root, the square of the root may be made to differ from the number proposed by a quantity smaller than any that can be assigned; so that you always have it in your power to approximate to the truth, as closely as you please. (97.) We must now return to our worked-out example at page 143, for I have some particulars to mention to you, as to the abridgment of the operation. You may, in general, take it for granted, unless the contrary plainly appear, that when a number containing decimals is given you to work upon, the final decimal of that number will be only approximately true. You may consider, therefore, that the number proposed in the example referred to, has, in its complete state, a long string of decimals, and that the decimals have been reduced to three, for convenience, or, because more than three decimals would encumber the value with an unnecessary degree of accuracy; consequently, when we have used the final decimal 2, we may conclude, that the final figures of the subsequent dividends will be affected with error, and our business is, to exclude the influence of this error from the figures of the root. It is plain, therefore, that after having arrived at the dividend, 43192, and its divisor, 47649, contracted division only should be used: you must see, too, that as each divisor differs from the next trial-divisor following, only by having the double of the new root-figure added to it, these successive additions can have no influence on any contracted divisor, because the figures that would be influenced become cut off. Hence, after the 42884 308 final decimal 2 has been used, the 4,7,6,4,9)43192(9064 remaining part of the operation is simply that of contracted division, which terminates of itself, as soon as all the root-figures that can be depended upon are obtained: the root may, therefore, be extended, as in the margin; and we may consider it to be correctly determined, as far as five places of decimals; its value being 238.29064: but as 22 19 3 the quotient-figure 4 is somewhat more below the truth than 5 is above it, it might be better to write 238.29065 for the root; if we knew that the final decimal in our proposed number were a little too small, we should affirm the latter to be the better approximation to the root; but, if we knew that H our proposed number were a little too great in its last decimal, we should prefer the former approximation: we cannot, therefore, be quite sure as to the last decimal, within a unit. 5,83,5274(241-5631 In the following example, contraction is used as soon as the decimals in the given number have been employed. 2. Extract the square-root of 58352.74. The work in the margin shows the root as far as four places of decimals to be 241-5631: you cannot depend upon it to any greater extent. (98.) I think you must now sufficiently see the manner in which you are to proceed with the extraction of the square-root, when you desire that root to be encumbered with no more decimals than you can place reliance upon. You can, of course, adopt the same method of contraction, when you have to approximate to the square-root of a whole number, in itself not a square as the unchecked, would process, left out end, you would fix a point at which restraint should 4 44)183 481) 752 481 4825)27174 4,8,3,0) 3049 2898 151 145 6 5 be put upon the increase of figures; and, by using contracted division from that step, bring the operation to a close: you can always easily see, from counting the number of figures in any divisor, how many root-figures, from that stage of the work, will be added on by the contracted di vision. After the first step in the extraction of the square-root, you need not take the trouble to multiply the root-figures by 2, in order to get the several trial, or incomplete divisors, since each trial-divisor is formed from merely adding twice the last root-figure to the preceding divisor; in practice, the several trial-divisors are always derived from one another in this way. I may also remark here, that some people mark off the periods, in the number proposed for extraction, by putting a dot over the last figure of each period: thus, the periods of the number in last example are distinguished in this way; 58352.74: you can, of course, use, in your own practice, whichever way you please. 76,80,76,96(8764 61 (99.) It is not easy to give a satisfactory proof of the foregoing method of extracting the square-root without the aid of algebra; but I will endeavour to explain the reason of the several steps of the operation by help of arithmetic only. To understand this explanation, you must first become convinced of the following property: namely, that if any number be separated into two parts by the sign plus, the square of that number will always be made up of the squares of the two parts plus twice the product of those parts. Take, for instance, the number 9, of which the square is 81; this square is made up of the squares of 4 and 5 (which together make 9) plus twice 4 x 5; it is also made up of the squares of the two parts 3 and 6 plus twice 3 x 6; or of the squares of the two parts 7 and 2 plus twice 7 × 2, and so on: that is, 92-42+2 (4 × 5) + 52 =32+2 (3 × 6) + 62 = 72 + 2 (7 × 2) + 22 = 82 + 2 (8 x 1) + 12, &c. And the same property holds, whatever be the number, and into whatever two parts it be separated. It is this general property that has suggested the rule for the square-root. Let us take any square number; the square of 8764, for instance; which is 76807696. If from having this square given, we wished to return to the root, we should readily foresee, that by dividing it into periods, as already explained, and then regarding only the first period 76, the leading figure 8 of the root could be at once discovered. As the local value of this 8 is 8000, we know, from the foregoing property, that the proposed number is made up of the following parts: namely, 80002+2 (8000 × 764) +7642, which is the same as 80002+ (2 × 8000+764) 17524) 70096 764. So that after having subtracted the square of the first root-figure, the remainder, or what has been called above the first dividend, is (2 × 8000 + 764) 764.