INVOLUTION AND EVOLUTION TABLE. By means of the following table the square, cube, square root, cube root, and reciprocal of any number may be obtained correct always to five significant figures, and in the majority of cases correct to six significant figures. In any number, the figures beginning with the first digit * at the left and ending with the last digit at the right, are called the significant figures of the number. Thus, the number 405,800 has the four significant figures 4, 0, 5, 8; and the number .000090067 has the five significant figures 9, 0, 0, 6, and 7. The part of a number consisting of its significant figures is called the significant part of the number. Thus, in the number 28,070, the significant part is 2807; in the number .00812, the significant part is 812; and in the number 170.3, the significant part is 1703. In speaking of the significant figures or of the significant part of a number, the figures are considered, in their proper order, from the first digit at the left to the last digit at the right, but no attention is paid to the position of the decimal point. Hence, all numbers that differ only in the position of the decimal point have the same significant part. For example, .002103, 21.03, 21,030, and 210,300 have the same significant figures 2, 1, 0, and 3, and the same significant part 2103. The integral part of a number is the part to the left of the decimal point. It will be more convenient to explain first how to use the table for finding square and cube roots. SQUARE ROOT. First point off the given number into periods of two figures each, beginning with the decimal point and proceeding to the left and right. The following numbers are thus pointed off: 12703, 1/27/03; 12.703, 12.70′30; 220000, 22′00'00; .000442, .00'04'42. * A cipher is not a digit. Having pointed off the number, move the decimal point so that it will fall between the first and second periods of the significant part of the number. In the above numbers, the decimal point will be placed thus: 1.2703, 12.703, 22, 4.42. If the number has but three (or less) significant figures, find the significant part of the number in the column headed n; the square root will be found in the column headed V or 10n, according to whether the part to the left of the decimal point contains one figure or two figures. Thus, 14.42 = 2.1024, and 22 = √10 × 2.204.6904. The decimal point is located in all cases by reference to the original number after pointing off into periods. There will be as many figures in the root preceding the decimal point as there are periods preceding the decimal point in the given number; if the number is entirely decimal, the root is entirely decimal, and there will be as many ciphers following the decimal point in the root as there are cipher periods following the decimal point in the given number. Applying this rule, 220000 469.04 and 1.000442 .021024. The operation when the given number has more than three significant figures is best explained by an example. EXAMPLE. (a) 13.1416 = ? (b) | 2342.9 = ? = SOLUTION. (a) Since the first period contains but one figure, there is no need of moving the decimal point. Look in the column headed n2 and find two consecutive numbers, one a little greater and the other a little less than the given number; in the present case, 3.1684 1.782 and 3.1329 = 1.772. The first three figures of the root are therefore 177. Find the difference between the two numbers between which the given number falls, and the difference between the smaller number and the given number; divide the second difference by the first difference, carrying the quotient to three decimal places and increasing the second figure by 1 if the third is 5 or a greater digit. The two figures of the quotient thus determined will be the fourth and fifth figures of the root. In the present example, dropping decimal points in the remainders, 3.1684-3.1329 = 355, the first difference; 3.14163.1329 or .25. (b) √2342.9 = ? Pointing off into periods we get 23'42.90; moving the decimal point we get 23 4290; the first three figures of the root are 484; the first difference is 23.5225 23.4256 = 969; the second difference is 23.4290 — 23.4256 == 34 969 .035+, or .04. 2342.9 = 48.404. 