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Unformatted text preview: 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 9 COMPLEX NUMBERS 9.1. Introduction It is easy to see that the equation x 2 + 1 = 0 has no solution x R . In order to solve this equation, we have to introduce extra numbers into our number system. Define the number i by writing i 2 + 1 = 0. We then extend the collection of all real numbers by adjoining the number i, which is then combined with the real numbers by the operations addition and multiplication in accordance with the rules of arithmetic for real numbers. The numbers a + b i, where a, b R , of the extended collection are then added and multiplied in accordance with the rules of arithmetic for real numbers, suitably extended, and the restriction i 2 + 1 = 0. Note that the number a + 0i, where a R , behaves like the real number a . Definition. We denote by C the collection of all complex numbers; in other words, the collection of all numbers of the form a + b i, where a, b R ....
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This note was uploaded on 10/19/2010 for the course MATHEMATIC Math123 taught by Professor Goh during the Spring '10 term at UCLA.
- Spring '10