* The divisor for this, which would supply accurately all the remaining figures of the root, namely the figures 764, is evidently 2 x 8000+764. But this divisor we cannot completely get, as the figures 764 which enter it are those of which we are in search; but we can get the greater part of it; namely, the part 2 × 8000, from knowing the already-found first figure 8, or in strictness 8000. We avail ourselves, therefore, of this, and call it our first trial, or incomplete divisor; this assists us in discovering the single figure 7, by which we are enabled to correct our trial divisor, by adding to it the part thus suggested, namely 700; and so corrected, we call 2 × 8000 + 700 our complete divisor, since it completely answers for 167)1280 1746) 11176 10476 70096 * The remainders, or dividends, are conceived to have the figures of the proposed number, afterwards brought down two at a time, to be actually appended to them; but, just as in long division, the remainders are kept free of these additional figures till they are wanted for use, in order to save unnecessary repetitions. It may be noticed here, that the notation above means that the whole of the quantity enclosed in the brackets is to be multiplied by 764. the figure 7 of the root, though not for the figures which follow. After multiplying this complete divisor by the local value, 700, of the quotientfigure 7, we get (2 x 8000+700) 700, which subtracted from the first dividend, leaves 2 x 8000 × 64 +7642-7002. But by the general property with which we started, 7642-7002+2 (700 × 64) + 642: consequently, the remainder spoken of is 2 x 8000 × 64 + 2 × 700 × 64 + 642; or, which is the same thing, 2 x 8700 x 64 + 642=(2 × 8700+64) 64. And this is our second dividend. Now just return to the preceding expression for the first dividend, and you will observe a perfect correspondence; the first dividend consists of twice the number already in the root+the number formed by the remaining figures, multiplied by that number; so here the second dividend consists of twice the number already in the root+the number formed by the remaining figures, multiplied by that number. Consequently, just as we got the second figure of the root out of the first dividend, so we may now get the third figure out of the second dividend; that is, the step by which the third figure is to be obtained must be exactly similar to that by which the second was obtained; and therefore, figure after figure is to be found by a series of uniform steps, as in the operation given at length in the margin, which you will find upon examination to be in strict accordance with what is here explained, when the several remainders or dividends are completed, by the latter figures of the original number being appended to them. In the work, these figures, to save repetitions, are not actually brought down till they are wanted. Exercises. Extract the square-root of each of the following numbers, those of them which terminate in decimals being only approximately true in the last figure, except where otherwise stated. 1. 31-782153. 3. 3236068. 5. 473256, to three decimals. 7, 9036890625. 2. 115.297356. 4. 11, to six decimals. 9. 32-398864, the last decimal true. 10. 000729, the last decimal true. 11. 784-375 12. 79.182. 13. 68-736, to nine decimals, the last, 6, being true. 14. 2951. 15. 104 16. 17%, to five decimals. 17. 15g, to six decimals. 18. 794, to eight decimals. 19. 34-867844. 20. The recurring decimal 7·6531 to seven places. (100.) There is another way of arranging the work for the square-root, which, although it presents to the eye more figures, and takes up more room, is nevertheless well worthy of your attention, on account of the very simple character of the several steps; and because, moreover, the operation, when thus arranged, becomes only a particular case of the general process for the extraction of any root, however high. The operation, too, is the same, whether the sought root be that of a number merely, as here, or that of a numerical equation; so that, if you ever advance to algebra, you will find a familiarity with the arrangement given below to be of much assistance to you in an important part of that subject. The work of the example last given, in the form here recommended to your notice, is as follows: By comparing this with the work of the same example at page 147, you will see that the two operations differ only in arrangement. In the latter form, 1, 0, and the given number, are written in a row; the 0 stands at the head of a column of work which gives the trial and true divisors; and the given number stands at the head of a column which furnishes the corresponding dividends; each trial-divisor, with the dividend to which it belongs, is marked (1), (2), &c, merely for the purpose of directing the eye to those numbers in the two columns which are directly concerned in the determination of the several root-figures; each true divisor is |