One of their most recent publications is CHAPTER 1 - The Foundations of Algebra. Which was published in journal .

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Bernard Kolman's Articles: (12)


Publisher SummaryThis chapter reviews some fundamentals of algebra and discusses the meaning and use of variables, algebraic expressions and polynomial forms, factoring, and operations with rational expressions or algebraic fractions. The rational and irrational numbers together form the real number system. In regard to the operations of addition and multiplication, the real number system has properties that are fundamental to algebra. There is a simple and very useful geometric interpretation of the real number system. The set of real numbers is identified with all possible points on a straight line and for every point on the line there is a real number and for every real number there is a point on the line. The line is called the real number line, and the number associated with a point is called its coordinate. A polynomial is an algebraic expression of a certain form, which plays an important role in the study of algebra, as many word problems translate into equations or inequalities that involve polynomials. The degree of the product of two nonzero polynomials is the sum of the degrees of the polynomials.


Publisher SummaryThis chapter discusses the concept of a function that has been developed as a means of organizing and assisting in the study of relationships. As graphs are powerful means of exhibiting relationships, one should begin with a study of the Cartesian, or rectangular, coordinate system. A function is a rule that, for each x in a set X, assigns exactly one y in a set Y. The element y is called the image of x. The set X is called the domain of the function and the set of all images is called the range of the function. The polynomial function of first degree f(x) = ax + b is called a linear function. The polynomial function of second degree f(x) = ax2 + bx + c, a ≠ 0 is called a quadratic function.


Publisher SummaryThis chapter discusses analytic geometry. Analytic geometry enables applying algebraic methods and equations to the solution of problems in geometry and, conversely, to obtain geometric representations of algebraic equations. It is also possible to obtain a formula for the coordinates (x, y) of the midpoint P of the line segment whose endpoints are P1 and P2. Passing lines through P and P2 parallel to the y-axis and a line through P1, parallel to the x-axis results in the similar right triangles P1AP and P1BP2. The conic sections provide an outstanding opportunity to illustrate the double-edged power of analytic geometry. A geometric figure defined as a set of points can often be described analytically by an algebraic equation. If a plane is passed through a cone at various angles, the intersections are called conic sections. In exceptional cases, the intersection of a plane and a cone is a point, a line, or a pair of lines.


Publisher SummaryThis chapter discusses the systems of equations and inequalities. Many problems in business and engineering require the solution of systems of equations and inequalities. In fact, systems of linear equations and inequalities occur with such frequency that mathematicians and computer scientists have devoted considerable energy to devising methods for their solution. The chapter discusses the methods of substitution and elimination that are applicable to all types of systems. It is possible for a system of equations to have no solutions. A system of equations may even have an infinite number of solutions. A system consisting only of equations that are of the first degree in x and y is called a system of linear equations or simply a linear system. The method of graphing has severe limitations as the accuracy of the solution depends on the accuracy of the graph. While solving a system of equations by graphing, one must estimate the coordinates of the point of intersection. The method of substitution provides exact answers but suffers from the disadvantage that it is difficult to program for use in a computer.


Publisher SummaryThis chapter describes different topics in algebra that involves the set of natural numbers. Mathematical induction provides a means of proving certain theorems involving the natural numbers that appear to resist other means of proof. Probability theory enables one to state the likelihood of occurrence of a given event and has obvious applications to games of chance. The theory of permutations and combinations, which enables counting the ways in which a set of objects or select a subset of the original set can be arranged, is necessary background to a study of probability theory. An infinite sequence is a function whose domain is the set of all natural numbers. The series associated with an arithmetic sequence is called an arithmetic series. The choice of which formula to use depends on the available information. In a geometric sequence, the terms between the first and last terms are called geometric means.


Publisher SummaryThis chapter focuses on the exponential and logarithmic functions. Exponential functions are useful in chemistry, biology, economics, mathematics, and engineering. The chapter describes the applications of exponential functions in calculating such quantities as compound interest and the growth rate of bacteria in a culture medium. Exponential functions occur in a wide variety of applied problems. The chapter reviews some problems dealing with population growth: predicting the growth of bacteria in a culture medium; radioactive decay, such as determining the half-life of strontium 90; and the interest earned when an interest rate is compounded. Logarithms can be viewed as another way of writing exponents. Logarithms have been used to simplify calculations; the slide rule, a device long used by engineers, is based on logarithmic scales. In today's world of inexpensive hand calculators, the need for manipulating logarithms is reduced.


Publisher SummaryThis chapter provides an overview of matrices and determinants. Their properties and applications are both extensive and important. The material on matrices and determinants presented in the chapter serves as an introduction to linear algebra, a mathematical subject that is used in the natural sciences, business and economics, and the social sciences. As matrix methods may require millions of numerical computations, computers have played an important role in expanding the use of matrix techniques to a wide variety of practical problems. The method of Gaussian elimination can be implemented using matrices. The chapter explains how matrix notation provides a convenient means for writing linear systems and how the inverse of a matrix can help solve such a system.

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