Question

(HugeInteger Class) Create a class Hugetnteger that uses a 40 -element array of digits to store integers as large as 40 digits each. Provide member functions input, output, add and substract. For comparing HugeInteger objects, provide functions isEqualTo, isNotEqualTo, isGreaterThan, isLessThan, isGreaterThan0rEqualTo and isLessThan0rEqualtoeach of these is a "predicate" function that simply returns TRue if the relationship holds between the two HugeIntegers and returns false if the relationship does not hold. Also, provide a predicate function is zero. If you feel ambitious, provide member functions multiply, divide and modulus.

Short answer

Define a `HugeInteger` class with methods for input, output, arithmetic operations, and comparisons.

Step-by-step solution

Define the Class Structure

Create a class named `HugeInteger`. This class will contain a private attribute to store the 40-element array, which represents a huge integer, and the public member functions for various operations.

Implement Constructor and Input Method

The constructor initializes the array with zeros. The `input` method accepts a string of up to 40 digits, validates it, and stores each digit as an element in the array.

Implement the Output Method

Create an `output` method that prints the huge integer. It iterates over the array of digits and constructs the number for display, omitting leading zeros.

Implement the Addition Method

Implement an `add` function that takes two `HugeInteger` objects. It performs digit-by-digit addition of the two arrays from the least significant digit, handling carry-over appropriately.

Implement the Subtraction Method

Develop a `subtract` function that performs subtraction between two `HugeInteger` objects, digit-by-digit. Ensure that the function handles borrowing and correctly results in a non-negative integer.

Implement Comparison Predicates

Implement the `isEqualTo`, `isNotEqualTo`, `isGreaterThan`, `isLessThan`, `isGreaterThanOrEqualTo`, and `isLessThanOrEqualTo` methods by directly comparing the digits of two `HugeInteger` objects.

Implement isZero Predicate

Create an `isZero` method that checks if all the elements in the array are zeros, which determines if the number is zero.

(Optional) Implement Multiplication Method

If desired, implement a `multiply` method that performs multiplication using an iterative approach, similar to how you would manually multiply large numbers.

(Optional) Implement Division and Modulus Methods

Consider implementing `divide` and `modulus` methods using repeated subtraction or more efficient algorithms to handle division and modulus operations for large integers.

Key concepts

Digit-by-Digit Operations

The HugeInteger class employs digit-by-digit operations to perform arithmetic such as addition and subtraction. To understand these operations, think about how you would add or subtract two numbers by hand. For example, when adding, you start with the least significant digit (rightmost) and consider carry-overs if the sum of two digits exceeds 9. In subtraction, you may need to "borrow" from more significant digits if the top digit is smaller than the bottom digit. In our HugeInteger implementation, a 40-element array stores each digit from right to left, reminiscent of traditional arithmetic processes.

• **Addition**: Iterate from the least significant digit to the most, adding corresponding digits and carrying over where necessary.
• **Subtraction**: Borrow from higher digits if the top number is smaller, moving left to right in the array.

Each digit is handled one at a time and adjusted for carries or borrows, ensuring accuracy up to 40 digits.

Comparison Predicates

Comparison predicates in the HugeInteger class are crucial for understanding relationships between large numbers. These predicates—such as `isEqualTo`, `isNotEqualTo`, `isGreaterThan`, and others—operate by comparing each digit between two HugeInteger objects, starting with the most significant digit. Think of them as a way of checking if one number is bigger, smaller, or equal to another, just like comparing the height of two people without a scale.

• `isEqualTo`: Returns true if all digits match between two objects.
• `isGreaterThan`: Compares each digit, starting from the left, to see if one number is bigger.
• `isLessThan`: Checks if one number has smaller digits at the earliest unmatched position.
• `isZero`: Simply checks if all digits in the array are zero, indicating the number zero.

These methods are simple but powerful tools for any arithmetic operation, providing the foundation needed for logic branching like `if` statements in programming.

Input and Output Methods

The input and output methods in the HugeInteger class are designed for manipulating and presenting large numbers conveniently. The `input` method is responsible for accepting a string of up to 40 digits, validating it, and storing each character in the array. This validation ensures that only valid numeric strings are processed, safeguarding against errors. The `output` method, on the other hand, reconstructs the number from the array of digits. This involves iterating through the array and displaying each digit sequentially, all while omitting leading zeros to avoid a false impression of magnitude. To illustrate:

• `input("12345")`: Converts each character to an integer and stores it in the array.
• `output()`: Reads the array and prints "12345", ignoring preceding zeros.

These methods make handling extremely large integers intuitive, allowing seamless interaction within programs.

Arithmetic Operations in Data Structures

Arithmetic operations like addition, subtraction, optionally multiplication, division, and modulus are integral parts of handling large integers within data structures like the HugeInteger class. By implementing these operations, you enable more complex calculations directly with large numbers.

• **Addition and Subtraction**: Covered in digit-by-digit operations, they lay the groundwork for more complex arithmetic.
• **Multiplication**: (optional) Involves iterative addition, utilizing concepts from manual multiplication of long numbers.
• **Division and Modulus**: (optional) Performed using techniques like repeated subtraction or more advanced algorithms to handle large integers efficiently.

These operations extend the capabilities of handling integers, making data structures like HugeInteger a composite tool for mathematical operations, transforming complex tasks into manageable processes.