Complex
Analysis. FreeTutorial
11.1
Division as multiplication and reciprocation
11.2 The modulus and argument
of the quotient
11.2.1
Example of division
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11.1 Division as multiplication and reciprocation
Let us replace division
z1/z2
with multiplication z1(1/z2),
where 1/z2
is the reciprocal of z2.
Step1: Construct conjugate
2
= (x2,
y2).
Step2: Construct reciprocal
.
Step3: Perform multiplication
z1(1/z2)
= z1/z2.
Such way the division can be compounded
from multiplication and reciprocation.
Figure 1.18 shows all steps.
Figure 1.18 Division
of the complex numbers z1/z2
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11.2 The modulus and argument of the quotient
Let us consider two complex numbers z1
and z2
in a polar form.
z1
= r1(cos+i
sin),
z2
= r2(cos+i
sin).
It was shown (1.18) that their product
z1z2
has the modulus r1r2
and the argument +.
We have replaced division z1/z2
with multiplication z1(1/z2).
Let 1/z2
= z3
.
z3
= r3(cos+i
sin).
Then
z1/z2
=
z1z3
=
r1r3[(cos(+)
+ i(sin(+)].
The modulus r3
is the modulus of
reciprocal 1/z2.
r3
= 1/z2.
The argument
is the argument of reciprocal 1/z2.
=
arg(1/z2)
=  arg(z2).
The quotient z1/z2
has the modulus z1/z2
and the argument {Arg(z1)
 Arg(z2)}.
z1/z2
= z1/z2,
(1.21)
Arg(z1/z2)
= Arg(z1)
 Arg(z2).
(1.22)
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11.2.1
Example of division
z1
= 2 + i,
z2=
2 + i.
Find modulus z1/z2
and arg(z1/z2).
Construct z1/z2
using multiplication and reciprocation.
z1
= 4,
z2
= ,
z1/z2
= z1/z2
= .
arg(z1)
= 60º,
arg(z2)
= 30º.
arg(z1/z2)
= arg(z1)
 arg(z2)
= 60º
 30º= 30º
z1/z2
= (cos(30º)+
i sin(30º)).
Let us construct z1/z2
using multiplication and reciprocation.
Step1: Construct conjugate
2.
2
= ,
arg(2)
= 30º.
Step2: Calculate z2^{}^{2}
and construct reciprocal of z2.
z2^{}^{2}
= ()^{2}
= 16/3,
Reciprocal
has polar coordinates (,
30º).
Step3: Perform multiplication
z1(1/z2)
= z1/z2.
z1(1/z2)
= (4, 60º)(,
30º)
= (,
30º).
z1/z2
= (cos(30º)+
i sin(30º)).
See Figure1.19.
Figure
1.19 Example z1/z2
by
Tetyana Butler
