Förlagets beskrivning
An Introductory Course of MATHEMATICAL ANALYSIS. CHAPTER I NUMBER 1. FUNDAMENTAL LAWS OF ALGEBRA 1. ALGEBRA may be described as the general science of arithmetic In arithmetic the processes of addition, subtraction, multiplication, division and other derived operations arise and are applied in con nection with particular numbers. It is seen that the methods employed, being applicable to all particular numbers encountered, with certain well-defined exceptions, must be capable of a general formulation. It is this general formulation which is the primary object of algebra. In order to express the truths of arithmetic in the general algebraic form it is necessary to employ some kind of symbolism. The choice of a symbolism is to some extent arbitrary, but not without effect on the development of the science the lack of an appropriate symbolism having caused in some instances a delay of hundreds of years in the progress of different branches of mathe matics. In the present matter of the algebraic statement of the facts of arithmetic the use of the letters of the alphabet to repre sent the unidentified numbers under consideration is singularly appropriate and has been universal since the sixteenth century, Its fertility is sufficiently apparent in the use of formulae in elementary algebra, and in all branches of science, for it to be un necessary here to enlarge upon it. It may be noted in passing however that the letters occurring in algebraic theorems do not always represent numbers with the same scope of generality. Thus in some theorems the letters or some of them may represent numbers belonging to the widest class contemplated, that of the real or complex numbers, including under this head, not only the whole numbers 1, 2, 3, etc., but also such entities as - 1, J, V2 V-1, which are defined below or they may be restricted to represent numbers of only a particular class, such as the whole numbers, or proper fractions, etc. On the other hand it is always the method of algebra to state theorems with the greatest possible generality, and the fewer the restrictions placed on the 2 NUMBER CH. I i. e. letters occurring in a theorem the more important in general will the theorem be. 2. Fundamental laws. The theorems on which algebra is built, the fundamental laws of algebra, are based on our intuitive ideas of counting, and therefore in the first instance are stated only for the ordinary whole numbers 1, 2, 3, etc. The laws are I The associative law for addition, viz. The terms of a sum of three numbers may be added together in any way preserving the original order without altering the sum or, symbolically, if a, 6, c are any three whole numbers then a - f b c a - f b 4-c. II The associative law for multiplication, viz. The terms of a product of three terms may be multiplied in any way preserving the original order without altering the product, or a x b x c a x b x c, where a, 6, c have the same significance as in I. III The commutative law for addition, viz. The terms in a sum of two numbers may be added together in either order without altering the sum, or a b b 4-a. IV The commutative law for multiplication, viz. The terms in a product may be multiplied in either order without altering the product, or a x b 6 x a. V Tlie distributive law, viz. The product of a sum of two num bers by any third number is the sum of the products of the separate terms of the sum by the common multiplier, or a 4-b x c a x c - f 6 x. c. In the statement of these laws we must be quite clear as to the meaning of the terms and symbols used. The simplest definition of the sum a b is to consider it as the number finally obtained if, having counted up to the number a
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An Introductory Course Of Mathematical Analysis
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