Question

Define a better version of make-rat that handles both positive and negative arguments. Make-rat should normalize the sign so that if the rational number is positive, both the numerator and denominator are positive, and if the rational number is negative, only the numerator is negative.

Short answer

Normalize the signs by ensuring the denominator is always positive.

Step-by-step solution

Identify the Problem

We need a function, make-rat, which ensures that rational numbers given as inputs are normalized. This means that if the number is positive, both numerator and denominator should be positive. If the number is negative, then only the numerator should be negative.

Define a Strategy

Our strategy is to ensure that the denominator is always positive. We will adjust the numerator's sign based on the sign of the resultant rational number. We can do this by multiplying both by -1 if the denominator is negative, thus normalizing the signs.

Implement the Solution

Write a function make-rat with inputs numerator (n) and denominator (d). Check the sign of the denominator. If the denominator is negative, multiply both numerator and denominator by -1. This will achieve the normalization by making the denominator positive and adjusting the numerator's sign accordingly.

Example Cases

Consider make-rat(5, -3): Since the denominator is negative, the function should return -5/3. For make-rat(-4, -2), it should return 2/1 after multiplying both by -1.

Key concepts

Normalization

Normalization in the context of rational numbers means adjusting the components so they follow specific rules, enhancing clarity and consistency. Here, we want our rational numbers to be presented in a standard form.
This standardization involves setting rules for how the signs of the numerator and denominator are represented:

• If a rational number is positive, both the numerator and the denominator should be positive.
• If a rational number is negative, only the numerator should carry the negative sign.

To achieve normalization, especially when input numbers do not already comply with these rules, we might need to perform operations such as multiplying by -1. This adjusts the signs appropriately, streamlining mathematical operations and comparisons between rational numbers. Consistent formats reduce error risk when rational numbers are used in calculations or communicated in other mathematical contexts.

Sign Adjustment

Sign adjustment is crucial in ensuring our rational numbers are displayed correctly. The task involves ensuring that regardless of how a rational number is input, it adheres to the normalization rules.
Let's break this down:

• Checking the Denominator: The denominator should always be positive. Thus, the first move is to examine its sign.
• Outcome-Based Adjustment: If the denominator is negative, multiply both the numerator and denominator by -1. This action flips the sign of the denominator while ensuring the overall value of the fraction remains unchanged.

For instance, consider the fraction \( \frac{5}{-3} \). After sign adjustment, it becomes \( \frac{-5}{3} \) fulfilling the condition of a positive denominator. Thus, sign adjustment plays an essential role in the seamless usage of rational numbers, making them uniformly approachable.

Numerator and Denominator

The numerator and denominator are the core components of any rational number. Understanding their roles helps in better manipulation and normalization of the rational number's sign.

• Numerator: Represents how many parts of the whole are considered. It can be positive or negative based on whether the rational number itself is positive or negative in value.
• Denominator: Denotes the total number of equal parts. In normalized forms, it's always positive to simplify both the definition and calculation of the rational number.

By ensuring the denominator remains positive, we naturally align the numerator's sign to reflect the number's true nature (positive or negative value). Thus, the interplay between the numerator and denominator is essential in creating rational numbers that are easy to read and compute accurately.