Chap 1 The Number System

X to see this draw rst of all the complex number z 1 i

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: and diagram: 0 p p p p pθ r pp p x p p p z y We shall study more carefully the triangle shown. By Pythagoras’s theorem, we have (6) Also (7) x = r cos θ and y = r sin θ. r2 = x2 + y 2 . Definition. Suppose that z = x + y i, where x, y ∈ R. We write |z | = x2 + y 2 and call this the modulus of z . On the other hand, any number θ ∈ R satisfying the equations (7) is called an argument of z , and denoted by arg z . Remarks. (1) Note that for a given z ∈ C, arg z is not unique. Clearly we can add any integer multiple of 2π to θ without affecting (7). We sometimes call a real number θ ∈ R the principal argument of z if θ satisfies the equations (7) and −π < θ ≤ π . Note that it follows from (7) that y/x = tan θ. However, even with this restriction on θ, it is not meaningful to write (8) θ = tan−1 y . x To see this, draw first of all the complex number z = 1 + i on the Argand diagram. Clearly the equations √ (7) are satisfied with r = 2 and θ = π/4. Further...
View Full Document

This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

Ask a homework question - tutors are online