Question

For an integer \(r \in \mathbb{N}_{>1}\), we consider the variable-length radix \(r\) representation \(\left(a_{0}, \ldots, a_{l-1}\right) 0\) a positive integer \(a\), with \(a=\sum_{0 \leq i

Short answer

The length of the radix representation is \(\lfloor \log_r a \rfloor + 1\).

Step-by-step solution

Understanding the Problem

We need to show that the length of the radix-\(r\) representation of an integer \(a\), noted as \(l\), is equal to $$\(\lfloor\log _{r} a\rfloor+1\). The radix representation consists of coefficients \(a_0, a_1, \ldots, a_{l-1}\), each between \(0\) and \(r-1\), for base \(r\), representing the integer \(a\) as a sum \(a = a_0 r^0 + a_1 r^1 + \cdots + a_{l-1} r^{l-1}\).

Express the Problem with Logarithm

Express the integer \(a\) in terms of its polynomial representation: \(a = a_0 r^0 + a_1 r^1 + \cdots + a_{l-1} r^{l-1}\). Since \(a_{l-1} eq 0\), \(a \geq r^{l-1}\). To maintain integrality, we have \(a < r^l\) if \(l\) is the least possible number of digits for the radix \(r\) representation.

Set Bounds Using Inequalities

From Step 2, we have \(r^{l-1} \leq a < r^l\). Taking the logarithm base \(r\) on all sides, $$we obtain the inequalities \(l-1 \leq \log_r a < l\). Since \(l\) is an integer, \(l = \lfloor \log_r a \rfloor + 1\) gives the shortest valid length \(l\) for the radix representation of \(a\).

Proving the Equation

The inequality \(l-1 \leq \log_r a < l\) implies the length \(l\) is the smallest integer greater than \(\log_r a\). Consequently, we can write \(l = \lfloor \log_r a \rfloor + 1\), which means the length of the variable-length radix-\(r\) representation is indeed this value.

Key concepts

Integer Representation

Integer Representation is a method of describing numbers using a specified numerical system or base, known as a radix. In a radix-r representation, a number a is expressed as a sum of terms. Each term is a digit multiplied by a power of the base r. This can be written as:

• \( a = a_0 r^0 + a_1 r^1 + \cdots + a_{l-1} r^{l-1} \)

Each a_i is a digit that can take any integer value from 0 to r-1. This representation is flexible and varies in length based on the size of a. The last non-zero coefficient a_{l-1} ensures the representation is minimal in size.
For example, in the decimal system (r = 10), the number 345 is represented as 3 hundreds, 4 tens, and 5 units. Thus, understanding integer representation across different bases is essential for grasping Radix Representation.

Logarithmic Functions

Logarithmic functions play a pivotal role in determining the length of a number's representation in different bases. The logarithm base r of a number a, expressed as log_r a, gives power to which base r must be raised to produce a.
More formally:

• The equation \(r^l = a\) implies \(l = log_r a\)

However, since the length l must be an integer, logarithms are used to establish bounds:

• The inequality \(l - 1 \leq log_r a < l\) is arranged.

The floor function rounds down log_r a to the greatest integer less than or equal to log_r a. This is added by 1, hence providing the shortest integer length l = \lfloorlog_r a\rfloor + 1. Logarithms thus help us bridge the understanding between numerical scale and digit count in different bases.

Base Conversion

Base conversion is the process of changing a number from one base to another. Understanding Radix Representation is particularly useful in base conversion.
The number a in base r is represented as:

• \( a = a_0 r^0 + a_1 r^1 + \cdots + a_{l-1} r^{l-1} \)

To convert this number into another base s, repeat the conversion process by recalculating powers and coefficients.
For example, to convert number 345 from base 10 to base 2, divide the number repeatedly by 2, recording remainders - these form the binary equivalent when read backward. Base conversion empowers us to work fluently across different systems.

Mathematical Proof

Mathematical proof involves constructing a logical argument that shows the truth of a given statement using established mathematical principles. Here, we have proved the representation length using inequalities and logarithmic identities.
The proof captures several foundational components:

• Identifying the upper and lower bounds with potentials like \(r^{l-1} \leq a < r^l\)
• Taking the logarithm base r to obtain \(l-1 \leq log_r a < l\)
• Simplifying result to find \(l = \lfloor log_r a \rfloor + 1\), the desired representation length.

This concise step-by-step logical explanation exemplifies how well-defined analytical processes underpin mathematical proofs, turning abstract problems into resolved certainties.