Problem 1: Algebraic manipulation of imaginary numbers The attached appendix B is from “Principles of Physics,” J.B. Marion and W.F. Hornyak, Saunders College Publishing, New York 1984 (see also Appendix D in Tipler, Ch. 3 in Hirose and Langren, class notes) Simplify expressions a) - d) and write the answer in the Cartesian (x+iy) form: a)

!

2 " i( ) " i 1" i 2( ) b)

!

2 " 3i( ) "2 + i( )

c)

!

1+ 2i

3" 4i+2 " i

5i

d)

!

i "1( )4

Find the value of the magnitude (ρ) and the argument (θ) for expressions f)-g) e) 1+i

f)

!

"2

1+ i 3

g)

!

3 " i( )6

Use the polar form (ρeiθ) to prove the equalities h) - j):

h)

!

5i

2 + i=1+ 2i

i)

!

"1+ i( )7

= "8 1+ i( ) j)

!

cos" + isin"( )n

= cos n"( ) + isin n"( ) {this equality is known as DeMoivre’s theorem} k) Without using a calculator, fill in the two smallest, positive values of the angles

(express as multiples of π) corresponding to the values of tan(θ) in the following table:

tan(θ) Θ (two values, expressed as multiples of π)

0

!

3 /3 1

!

3

!

±" -

!

3 -1

-

!

3 /3