PF: PF: Reduced denominator's prime factorization is only 2s and 5s ↔ terminating decimal.
A fraction is reducible if its numerator and denominator have any common factor>1, i.e. their prime factorizations are not disjoint, the numerator and denominator are not coprime, GCF≠1. Irreducible if prime factorizations are disjoint, the numerator and denominator are coprime, GCF=1. 1/d and d-1/d (and improper d+1/d) are not reducible. [A pair of consecutive numbers is coprime.] A proper fraction can be written as a sum of distinct unit fractions (1/n), i.e. as an Egyptian fraction. E.g. 5/8=1/2+1/8 43/48=1/2+1/3+1/16 3/7=1/3+1/11+1/231 Egyptian denominators: (for proper fractions only)
Experiment: generate all fractions of denominators 2-D. Count #, #reducible, #terminating decimals. D: #total fractions: - #reducible: = #actual/unique fractions: #total terminating decimals: - #terminating reducibles: = actual # terminating decimals:
All fractions (reduced): Show? Yes No A reduced fraction with a prime denominator p, the numerator can be all numbers from 1 to p-1.
D=10 : 45 fractions, 14 reducible, 25 with terminating decimals. D=100 : 4950 fractions, 1907 are reducible, 660 with terminating decimals. D=1000 : 499500 fractions, 195309 reducible, 12608 with terminating decimals.