set will divide evenly. The LCM of 2, 3, 4, and 6, for example, is 12. The LCM can be calculated by factoring numbers into their prime components. The LCM is the product of the highest power of each prime factor of the given numbers. For example, to find the LCM for the three numbers 27, 63, and 75, each number is first factored: 27 = 33, 63 = 32·7, and 75 = 3·52. The prime factors of these three numbers are 3, 5, and 7, and the highest powers of those three factors are 33, 52, and 7. The LCM, therefore, is 33·7·52 = 4,725; 4,725 is the smallest number into which 27, 63, and 75 will all divide evenly. Given several algebraic expressions, the least common multiple is the expression of lowest degree and least coefficient that can be divided evenly by each of the expressions. To find a common multiple of the terms 2x2y, 15x2y2, and 6ay3, all three expressions could simply be multiplied together: (2x2y)(15x2y2)(6ay3) = 180ax4y6. However, 180ax4y6 is not the least common multiple. To determine which is the least, each of the terms is reduced to its prime factors. For the numerical coefficients 2, 15, and 6, the prime factors are 2, 3·5, and 2·3, respectively; the least common multiple for the numerical coefficients is therefore 2·3·5, or 30. Similarly, because the constant a appears only once, it too must be a factor. Of the variables, x2 and y3 are required because they are the highest powers of the two variables that appear in any of the expressions. The LCM of the three terms, therefore, is 30ax2y3. Each term will evenly divide this expression: IV SOLVING EQUATIONS The means of manipulating equations outlined in the previous section can be employed to solve equations. Given an equation, algebra supplies solutions based on the general idea of the identity a = a. As long as the same arithmetic or algebraic procedure is applied simultaneously to both sides of the equation, the equality remains unaffected. The basic strategy is to isolate the unknown term on one side of the equation and the solution on the other. Suppose eight people each have an equal but unknown number of pennies. The people have a scale with 12 pennies on one side and 6 on the other. They start putting their pennies on this unbalanced scale, trying to make it balance. The scale balances once five of the people put all of their pennies on the side that already has six and the other three people put all of their pennies on the side that already has 12. The number of pennies each person originally had can be determined by turning this situation into a linear equation with one unknown: 5 x + 6 = 3x + 1 2 To solve this equation, the variable terms are isolated on one side and the constant terms on the other. The term 3x can be removed from the right side by subtracting; 3 x must then be subtracted from the left side as well: The number 6 is then subtracted from both sides: To isolate x on the left side, both sides of the equation are divided by 2: The solution then follows directly: x = 3. Each person, then, originally had three pennies. This can easily be verified by substituting the solution value x = 3 back into the original equation: 5 x + 6 = 3x + 1 2 5(3) + 6 = 3(3) + 12 15 + 6 = 9 + 12 21 = 21 A Factoring Quadratic Equations It is not so easy to isolate the variable on one side of an equation in which more than one degree of the variable appears. Quadratic equations are the simplest such equations. Given any quadratic equation of the general form ax2 + bx + c = 0 a number of approaches are possible depending on the specific nature of the equation in question. If the equation can be factored, then the solution is straightforward. For instance, if a = 1, b = -3, and c = -10, then x2 - 3x - 10 = 0 can be factored as follows: (x - 5)(x + 2) = 0 The only way to get 0 when multiplying numbers together is when one of the numbers is 0. Therefore this equation can only be true when one or the other of the individual factors is equal to zero--that is, when x - 5 = 0 or x + 2 = 0. Thus the equation has two solutions: x = 5 and x = -2. That these are the solutions to the equation may again be verified by substituting them back into the original equation: 5 2 - 3(5) - 10 = 25 - 15 - 10 = 0 and (-2)2 - 3(-2) - 10 = 4 + 6 - 10 = 0. B Completing the Square If, on inspection, no obvious means of factoring the equation directly can be found, an alternative might exist. For example, in the equation 4 x2 + 12x = 7 the expression 4x2 + 12x could be factored as a perfect square if it were 4x2 + 12x + 9, which equals (2x + 3)2. This can easily be achieved by adding 9 to both sides of the equation, completing the square: 4 x2 + 12x + 9 = 7 + 9 Factoring and simplifying yields: (2x + 3)2 = 16 By taking the square root of both sides, this can be reduced to (2x + 3) =  which is the same as 2 x + 3 = 4 and 2x + 3 = -4 because  has two values, positive 4 and negative 4. The first equation leads to the solution x = y because 2x + 3 = 4 becomes 2x = 1 after subtracting 3 from both sides, and 2x = 1 