A rational n/d
n = q₀·d+r₀ where q,r∈ℤ and 0≤r<d
10·q₀ = q₁·d+r₁
10·q₁ = q₂·d+r₂
10·q₂ = q₃·d+r₃
...
If there is an i such that rᵢ=0, n/d is a terminating decimal.
If 0<rᵢ for all i, then there must be a j such that rᵢ=rⱼ because rᵢ<d.
rᵢ=rⱼ ⇒ 10·rᵢ=10·rⱼ ⇒ qᵢ₊ₖ=qⱼ₊ₖ ∧ rᵢ₊ₖ=rⱼ₊ₖ
This means n/d ia a recurring decimal.
