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.

| | | | | created: 2015-02-18 13:58:52 | modified: 2017-04-14 11:53:02 | by: bookofproofs | references: [581]

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