Arnold Shapiro

MATRICES AND DETERMINANTS

This chapter discusses matrices and determinants. By conventional agreement, an m × n matrix has m rows and n columns. A square matrix has as many columns as rows. From a system of linear equations, a coefficient matrix can be obtained by using the coefficients of the variables as entries. An augmented matrix, which includes the right-hand sides of the equations, can also be obtained by this system of linear equations. The methods of Gaussian elimination—now called elementary row operations—can be applied to an augmented matrix of a linear system of equations to obtain the solution. Two matrices are equal if they have the same dimension and if their corresponding elements are equal. Matrices are added by adding corresponding element. The product of an m × n matrix times an n × r matrix is a matrix of dimension m × r. Multiplication of matrix is associative and not cumulative. A square n × n matrix A is invertible if an n × n matrix B can be found such that AB = BA = I, with I being the identity matrix of order n. In such a case, B is the inverse of A. If AX = B is a linear system, with A being the coefficient matrix, and if the inverse A −1 of A exists, then the system has exactly one solution X = A −1 B.

ROOTS OF POLYNOMIALS

This chapter provides an overview of roots of polynomials. Dividing a polynomial by another polynomial is similar to long division in arithmetic in that there is a quotient that may or may not contain a remainder. Symbolically, P(x) = Q(x) • D(x) + R(x), where the degree of R(x) is less than that of P(x). Synthetic division is the name given to a “shortcut” process of long division of polynomials. From the remainder theorem of polynomials, if a polynomial P(x) is divided by (x − r), then the remainder is P(r). The remainder theorem, coupled with synthetic division, is an easy way to obtain (x, y) pairs for a polynomial. It is used to assist in plotting points in graphing a polynomial. In factor theorem, a polynomial P(x) has a factor (x − r) if and only if P(r) = 0. The conjugate of a complex number a + bi is a − bi and the product of the two conjugate complex numbers is a real number—for example, (a + bi) (a − bi) = a 2 + b 2 . The quotient of two complex numbers is found by multiplying the dividend and the divisor by the conjugate of the divisor. From the fundamental theorem of algebra, every polynomial P (x) of degree ≥1 has at least one complex root.

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

This chapter discusses exponential function and its inverse and the logarithmic function. Exponential functions apply in nature and are useful in chemistry, biology, and economics as well as in mathematics and engineering. The chapter discusses the applications of exponential functions in calculating such quantities as compound interest and the growth rate of bacteria in a culture medium. In many business transactions, the interest that is added to the principal at regular time intervals also earns interest. This is called the compound interest process. The time period between successive additions of interest is known as the conversion period. Thus, if interest is compounded quarterly, the conversion period is three months; if interest is compounded semiannually, the conversion period is six months. Logarithms can be viewed as another way of writing exponents. Historically, logarithms have been used to simplify calculations; in fact, the slide rule, a device long used by engineers, is based on logarithmic scales. In todays world of inexpensive hand calculators, the need for manipulating logarithms is reduced. The chapter presents computing with logarithms that provide enough background to allow one to use this powerful tool but omits some of the detail found in older textbooks.

ANGLES AND TRIANGLES

This chapter focuses on the angles and triangles in applied mathematics. An angle is the result of rotating a half line about its end point. It is said to be positive if the rotation is counterclockwise and negative if the rotation is clockwise. Angles are measured in two basic ways: (1) in degrees, a degree being of a complete revolution; and (2) in radians, a radian being of a complete revolution. The circumference of a unit circle is 2π, so it can be seen that the radian measure is a measure of the arc an angle intercepts. The reference angle θ of an angle θ is the acute angle formed by the terminal side of θ and the x-axis. If an angle θ subtends an arc of length s in a circle of radius r, then the radian measure of θ = s/r. The reference angle concept allows the application of only acute angles when finding the trigonometric values of any angle. The trigonometric functions of acute angles can be defined in terms of the sides of a right triangle. Right triangle trigonometry can be used to solve many practical problems in areas as diverse as carpentry and navigation.

