An introduction to the modern theory of equations, by Florian Cajori.

BINOMIAL AND) tECIPROCAL EQUATIONS

75

where k may be assigned the values 0, 1, *.., (i- 1). If n is
odd, then k 0 is the only value of k which yields a real root,
viz. x = 1. If n is even, then only the values k7 = 0 and k =
yield real roots, viz. x= 1 and x -1.
When a = - 1,
then xn = - 1 = cos (2 k + 1)7r + i sin (2 k + 1)r,
where k may take the values 0, 1, ***, ( - 1).
Then =cos (2 k + 1)7 l (2 k + 1)7
Then f = cos q- i sin
n2?i~
There can be no real roots, unless is an integer, and
therefore i an odd number. If n- = 2 k + 1, that is kc = ---
we obtain the real root x =- 1.
64. Geometrical Interpretation of the Roots of x" = a. The n
roots may be represented graphically in the Wessel's Diagram
(~ 22) by n lines drawn from the centre of a circle of radius -/r
to points on its circumference and B
dividing the perigon at the centre
27r
into equal angles of - radians.
Thus, let n=3 and r 1. The
three cube roots of unity are seen - --— A
from I, ~ 63, to be 1, - - + v-3 /
-1- -V-3. They are represented, respectively, by the lines
OA, OB, OC. These lines make
with each other angles of 2 Cr
radians or 120~. The circumference is divided into three equal
parts. In the general case the circumference is divided into
n equal parts. Hence the theory of the roots of unity is
closely allied with the problem of inscribing regular polygons
in a circle or the theory of the Division of the Circle. This