Chap 1 The Number System

# Definition suppose that z x y i where x y r the real

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview:  d)i. For the division rule, suppose that c + di = 0, so that c = 0 or d = 0, whence c2 + d2 = 0. If a + bi = x + y i, c + di where x, y ā R, then a + bi = (c + di)(x + y i) = (cx ā dy ) + (cy + dx)i. It follows that a = cx ā dy, b = cy + dx. This system of simultaneous linear equations has the unique solution x= so that a + bi ac + bd bc ā ad =2 +2 i. c + di c + d2 c + d2 The special case a = 1 and b = 0 gives 1 c ā di . =2 c + di c + d2 This can also be obtained by noting that (c + di)(c ā di) = c2 + d2 , so that 1 c ā di c ā di . = =2 c + di (c + di)(c ā di) c + d2 It is also useful to note that in has exactly four possible values, with i2 = ā1, i3 = āi and i4 = 1. Definition. Suppose that z = x + y i, where x, y ā R. The real number x is called the real part of z , and denoted by x = Rez . The real number y is called the imaginary part of z , and denoted by y = Imz . The set C = {z = x + y i : x, y ā R} is called the set of all complex numbers. Chapter 1 : The Number Sy...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online