complex_numbers

complex_numbers - Basics of Complex Numbers(I 1 General i-1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Basics of Complex Numbers (I) 1. General i - 1, so i 2 = - 1, i 3 = - i , i 4 = 1 and then it starts over again ( i 5 = i , i 6 = - 1,. . . ). Any complex number z can be written as the sum of a real part and an imaginary part: z = [Re z ] + i [Im z ] , where the numbers or variables in the []’s are real . So z = x + y i with x and y real is in this form but w = 1 / ( a + b i ) is not (see ”Rationalizing” below). Thus, Im z = y , but Re w 6 = 1 /a . Complex Conjugate: The complex conjugate of z , which is written as z * , is found by changing the sign of every i in z : z * = [Re z ] - i [Im z ] so if z = 1 a + b i , then z * = 1 a - b i . Note: There may be “hidden” i ’s in the variables; if a is a complex number, then z * = 1 / ( a * - b i ). Magnitude: The magnitude squared of a complex number z is: zz * ≡ | z | 2 = [Re z ] 2 - ( i ) 2 [Im z ] 2 = [Re z ] 2 + [Im z ] 2 0 , where the last equality shows that the magnitude is positive (except when z = 0). Basic rule: if you need to make something real, multiply by its complex conjugate.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/03/2011 for the course H 133 taught by Professor Furnstahl during the Spring '11 term at Ohio State.

Page1 / 2

complex_numbers - Basics of Complex Numbers(I 1 General i-1...

This preview shows document pages 1 - 2. Sign up to view the full document.

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