Exploring Unit Fractions

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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. 5푚 5푛−푚 2푛. 5푛 10푛

Since the denominator is 10푛 it follows that the fraction contains 푛 terminating decimals. Similar arguments hold for 푚 > 푛 (where there will be 푚 terminating decimals) as well as for 푚 = 푛.

1 Let us now consider fractions of the form . In line with the above, fractions of this form can be 5푚 2푚 converted to the form from which it is then possible to write down the terminating decimal expansion 10푚 1 directly since the last of the 푚 decimals will simply be the 푚th power of 2. Thus will have 4 terminating 54 1 1 decimals, the final digits of which will be 16 (since 24 = 16), i.e. = 0,0016. Similarly, will have 8 54 58 1 terminating decimals, the final digits of which will be 256 (since 28 = 256), i.e. = 0,00000256. 58

6 http://www.amesa.org.za/LTM%2016_2.pdf

Learning and Teaching Mathematics, No. 25, 2018, pp. 36-39

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1 Note that a similar argument holds for unit fractions of the form . In this case the terminating decimal 2푚 1 expansion will have 푚 digits, and the last of these 푚 decimals will be the 푚th power of 5. Thus will 24 have 4 terminating decimals, the final digits of which will be 625 (since 54 = 625). We can thus immediately 1 write = 0,0625. 24

NON-TERMINATING UNIT FRACTIONS If the prime factorisation of the denominator of a unit fraction contains factors other than 2 and/or 5, then the decimal expansion of the fraction will be non-terminating. More specifically, the decimal expansion will 1 be cyclic – i.e. it will have a repeating structure. Consider the unit fraction which can be expressed in 540 1 1 1 decimal form as 0,00185̅̅̅̅̅. Since 540 = 22. 5. 33 we can write in the form × , i.e. the product 540 22.5 27 of a terminating fraction and a non-terminating fraction. Note that the larger exponent in the numerator 1 1 of is 2, while = 0, 037̅̅̅̅̅. We thus notice that the larger exponent in the denominator of the 22.5 27 1 terminating fraction equates to the number of non-repeating digits in , i.e. two (00), while the number 540 1 1 of repeating digits in the non-terminating fraction equates to the number of repeating digits in , i.e. 27 540 three (185). 1 As a further example consider the unit fraction which can be expressed in decimal form 2625 1 1 1 as 0,000380952̅̅̅̅̅̅̅̅̅̅. Since 2625 = 53. 3.7 we can write in the form × , i.e. the product of a 2625 53 21 1 terminating fraction and a non-terminating fraction. Note that = 0,008 with three terminating digits as 53 1 expected, and = 0, 047619̅̅̅̅̅̅̅̅̅̅ with six repeating digits. As anticipated, the exponent in the denominator of 21 1 the terminating fraction equates to the number of non-repeating digits in , i.e. three (000), while the 2625 1 number of repeating digits in the non-terminating fraction equates to the number of repeating digits 21 1 in , i.e. six (380952). 2625

This suggests we can make the following general statement:

Statement 1: 1 1 If is a terminating decimal fraction and is a fully/purely repeating decimal fraction, 푎 푏 1 1 then will have a non-periodic decimal length equivalent to that of and a repeating 푎푏 푎 1 decimal length equivalent to that of . 푏

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1 It is important to note that this only holds true if the repeating decimal of commences directly after the decimal 푏 1 1 comma, i.e. must be a fully/purely repeating decimal fraction. Consider for example = 0,00260416̅ 푏 384 1 1 1 1 1 which can be written in the form × where = 0,0625 and = 0,0416̅. Clearly is not 24 24 24 24 24 purely repeating, but has a combination of three non-periodic digits (041) and one repeating digit (6). We 1 know that has four terminating decimals, and adding these four to the three non-periodic digits 24 1 1 in gives a total of seven, the number of non-periodic decimals in (i.e. 0026041). This suggests we 24 384 can make the following more general statement:

Statement 2: 1 1 If is a terminating decimal fraction of length 푚, and is a non-terminating decimal fraction 푎 푏 1 comprising 푝 non-periodic digits and 푛 repeating digits, then will have 푚 + 푝 non- 푎푏 periodic digits and 푛 repeating digits where 푚 and 푛 are positive integers and 푝 is a non- negative integer (i.e. 푝 can equal zero).

