Java Reference
In-Depth Information
You can also obtain the absolute value for a complex number using the fol-
lowing formula:
a
2
b
2
a
+
bi
= 2
+
(A complex number can be interpreted as a point on a plane by identifying the
(
a,b
) values as the coordinates of the point. The absolute value of the complex
number corresponds to the distance of the point to the origin, as shown in Figure
15.12b.)
Design a class named
Complex
for representing complex numbers and the
methods
add
,
subtract
,
multiply
,
divide
, and
abs
for performing complex-
number operations, and override
toString
method for returning a string
representation for a complex number. The
toString
method returns
(a + bi)
as a string. If
b
is
0
, it simply returns
a
.
Provide three constructors
Complex(a, b)
,
Complex(a)
, and
Complex()
.
Complex()
creates a
Complex
object for number
0
and
Complex(a)
creates a
Complex
object with
0
for
b
. Also provide the
getRealPart()
and
getImaginaryPart()
methods for returning the real
and imaginary part of the complex number, respectively.
Write a test program that prompts the user to enter two complex numbers
and displays the result of their addition, subtraction, multiplication, and divi-
sion. Here is a sample run:
Enter the first complex number:
Enter the second complex number:
(3.5 + 5.5i) + (-3.5 + 1.0i) = 0.0 + 6.5i
(3.5 + 5.5i) - (-3.5 + 1.0i) = 7.0 + 4.5i
(3.5 + 5.5i) * (-3.5 + 1.0i) = -17.75 + -15.75i
(3.5 + 5.5i) / (-3.5 + 1.0i) = -0.5094 + -1.7i
|(3.5 + 5.5i)| = 6.519202405202649
3.5 5.5
-3.5 1
**15.20
(
Mandelbrot fractal
) Mandelbrot fractal is a well-known image created from a
Mandelbrot set (see Figure 15.13a). A Mandelbrot set is defined using the fol-
lowing iteration:
z
n
+
z
n
+
1
=
c
c is a complex number and the starting point of iteration is For a given
c
, the iteration will produce a sequence of complex numbers:
It can be shown that the sequence either tends to infin-
ity or stays bounded, depending on the value of
c
. For example, if
c
is
0
, the
sequence is which is bounded. If
c
is
i
, the sequence is
which is bounded. If
c
is the
sequence is which is unbounded. It is known that if
the absolute value of a complex value in the sequence is greater than 2, then
the sequence is unbounded. The Mandelbrot set consists of the
c
value such
that the sequence is bounded. For example, 0 and
i
are in the Mandelbrot set. A
Mandelbrot image can be created using the following code:
z
0
=
0.
{
z
0
,
z
1
,
,
z
n
,
}.
c
c
{0, 0,
},
c
{0,
i
,
-
1
+
i
,
-
i
,
-
1
+
i
,
i
,
},
1
+
i
,
c
{0, 1
+
i
, 1
+
3
i
,
},
c
z
i
1
class
MandelbrotCanvas
extends
JPanel {
2
final static int
COUNT_LIMIT =
60
;
