COMPETENCY 1.0 ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings) Algebra must be systematically organized in order to create the axioms for sets of numbers. The binary relation symbols +,, and are operators that require two inputs and obey the algebraic axioms of field theory. Any set that includes at least two nonzero elements that satisfies the field axioms for addition and multiplication is a field. The real numbers, R, as well as the C, are each a field, with the real numbers being complex numbers, a subset of the complex numbers. The field axioms are summarized below. Addition: Commutativity a+ b = b+ a Associativity a+ ( b+ c) = ( a+ b) + c Identity a+ 0 = a Inverse a+ ( a) = 0 Multiplication: Commutativity ab = ba Associativity a( bc) = ( ab) c Identity a 1= a 1 Inverse a = 1 a ( a 0 ) Addition and multiplication: Distributivity a( b+ c) = ( b+ c) a = ab+ ac Note that both the real numbers and the complex numbers satisfy the axioms summarized above.
A ring is an integral domain with two binary operations (addition and multiplication) where, for every non-zero element a and b in the domain, the product ab is non-zero. A field is a ring where multiplication is commutative, or a b = b a, and all non-zero elements have a multiplicative inverse. The set Z (integers) is a ring that is not a field in that it does not have the multiplicative inverse; therefore, integers are not a field. A polynomial ring is also not a field, as it also has no multiplicative inverse. Furthermore, matrix rings do not constitute fields because matrix multiplication is not generally commutative. Note: Multiplication is implied when there is no symbol between two variables. Thus, a b can be written ab. Multiplication can also be indicated by a raised dot ( ).
1.1b. Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc) Basic Properties of Real and Imaginary/Complex Numbers Real Numbers Irrational Numbers Natural Numbers Whole Numbers Integers Rational Numbers Real numbers are denoted by R and are numbers that can be shown by an infinite decimal representation such as 3.8675347... Real numbers include rational numbers, such as 4 and 3/19, and irrational numbers, such as the and π, can be represented as points along an infinite number line. Real numbers are also known as the unique complete Archimedean ordered field. Real numbers are to be distinguished from imaginary numbers. Real numbers are classified as follows: A. Natural numbers, denoted by N: the counting numbers, 1,, 3,. B. Whole numbers: the counting numbers along with zero, 0, 1,, 3,. C. Integers, denoted by Z: the counting numbers, their negatives, and zero,,, 1, 0, 1,,.
D. Rationals, denoted by Q: all of the fractions that can be formed using whole numbers. Zero cannot be the denominator. In decimal form, these numbers will either be terminating or repeating decimals. Simplify square roots to determine if the number can be written as a fraction. E. Irrationals: Real numbers that cannot be written as a fraction. The decimal forms of these numbers are neither terminating nor repeating. Examples include π, e and. Imaginary and complex numbers are denoted by. The set is defined as { a+ bi : a, b } ( means element of ). In other words, complex numbers are an extension of real numbers made by attaching an imaginary number i, which satisfies the equality i = 1. Complex numbers are of the form a + bi, where a and b are real numbers and i = 1. Thus, a is the real part of the number and b is the imaginary part of the number. When i appears in a fraction, the fraction is usually simplified so that i is not in the denominator. The set of complex numbers includes the set of real numbers, where any real number n can be written in its equivalent complex form as n + 0i. In other words, it can be said that (or is a subset of ). Complex numbers Real numbers The number 3i has a real part 0 and imaginary part 3; the number 4 has a real part 4 and an imaginary part 0. As another way of writing complex numbers, we can express them as ordered pairs: Complex number Ordered pair 3+i (3, ) 3+ 3i ( 3, 3 ) 7i (0, 7) 6+i 6, 7 7 7
These properties of real and complex numbers can be applied to the construction of various mathematical arguments. A mathematical argument proves that a proposition is true. Example: Prove that for every integer y, if y is an even number, then y is even. The definition of even implies that for each integer y there is at least one integer x such that y = x. y = x y = 4x Since 4x is always evenly divisible by two ( x is an integer), y is even for all values of y. Example: If a, b and c are positive real numbers, prove that c a+ b = b+ a c. ( ) ( ) Use the properties of the set of real numbers. c( a+ b) = c( b+ a) Additive commutativity = cb + ca Distributivity = bc + ac Multiplicative commutativity = b+ a c Distributivity ( ) Example: Given real numbers a, b, c and d where ad = bc, prove that (a + bi)(c + di) is real. Expand the product of the complex numbers. ( a + bi )( c + di ) = ac + bci + adi + bdi Use the definition of i. ( a + bi )( c + di ) = ac bd + bci + adi Apply the fact that ad = bc. ( a + bi )( c + di ) = ac bd + bci bci = ac bd Since a, b, c and d are all real, ac bd must also be real.
