Express each of the following complex numbers in the form re

Express each of the following complex numbers in the form re^i theta where r and theta are real numbers, with r > 0 and 0

Solution

a)-1+3 i

multiply and divide by [(-1)²+(3)²]=4 =2

2(-1+3 i)/2

=2((-1/2)+(3 /2) i)

=()-tan^-1((3 /2)/(1/2))

=()-tan^-1(3)

=()-(/3)

=2/3

-1+3 i=2e^i2/3

b)1+i

multiply and divide by [(1)²+(1)²]=2

2 (1+ i)/2

=2 ((1/2 )+(1/2 ) i)

=tan^-1((1/2 )/(1/2))

=tan^-1(1)

=/4

1+ i=2 e^i/4

C)(1+i)(2+i)(3+i)

=(2+2i+i+i²)(3+i)

=(2+3i-1)(3+i)

=(3i+1)(3+i)

=9i-3+3+i

=10i

r =10 , =/2

10i =10e^i/2

d)(1+i)⁴

from (b) 1+i =(2 e^i/4)

( 1+i)⁴ =(2 e^i/4)⁴

( 1+i)⁴ =(2)⁴ e^(i/4)*4

( 1+i)⁴ =4eⁱ