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Introduction to Complex Numbers

Basic Mathematics Introduction to Complex Numbers Martin Lavelle The aim of this package is to provide a short study and self assessment programme for stu- dents who wish to become more familiar with Complex Numbers . Copyright c 2001 Last Revision Date: June 11, 2004 Version Table of Contents 1. The Square Root of Minus One! 2. Real, Imaginary and Complex Numbers 3. Adding and Subtracting Complex Numbers 4. Multiplying Complex Numbers 5. Complex Conjugation 6. Dividing Complex Numbers 7. Quiz on Complex Numbers Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. Section 1: The Square Root of Minus One! 3. 1. The Square Root of Minus One! If we want to calculate the square root of a negative number , it rapidly becomes clear that neither a positive or a negative number can do it.

Section 6: Dividing Complex Numbers 10 6. Dividing Complex Numbers The trick for dividing two complex numbers is to multiply top and bottom by the complex conjugate of the denominator: z 1 z 2 = z 1 z 2 = z 1 z 2 × z∗ 2 z∗ 2 = z 1z∗ 2 z 2z∗ 2 The denominator, z 2 z∗, is now a real number. Example 4 1 i = 1 i × −i −i = −i i× ...

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