1.1c. Know that the rational numbers and real numbers can be ordered and that the complex numbers cannot be ordered, but that any polynomial equation with real coefficients can be solved in the complex field The previous skill section reviews the properties of both real and complex numbers. Based on these properties, it can be shown that real and rational numbers can be ordered but complex numbers cannot be ordered. Real numbers are an ordered field and can be ordered. As such, an ordered field F must contain a subset P (such as the positive numbers) such that if a and b are elements of P, then both a + b and ab are also elements of P. (In other words, the set P is closed under addition and multiplication.) Furthermore, it must be the case that for any element c contained in F, exactly one of the following conditions is true: c is an element of P, c is an element of P or c = 0. Likewise, the rational numbers also constitute an ordered field. The set P can be defined as the positive rational numbers. For each a and b that are elements of the set (the rational numbers), a + b is also an element of P, as is ab. (The sum a + b and the product ab are both rational if a and b are rational.) Since P is closed under addition and multiplication, constitutes an ordered field. Complex numbers, unlike real numbers, cannot be ordered. Consider the number i = 1 contained in the set of complex numbers. Assume that has a subset P (positive numbers) that is closed under both addition and multiplication. Assume that i > 0. A difficulty arises in that i = 1< 0, so i cannot be included in the set P. Likewise, assume i < 0. The problem once again arises that 4 i = 1> 0, so i cannot be included in P. It is clearly the case that i 0, so there is no place for i in an ordered field. Thus, the complex numbers cannot be ordered. Polynomial equations with real coefficients cannot, in general, be solved using only real numbers. For instance, consider the quadratic function given below: ( ) x f x = + 1
There are no real roots for this equation, since ( ) 0 1 f x = = x + x = 1 The Fundamental Theorem of Algebra, however, indicates that there must be two (possibly non-distinct) solutions to this equation. (See Skill 1.c for more on the Fundamental Theorem of Algebra.) Not that if the complex numbers are permitted as solutions to this equation, then x =± i Thus, generally, solutions to any polynomial equation with real coefficients exist in the set of complex numbers.
SKILL 1. 1.a. POLYNOMIAL EQUATIONS AND INEQUALITIES Know why graphs of linear inequalities are half planes and be able to apply this fact (e.g., linear programming) Linear Inequalities To graph a linear inequality expressed in terms of y, solve the inequality for y. This renders the inequality in slope-intercept form (for example: y < mx + b). The point (0, b) is the y-intercept and m is the slope of the line. If the inequality is expressed only in terms of x, solve for x. When solving the inequality, remember that dividing or multiplying by a negative number will reverse the direction of the inequality sign. An inequality that yields any of the following results in terms of y, where a is some real number, then the solution set of the inequality is bounded by a horizontal line: y < a y a y > a y a If the inequality yields any of the following results in terms of x, then the solution set of the inequality is bounded by a vertical line: x < a x a x > a x a When graphing the solution of a linear inequality, the boundary line is drawn in a dotted manner if the inequality sign is < or >. This indicates that points on the line do not satisfy the inequality. If the inequality sign is either or, then the line on the graph is drawn as a solid line to indicate that the points on the line satisfy the inequality. The line drawn as directed above is only the boundary of the solution set for an inequality. The solutions actually include the half plane bounded by the line. Since, for any line, half of the values in the full plane (for either x or y) are greater than those defined by the line and half are less, the solution of the inequality must be graphed as a half plane. (In other words, a line divides the plane in half.) Which half plane satisfies the inequality can be found by testing a point on either side of the line. The solution set can be indicated on a graph by shading the appropriate half plane.
For inequalities expressed as a function of x, shade above the line when the inequality sign is or >. Shade below the line when the inequality sign is < or. For inequalities expressed as a function of y, shade to the right for > or. Shade to the left for < or. The solution to a system of linear inequalities consists of the portion of the graph where the shaded half planes for all the inequalities in the system overlap. For instance, if the graph of one inequality was shaded with red, and the graph of another inequality was shaded with blue, then the overlapping area would be shaded purple. The points in the purple area would be the solution set of this system. Example: Solve by graphing: x+ y 6 x y 6 Solving the inequalities for y, they become the following: y x+ 6 (y-intercept of 6 and slope of 1) 1 y x 3 (y-intercept of 3 and slope of 1/) A graph with the appropriate shading is shown below: x + y 6 10 8 6 4 0-10 -8-6 -4 - - 0 4 6 8 10-4 x y 6-6 -8-10
Linear programming, or linear optimization, involves finding a maximum or minimum value for a linear function subject to certain constraints (such as other linear functions or restrictions on the variables). Linear programming can be used to solve various types of practical, real-world word problems. It is often used in various industries, ecological sciences and governmental organizations to determine or project, for instance, production costs or the amount of pollutants dispersed into the air. The key to most linear programming problems is to organize the information in the word problem into a chart or graph of some type. By plotting the inequalities that define the problem, for instance, the range of possible solutions can be shown visually. Example: A printing manufacturer makes two types of printers: a Printmaster and a Speedmaster printer. The Printmaster requires 10 cubic feet of space, weighs 5,000 pounds and the Speedmaster takes up 5 cubic feet of space and weighs 600 pounds. The total available space for storage before shipping is,000 cubic feet and the weight limit for the space is 300,000 pounds. The profit on the Printmaster is $15,000 and the profit on the Speedmaster is $30,000. How many of each machine should be stored to maximize profitability and what is the maximum possible profit? First, let x represent the number of Printmaster units sold and let y represent the number of Speedmaster units sold. Then, the equation for the space required to store the units is the following. 10x+ 5y 000 x+ y 400 Since the number of units for both models must be no less than zero, also impose the restrictions that x 0 and y 0. The restriction on the total weight can be expressed as follows. 5000x+ 600y 300000 5x+ 3y 1500 The expression for the profit P from sales of the printer units is the following. P = $15,000 x+ $30,000y