becomes x = y after dividing both sides by 2. The second equation leads to the solution x = -72, or x = -3y. Both solutions can be verified by substituting the two solutions in question back into the original equation, 4x2 + 12x + 9 = 7 + 9: 4(y)2 + 12(y) + 9 = 7 + 9 4(? + 6 + 9 = 16 ) 1 + 6 + 9 = 16 16 = 16 4(-3y)2 + 12(-3y) + 9 = 7 + 9 4(12? - 42 + 9 = 16 ) 49 - 42 + 9 = 16 16 = 16 C The Quadratic Formula Sometimes it is not possible or readily apparent how to factor an equation or complete its square. However, all quadratic equations that can be put into the form ax2 + bx + c = 0 can be solved using the quadratic formula: For example, to find the roots of x2 - 4x = -3 the equation is first put into the standard form x2 - 4x + 3 = 0 In this equation a = 1, b = -4, and c = 3. These terms are then substituted into the quadratic formula: D Solving Simultaneous Equations Solving Simultaneous Equations When solving more than one equation at a time, we are interested in finding the set of all solutions that will satisfy both equations. An easy way to find this set of solutions for the linear equations 3x + 4y = 10 and 2x + y = 5 is to express one variable in terms of the other. In this case, the variable y is isolated in the second equation, which may be rewritten y = 5 - 2x. If we substitute this new expression of y into the first equation, we get: 3x + 4(5 - 2x) = 10. Now there is only one variable and the equation may be solved. The solution, x = 2, may then be substituted into both equations, which yields a value of y = 1. Thus, the set of solutions which satisfies both equations is (2,1). Graphically, any values that satisfy both equations will result in an intersection of the lines (see graph). © Microsoft Corporation. All Rights Reserved. Algebra is frequently used to solve not just a single equation with a single unknown but also several equations involving multiple unknowns at the same time. The problem is to find the set of all solutions that simultaneously satisfies all of the equations. The equations to be solved are called simultaneous equations, and specific algebraic techniques can be used to solve them. For example, given the following two linear equations with two unknowns: (1) 3x + 4y = 10 (2) 2x + y = 5 a simple solution exists. The variable y in equation (2) is isolated (y = 5 - 2x), then this value of y is substituted into equation (1): 3 x + 4(5 - 2x) = 10 This reduces the problem to one involving the single linear unknown x, and it follows that 3 x + 20 - 8x = 1 0 or -5x = -10 so x=2 When this value is substituted into either equation (1) or (2), it follows that y=1 A faster method of solving simultaneous equations is to multiply both sides of one equation by a number that will give one of the unknown terms the same coefficient as it has in the other equation. This multiplication allows one equation to be subtracted from the other, completely canceling out one of the unknowns. For example, if both sides of equation (2) are multiplied by 4, then 8 x + 4y = 2 0 If equation (1) is then subtracted from equation (2) the y term cancels: The remaining equation, 5x = 10, is easy to solve, giving x = 2. This procedure leads to another mathematical tool, matrices, which help to produce solutions for any set of linear equations in any number of unknowns (See Matrix Theory and Linear Algebra). V HISTORY The word algebra comes from the Arabic al-jabr, meaning restoration. During the 9th century Islamic mathematicians systemized algebra, which they called the science of restoration and balancing. Algebraic techniques, however, have been employed to solve simple equations for thousands of years. A Ancient Origins Babylonian Mathematical Tablet The markings on this tablet are mathematical symbols that were used by the Babylonian civilization. The Babylonian system of counting was based on the number 60, much like the system used today is based on the number 10. The Babylonians used their system to compute the positive roots of quadratic equations, and in geometry they were able to find the areas and volumes of various geometrical shapes. Scala/Art Resource, NY The ancient Egyptians and Babylonians used algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago. The ancient Babylonians solved quadratic equations using essentially the same procedures taught today. First century AD mathematician Hero of Alexandria and 3rd century AD mathematician Diophantus continued the traditions of ancient Egypt and Babylon. Diophantus's book Arithmetica is on a much higher level, however, giving many solutions to difficult indeterminate equations. This knowledge of solutions of equations in turn found a home in the Islamic world. In the 9th century Arab mathematician al-Khwârizmî wrote one of the first systematic explanations of the basic theory of equations and included examples and proofs. Proofs are lines of reasoning used to show that mathematical statements are true. By the end of the 9th century, Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and