The natural numbers
An Introduction to Real Analysis
2025-10-06
This section begins “The natural numbers are the positive integers: \(1, 2, 3, \dots\) .” However, the symbols \(1, 2, 3, \dots\) are the natural numbers in their radix-\(10\), or base-\(10\), representation. Most of the properties of natural numbers, and indeed real numbers, as proven by Katznelson and Katznelson (2024, chaps. 1–2) and Schröder (2008, chap. 1) are independent of the choice of radix; the numbers in any given proposition or theorem are typically represented by letters and without reference to any radix-\(b\) representation.
A natural number \(a \in \symbb{N}\) and its radix-\(b\) representation are related yet distinct concepts, and this distinction ought to be clarified. However, doing so requires the integers \(\symbb{Z}\), which are not introduced by Katznelson and Katznelson (2024) until Section 1.2. This is in part because the definition of the radix-\(b\) representation of \(a\) involves \(0\), which is not available when constrained to \(\symbb{N}\). Furthermore, the notes for Section 1.2 of Katznelson and Katznelson (2024) are primarily concerned with constructing \(\symbb{Z}\) from \(\symbb{N}\) and developing the properties of the former. Therefore the exposition of radix-\(b\) representations of natural numbers is located in the Section 1.4 notes for Schröder (2008), which is also concerned with the natural numbers and integers.
In contrast to these notes for Katznelson and Katznelson (2024), which take a bottom-up approach by starting with \(\symbb{N}\) and successively constructing \(\symbb{Z}\), \(\symbb{Q}\) and \(\symbb{R}\), Schröder (2008) takes \(\symbb{R}\) as axiomatic and introduces \(\symbb{N}\), \(\symbb{Z}\) and \(\symbb{Q}\) as subsets thereof, so no construction is necessary.