184 Algebra of Complex Numbers
EXERCISE 4.1.3.C Using the suitable trigonometric identities, verify that if
z\ = n (cos #1 + i sin 6) and z 2 = r 2 (cos 82 4- i sin 62), then
z\Z2 = rir2(cos(0j + 02 ) + ism(9i + 0 2 )),^ = rM cos(0! - B 2 ) + i sin(0i - 92 )) • (4'L3)
z 2 r 2 \ >Some sources call the left picture in Figure 4.1.1 the Argand diagram,
in honor of the Swiss-born non-professional mathematician JEAN-ROBERT
ARGAND (1768-1822), who published the idea of the geometric interpre-
tation of complex numbers in 1806, while managing a bookstore in Paris.
Mathematical formulas are neither copyrightable nor patentable, and the
names of those formulas are often assigned in an unpredictable way. Ar-
gand's diagram is one such example: in 1685, when complex number were
much less popular, a similar idea appeared in a book by the English math-
ematician JOHN WALLIS (1616-1703).
The following result is known as Euler's formula:
cos6 + isin0 = ei9. (4.1.4)In this case, the name is true to the fact: the result was first published
by L. Euler in 1748. At this point, we will take (4.1.4) for granted and
only mention that (4.1.4) is consistent with (4.1.3); later on, we will prove
(4.1.4) using power series. A particular case of (4.1.4), em + 1 = 0, collects
the five most important numbers in mathematics, 0,1, e, n, i, in one simple
equality.
An immediate consequence of (4.1.4) is that multiplication of a complex
number z by el$ is equivalent to a rotation of z in the complex plane by 0
radians counterclockwise. Note also that e%e = el(s+2n). As a result, a time-
varying periodic quantity A is conveniently represented as A(t) = AoeluJt,
where AQ is the amplitude, and u is the angular frequency (so that 27r/w
is the period). Below, we will use such representations in the analysis of
electrical circuits and planar electromagnetic waves.
EXERCISE 4.1.4? Let z\ = x\ +iyi, 22 = ^2 +iy2 be two complex numbers,
and ri = xii + yij, r 2 = X2 i + 2/2 J, the corresponding vectors, (a) Verify
that Z1Z2 = (7*1 • rg) + i((ri x r 2 ) • k), where, as usual, k = i x j. (b)
Express the angle between the vectors r\ and r 2 in terms of Arg(z) and
Arg[z,2)- Hint: draw a picture; the angle is always between 0 and n.
One application of (4.1.4) is computing roots of complex numbers.