Answer s
Section 1 .1. The Origi n of Complex Numbers: page 6
1. Mimic the argument the text gives in showing 2 + A = V2 + ./-121.
3a. The roots are x1 = -~, x2 = -~, x3 = ~·5a. Use Formula (1-3) to get x = Vis+ 26H + vis -26A. Assume, as
Bombelli did , that this expression can be put in t he form (u+vA} + (u-
vA), where u and v are integers. Next, imitate the argument in the text
that leads to equations (1-4), (1-5), and (1-6) to get u(u^2 - 3v^2 ) + iv(3u^2 -v^2 ) = lS + 26i. The only factors of lS are 1, 2, 3, 6, 9, and lS, so you can
deduce (explain your reasoning) t hat u = 3 and v = 1 solve this system.
Thus, one solution to x^3 - 30x - 36 = 0 is x = 6. Divide x^3 - 30x - 36
by x - 6 and solve the resulting quadratic to get the remaining solutions:
x = -3± ./3.5c. Proceed as with part a. The solutions are x = S, x = - 4 ± 2./3.Section 1.2. The Algebra of Complex Numbers: page 14
1 a. i ·27< " = ( i •2)^137 i. = ( -i)l^37 i. = - i..le. O.le. 2 + 2i.lg. 3.
•· - 27 T + 5 11. i.
Let z = x +iy be an arbitrary complex number. Then zz = (x+iy)(x- iy) =
x2 + y^2 , which is obviously a real number.
5Sl