Chapter 13, Problem 13.17PE

(Math: The Complex class) A complex number is a number in the form a + bi, where a and b are real numbers and i is $\sqrt{-1}.$ The numbers a and b are known as the real part and imaginary part of the complex number, respectively. You can perform addition, subtraction, multiplication, and division for complex numbers using the following formulas:

a + bi + c + di = (a + c) + (b + d)i

a + bi – (c + di) = (a – c) + (b – d)i

(a + bi) * (c + di) = (ac — bd) + (bc + ad)i

(a + bi) / (c + di) = (ac + bd) / (c² + d²) + (bc – ad)i / (c² + d²)

You can also obtain the absolute value for a complex number using the following formula:

     $\left|a+bi\right|=\sqrt{a^2+b^2}$

(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 13.10.)

  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. Your Complex class should also implement Cloneable and Comparable. Compare two complex numbers using their absolute values.

  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 part and the imaginary part of the complex number, respectively.

  Draw the UML class diagram and implement the class. Write a test program that prompts the user to enter two complex numbers and displays the result of their addition, subtraction, multiplication, division, and absolute value. Here is a sample run:

Enter the first complex number: 3.5 5.5 Enter the second complex number: –3.5 1 (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