Question

Create a class RationalNumber (fractions) with the following capabilities: a. Create a constructor that prevents a 0 denominator in a fraction, reduces or simplifies fractions that are not in reduced form and avoids negative denominators. b. Overload the addition, subtraction, multiplication and division operators for this class. c. Overload the relational and equality operators for this class.

Short answer

Define a RationalNumber class with overloaded arithmetic and comparison operators, ensuring zero denominator prevention, simplification, and positive denominators.

Step-by-step solution

Define the RationalNumber Class

Start by defining a class called `RationalNumber`. This class should contain two private integer members: `numerator` and `denominator` to represent the fraction.

Implement the Constructor

In the constructor of `RationalNumber`, add logic to prevent the denominator from being zero. If the denominator is negative, multiply both the numerator and the denominator by -1 to make the denominator positive. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Create the GCD Function

Define a private helper function within the class to calculate the greatest common divisor of two numbers, using the Euclidean algorithm. Use this function in the constructor to reduce fractions.

Overload the Addition Operator

To overload the `+` operator, create a member function `operator+` that takes another `RationalNumber` object as a parameter. Add the fractions by: \[ \text{new numerator} = a \times d + b \times c \]\[ \text{new denominator} = b \times d \]Where \((a/b)\) is the current object and \((c/d)\) is the parameter object. Simplify the result before returning it.

Overload the Subtraction Operator

Similarly, create an `operator-` function. Subtract two fractions by:\[ \text{new numerator} = a \times d - b \times c \]\[ \text{new denominator} = b \times d \]Simplify and return the resulting `RationalNumber`.

Overload the Multiplication Operator

Implement `operator*` which multiplies two fractions:\[ \text{new numerator} = a \times c \]\[ \text{new denominator} = b \times d \]Return the simplified result as a new `RationalNumber` object.

Overload the Division Operator

For division, write `operator/`, where you:\[ \text{new numerator} = a \times d \]\[ \text{new denominator} = b \times c \]Ensure that division by zero is handled. Simplify the fraction before returning the result.

Overload the Equality Operators

To compare two `RationalNumber` objects, implement `operator==` and `operator!=`. Compare the numerators and denominators after both fractions are simplified.

Overload the Relational Operators

For relational operators like `<`, `>`, `<=`, and `>=`, convert fractions to a common denominator before comparing them. Implement these overloads by cross-multiplying to avoid floating-point inaccuracies.

Key concepts

Rational Numbers

Rational numbers are fractions composed of two integers: a numerator and a non-zero denominator. They can represent both whole and fractional values. For instance, the fraction \( \frac{1}{2} \) is a rational number. When dealing with rational numbers in programming, it's important to ensure fractions are simplified. This means dividing both the numerator and denominator by their greatest common divisor (GCD). Simplified fractions are easier to work with and compare.
A rational number is typically represented as \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. The challenge with rational numbers is handling division by zero, which is undefined. Also, fractions should have positive denominators. A negative denominator can be converted into a positive one by multiplying both numerator and denominator by \(-1\).
In object-oriented programming, rational numbers are often managed by a class, like `RationalNumber`, which ensures that every fraction is correctly represented and simplified. This class also provides a foundation for operations such as addition, subtraction, multiplication and division, all of which need rational number arithmetic rules.

Operator Overloading

Operator overloading in object-oriented programming allows developers to redefine how operators behave for user-defined types, such as classes. This concept is especially useful for arithmetic operations on classes like `RationalNumber`. Overloading gives meaning to operators like `+`, `-`, `*`, and `/` when they are used with rational number objects, enabling typical arithmetic functionality.
For instance, to overload the addition operator for rational numbers, one must ensure that the resulting rational number is properly calculated and simplified. Consider two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \). Their sum is computed using:

• New numerator: \( a \times d + b \times c \)
• New denominator: \( b \times d \)

Simplifying the resulting fraction is crucial. Similar steps are followed for the subtraction, multiplication, and division by appropriately modifying the formulae and ensuring all results are in simplified form.
Apart from arithmetic operators, relational and equality operators can also be overloaded. This means you can directly compare rational numbers using `==`, `!=`, `>`, `<`, `>=`, and `<=` after converting them to a common denominator, effectively streamlining comparisons without extra coding burden.

Class Constructor

A class constructor in object-oriented programming is a special method that initializes new objects. For a `RationalNumber` class, the constructor ensures that each rational number object is valid immediately upon creation.
Important tasks of the constructor include:

• Preventing zero denominators: Check if a given denominator is zero and handle it accordingly, often by setting a default value or raising an error.
• Converting negative denominators: If the denominator is negative, multiply both the numerator and the denominator by \(-1\) to normalize the fraction.
• Simplifying fractions: Use a private helper function to calculate the greatest common divisor (GCD) of the numerator and denominator. Simplify the fraction by dividing each by the GCD.

Performing these tasks ensures that each rational number starts its life cycle in a standardized, simplified, and valid form. This reduces errors and makes operator overloading implementations straightforward. Constructors are the backbone of ensuring that objects maintain consistent state within a program. By addressing potential initialization issues up front, constructors greatly enhance program reliability.