- 4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 29
•EXAMPLE 1.13 If z = 1 + i , then r = lzl = viz and (} = Argz = ~·
Therefore, z-^1 = ~ [cos ( - 4 " ) + i sin ( -;r)] = ~ [ ~ -i~] and has modulus
72-^1 - .ill 2.
• EXAMPLE 1.14 If z1 = Bi and z2 = 1 + iv'3, then representative polar
forms for these numbers are z 1 = 8 (cos~ + i sin I) and z 2 = 2 (cos~ + i sin! ).
Hence
-------~EXERCISES FOR SECTION 1.4- Find Arg z for t he following values of z.
(a) I - i.
(b) -J3+i.
(c) (- 1 -iv'3)^2.
( d) (1 - i )^3.(e) i+~Ja·
(f) i~l.() g 1H1 (l+i).
(h) (1 +iJ3) (1 + i).- Use exponential notation to show that
(a) ( J3 -i) (1 + iv'3) = 2)3 + 2i.
(b) (1 + i)^3 = - 2 + 2i.
(c) 2i(v'3+ i) (l+i/3) = - 8.
(d) i!i = 4 - 4i.- Represent t he following complex numbers in pola.r form.
(a) - 4.
(b) 6 - 6i.
(c) - 7i.