Decimal Expansion

Terminating Decimal Expansion

A terminating decimal expansion is one of the form

0. a_{1} a_{2} a_{3} \dots a_{k} 000 \dots

where

k \in ℕ_{1}

and

a_{i} \in ℕ_{0}

Finite Decimal Expansion Implies Restriction on Denominator

Let

m, n \in ℕ_{1}

such that

\gcd (m, n) = 1

, if

\frac{m}{n}

has a finite decimal expansion then

n = 2^{a} \cdot 5^{b}

for some

a, b \in ℕ_{0}

Suppose that

\frac{m}{n} = 0. a_{1} a_{2} \dots a_{k} 000 \dots

thus we have

10^{k} (\frac{m}{n}) = a_{1} a_{2} \dots a_{k}

therefore

10^{k} (\frac{m}{n}) \in ℕ_{1}

so that

10^{k} m = n j

j \in ℤ

therefore

n | 10^{k} m

but since

\gcd (m, n) = 1

n | 10^{k}

so that

n = 2^{a} \cdot 5^{b}

for some

a, b \in ℤ