4_Complex_(notes)

# 4_Complex_(notes) - z = re i . ex. Write z 1 = ! 8 and z 2...

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Complex Numbers Complex numbers have the form z = x + iy (z = x + yi) x is the real part and y is the imaginary part. ex. 3 + 3i 5 - i 3 i = -1 i 2 = ( -1) 2 = -1 The conjugate of z = a + bi is z = a ! bi . The conjugate of z = a ! bi is z = a + bi . Operations/Definitions 1. a + bi = c + di ! a = c and b = d 2. a + bi ( ) + c + di ( ) = a + c ( ) + b + d ( ) i 3. a + bi ( ) c + di ( ) = ac + adi + bci + bdi 2 = ( ac " bd ) + ( ad + bc ) i 4. a + bi ( ) c + di ( ) = a + bi ( ) c " di ( ) c + di ( ) c " di ( ) = ac " adi + bci " bdi 2 ( ) c 2 " d 2 i 2 = ac + bd ( ) + bc " ad ( ) i c 2 + d 2

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ex. x + iy = 3 + 4 i ex. (2 ! 3 i ) + (5 + i ) ex. (2 ! 3 i )(5 + i ) ex. 5 + i 2 ! 3 i
ex. 1 ! 4 + 3 i Graphing The modulus or absolute value, z , of a complex number is its distance to the origin. z = x 2 + y 2 If z = 2 ! 3 i , then z = 2 2 + ! 3 ( ) 2 = 13 . z = r .

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Since z = x + iy z = r cos ! + ir sin z = r cos + i sin ( ) Euler’s formula: e i = cos + i sin So z = re i (polar form of a complex number) z = x + iy (Cartesian form of a complex number) ex. Write z = ! 1 + i in polar form

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Unformatted text preview: z = re i . ex. Write z 1 = ! 8 and z 2 = 8 i in polar form. ex. Write z = 5 e i 3 ! 2 " # \$ % & in Cartesian form. ex. Evaluate z = 1 ! i 3 ( ) 10 . Write it in both Cartesian and polar forms. Roots ex. Find the fifth roots of 32. 1. Write the number in polar form. 2. Find one fifth root. 3. The other roots are spaced evenly around a circle at intervals of 2 ! # of roots = 2 5 ex. Find the fourth roots of z = ! 81 2 ! 81 3 2 i . Do: 1. Let z = 3 + i , w = 2 i Find z 1 w ! " # \$ 2. Write (1.) in polar form. 3. Write 3 e i ! 6 " # \$ % & in Cartesian form. 4. Find the cube roots of 64i. Remember that r 0....
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## This note was uploaded on 03/29/2008 for the course MATH 1224 taught by Professor Dontremember during the Spring '08 term at Virginia Tech.

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4_Complex_(notes) - z = re i . ex. Write z 1 = ! 8 and z 2...

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