Question

Show that we can represent pairs of nonnegative integers using only numbers and arithmetic operations if we represent the pair \(a\) and \(b\) as the integer that is the product \(2^{a} 3^{b}\). Give the corresponding definitions of the procedures cons, car, and cdr.

Short answer

Pair \((a, b)\) is represented as \(2^a \times 3^b\); use \(\text{cons}(a, b) = 2^a \cdot 3^b\), \(\text{car}(n)\) to find \(a\), and \(\text{cdr}(n)\) to find \(b\).

Step-by-step solution

Understand the Representation

In this problem, we are representing a pair of nonnegative integers \(a\) and \(b\) by the number \(n = 2^a \times 3^b\). This means that each pair \((a, b)\) corresponds to a unique integer \(n\). The goal is to define procedures (functions) for constructing this number \(n\) from \(a\) and \(b\), and for recovering \(a\) and \(b\) from \(n\).

Define the 'cons' Procedure

The 'cons' procedure is used to create the integer representation of the pair. It takes two inputs \(a\) and \(b\) and outputs \(2^a \times 3^b\). The definition of cons is:\[ \text{cons}(a, b) = 2^a \cdot 3^b \]

Define the 'car' Procedure

The 'car' procedure retrieves the first element \(a\) from the integer representation \(n = 2^a \times 3^b\). To do this, we divide \(n\) by the largest power of 3 possible, leaving us with \(2^a\). We can then determine \(a\) by counting the factors of 2 in \(2^a\).The definition of car is:\[ \text{car}(n) = \text{the number of times 2 divides } n \]

Define the 'cdr' Procedure

The 'cdr' procedure retrieves the second element \(b\) from the integer representation \(n = 2^a \times 3^b\). We repeatedly divide \(n\) by 2 until it is no longer divisible by 2, leaving us with \(3^b\). We then determine \(b\) by counting the factors of 3 in \(3^b\).The definition of cdr is:\[ \text{cdr}(n) = \text{the number of times 3 divides } n \]

Key concepts

Arithmetic Operations

Arithmetic operations are basic mathematical calculations used in everyday life to manipulate numbers. They are essential in computer science and mathematics to solve complex problems and perform calculations. The primary arithmetic operations are addition, subtraction, multiplication, and division.

In our exercise, we focus on multiplication and division. We use them to construct and deconstruct a unique number representation of a pair of integers. Specifically, we start with the representation of a pair \((a, b)\) as an integer \(n = 2^a \times 3^b\). The multiplication operation \(2^a \times 3^b\) combines our integers into one, while division helps us separate them later. For instance, dividing by the largest power of 2 or 3 allows us to extract the original integer values \(a\) and \(b\).

The arithmetic operations used in this context help in transforming and retrieving integer values efficiently. They demonstrate how mathematical tools can be used to encode and decode information in a systematic way.

Integer Representation

Integer representation is a method of displaying numbers in a specific form that computers and humans can understand. It's crucial in computing as it allows us to manipulate and store numerical data efficiently.

In the given exercise, integer representation takes place by expressing a pair \((a, b)\) as a single integer \(n\), calculated through \(2^a \times 3^b\). This is a unique way of encoding two non-negative integers into one. The use of powers of 2 and 3 ensures no overlap exists between different pairs. This uniqueness occurs because 2 and 3 are prime numbers. Therefore, for any given \(n\), there is only one pair of \((a, b)\) such that \(2^a \cdot 3^b = n\).

Understanding integer representation is key to grasping how data can be compactly represented and efficiently processed within algorithms and digital systems. It allows for space-saving techniques, which is essential in computing where resources are often limited.

Pair Representation

Pair representation involves the use of mathematical techniques to encode two numbers as a single entity. This is particularly useful in programming and algorithm design, where complex data structures need simple underlying representations.

In this exercise, pairs \((a, b)\) are represented using the mathematical expression \(2^a \times 3^b\). This encoding scheme makes use of the unique factorization property of numbers, allowing each pair to correspond to a unique integer value \(n\). The procedures used for encoding and decoding are called 'cons', 'car', and 'cdr':

• cons(a, b): Multiplies powers of 2 and 3 to create a numeric representation.
• car(n): Determines the value of \(a\) by counting how many times 2 divides \(n\). This recovers the first element of the pair.
• cdr(n): Determines the value of \(b\) by counting how many times 3 divides \(n\). This recovers the second element of the pair.

Pair representation through these methods offers a compact and efficient means of handling data, making it a powerful concept in computer science.