Page 36 Exploring Unit Fractions Alan Christison
[email protected] INTRODUCTION In their article “The Beauty of Cyclic Numbers”, Duncan Samson and Peter Breetzke (2014) explored, among other things, unit fractions and their decimal expansions6. They showed that for a unit fraction to terminate, its denominator must be expressible in the form 2푛. 5푚 where 푛 and 푚 are non-negative integers, both not simultaneously equal to zero. In this article I take the above observation as a starting point and explore it further. FURTHER OBSERVATIONS 1 For any unit fraction of the form where 푛 and 푚 are non-negative integers, both not 2푛.5푚 simultaneously equal to zero, the number of digits in the terminating decimal will be equal to the larger 1 1 of 푛 or 푚. By way of example, has 4 terminating decimals (0,0625), has 5 terminating decimals 24 55 1 (0,00032), and has 6 terminating decimals (0,000125). The reason for this can readily be explained by 26.53 1 converting the denominators into a power of 10. Consider for example which has six terminating 26.53 decimals. In order to convert the denominator into a power of 10 we need to multiply both numerator and denominator by 53. Thus: 1 1 53 53 53 125 = × = = = = 0,000125 26. 53 26. 53 53 26. 56 106 1000000 1 More generally, consider a unit fraction of the form where 푛 > 푚. Multiplying both the numerator 2푛.5푚 and denominator by 5푛−푚 gives: 1 1 5푛−푚 5푛−푚 5푛−푚 = × = = 2푛. 5푚 2푛.