Suppressing 8 in the last divisor, 568, and proceeding in the manner already described; 253÷56=4+ a remainder. 56 × 4+3=227 (the 3 being tens carried from the product of 8, the last suppressed figure of the divisor by the quotient figure 4). And 253-227=26. Lastly, 26÷54: 5×4+2=22. And 26-22=4 rem". ..75744 is the quotient of the proposed numbers, truc, to within less than unity, if the exact quotient. By combining the rule for division of decimal fractions (Art. 115.) with the process illustrated in the last example, the quotient of two decimal fractions may be obtained with any required degree of approximation. For example, let it be required to divide 1234.569 by 27-35894; and to give a quotient true to thousandths. Annexing 3 zeros to the dividend, because the quotient is to be carried to thousandths, and proceeding as in the last example, except in so far that two figures of the divisor must be suppressed before the first partial division can be effected, it is found that the quotient is 45 124. It will be observed that the product of 27358 by 4 is increased by 4: the reason is, that the product of the suppressed figures 94 by 4-376. This number being nearer to 400 than 300, 4 is carried as the nearer approximation. When, in similar cases, the second figure is greater than 5, the error committed by carrying the first figure augmented by 1 is less than that made by carrying it without this augmentation. Abridged calculation. 2735894)123456|900000(45.124 109436 14020 13679 341 273 68 55 13 11 2 ... POWERS AND ROOTS OF NUMBERS. ... 123. When a number, multiplied by itself, is made 2, 3, 4,... n, times a factor, the product is called the 2nd, 3rd, 4th, nth power of the number; and the number, itself, the 2nd, 3rd, 4th, nth root of the corresponding power or product. (Art. 42.) The second and third powers of a number are called, respectively, the square and the cube of the number; the number itself is termed the square root of its second power, and the cube root of its third power. The fourth power and root are sometimes called biquadrate; but, generally, higher powers and roots than the third are called the 4th power, the 4th root; the 5th power, the 5th root, &c. The square of a number is obtained by one multiplication; the cube, by two; and, generally, any power by as many multiplications, less one, as there are ones in the exponent of the power. A high power may, however, be formed otherwise than by continual multiplication of the consecutive powers by the root. For example, ao=a3 × a3 × a3=a6 × a3 (Art. 42. ); and a12=a3 × a3 × a3 × a3=aa × aa × a1=a6 × a6. The subjoined Table gives the nine first powers of the numbers from 2 to 9. The symbol, called the radical sign, written on the left of a number, indicates that the square root of that number is to be extracted: thus a signifies that the square root of a is to be taken. The same symbol, with an appropriate figure, indicates the extraction of any other root: a, signifies that the cube root of a is to be extracted; in like manner a, indicates the 4th root of a; anda, the nth root of a. The processes for extraction of the square and cube roots of numbers are usually reckoned among the elementary operations of arithmetic. But for the extraction of roots with higher indices than 3 (excepting that of the 4th by means of two extractions of the square root) the calculations are very complicated and tedious. In such cases, and even in that of the cube root, it is easier to extract roots by means of a Table of the Logarithms of Numbers than by any elementary method. EXTRACTION OF THE SQUARE ROOT. 124. The product of two equal factors is formed in the same manner as that of two unequal factors, whether these be whole or fractional numbers. But the reverse operation, or that of returning from the product to the factors, is not the same in both cases. When the factors are unequal the product and one factor are given to find the other factor; when equal, the product of one factor by itself is given to find that factor. The first is division, the second, extraction of the square root. Extraction of the square root being a reverse process, the method to be followed must be sought by examining the composition of the square of a number. ... Now, denoting by a, b, c, d, the figures of a number, taken from left to right (a+b)2=a2 + 2ab+b2=a2 +(2a+b)b. (Art. 43.) =a2+2ab+2 ac+b2+2bc+c2. (a+b+c)2=a2 + 2ab+b2 + 2 (a + b) c + c2. (a + b + c + d)2 { And so on. [=a2+(2a + b) b + (2 (a + b) + c) c. [=a2+2ab+b2 + 2 (a + b) c + c2 + 2 (a + b + c) d + d2 the highest figure of the number (or the first figure reckoning from the left) plus the product of the second figure by the sum of twice the first and once the second figure, plus the product of the third figure by the sum of twice the first, twice the second, and once the third figures, plus the product of the fourth figure by the sum of twice the first, twice the second, twice the third, and once the fourth figures, plus &c. If a number is expressed by three figures (for example) denoted by a, b, c; the number, taking the relative values of the figures into consideration, is 100a +10b+c; and the preceding formula becomes (100a + 10b+c)2 = 10000 a2 + (2 × 100a + 10b) 10b + {2 (100 a + 10b) +c}c. This formula gives the means of determining, within certain limits, those figures of the power which contain the significant figures arising from the square of each figure of the root. c, denoting the last figure, c2 is expressed by one or by two figures (Art. 38.): therefore the numerical value of the square of the simple units of the root is contained in the tens and units of the power or in the units only. (106)2, or 100 b2 or b2 × 100, is expressed by three or by four figures, of which the two last are zeros. Whence the square of the tens of the root gives zeros to the simple units and tens of the power and significant figures to the hundreds and thousands or to the hundreds only. The significant figures of the square of the tens cannot therefore fall so low as the second or so high as the fifth figure of the power; that is, they are comprehended in the two figures on the left of the period containing the square of the simple units of the root. For the same reasons, the significant figures of the square of the third figure of the root (100a) are comprehended in the fifth and sixth figures or in the fifth figure of the power; the significant figures of the square of the fourth figure of the root are comprehended in the seventh and eight figures, or in seventh figure of the power, &c. Whence if the number expressing the square is divided into periods of two figures each, beginning from the place of simple units, the significant figures of the square of the simple units of the root are contained in the lowest or right-hand period; the significant figures of the square of the tens of the root in the second period; of the hundreds, in the third period, &c. Since the square of one figure may be expressed by two figures or by one, the highest period on the left may be composed of two figures or of one only. To show the manner of applying these principles, let it be required to extract the square root of 65536. This number consists of three periods, 6, 55, and 36; therefore its square root is composed of three figures. The first period, 6, is equal to a2+ the numbers carried from the lower terms. Also, the square number nearest to 6 and less than 6 is 4, which is the square of 2. ... a=2; a2=4; 10000 a2=40000. And 65536-40000=25536. ... 25536=(200a +10b)10b+ &c.; b being unknown. Again, 200 a × 10=200 × 2 × 10=4000. And 25536÷4000=6=b. ... (200a + 10b) 10b=(200 × 2 + 10 × 6) 10 × 6=(400+60) 60=27600. 27600 being greater than 25536, the value of b must be less than 6. Trying the next lower number, 5, (200 a + 10b)10b=(400 + 50)50=450 × 50 =22500. And 25536-225003036. .°. b=5 and 3036 = {2(100 a + 10b)+c}c; c being unknown. Lastly, 2(100a + 10b)=2(200+50)=500. And 3036500=6=c. {2(100 a + 10b) + c}c=506 x 6=3036. And 3036-3036=0. The last remainder being zero, the number is a perfect square. ... 100a10b+c=256 65536. = The process may be represented in either of the following forms, of which the second is that generally adopted. In the second or abridged form of calculation, the zeros in the square of the highest figure of the root are omitted; the significant figures of the square are subtracted from the first period of the power; and the second period is annexed to the remainder in the same manner as a figure of a dividend is |