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ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 1 BASIC ALGEBRA 1.1. The Real Number System We shall assume that the reader has a knowledge of the real numbers. Some examples of real numbers are 3, 1 / 2, π , 23 and 3 5. We shall denote the collection of all real numbers by R , and write x R to denote that x is a real number. Among the real numbers are the collection N of all natural numbers and the collection Z of all integers. These are given by N = { 1 , 2 , 3 ,... } and Z = { ..., 3 , 2 , 1 , 0 , 1 , 2 , 3 } . Another subcollection of the real numbers is the collection Q of all rational numbers. To put it simply, this is the collection of all fractions. Clearly we can write any fraction if we allow the numerator to be any integer (positive, negative or zero) and insist that the denominator must be a positive integer. Hence we have Q = ½ p q : p Z and q N ¾ . It can be shown that the collection Q contains precisely those numbers which have terminating or repeating decimals in their decimal notations. For example, 3 4 =0 . 75 and 18 13 = 1 . 384615 are rational numbers. Here the digits with the overline repeat. It can be shown that the number 2 is not a rational number. This is an example of a real number which is not a rational number. Indeed, any real number which is not a rational number is called an irrational number. It can be shown that any irrational number, when expressed in decimal notation, has This chapter was written at Macquarie University in 1999.
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1–2 W W L Chen and X T Duong : Elementary Mathematics non-terminating and non-repeating decimals. The number π is an example of an irrational number. It is known that π =3 . 14159 .... The digits do not terminate or repeat. In fact, a book was published some years ago giving the value of the number π to many digits and nothing else–avery uninteresting book indeed! On the other extreme, one of the states in the USA has a law which decrees that π = 3, no doubt causing a lot of problems for those who have to measure the size of your block of land. 1.2. Arithmetic In mathematics, we often have to perform some or all of the four major operations of arithmetic on real numbers. These are addition (+), subtraction ( ), multiplication ( × ) and division ( ÷ ). There are simple rules and conventions which we need to observe. SOME RULES OF ARITHMETIC. (a) Operations within brackets are performed Frst. (b) If there are no brackets to indicate priority, then multiplication and division take precedence over addition and subtraction.
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