Terminating Decimal Expansion
A terminating decimal expansion is one of the form \[ 0 . a _ 1 a _ 2 a _ 3 \ldots a _ k 0 0 0 \ldots \] where \( k \in \mathbb{ N } _ 1 \) and \( a _ i \in \mathbb{ N } _ 0 \)
Finite Decimal Expansion Implies Restriction on Denominator
Let \( m, n \in \mathbb{ N } _ 1 \) such that \( \gcd \left( m, n \right) = 1 \), if \( \frac{m}{n} \) has a finite decimal expansion then \( n = 2 ^ a \cdot 5 ^ b \) for some \( a, b \in \mathbb{ N } _ 0 \)
Suppose that \( \frac{m}{n} = 0.a _ 1 a _ 2 \ldots a _ k 0 0 0 \ldots \) thus we have \[ 10 ^ k \left( \frac{m}{n} \right) = a _ 1 a _ 2 \ldots a _ k \] therefore \( 10 ^ k \left( \frac{m}{n} \right) \in \mathbb{ N } _ 1 \) so that \[ 10 ^ k m = n j \] \( j \in \mathbb{ Z } \) therefore \( n \mid 10 ^ k m \) but since \( \gcd \left( m, n \right) = 1 \) then \( n \mid 10 ^ k \) so that \( n = 2 ^ a \cdot 5 ^ b \) for some \( a, b \in \mathbb{ Z } \)