PROOF 05
Prove.
Every repeating decimal can be written as a fraction
On the reals ℝ.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
The value of 0.ḂḂ… is the limit of the sequence of partial sums (Sₙ).
Proof.
(1)B is the repeating block of k digits, N is the integer those digits form, and Sₙ = N·10⁻ᵏ + N·10⁻²ᵏ + … + N·10⁻ⁿᵏ.convention
(2)That this sequence approaches a single real number is guaranteed by A1.A1
∎
Lemma L1
For all n, (10ᵏ − 1)Sₙ = N(1 − 10⁻ⁿᵏ)
Proof.
(1)Sₙ = N·10⁻ᵏ + N·10⁻²ᵏ + … + N·10⁻ⁿᵏconvention
(2)10ᵏSₙ = N + N·10⁻ᵏ + … + N·10⁻⁽ⁿ⁻¹⁾ᵏ(1), A5, A6, E
(3)10ᵏSₙ = N + Sₙ − N·10⁻ⁿᵏ(1), (2)
(4)(10ᵏ − 1)Sₙ = N − N·10⁻ⁿᵏ(3), A6, E
(5)(10ᵏ − 1)Sₙ = N(1 − 10⁻ⁿᵏ)(4), A6
∎
Theorem T1
0.ḂḂ… = N / (10ᵏ − 1)
Proof.
(1)x = 0.ḂḂ… = lim SₙD1
(2)(10ᵏ − 1)Sₙ = N − N·10⁻ⁿᵏL1, A6
(3)lim[(10ᵏ − 1)Sₙ] = (10ᵏ − 1)x(1), A3
(4)lim[N − N·10⁻ⁿᵏ] = N − N·lim 10⁻ⁿᵏA2, A3
(5)lim 10⁻ⁿᵏ = 0A4
(6)lim[N − N·10⁻ⁿᵏ] = N(4), (5), A7
(7)(10ᵏ − 1)x = N(2), (3), (6), E
(8)The length k of the repeating block is at least 1, so 10ᵏ − 1 ≥ 9convention
(9)x = N / (10ᵏ − 1)(7), (8), E
∎
Corollary C
A decimal whose repetition begins after m digits can also be written as a fraction.
Proof.
(1)Suppose the repetition of x begins after the m-th decimal digit.assumption
(2)10ᵐx = lim(10ᵐSₙ)D1, A3
(3)10ᵐSₙ = A + Tₙ, where A is the integer formed by the first m digits and Tₙ is the partial sum of the repeating part.A5, A6, convention
(4)10ᵐx = A + y, y = lim Tₙ(2), (3), A2, D1
(5)y = N / (10ᵏ − 1)T1
(6)x = (A(10ᵏ − 1) + N) / (10ᵐ(10ᵏ − 1))(4), (5), E
(7)Numerator and denominator are both integers, and the denominator is nonzero.(6)
∎
Remark
Can A1 be removed from this proof?
The common method sets x = 0.333… and computes 10x − x. Which axiom is that method using silently?