# Decimal fractions1121

Finite decimal numbers are called decimal fractions, because they are a different representation for fractions with powers of ten in the denominator. So is:

$$\frac{z}{n} = \frac{q_1}{1} + \frac{q_2}{10} + \dots + \frac{q_k}{10^k}$$

with $$k \in \mathbb{N}$$ and $$q_k$$ the $$k-1$$ -th place to the right after the comma.

Now is:

$$\frac{z}{n} = \frac{10^k \cdot q_1 + 10^{k-1} q_2 + \dots + q_k}{10^k} = \frac{10^k \cdot q_1 + 10^{k-1} q_2 + \dots + q_k}{2^k \cdot 5^k}$$

This means: If the denominator for a general fraction can be extended to $$2^k \cdot 5^k$$ in a completely abbreviated form $$\frac{z}{n}$$ , it is a finite decimal fraction. If we consider the prime factorization of the denominator $$n = p_1^{l_1} \cdot \, \dots \, \cdot p_j^{l_j}$$ , then according to the fundamental theorem of arithmetic, this can be expressed as $$f = 2^{km} \cdot 5^{kn}$$ to $$2^k \cdot 5^k$$ if $$n = 2^m \cdot 5^n$$ . This applies:

Only fractions whose denominators have no prime factors other than 2's or 5's when fully abbreviated result in a finite decimal fraction.

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