Annexing 2 to 35, 352 is assumed to express the cube root of the given number. Since 3523-43614208, 43725658-43614208=111450; and 111450<3 × 3522+3 × 352+1, or 372769, it follows that 352 is the cube root of 43725658 to the nearest unit. If any divisor is not contained in the corresponding dividend, zero is written as the proper figure of the root; three figures of the given number are brought down to form a new dividend, and the process of calculation is continued in the same manner as if the zero had been a significant figure. The process cannot be commenced by finding the units of the cube root of a given number; for the cube of the units may give tens and hundreds, which cannot be distinguished from tens and hundreds arising from 3a2b and 3ab2. 274. To extract the cube root of a whole number: Rule. Separate the number into periods of three figures each (taking the figures from right to left) till a last period of one, two, or three figures is obtained. The number of periods is equal to the number of figures of the root. Extract the cube root of the greatest cube number contained in the first period of the given number (the period, namely, on the left), and subtract this cube from the first period; to the remainder annex the first figure of the second period; divide the number thus formed by three times the square of the figure of the root already found; annex the quotient to this first figure as the second figure of the root; form the cube of the number expressed by the first and second figures of the root, and subtract it from the first and second periods of the given number. But if the cube of the first and second figures is greater than the number expressed by the first and second periods of the given number, diminish the second figure of the root by one or more units, till a cube which can be subtracted from the first and second periods of the given number is obtained. The subtraction made, annex to the 2d remainder the 1st figure of the 3d period; divide the number thus formed by three times the square of the number expressed by the 1st and 2d figures of the root; the quotient figure (which, if not too great, is the third figure of the root) ought to be such that, annexing it to the number expressed by the 1st and 2d figures of the root, and forming the cube of the number thus obtained, this cube shall be capable of subtraction from the number composed of the 1st, 2d, and 3d periods of the given number. This series of operations is continued till all the periods of the given number are brought down. When, in the course of the operation, it is suspected that one of the quotient figures is too great, and it is in consequence diminished by one or more units, if, after forming the cube of the assumed root and subtracting it from the corresponding periods of the given number, the remainder is so considerable that it appears doubtful whether the last figure of the root is not too small, the following test can be employed: if the remainder is equal to or greater than three times the square of the assumed root plus three times the root plus unity, the root is too small; but if less, the root is accurate to within less than unity of the order of its last figure. Exercises in extraction of the cube root: 1st. The cube root of 103823 ?.. 8th. 21035?... 48228544? 485613?.. 5159780352?.. 343147021001 ?.... ...Ans. 47. .Ans. 27+. .Ans. 364. ...Ans. 234. ..Ans. 78+. ....Ans. 1728. ...Ans. 7011. 673373097125 ?.........................................Ans. 8765. 275. The following are tests by means of which it may be ascertained whether a number is a perfect cube: 1st. Every even number being expressible by 2n, and its cube by 8n3, the cube of every even number is divisible by 8. Every even number, therefore, which is not divisible by 8 is not a perfect cube. 2d. Every number which involves a prime factor, a, and which cannot be divided by a3, is not a perfect cube; for, if the cube root of a number which involves the prime factor a is a whole number, it must be of the form an, of which the cube a3n3 is divisible by a3. Hence a number divisible by 7 or by 11 must be divisible also by 343 or by 1331, in order that it may be a cube number. 3d. Every number terminated by a number of zeros not expressible by 3n (n being any number of the series 1, 2, 3 .. .) cannot be a cube number; for, if the cube root of such a number is exact, it must be a whole number terminated by one or more zeros. The cube of this cube root must contain three times as many zeros as are contained in the root, and consequently a number expressed by 3n, which is contrary to the supposi tion. 276. Extraction of the cube root by approximation. When the given number is not the cube of another whole number, the preceding operation gives only the integral part of the root. As to the fraction which ought to complete the root, it has been already proved (Art. 272) that this cannot be obtained exactly, but another fraction may be found which shall differ from it by as small a quantity as is desired. Let it be required to extract the cube root of a (a representing any whole number which is not a perfect cube) to within the th part of unity. 1 Then, if r is the root of the greatest cube number contained in a×n3 that is, the root of an3 to the nearest unit, the number an3 or a is compren3 Wherefore also the cube root of a is com prehended between the cube roots of these two numbers, or between and 1 n 1 n n +1. Consequently is the expression of the cube root of a to within the n th part of unity. Whence, to extract the cube root of a number to within the th part of unity (n denoting any whole number): Rule. Multiply the given number by the cube of the denominator n, extract the cube root of the product to the nearest unit, and divide the result by n. Example. Extract the cube root of 15 to within theth part of unity: 123 12 x 12 × 12=1728, 15 × 123=15 × 1728=25920, 3/25920=29+a fraction less than unity. Whence the required root is 277. Approximation in decimals is a particular case of the method of Article 276. Suppose that it is required to extract the cube root of 25 to within a smaller part of unity than that which is expressed by the decimal 001, that is, to within the oth part of unity; in this instance it is necessary to multiply 25 by 1000 (or to annex nine zeros to 25), to extract the cube root of the product, and to divide the result by 1000. Now 25000000000=2924+a fraction less than 1. Whence 2824, or 2.924, is the root required. To obtain the approximate value in decimals of a whole number which is not a perfect cube: Rule. Annex to the given number three times as many zeros as the root is required to contain decimal figures; extract the cube root of this number to the nearest unit, and point off the required number of decimal figures from the result. 