1 Notice that in the above example we can force 푝 to be zero as follows. Instead of writing in the 384 1 1 1 1 form × we can write it in the form × . We can now apply Statement 1 to conclude 24 24 27 3 1 1 that has seven non-repeating digits (from the exponent in 27 ) and one repeating digit (since = 0, 3̅). 384 3

ODD NUMBERS 1 Let us now consider unit fractions where 푥 is an odd positive integer. If 푥 is a multiple of 5 then we 푥 1 can rewrite 푥 in the form 5푏. 푁 where 푁 is odd. We can thus write the unit fraction in the modified 푥 1 1 1 form × . From the previous discussion it should be clear that relates to the number of non- 5푏 푁 5푏 1 1 repeating decimal digits while relates to the number of repeating digits in the decimal expansion of . 푁 푥 1 1 Note that if we have 푏 = 0 then = and the repeating decimal digits commence directly after the decimal 푥 푁 comma. This leads to the following statement:

Statement 3: Any odd number greater than 1, and not divisible by 5, when expressed as a unit fraction, will have repeating decimal digits that commence directly after the decimal comma.

EVEN NUMBERS Statement 3 suggests that we can also make the following general claim:

Statement 4: Any even number expressed as a unit fraction and having some repeating decimal(s) must have leading non-repeating decimal digits.

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1 We can prove Statement 4 as follows. Let have 푚 repeating decimal digits directly after the comma, in 푥 other words: 1 = 0, 푑 푑 … 푑 푑 푑 … 푥 1 2 푚 1 2

Multiplying through by 10푚 gives:

10푚 = 푑 푑 … 푑 , 푑 푑 … 푑 푑 푑 … 푥 1 2 푚 1 2 푚 1 2 We can thus see that:

10푚 1 10푚 − 1 − = = 푑 푑 … 푑 푥 푥 푥 1 2 푚

푚 If we let 푑1푑2 … 푑푚 = 푦 then 10 − 1 = 푥푦. Assuming 푥 to be even, then 푥 can be expressed as 2푡 where 푡 is a positive integer. It then follows that 10푚 − 1 = 2푡푦, which is impossible since the left- hand side, 10푚 − 1, is clearly odd. Our assumption is thus false, and any even number expressed as a unit fraction and having some repeating decimal(s) must have leading non-repeating decimal digits. A few examples relating to Statement 3 and Statement 4 are illustrated in the following table:

Unit fraction Statement 3 Statement 4 Comments 1 27 is odd. The unit fraction has repeating 0, ̅037̅̅̅̅ decimal digits that commence directly after the 27 decimal comma. 1 189 is odd. The unit fraction has repeating 0, ̅005291̅̅̅̅̅̅̅̅̅ decimal digits that commence directly after the 189 decimal comma. 1 105 is odd but has a factor of 5. There is a 0,0̅095238̅̅̅̅̅̅̅̅̅ leading non-repeating decimal digit after the 105 decimal comma. 1 12 is even. There are leading non-repeating 0,083̅ 12 decimal digits after the decimal comma. 1 110 is even. There are leading non-repeating 0,0090̅̅̅̅ 110 decimal digits after the decimal comma.

CONCLUDING COMMENTS The behaviour of decimal digits relating to unit fractions makes for an interesting investigation. The above exploration can readily be extended into a number of other interesting areas, for example cyclic numbers, their link to primes (specifically full reptent primes) and the longest string of repeating decimal digits that these primes may exhibit.

REFERENCES Samson, D. & Breetzke, P. (2014). The beauty of cyclic numbers. Learning & Teaching Mathematics, 16, 24-29.

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