A data type for rational numbers can be defined with the...

A data type for rational numbers can be defined with the constructor makeRational $[n, d]$ and the selectors numerator $[r]$ and denominator $[r]$. The representation of a data element is simply rational $[n, d]$. It should represent the rational number $n / d$. In this exercise, you should not use the built-in rational numbers; use only integers. 1. Give the definitions of makeRational, numerator, and denominator. Complete the right side: $$ \begin{aligned} & \text { makeRational }\left[n_{-}, d_{-}\right]:=\ldots \\ & \text { numerator }\left[\text { rational }\left[n_{-}, d_{-}\right]\right]:=\ldots \\ & \text { denominator }\left[\text { rational }\left[n_{-}, d_{-}\right]\right]:=\ldots \\ & \end{aligned} $$ 2. The representation rational $[n, d]$ for rational numbers is not yet unique. For example, the expressions rational $[-1,2]$, rational $[1,-2]$, and rational $[-2,4]$ describe the same number. The representation becomes unique if we demand that the denominator is always positive and the numerator and denominator are relatively prime; that is, they have no factors in common. Find two rules for rational $\left[n_{-}, d_{-}\right]$that ensure that data elements are transformed into a unique form. 3. Give definitions for addition and multiplication of rational numbers that work exclusively with constructors and selectors. Complete the right side: rational/: a_rational + b_rational $:=\ldots$ rational/: a_rational $*$ b_rational $:=\ldots$

A data type for rational numbers can be defined with the constructor makeRational $[n, d]$ and the selectors numerator $[r]$ and denominator $[r]$. The representation of a data element is simply rational $[n, d]$. It should represent the rational number $n / d$. In this exercise, you should not use the built-in rational numbers; use only integers.
1. Give the definitions of makeRational, numerator, and denominator. Complete the right side:
$$
\begin{aligned}
& \text { makeRational }\left[n_{-}, d_{-}\right]:=\ldots \\
& \text { numerator }\left[\text { rational }\left[n_{-}, d_{-}\right]\right]:=\ldots \\
& \text { denominator }\left[\text { rational }\left[n_{-}, d_{-}\right]\right]:=\ldots \\
&
\end{aligned}
$$
2. The representation rational $[n, d]$ for rational numbers is not yet unique. For example, the expressions rational $[-1,2]$, rational $[1,-2]$, and rational $[-2,4]$ describe the same number. The representation becomes unique if we demand that the denominator is always positive and the numerator and denominator are relatively prime; that is, they have no factors in common. Find two rules for rational $\left[n_{-}, d_{-}\right]$that ensure that data elements are transformed into a unique form.
3. Give definitions for addition and multiplication of rational numbers that work exclusively with constructors and selectors. Complete the right side:
rational/: a_rational + b_rational $:=\ldots$
rational/: a_rational $*$ b_rational $:=\ldots$

Key Concepts

Normalization and Canonical Representation

Normalization is the process of converting data into a standardized or canonical form. For rational numbers, this means ensuring that the denominator is always positive and that the numerator and denominator are reduced to their smallest possible values (i.e., they are relatively prime). This canonical form removes ambiguity, ensuring that different representations of the same mathematical rational number are unified.

Arithmetic Operations on Abstract Data

Arithmetic operations, such as addition and multiplication, can be defined purely in terms of the abstract data type’s interface. By operating exclusively with constructors and selectors, these operations manipulate the rational number components without revealing or depending on the data type's underlying representation, thereby promoting data encapsulation and modular design.

Abstract Data Types

Abstract data types provide a way to encapsulate data and operations on that data, hiding the implementation details from the user. In the context of rational numbers, the data type is defined solely by its constructor (for creating instances) and selectors (for accessing parts of the data), enabling a clean separation between the interface and the implementation.

Constructors and Selectors

Constructors are functions that create instances of abstract data types, while selectors are functions that retrieve components of those instances. In this rational number example, the constructor (makeRational) builds a rational number from a numerator and denominator, and the selectors (numerator and denominator) extract the corresponding values. This abstraction provides a uniform approach to handling complex data structures.

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