There is a useful method of finding fractions which shall be nearly equal to a given fraction, and with which the computer ought to be acquainted. Proceed as in the rule for finding the greatest common measure of the numerator and denominator, and bring all the quotients into a line. Then write down,
| 1 | 2nd Quot. | |
|---|---|---|
| 1st Quot. | 1st Quot. × 2d Quot. + 1 |
Then take the third quotient, multiply the numerator and denominator of the second by it, and add to the products the preceding numerator and denominator. Form a third fraction with the results for a numerator and denominator. Then take the fourth quotient, and proceed with the third and second fractions in the same way; and so on till the quotients are exhausted. For example, let the fraction be ⁹¹³¹/₁₃₁₂₈.
- 9131)13128(1, 2
- 1137 3997(3, 1
- 551 586(1, 15
- 201 35(1, 2
- 26 9(1, 8
- 8 1
This is the process for finding the greatest common measure of 9131 and 13128 in its most compact form, and the quotients and fractions are:
| 1 | 2 | 3 | 1 | 1 | 15 | 1 | 2 | 1 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 7 | 9 | 16 | 249 | 265 | 779 | 1044 | 9131 |
| 1 | 3 | 10 | 13 | 23 | 358 | 381 | 1120 | 1501 | 13128 |
It will be seen that we have thus a set of fractions ending with the original fraction itself, and formed by the above rule, as follows:
| 1st Fraction = | 1 | = | 1 | |||
|---|---|---|---|---|---|---|
| 1st Quot. | 1 | |||||
| 2d Fraction = | 2d Quot. | = | 2 | |||
| 1st Quot. × 2d Quot. + 1 | 3 | |||||
| 3d Fraction = | 2d Numʳ. × 3d Quot. + 1st Numʳ. | = | 2 × 3 + 1 | = | 7 | |
| 2d Denʳ. × 3d Quot. + 1st Denʳ. | 3 × 3 + 1 | 10 | ||||
| 4th Fraction = | 3d Numʳ. × 4th Quot. + 2d Numʳ. | = | 7 × 1 + 2 | = | 9 | ; |
| 3d Denʳ. × 4th Quot. + 2d Denʳ. | 10 × 1 + 3 | 13 |