Definition D1
x is a terminating decimal if x = m / 10ⁿ for some integer m and some integer n ≥ 0.
Proof.
A notation ending at the n-th place after the point means the same as a fraction with denominator 10ⁿ.convention
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Definition D2
p/q is in lowest terms if p and q share no prime factor. Below, p and q are natural numbers.
Proof.
That is, no prime divides both p and q.convention, A3
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Lemma L1
If N = d·e, then the prime factors of N are exactly the prime factors of d together with those of e.
Proof.
(1)If d > 1, then d is a product of primes. If d = 1, regard it as the empty product.A1, convention
(2)The same holds for e.A1, convention
(3)Concatenating the two products writes N = d·e as a product of primes.(1), (2), A6, A7, E
(4)If N > 1, that decomposition is unique, so (3) is the prime factorization of N.(3), A1
(5)Therefore the prime factors of N are exactly the prime factors of d and of e.(4)
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Lemma L2
For every integer n ≥ 0, the only prime factors of 10ⁿ are 2 and 5.
Proof.
(1)If n = 0, then 10⁰ = 1 and there are no prime factors.convention
(2)10 = 2 · 5convention
(3)10ⁿ = (2·5)ⁿ = 2ⁿ5ⁿ(2), A5, E
(4)2ⁿ5ⁿ is a product of primes composed of 2 and 5 only.(3)
(5)By uniqueness this is the prime factorization of 10ⁿ.(4), A1
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Theorem T1
If q = 2ᵃ5ᵇ, then p/q is a terminating decimal.
Proof.
(1)Set n = max(a, b); then n − a ≥ 0, n − b ≥ 0convention
(2)k = 2ⁿ⁻ᵃ · 5ⁿ⁻ᵇ is a natural number.(1)
(3)q·k = 2ᵃ5ᵇ · 2ⁿ⁻ᵃ5ⁿ⁻ᵇ = 2ⁿ5ⁿ(2), A4, A6, A7
(4)2ⁿ5ⁿ = (2·5)ⁿ = 10ⁿA5
(5)q·k = 10ⁿ(3), (4)
(6)p · 10ⁿ = p(qk) = q(pk)(5), A6, A7, E
(7)p/q = (pk) / 10ⁿ(6), A2
(8)Therefore p/q is a terminating decimal.(7), D1
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Theorem T2
If a fraction p/q in lowest terms is a terminating decimal, then the only prime factors of q are 2 and 5.
Proof.
(1)There are a natural number m and an integer n ≥ 0 with p/q = m / 10ⁿ.assumption, D1
(2)p · 10ⁿ = m · q(1), A2
(3)q | p · 10ⁿ(2), A3, A7
(4)Let r be a prime factor of q.assumption
(5)There is a natural number e with p · 10ⁿ = q · e.(3), A3
(6)r is a prime factor of q, hence a prime factor of p · 10ⁿ.(4), (5), L1
(7)A prime factor of p · 10ⁿ is a prime factor of p or of 10ⁿ.L1
(8)The only prime factors of 10ⁿ are 2 and 5.L2
(9)If r were a prime factor of p, then r would be a common prime factor of p and q, contradicting lowest terms.(4), D2
(10)Therefore r = 2 or r = 5.(6), (7), (8), (9)
(11)The only prime factors of q are 2 and 5, so the prime factorization of q has the form q = 2ᵃ5ᵇ.(10), A1
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Corollary C
For a fraction p/q in lowest terms, being a terminating decimal and having q = 2ᵃ5ᵇ are equivalent.
Proof.
(1)T1 gives one direction.T1
(2)T2 gives the other direction.T2
(3)Both directions hold, so the two conditions are equivalent.(1), (2)
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