Defining integer and fractional parts consistently with positional notation

Release 1.4: This paper introduces rigorous and consistent definitions for the integer and fractional parts of a real number 𝑥 within the framework of positional notation. Building on the positional series representation, I analyze the three main methods: (1) Method 1 (Graham, Knuth, & Patashnik, 1992) [4] satisfies the Regularity Theorem, ensuring that the integer part increases by 1 for each increment of 𝑥 by 1. However, achieving this regularity for negative numbers requires offsets that diverge from positional notation principles, leading to inconsistencies in reconstructing the standard positional representation. (2) Method 2a (Daintith, 2004) [8] is the only method that fully aligns with both the standard positional number representation and the positional series decomposition, ensuring symmetry for positive and negative values. This approach treats the sign as a global attribute, enabling compliance with IEEE 754 standards and offering a mathematically rigorous, interoperable solution for both theoretical and computational applications. (3) Method 2b (Weisstein, MathWorld) [3] embeds the sign directly into both the integer and fractional parts, simplifying computational implementation. However, this introduces structural inconsistencies for negative numbers, as re-concatenating the parts results in an additional negative sign, diverging from the standard positional representation. Of these, Method 2a stands out as the most robust approach, combining mathematical rigor, theoretical precision, and practical compatibility. By adopting Method 2a, positional representations of real numbers can achieve consistency, precision, and cross-platform interoperability.

Pi can be define as an Unbounded Rational Number

This article extends the definition of unbounded rational numbers established in my previous work, Every Real Number Can Be Expressed as an Unbounded Rational Number, to accommodate any chosen method for defining the integer part. While 𝜋 is typically defined as the ratio of a circle’s circumference to its diameter [1], it can also be expressed as an unbounded rational, inviting reconsideration of its traditional classification as irrational.

This work is licensed under CC BY-NC-SA 4.0