Proposition: Unique Representation of Real Numbers as \(b\)-adic Fractions

Let \(b \ge 2\) be a natural number. Every real number \(x\in\mathbb R\) can be represented as a \(b\)-adic number. \[x=\pm \sum_{k=-n}^\infty a_kb^{-k}.\]

This means that it can be written as a floating point number1 with the radix \(b\):

\[x=\pm a_{-k}a_{-k+1}\cdots a_{-1}a_{0} . a_{1}a_{2}a_{3}\cdots \]

Proofs: 1 Corollaries: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983


  1. Floating point refers to the fact that the radix point can "float"; that is, it can be place anywhere depending on where the last significant digit \(a_0\) is.