THE TRIGONOMETRIC FUNCTIONS

The word trigonometry is derived from Greek, meaning measurement of triangles. The conventional approach to the subject matter of trigonometry deals with relationships among the sides and angles of a triangle, reflecting the important applications of trigonometry in fields such as navigation and surveying. The modern approach to trigonometry emphasizes the function concept. This has become the accepted approach, because it demonstrates the unifying influence of the function concept. This chapter defines this important class of functions and discusses their fundamental properties and graphs. It discusses trigonometry in terms of functions of angles and real numbers. It also discusses the inverse trigonometric functions.

THEORY OF POLYNOMIALS

This chapter presents some of the fundamental issues concerning zeros of a polynomial, which have attracted the attention of mathematicians for centuries: (1) whether a polynomial always has a zero; (2) the total number of zeros of a polynomial of degree n ; (3) the number of zeros of a polynomial that are real numbers; (4) the number of the zeros that are rational numbers in case if the coefficients of a polynomial are integers; (5) whether there is a relationship between the zeros and the factors of a polynomial; and (6) whether a formula can be derived for expressing the zeros of a polynomial in terms of the coefficients of the polynomial. Some of these questions are tough mathematical problems. This chapter explores them and presents the answers in the course of it. To find the zeros of a polynomial, it is necessary to divide the polynomial by a second polynomial. There is a procedure for polynomial division that parallels the long division process of arithmetic. The chapter discusses polynomial division and synthetic division, the remainder and factor theorems, factors and zeros, real and rational zeros, and rational functions and their graphs.

SEQUENCES AND SERIES

This chapter discusses the operation and approach that involve the set of natural numbers. The sequences lead to considerations of series, and the underlying concepts of infinite series can be used as an introduction to calculus. The chapter also describes mathematical induction, which provides a means of proving certain theorems involving the natural numbers that appear to resist other means of proof. It presents the use of mathematical induction to prove that the sum of the first n consecutive positive integers is n ( n + l)/2. The chapter discusses binomial theorem, which gives one a way to expand the expression ( a + b ) n where n is a natural number.

FUNCTIONS AND GRAPHS

This chapter discusses the concepts of functions and graphs. The concept of a function has been developed as a means of organizing and assisting in the study of relationships. If one thinks of the x-axis as a real number line, one may mark off some convenient unit of length, with positive numbers to the right of the origin and negative numbers to the left of the origin. Similarly, one may think of the y-axis as a real number line. The x and y axes are called coordinate axes and together they constitute a rectangular or Cartesian coordinate system. The coordinate axes divide the plane into four quadrants. The chapter discusses the graphs of functions. The information available at a glance from a graph is so impressive that it is vital for a student planning to study advanced mathematics to be familiar with techniques for quickly sketching the graphs of those functions that occur most frequently.

LINEAR EQUATIONS AND INEQUALITIES

This chapter presents ways to learn to solve the most basic forms of equations and inequalities. An equation states that two algebraic expressions are equal. The expression to the left of the equal sign is called the left-hand side of the equation while the expression to the right of the equal sign is called the right-hand side. The task is to find values of the variable for which the equation holds true. These values are called solutions or roots of the equation and the set of all solutions is called the solution set. To solve an equation, all the solutions or roots are to be found. If an equation can be replaced by another, simpler equation that has the same roots, this will provide an approach to solving equations. Equations having the same roots are called equivalent equations. There are two important rules that allow for the replacement an equation by an equivalent equation—addition or subtraction of a number or expression to both sides of the equation and multiplication or division of both sides of the equation by a number other than 0.

EXPONENTS, RADICALS, AND COMPLEX NUMBERS

This chapter presents a method that helps to expand the capability to handle any exponent, whether zero, a positive integer, a negative integer, or a rational number. Radicals are an alternate way of writing rational exponent forms. As solutions of polynomial equations frequently involve radicals, the chapter presents ways to manipulate and simplify radical forms as background to the study of polynomial equations of degree greater than 1. The real number system is inadequate to provide a solution to all polynomial equations. It is necessary to create a new type of number, called a complex number.