(Math: The Complex class) A complex number has the form a + bi , where a and b are real nu...

(Math: The Complex class) A complex number has the form a + bi , where a and b are real numbers and i is . 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:

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

(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.)

Figure A complex number can be interpreted as a point in a plane.

Design a class named Complex for representing complex numbers and the functions add , subtract , multiply , divide , abs for performing complex-number operations, and the toString function for returning a string representation for a complex number. The toString function 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() functions for returning the real and imaginary part of the complex number, respectively.

Overload the operators + , − , * , / , += , −= , *= , /= , [] , unary + and − , prefix ++ and -- , postfix ++ and −− , << , >>.

Overload the operators + , − , * , / as nonmember functions. Overload [] so that [0] returns a and [1] returns b.

Write a test program that prompts the user to enter two complex numbers and display the result of their addition, subtraction, multiplication, and division. Here is a sample run: