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by Someone
1467 days ago
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That’s not a very good explanation. That second sentence > Any fraction with a denominator that is a product of prime factors of 10 can be turned into a fraction with a denominator that is a power of 10. isn’t sufficient. You also have to argue that any fraction with a denominator that is not a product of the prime factors of 10 cannot be written as a finite decimal. If you do that you end up with a tautology. A better and more rigid explanation would start with a rational p/q, with p and q relatively prime that can be written as some integer divided by 10^n, and reason from there. |
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You almost gave a really good explanation yourself on why this should be the case but stopped short of it. Don't know why. Any terminating decimal can be written in the form a/10^{n} where a and n are both integers. After we have our p/q, there exists an integer c such that p/q · c/c = p/10^{n} if and only if q consists solely of the primes 2 or 5 (or both). And if not, then there isn't an integer c to satisfy said condition, thus making it impossible to write our p/q in the form of a/10^{n} (white still keeping the numerator an integer at least) so it's an infinitely repeating decimal.