Conquering Mathematics: From Arithmetic to Calculus by Lloyd Motz

By Lloyd Motz

We have designed and written this e-book. now not as a textual content nor for the pro mathematician. yet for the overall reader who's clearly drawn to arithmetic as an excellent intellec­ tual problem. and for the specific reader whose paintings calls for him to have a deeper figuring out of arithmetic than he obtained at school. Readers within the first workforce are attracted to psychological leisure actions comparable to chess. bridge. and numerous different types of puzzles. yet they often don't reply enthusiastically to arithmetic as a result of their unsatisfied studying studies with it in the course of their college days. The readers within the secondgrouptum to arithmetic as a need. yet with painful resignation and massive apprehension relating to 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 frequently stem from their prior difficult encounters with it. vii viii PREFACE This booklet will express those readers that those fears, frustrations, and normal antipathy are unwarranted, for, as said, it's not a textbook jam-packed with lengthy, dull proofs and 1000s of difficulties, fairly it really is an highbrow event, to be learn with excitement. It used to be written to be simply obtainable and with drawback for the psychological tranquilityofthe reader who willexperience significant achievement whilst he/she sees the simplicity of easy arithmetic. The emphasis all through this ebook is at the transparent clarification of mathematical con­ cepts.

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This new sequence of points is 1/2, 2/2, 3/2,4/2, etc. We note immediately that the point labeled 2/2 is the same as that labeled 1 by the second number in our integer sequence. We therefore write 1 = 2/2; this expression shows us that a fraction equals the number we get when we divide the numerator of the fraction by its denominator (the number that is below the line of the fraction) . 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.

Thus, 4 1/ 2 = J4 = 2, which is a special case of the following general rule: If the exponent to which a number is raised is a fraction, the numerator of this fraction is the integer power to which the number is raised and the denominator is the root of the number. The power exponent 2/3 means square the number and then take its cube root or take its cube root first and then square it as in the example 82/ 3 = (82 ) 1/ 3 = 64 1/ 3 = ~ = 4 THE NUMBER SYSTEM = (81/ 3)2 = (~)2 = 22 = 4. Jf/J4 = 1/2> 1/4.

We see at once that 3/3 = 2/2 = 1, 6/3 = 4/2 = 2,4/3 = (3 + 1}/3 = 1 + 1/3,9/3 = 3, and so on. From this expression we deduce the basic rule that if the numerator of a fraction equals its denominator, the fraction equals 1. This conclusion leads us to the very important rule that multiplying any fraction by a fraction whose numerator and denominator are equal does not change the value of the original fraction since this operation is equivalent to multiplying it by 1. Thus, as an example, 2/3 = 2/3 x 1 = 2/3 x 4/4 = (2 x 4)/(3 x 4) = 8/12.

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