Definition: \(b\)-Adic Fractions

Let \(b \ge 2, k\ge 0\) be natural numbers. A \(b\)-adic fraction is a real infinite series of the form

\[\pm \sum_{k=-n}^\infty a_kb^{-k},\]

where for all \(0\le a_k < b \) for all \(k\). The number \(b\) is called the basis or the radix of the \(b\)-adic fraction.

If the basis is known, we can write the b-adic fraction also explicitly in the form

\[\begin{array}{rcl} \pm a_{-k}a_{-k+1}\cdots a_{-1}a_{0} & . & a_{1}a_{2}a_{3}\cdots \\ &\uparrow&\\ \end{array}\]

The dot in the middle denotes the position of the \(0\)-th element of the sequence.

  1. Proposition: \(b\)-Adic Fractions Are Real Cauchy Sequences
  2. Proposition: Unique Representation of Real Numbers as \(b\)-adic Fractions
  3. Definition: Decimal Representation of Real Numbers

Chapters: 1
Definitions: 2
Proofs: 3
Propositions: 4 5

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