The repeating quotient, written as above, has only 5 figures, and 5 measures 41 - 1.
0₁₀2₁₈4₁₆3₃₇9₁
Now, this period, it will be found, has its figures merely transposed, if we multiply by 10, 18, 16, or 37. But if we multiply by any other number under 41, we convert this period into the period of another
fraction whose denominator is 41. The following are 8 periods which may be found.
| 0₁₀2₁₈4₁₆3₃₇9₁ | 1₉2₈1₃₉9₂₁5₅ |
|---|---|
| 0₂₀4₃₆8₃₂7₃₃8₂ | 1₁₉4₂₆6₁₄3₁₇4₆ |
| 0₃₀7₁₃3₇1₂₉7₃ | 2₂₈6₃₄8₁₂2₃₈9₁₁ |
| 0₄₀9₈₁7₂₃5₂₅6₄ | 3₂₇6₂₄5₃₅8₂₂5₁₅ |
To find m/41, look out for m among the remainders, and take the period in which it is, beginning after the remainder. Thus, ³⁴/₄₁ is ·8292682926, &c., and ¹⁵/₄₁ is ·3658536585, &c. These periods are complemental, four and four, as 02439 and 97560, 07317 and 92682, &c. And if the first number, 02439, be multiplied by any number under 41, look for that number among the remainders, and the product is found in the period of that remainder by beginning after the remainder. Thus, 02439 multiplied by 23 gives 56097, and by 6 gives 14634.
The reader may try to decipher for himself how it is that, with no more figures than the following, we can extend the result of our division. The fraction of which the period is to be found is ¹/₈₇.
- 87)100(01149425
- 130
- 430
- 82001149425 × 25
- 37028735625 × 25
- 220718390625 × 25
- 46017959765625 × 25
- 25448994140625
- 0114942528735625
- 718390625
- 1795976 5625
- 448994
- 0114942528735632183908045977|011494
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