278. Extraction of the cube root of a vulgar fraction: n If represents a fractional expression whose cube root is required, multiply the terms of the fraction by the square of the denominator; then n nd2 d d3• r Representing by r the integral part of the cube root of nd2, it follows, from a process of reasoning analogous to that of Article 276, that a is the If a more exact degree of approximation is required, it becomes necessary to approximate more nearly to the value of nd2, and to divide the result by d. When the denominator, without being already a perfect cube, contains factors of which some are perfect cubes and others perfect squares, the reduction to be made upon the fraction is more simple. Let it be required, as an illustration, to extract the cube root of 1. The denominator 360 can be decomposed into the prime factors 23, 32, and 5; whence, if the terms of the given fraction are multiplied by 113 × 75 3x52=75, it becomes 8475 23 X 33 X 53 303 · Since 8475=20+a fraction less than 1, and 3/113 3/360 303=30, 3/8475 If the denominator is a perfect cube, but the numerator not, the cube root of the number, approximated to any required degree of accuracy, is divided by the cube root of the denominator for the cube root of a given fractional expression. 279. Extraction of the cube root of a decimal fraction: Let it be required to find the cube root of 3.1415. The denominator of 3.1415, which is 10000, is not a perfect cube, being equal to 1000 × 10, or 103 × 10; it is rendered a perfect cube by the multiplication of the terms of the fraction by 102 or 100; this is accomplished by annexing two zeros to the terms of the fraction, which thus becomes 3141500 3141500 Now 3141500=146+a fraction less than 1, and 3/100=100, 146 1 ..3.1415= =146, a value true to within the 100th part of unity. 100 If a more exact degree of approximation is required, it is necessary to annex to the number three zeros for each additional decimal figure required in the root. To extract the cube root of a decimal fraction: Rule. Render the number of decimal figures a multiple of 3, and equal to three times the number of decimal figures required in the root; make abstraction of the decimal point in this number, and extract its cube root to the nearest unit; lastly, point off from the right of the root the required number of decimal figures. To obtain in decimals the cube root of a vulgar fraction, reduce the vulgar fraction to a decimal, and extract the cube root of the latter. 280. Exercises in extraction of the cube root of fractional expressions, and of numbers which are not perfect cubes: 1st. The cube root of 2 ?....................... 42.875?.......... Ans. 1.25992+. ...... Ans. 3.5. ....Ans. 3'06. Ans. 24. .Ans. 4. ...Ans. ·829. ....Ans. 21. ...Ans. 34. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th. 28 ?...... 11th. 85?............ ........Ans. 2·092. ......Ans. 7%. ...Ans. 2·057+. .......Ans. 5'03. 150 .....Ans. ·824+. 20th. 0034567?. .Ans. 1512+. SECTION IX. OF DIFFERENT SYSTEMS OF NUMERATION. 281. It has been explained in the Section on Numeration that it is possible to express any finite number by means of ten characters or figures, and of the conventional principle that any figure placed to the left of another shall express units ten times greater than the units expressed by that other figure. From the number of distinct characters employed, and the principle which fixes the relative values of these characters, this system of numeration has obtained the name of the decimal or denary system. The decimal is the system used by all civilized nations. The names of numbers, the figures employed to express them, and all the details of arithmetic are so identified with this system, that it may seem difficult to contemplate number otherwise than through its means. Yet innumerable other systems are equally capable of serving the same purposes. The determining cause of the first preference given to the decimal was probably the ready and constant facility afforded by the fingers of both hands to the breaking down of numbers into equal parcels, and the consequent more easy comparison of large numbers with unity or with each other. Some articles are enumerated by twos or pairs, others by threes or leashes, twelves or dozens, twentys or scores, &c., but in these instances the number of pairs, leashes, dozens, scores, &c. is always expressed by the ordinary names and characters of numbers. It is, however, possible to express all numbers by means of fewer or of more than ten figures, but there ought to be not fewer than two, and of the number, whatever it is, zero should be one. 282. The number of figures employed is called the base of a system of numeration. The system in which two figures are employed, or in which the number two three four five ternary, quaternary, quinary, ten twelve denary, duodenary. In every system of numeration like the denary the convention must be made that any figure placed on the left of another shall express units as many times greater, with regard to the units of that other figure, as there are units in the base, or as there are figures in the system. Thus, in the binary system each figure obtains a value from two to two times greater, as it is moved one, two, three... places towards the left; in the ternary system the value of a figure is from three to three times greater; in the quinary, from five to five times; in the duodenary, from twelve to twelve times greater; and, in general, in a system of which the base is b, each figure acquires a value from b to b times greater. |