12.2.66 Number Theory

Let the rational number (m / n) are less than one. Prove that (i-) the figure of its decimal recording (0, overline {a_ {0} a_ {1} a_ {2} a_ {3} cdots}) is calculated by the formula: (a_ {i} = left (left (m cdot 10 {i} {i} ight) mod n IGHT) CDOT 10 ight) / n, quad) where the last division operation is implied by integer. Prove that when dividing the corner, the final decimal fraction or periodic endless will always turn out. Come up with an algorithm for converting an arbitrary periodic decimal fraction into a rational number.