Question

\((\) Complex Numbers) Create a class called Complex for performing arithmetic with complex numbers. Complex numbers have the form \\[ \text {reallart }+\text { imaginaryPart }^{*} \\] where \(i\) is \\[ \sqrt{-1} \\] Write a program to test your class. Use floating-point variables to represent the private data of the class. Provide a constructor that enables an object of this class to be initialized when it is declared. Provide a no-argument constructor with default values in case no initializers are provided. Provide public methods that perform the following operations: a) Add two Complex numbers: The real parts are added together and the imaginary parts are added together. b) Subtract two Complex numbers: The real part of the right operand is subtracted from the real part of the left operand, and the imaginary part of the right operand is subtracted from the imaginary part of the left operand. c) Print Complex numbers in the form (a, b), where a is the real part and b is the imaginary part.

Short answer

Create a `Complex` class with methods for addition, subtraction, and printing.

Step-by-step solution

Define the Complex Class

Create a class named `Complex`. This class should have two private floating-point attributes: `realPart` and `imaginaryPart` to represent the real and imaginary components of the complex number.

Implement Constructor Methods

Define two constructors: one with parameters to set the initial values of `realPart` and `imaginaryPart`, and a no-argument constructor that initializes these attributes to default values, such as 0.0.

Create Add Method

Implement a public method called `add` which takes another `Complex` object as a parameter. It should add the real parts together and the imaginary parts together, returning a new `Complex` object with the resulting values. For example, if added to \(c1 (a, b)\) and \(c2 (c, d)\), the result would be \( (a+c, b+d) \).

Create Subtract Method

Implement a public method named `subtract` which accepts another `Complex` object as an argument. Subtract the real part of the argument from the current object’s real part, as well as the imaginary part from the current object's imaginary part. Return a new `Complex` object with these resulting values. If \(c1 (a, b)\) and \(c2 (c, d)\), then the result would be \( (a-c, b-d) \).

Implement Print Method

Create a `toString` or `print` method that displays the complex number in the form `(a, b)`, where `a` is the real part and `b` is the imaginary part. This method should format these values appropriately and return a string representation.

Test the Complex Class

Write a main function or testing script that creates instances of the `Complex` class using both constructors. Test the `add` and `subtract` methods by creating combinations of complex numbers and print their results using the `print` method to verify correctness.

Key concepts

Java Arithmetic Operations

In Java, complex numbers rely on basic arithmetic operations to perform tasks, such as addition and subtraction. These operations are similar to regular arithmetic but are applied separately for both real and imaginary components. When you perform arithmetic operations on complex numbers, you typically work with:

• Real Parts: These are the non-imaginary components of the complex number. For instance, in (3 + 4i), 3 is the real part.
• Imaginary Parts: These parts are associated with the imaginary unit i, like 4i in the example.

To add two complex numbers, you simply add their real parts and their imaginary parts together. For subtraction, subtract each component separately, namely, the real parts and the imaginary parts. These operations are straightforward because they transform into regular addition and subtraction operations within each component.
For example, given two complex numbers, say (a + bi) and (c + di), when added, the result becomes (a + c) + (b + d)i. This aspect of arithmetic provides the underpinning for operations in complex classes in Java.

Object-Oriented Programming Concepts

Object-oriented programming (OOP) is crucial when working with complex numbers in Java since it allows for encapsulation and modular code design. By using a class such as `Complex`, you can encapsulate the behavior and properties of a complex number. This encapsulation makes it easier to manage and more resilient to changes. In OOP:

• Encapsulation: Both data (real and imaginary parts) and related functions (add, subtract) are bundled within a single unit—the class.
• Abstraction: Users interact with complex numbers at a high level through public methods, without needing to understand the implementation details.
• Inheritance (not used here but important): It allows you to create new classes based on existing ones, extending functionality.
• Polymorphism (not used here either): Allows for methods to do different things based on the object they're acting upon.

OOP makes manipulating complex numbers straightforward and intuitive. Each complex number is an object that can store its own state (i.e., the values of the realPart and imaginaryPart) and expose methods that operate on its internal state.

Java Class Implementation

Implementing a class in Java, like the `Complex` class, involves encapsulating data attributes (real and imaginary parts) and methods to manipulate these attributes. For complex numbers, we begin by defining a class that represents the key components of a complex number:
- **Attributes:** Define private data members for the `realPart` and `imaginaryPart`, both as float types for precision.
- **Constructors:** Create at least two constructors:

• Parameterized Constructor: Initializes an object with specific real and imaginary values.
• No-Argument Constructor: Initializes the object with default values, typically set to 0.0 for both components.

- **Methods:** Add methods such as `add`, `subtract`, and `toString` or `print` to perform operations and provide a string representation.
- **Access Modifiers:** Use private access modifiers for data and public modifiers for methods to enforce encapsulation.

Java encapsulation ensures that objects manage their data internally and interact through a well-defined interface. This principle keeps the code clean and modular, making reuse and maintenance more manageable.

Floating-Point Precision in Java

Floating-point precision in Java is an important consideration, especially when dealing with real and imaginary parts of complex numbers. Floating-point numbers in Java are generally of type `float` or `double`, the latter offering greater precision. In the context of the `Complex` class:

• **Precision:** `float` provides sufficient precision for most applications involving complex numbers, but if extremely high precision is required, consider using `double`.
• **Rounding Errors:** Be aware of potential rounding errors that arise from operations like addition and subtraction. This occurs due to how floating-point numbers are stored in memory.
• **Performance:** `float` is faster to process because it uses less memory compared to `double`, which might be preferable for certain performance-sensitive applications.

Using floating-point numbers allows the `Complex` class to represent non-integer components accurately. Still, always consider the potential trade-offs regarding precision and performance based on the specific needs of your application. Handling floating-point numbers requires careful consideration, especially when exact calculations are necessary or when operations are cumulative.