PROOF 72
Prove.
A decimal is a fraction whose denominator is a power of 10
On the rational numbers ℚ.
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
0.d₁d₂…dₙ means d₁·10⁻¹ + d₂·10⁻² + … + dₙ·10⁻ⁿ.
Proof.
(1)Each dᵢ is an integer from 0 to 9. We set 10⁰ = 1 by convention.convention
(2)10⁻ⁱ means the multiplicative inverse of 10ⁱ.convention
(3)Since 10ⁱ ≠ 0, that inverse exists. This is why the definition has meaning.A2, A3
∎
Definition D2
When b has a multiplicative inverse, a/b means a · b⁻¹.
Proof.
To write a decimal as a fraction is to write it in this form.convention
∎
Lemma L1
If i ≤ n, then 10ⁿ · 10⁻ⁱ = 10ⁿ⁻ⁱ
Proof.
(1)10ⁿ = 10ⁿ⁻ⁱ · 10ⁱA1
(2)10ⁿ·10⁻ⁱ = (10ⁿ⁻ⁱ·10ⁱ)10⁻ⁱ(1), E
(3) = 10ⁿ⁻ⁱ(10ⁱ·10⁻ⁱ)A4
(4)10ⁱ · 10⁻ⁱ = 1D1, A2, A3
(5)10ⁿ·10⁻ⁱ = 10ⁿ⁻ⁱ(2), (3), (4), A6
∎
Theorem T1
0.d₁d₂…dₙ = N / 10ⁿ, where N = d₁·10ⁿ⁻¹ + d₂·10ⁿ⁻² + … + dₙ.
Proof.
(1)x = d₁·10⁻¹ + … + dₙ·10⁻ⁿD1
(2)10ⁿ·x = 10ⁿ(d₁·10⁻¹) + … + 10ⁿ(dₙ·10⁻ⁿ)(1), A7, A5, E
(3)10ⁿ(dᵢ·10⁻ⁱ) = dᵢ(10ⁿ·10⁻ⁱ)A4, A5
(4) = dᵢ · 10ⁿ⁻ⁱL1
(5)10ⁿ·x = d₁·10ⁿ⁻¹ + … + dₙ·10⁰ = N(2), (3), (4), D1, A6
(6)The exponents n−i are all at least 0 and the dᵢ are integers, so N is an integer.A8
(7)Since 10ⁿ ≠ 0, (10ⁿ)⁻¹ exists.A2, A3
(8)x = (10ⁿ·x)(10ⁿ)⁻¹ = N·(10ⁿ)⁻¹(5), (7), A4, A5, A6, E
(9)x = N / 10ⁿ(8), D2
∎
Corollary C
Every terminating decimal is a rational number.
Proof.
(1)A rational number is a number that can be written p/q with p an integer and q a nonzero integer.convention
(2)N is an integer.T1, A8
(3)10ⁿ is an integer and is nonzero.A8, A2
(4)x = N / 10ⁿT1
(5)Therefore x is rational. The denominator is always a power of 10, and the number of decimal places is its exponent.(1), (2), (3), (4)
∎
Remark
Can A3 be removed from this proof?
Begin by checking which number the notation 0.3 refers to.