168. For example, what is the square root of (1⅜) to five places of decimals? This is (145) 1·375, and the process is the first example over leaf. The second example is the extraction of the root of ·081 to seven places, the first period being 08, from which the cipher is omitted as useless.
- 1,37,5(1·17260
- 1
- 21) 37
- 21
- 227) 1650
- 1589
- 2342) 6100
- 4684
- 23446) 141600
- 140676
- 23452) 92400
- 8,1(·2846049
- 4
- 48)410
- 384
- 564) 2600
- 2256
- 5686) 34400
- 34116
- 569204) 2840000
- 2276816
- 569208) 56318400
- ·000002413672221(·001553599
- 1
- 25) 141
- 125
- 305) 1636
- 1525
- 3103) 11172
- 9309
- 31065) 186322
- 155325
- 310709) 3099710
- 2796381
- 30332900
169. When more than half the decimals required have been found, the others may be simply found by dividing the dividend by the divisor, as in (155). The extraction of the square root of 12 to ten places, which will be found in the next page, is an example. It must, however, be observed in this process, as in all others where decimals are obtained by approximation, that the last place cannot always be depended upon: on which account it is advisable to carry the process so far, that one or even two more decimals shall be obtained than are absolutely required to be correct.
- A
- 12(3·46410161513
- 9
- 64) 300
- 256
- 686) 4400
- 4116
- 6924) 28400
- 27696
- 69281) 70400
- 69281
- 6928201) 11190000
- 6928201
- 69282026) 4261799|00
- 4156921|56
- 692820321) 104877|4400
- 69282|0321
- 6928203225) 35595|407900
- 34641|016125
- 69282032301) 954|39177500
- 692|82032301
- 692820323023) 261|5714519900
- 207|8460969069
- 53|7253550831
- B
- 692820323026)537253550831(77545870549
- 484974226118
- 52279324713
- 48497422611;
- 3781902102
- 3464101615
- 317800487
- 277128129
- 40672358
- 34641016
- 6031342
- 5542562
- 488780
- 484974
- 3806
- 3464
- 342
- 277
- 65
- 62
- 3
If from any remainder we cut off the ciphers, and all figures which would come under or on the right of these ciphers, by a vertical line, we find on the left of that line a contracted division, such as those in (155). Thus, after having found the root as far as 3·464101, we have the remainder 4261799, and the divisor 6928202. The figures on the left of the line are nothing more than the contracted division of this remainder by the divisor, with this difference, however, that we have to begin by striking a figure off the divisor, instead of using the whole divisor once, and then striking off the first figure. By this alone we might have doubled our number of decimal places, and got the additional figures 615137, the last 7 being obtained by carrying the contracted division one step further with the remainder 53. We have, then, this rule: When half the number of decimal places have been obtained, instead of annexing two ciphers to the remainder, strike off a figure from what would be the divisor if the process were continued at length, and divide the remainder by this contracted divisor, as in (155).
As an example, let us double the number of decimal places already obtained, which are contained in 3·46410161513. The remainder is 537253550831, the divisor 692820323026, and the process is as in (B). Hence the square root of 12 is,
3·4641016151377545870549;
which is true to the last figure, and a little too great; but the substitution of 8 instead of 9 on the right hand would make it too small.
EXERCISES.
| Numbers. | Square roots. |
| ·001728 | ·0415692194 |
| 64·34 | 8·02122185 |
| 8074 | 89·8554394 |
| 10 | 3·16227766 |
| 1·57 | 1·2529964086141667788495 |