Il existe 8 périodes de 11 chiffres pour n/89 en base 2.

Pour toutes les fractions de n/89 en base 2, la période de 1/89 revient alors 11 fois (en orange)

1/89=0,00000010111...

2/89=0,00000101110...

3/89=0,00001000101...

4/89=0,00001011100...

5/89=0,00001110011...

6/89=0,00010001010...

7/89=0,00010100001...

8/89=0,00010111000...

9/89=0,00011001111...

10/89=0,00011100110...

11/89=0,00011111101...

12/89=0,00100010100...

13/89=0,00100101011...

14/89=0,00101000010...

15/89=0,00101011001...

16/89=0,00101110000...

17/89=0,00110000111...

18/89=0,00110011110...

19/89=0,00110110101...

20/89=0,00111001100...

21/89=0,00111100011...

22/89=0,00111111010...

23/89=0,01000010001...

24/89=0,01000101000...

25/89=0,01000111111...

26/89=0,01001010110...

27/89=0,01001101101...

28/89=0,01010000100...

29/89=0,01010011011...

30/89=0,01010110010...

31/89=0,01011001001...

32/89=0,01011100000...

33/89=0,01011110111...

34/89=0,01100001110...

35/89=0,01100100101...

36/89=0,01100111100...

37/89=0,01101010011...

38/89=0,01101101010...

39/89=0,01110000001...

40/89=0,01110011000...

41/89=0,01110101111...

42/89=0,01111000110...

43/89=0,01111011101...

44/89=0,01111110100...

45/89=0,10000001011...

46/89=0,10000100010...

47/89=0,10000111001...

48/89=0,10001010000...

49/89=0,10001100111...

50/89=0,10001111110...

51/89=0,10010010101...

52/89=0,10010101100...

53/89=0,10011000011...

54/89=0,10011011010...

55/89=0,10011110001...

56/89=0,10100001000...

57/89=0,10100011111...

58/89=0,10100110110...

59/89=0,10101001101...

60/89=0,10101100100...

61/89=0,10101111011...

62/89=0,10110010010...

63/89=0,10110101001...

64/89=0,10111000000...

65/89=0,10111010111...

66/89=0,10111101110...

67/89=0,11000000101...

68/89=0,11000011100...

69/89=0,11000110011...

70/89=0,11001001010...

71/89=0,11001100001...

72/89=0,11001111000...

73/89=0,11010001111...

74/89=0,11010100110...

75/89=0,11010111101...

76/89=0,11011010100...

77/89=0,11011101011...

78/89=0,11100000010...

79/89=0,11100011001...

80/89=0,11100110000...

81/89=0,11101000111...

82/89=0,11101011110...

83/89=0,11101110101...

84/89=0,11110001100...

85/89=0,11110100011...

86/89=0,11110111010...

87/89=0,11111010001...

88/89=0,11111101000...

On remarque que le produit du nombre de périodes (8) et de leurs longueurs (11) est égal à 88 et donc au premier -1.