8. complex - Complex Numbers and Exponentials Definition...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

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

Unformatted text preview: Complex Numbers and Exponentials Definition and Basic Operations A complex number is nothing more than a point in the xy plane. The sum and product of two complex numbers ( x 1 , y 1 ) and ( x 2 , y 2 ) is defined by ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) ( x 1 , y 1 ) ( x 2 , y 2 ) = ( x 1 x 2- y 1 y 2 , x 1 y 2 + x 2 y 1 ) respectively. It is conventional to use the notation x + iy (or in electrical engineering country x + jy ) to stand for the complex number ( x, y ). In other words, it is conventional to write x in place of ( x, 0) and i in place of (0 , 1). In this notation, the sum and product of two complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 is given by z 1 + z 2 = ( x 1 + x 2 ) + i ( y 1 + y 2 ) z 1 z 2 = x 1 x 2- y 1 y 2 + i ( x 1 y 2 + x 2 y 1 ) The complex number i has the special property i 2 = (0 + 1 i )(0 + 1 i ) = (0 - 1 1) + i (0 1 + 1 0) =- 1 For example, if z = 1 + 2 i and w = 3 + 4 i , then z + w = (1 + 2 i ) + (3 + 4 i ) = 4 + 6 i zw = (1 + 2 i )(3 + 4 i ) = 3 + 4 i + 6 i + 8 i 2 = 3 + 4 i + 6 i- 8 =- 5 + 10 i Addition and multiplication of complex numbers obey the familiar algebraic rules z 1 + z 2 = z 2 + z 1 z 1 z 2 = z 2 z 1 z 1 + ( z 2 + z 3 ) = ( z 1 + z 2 ) + z 3 z 1 ( z 2 z 3 ) = ( z 1 z 2 ) z 3 0 + z 1 = z 1 1 z 1 = z 1 z 1 ( z 2 + z 3 ) = z 1 z 2 + z 1 z 3 ( z 1 + z 2 ) z 3 = z 1 z 3 + z 2 z 3 The negative of any complex number z = x + iy is defined by- z =- x + (- y ) i , and obeys z + (- z ) = 0. Other Operations The complex conjugate of z is denoted z and is defined to be z = x- iy . That is, to take the complex conjugate, one replaces every i by- i . Note that z z = ( x + iy )( x- iy ) = x 2- ixy + ixy + y 2 = x 2 + y 2 is always a positive real number. In fact, it is the square of the distance from x + iy (recall that this is the point ( x, y ) in the xy plane) to 0 (which is the point (0 , 0)). The distance from z = x + iy to 0 is denoted | z | and is called the absolute value, or modulus, of z . It is given by | z | = p x 2 + y 2 = z z December 17, 2007 Complex Numbers and Exponentials 1 Since z 1 z 2 = ( x 1 + iy 1 )( x 2 + iy 2 ) = ( x 1 x 2- y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ), | z 1 z 2 | = p ( x 1 x 2- y 1 y 2 ) 2 + ( x 1 y 2 + x 2 y 1 ) 2 = q x 2 1 x 2 2- 2 x 1 x 2 y 1 y 2 + y 2 1 y 2 2 + x 2 1 y 2 2 + 2 x 1 y 2 x 2 y 1 + x 2 2 y 2 1 = q x 2 1 x 2 2 + y 2 1 y 2 2 + x 2 1 y 2 2 + x 2 2 y 2 1 = q ( x 2 1 + y 2 1 )( x 2 2 + y 2 2 ) = | z 1 || z 2 | for all complex numbers z 1 , z 2 ....
View Full Document

This note was uploaded on 01/27/2011 for the course MATH 267 taught by Professor Yilmaz during the Spring '10 term at The University of British Columbia.

Page1 / 6

8. complex - Complex Numbers and Exponentials Definition...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online