48 Chapter^1
w yx
Fig. 1.13
it is clear that the nth roots of w can be obtained by multiplying the
principal. root of w by each of the nth roots of unity.
Geometrical interpretation. Since all the nth roots of w = reio have
the same modulus f.(i, the corresponding affixes in the complex plane all
lie on a circle with radius f.(i and center at the origin. One of the roots
has argument () /n and the others are equally spaced around the circumfer-
ence, their arguments being obtained by adding multiples of 27r/n to O/n.
Figure 1.13 shows the geometric representation of the sixth roots of w.
To the nth roots of unity there correspond n equidistant points along
the unit circle, the first of which is the point (1, 0) on the real axis.
It is cl.ear that the problem of finding the nth roots of a complex number
corresponds geometrically to the problem of constructing a regular polygon
of n sides inscribed in a given circle. C. F. Gauss showed in his Disquisi-
tiones A·rithmeticae (Leipzig, 1801) that this cannot be accomplished with
ruler and compass unless n is of the form
n = 2m(22m1 + l)(22m2 + 1) ... (22mk + 1)
where m.i are distinct positive integers and each 22 m; + 1 is a prime number.
EXERCJ(SES 1.5- Compute the following powers.
(a) (4 + 3i)^2
(c) (5 + 2i)^2
(b) (-3+i)^2
(d) (2 + i)^3