34; Hence, 87, the second difference; 87 ÷ 355 = .245+, Hence, 3.1416 = 1.7725. CUBE ROOT. The cube root of a number is found in the same manner as the square root, except the given number is pointed off into periods of three figures each. The following numbers would be pointed off thus: 3141.6, 3'141.6; 67296428, 67′296'428; 601426.314, 601/426.314; .0000000217, .000'000'021'700. Having pointed off, move the decimal point so that it will fall between the first and second periods of the significant part of the number, as in square root. In the above numbers the decimal point will be placed thus: 3.1416, 67.296428, 601.426314, and 21.7. If the given number has but three (or less) significant figures, find the significant part of the number in the column headed n; the cube root will be found in the column headed, 10n, or 100n, according to whether one, two, or three figures precede the decimal point after it has been moved. Thus, the cube root of 21.7 will be found opposite 2.17, in column headed 10n, while the cube root of 2.17 would be found in the column headed ñ, and the cube root of 217 in the column headed 100n, all on the same line. If the given number contains more than three significant figures, proceed exactly as described for square root except that the column headed n3 is used. EXAMPLE. (a) .0000062417 = ? (b) 50932676 ? SOLUTION. (a) Pointing off into periods, we get 000/006'241/700; moving the decimal point, we get 6.2417. The number falls between 6.22950 = 1.843 and 6.33163 1.853; the first difference = 10213; the second difference is 50932676 6.24170-6.22950 =1220; 1220 ÷ 10213= .119+, or .12, the fourth and fifth figures of the root. The decimal point is located by the rule previously given; hence, .0000062417 = .018412. (b) ? As the number contains more than six significant figures, reduce it to six significant figures by replacing all after the sixth figure with ciphers, increasing the sixth figure by 1 when the seventh is 5 or a greater digit. In other words, the first five figures of 50932700 and of 50932676 are the same. Pointing off into periods, we get 50′932/700; moving the decimal point, we get 50.9327, which falls between 50.6530 = 3.703 and 51.0648 = 3.713; the first difference is 4118; the second difference is 2797; 2797 ÷ 4118 = .679+, or .68. The integral part of the root evidently con tains three figures; hence, 50932676 = 370.68, correct to five figures. = SQUARES AND CUBES. If the given number contains but three (or less) significant figures, the square or cube is found in the column headed n2 or n3, opposite the given number in the column headed n. If the given number contains more than three significant figures, proceed in a manner similar to that described for extracting roots. To square a number, place the decimal point between the first and second significant figures and find in the column headed √n or 1/10 n two consecutive numbers, one of which shall be a little greater and the other a little less than the given number. The remainder of the work is exactly as heretofore described. To locate the decimal point, employ the principle that the square of any number contains either twice as many figures as the number squared or twice as many less one. If the column headed 10n is used, the square will contain twice as many figures, while if the column headed √n is used, the square will contain twice as many figures as the number squared, less one. If the number contains an integral part, the principle is applied to the integral part only; if the number is wholly decimal, there will be twice as many ciphers following the decimal in the square or twice as many plus one as in the number squared, depending on whether 10n or n column is used. For example, 273.422 will contain five figures in the integral part; 4516.22 will contain eight figures in the integral part, all after the fifth being denoted by ciphers; .00294532 will have five ciphers following the decimal point; .0524362 will have two ciphers following the decimal point. EXAMPLE. (a) 273.422 = ? (b) .0524362 = ? SOLUTION. (a) Placing the decimal point between the first and second significant figures, the result is 2.7342; this number occurs between 2.73313 7.47 and 2.73496 = 7.48 in the column headed. The first difference is 2.73496 2.73313 183; the second difference is 2.73420 2.73313 = 107; and 107 183 .584+, or .58. Hence, 273.422 = 74,758, correct to five significant figures. (b) Shifting the decimal point to between the first and second significant figures, we get the number 5.2436, which falls between 5.23450 = 27.4 and 5.24404 = 27.5. The first difference is 954; the second difference is 910; 910 954 = .953+, or .95. Hence, .0524862 = .0027495, to five significant figures. A number is cubed in exactly the same manner, using the column headed in, 10 n, or 100n, according to whether the first period of the significant part of the number contains one, two, or three figures, respectively. If the number contains an integral part, the number of figures in the integral part of the cube will be three times as many as in the given number if column headed 100 n is used; it will be three times as many less 1 if the column headed 10n is used; and it will be three times as many less 2 if the column headed is used. If the given number is wholly decimal the cube will have either three times, three times plus one, or three times plus two, as many ciphers following the decimal as there are ciphers following the decimal point in the given number. EXAMPLE. (a) 129.6843 = ? (b) .764423 = ? (c), .0324253 SOLUTION. (a) Placing the decimal point between the = =? = = |