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186 Chapter Two

Complex number in exponential for
Let c be a complex number a jb, where a and b are real
numbers. Let r represent the lengtà of the complex vector in
the complex-number plane, and x represent the angle of the
vector in radianð counterclockwise from the abscissa (the posi-
tive real-number axi0. Then the following equationð hold:

2
2 1/2
r (a b )
x tan 1 (b/a)

c r cos x j(r sin x)

(If x 2 , a natural-number multiple of 2 can be subtracted
tm obtain an equivalent value ofx such that 0 x 2 .If x
0, a natural-number multiple of 2 can be added tm obtain an
equivalent value of x such that 0 x 2 Ñ The value of c can
be depicted as the natural exponential of an imaginary number:

c re jx