had solved complicated problems such as finding x, y, and z such that x + y + z = 10, x2 + y2 = z2, and xz = y2. B Algebra in the Middle Ages and Renaissance Leonardo Fibonacci Italian mathematician Leonardo Fibonacci made advances in number theory and algebra. He is especially known for his work on series of numbers, including the Fibonacci series. Each number in a Fibonacci series is equal to the sum of the two numbers that came before it. Corbis Ancient civilizations wrote out algebraic expressions using only occasional abbreviations. These longhand notations were cumbersome. The exponent x6, for example, required notation equivalent to x · x · x · x · x · x. By medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown variable x, and work out the basic algebra of polynomials (although they did not yet use modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as knowledge of the binomial theorem, which describes how to raise a binomial to an arbitrarily high power. Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations using line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of al-Khw?rizm?'s algebra text appeared in the 12th century. In the early 13th century, the great Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the specific cubic equation x3 + 2x2 + cx = d . Because Fibonacci had traveled in Islamic lands, he probably used an Arabic method of successive approximations to reach his solution. René Descartes The first modern philosopher, René Descartes believed science and mathematics could explain and predict events in the physical world. Descartes developed the Cartesian coordinate system for graphing equations and geometric shapes. Modern maps use a grid system that can be traced back to Cartesian graphing techniques. Hulton Deutsch Early in the 16th century, Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano found a general solution for cubic equations. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree, and as a result, mathematicians for the next several centuries tried to find a formula for the roots of equations of degree five and higher. Early in the 19th century, however, Norwegian mathematician Niels Abel and French mathematician Évariste Galois proved that no such formula exists. An important development in algebra in the 16th century was the introduction of modern symbols for unknowns, algebraic powers, and algebraic operations. As a result of this development, Book III of La géometrie (1637), written by French philosopher and mathematician René Descartes, looks much like a modern algebra text. Descartes's most significant contribution to mathematics, however, was his discovery of analytic geometry, which allows problems in geometry to be solved by solving algebraic equations. His geometry text also contained the essentials of a course on the theory of equations, including his so-called rule of signs for counting the number of what Descartes called the "true" (positive) and "false" (negative) roots of an equation. Work continued through the 18th century on the theory of equations. In 1799 German mathematician Carl Friedrich Gauss published a proof showing that every polynomial equation has at least one root that is a complex number. Complex numbers are the sum of a real number and an imaginary number. C Modern Algebra By the beginning of the 19th century, algebra had entered its modern phase. Attention shifted from numbers and solving polynomial equations to studying the structure of abstract mathematical systems whose laws are based on the behavior of mathematical objects. Groups, sets of elements and operations that take any two elements of a set and form another element of the set, are an example of such a system. Groups share some of the properties of number systems but also depart from them in important ways. Groups became one of the chief unifying concepts of 19th-century mathematics. French mathematicians Galois and Augustin Cauchy, British mathematician Arthur Cayley, and Norwegian mathematicians Abel and Sophus Lie made important contributions to the study of groups. German mathematician Hermann Grassmann laid the foundations of another important branch of modern algebra, vector analysis, during the 1840s. Vectors are mathematical quantities that have both magnitude and direction. Despite the abstract character of vector analysis, American physicist J. Willard Gibbs later recognized that it could be extremely useful for physicists. The velocity of a car, for example, can be considered a vector because it has magnitude (speed) and direction. The widespread influence of vector analysis led British mathematician George Boole to write An Investigation of the Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra--also called abstract algebra--has continued to develop. Important new results have been discovered, and algebra has found applications in all branches of mathematics as well as in many of the physical sciences. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
