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6.4.1 New Insights into Multiplication and Division of Complex Numbers
Consider the two complex numbers z and z written in polar form:
2
1
z = z (cos( )θ + j sin( ))θ (6.17)
1 1 1 1

z = z (cos(θ ) + j sin(θ )) (6.18)
2 2 2 2

Their product z z is given by:
1 2
(cos( )cos(θ θ ) − sin( )sin(θ θ )) 
zz = z z  1 2 1 2  (6.19)
12 1 2 
))
 +  j(sin( )cos(θ 1 θ 2 ) + cos( )sin(θ 1 θ 2 
But using the trigonometric identities for the sine and cosine of the sum of
two angles:

cos(θ + θ ) = cos( )cos(θ θ ) sin( )sin(θ− θ ) (6.20)
1 2 1 2 1 2

sin(θ + θ ) = sin( )cos(θ θ ) cos( )sin(θ+ θ ) (6.21)
1 2 1 2 1 2
the product of two complex numbers can then be written in the simpler form:

zz = z z [cos(θ + θ ) + j sin(θ + θ )] (6.22)
1 2 1 2 1 2 1 2

That is, when multiplying two complex numbers, the modulus of the product
is the product of the moduli, while the argument is the sum of arguments:

zz = z z (6.23)
12 1 2

arg(zz ) = arg( ) arg( )z + z (6.24)
12 1 2
The above result can be generalized to the product of n complex numbers
and the result is:

zz … z = z z … z (6.25)
12 n 1 2 n

arg(zz … z ) = arg( ) arg( )z + z +…+ ( )z (6.26)
12 n 1 2 n
A particular form of this expression is the De Moivre theorem, which states
that:

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