Lloyd Motz's Conquering Mathematics: From Arithmetic to Calculus PDF

By Lloyd Motz

We have designed and written this publication. now not as a textual content nor for the pro mathematician. yet for the final reader who's evidently interested in arithmetic as a very good intellec­ tual problem. and for the distinct reader whose paintings calls for him to have a deeper knowing of arithmetic than he got in class. Readers within the first workforce are interested in psychological leisure actions resembling chess. bridge. and numerous varieties of puzzles. yet they typically don't reply enthusiastically to arithmetic due to their unsatisfied studying reviews with it in the course of their tuition days. The readers within the secondgrouptum to arithmetic as a need. yet with painful resignation and massive apprehension concerning their skills to grasp the department ofmathematics they want of their paintings. In both case. the terror of and revulsion to arithmetic felt through those readers often stem from their previous troublesome encounters with it. vii viii PREFACE This ebook will express those readers that those fears, frustrations, and normal antipathy are unwarranted, for, as acknowledged, it's not a textbook jam-packed with lengthy, dull proofs and hundreds of thousands of difficulties, relatively it's an highbrow experience, to be learn with excitement. It used to be written to be simply available and with situation for the psychological tranquilityofthe reader who willexperience substantial achievement whilst he/she sees the simplicity of uncomplicated arithmetic. The emphasis all through this booklet is at the transparent rationalization of mathematical con­ cepts.

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We might write this extremely large but finite number as 1080080lp le x which would be more convenient than trying to write a googolplex of zeros after the number one. We could carry this nonsense even further by raising 10 to the googolplexth power which is in tum raised to the googolplexth power and so on without limit but this would become a fairly pointless exercise after a while because we have no known physical application for such huge numbers nor can they be easily comprehended. Moreover, it is not always a good idea to walk around muttering about raising a googolplex to the googolplexth power because such behavior will cause your family and friends to think you have lost your mind and encourage them to put you in a room with padded walls.

6 Newman, op. cit, p. 2067. 7 Kasner and Newman, op. , p. 80. , p. 87. , pp . 37-3 8. , p. 38. I CHAPTER 2 Irrational Numbers, Imaginary Numbers, and Other Curiosities Round numbers are always false. -SAMUEL JOHNSON We devoted most of the first chapter to a discussion of the rules of arithmetic and their application to integers and fractions and we used the points on a line to help us understand the rules of arithmetic. We now discuss some features of arithmetic operations tha t will lead us to the irrational numbers and to complex numbers.

This is to be expected since the word "fraction" itself means dividing a given quantity of anything into a sum of equal parts and taking one of these parts. In chemistry the word "fractionate" means separating a mixture of chemical compounds into its individual constituents or components but these are not necessarily equal in magnitude. ). Returning now to the sequence 1/2,2/2,3/2,4/2, ... we see that 3/2 labels the point midway between 1 and 2; it may therefore be labeled, equivalently, as 1 + 1/2 so that 3/2 = 1 + 1/2, which we see immediately if we write 3/2 as (2 + 1)/2 = 2/2 + 1/2.

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