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It is easy
the fourth root,
and so on, according to a very obvious system. (See page 65.) There is another notation of roots, which consists in using fractional indices, instead of radical signs. Thus,
is denoted also by 10
which may be read, « 10 to the power of one-third." This mode of representing roots, although convenient and natural, is, however, seldom adopted in arithmetic.
Simple as is the process of involution, or the raising of a number to any required power, the opposite process—that of obtaining the number of which a certain power is given-must be regarded at somewhat complex and tedious, especially when a root of higher degree than the square root is sought to be extracted. enough to multiply 14 three times consecutively into itself, and thus produce its fourth power, 38416. But if 38416 be proposed as a number of which the unknown fourth root is to be ascertained, it is not likely that a method of solving the question, apart from a series of trials or guesses, would suggest itself to any person unacquainted with algebra.
Roots in general may be extracted by an application of the ingenious and beautiful system known as “ Horner's method of resolving numerical equations."* As however, it is rarely necessary in practice to compute any root except the square root, and as roots of all degrees can be found when requisite with great facility and an exactness sufficient for most purposes, by the aid of logarithms, as will be shown in another part of this work, it is not designed in the present place to treat of the subject of evolution further than to point out the details and principle of the operation specially used in the extraction of the square root, with a brief notice of a useful rule for finding approximately the cube root.
It is not every number which has an exact square root. 7 x 7 = 49, but no number can be found, either whole or partly whole and fractional, which multiplied by itself will exactly produce 7. Indeed comparatively few numbers have the property of being the precise squares of other numbers. The truth of this statement as regards the smaller numbers will be evident on looking over an ordinary multiplication table. Although in many instances square roots do not exist, still it is always possible by means of extended decimals to obtain a number which when raised to its second power shall give a product differing from the number of which the square root is required by as small a value as we please. 2, for example, has no exact square root, but the square of 1.4142, differs from 2 by less than 10000 of unity, and a nearer and nearer approach to 2 may be had on continuing the proper process indefinitely.
Extraction of the Square Root. The number of figures in the square root of any number may readily be inferred from the following considerations : The Square-root of
is 1 100
&c. It is thus seen that the square root of any number between 1 and 100 must be between 1 and 10, and have therefore but one figure in its integral part,--that the
* For an account of this process, with examples worked out at length, see amongst other works, Thomson's Arithmetic 34th Edition, page 190, and also a note at the
end of the volume; or De More gan's Arithmetic, 5th Edition, Appendix, page 210.
square root of every whole number between 100 and 10,000, must lie between 10 and 100, or consist of two figures,-of every whole number between 10,000 and 1,000,000 must be between 100 and 1,000 or consist of three figures, and so on. If then beginning from the right hand or the place of units, we divide any whole number of which the square root is required, into periods of two figures each, (ending with a final period on the left of one figure only, should the given number be odd,) it is evident that the number of periods thus formed will indicate the number of figures in the square root sought. According to this system of partition into periods, or pointing as it is termed, a one or a two-figure number is readily seen to have a square root of one figure ; a three or a four-figure number to have a square-root of two figures ; &c. &c.
The entire process of the extraction of the square root of whole numbers is governed by the following rule.
RULE.First point the number by placing a point or dot over the units' figure, and thence over every second figure to the left ; or otherwise mark off in the same direction periods of two figues each as often as possible. The number of such periods, together with the period of one figure left in the case of odd numbers, corresponds to the number of figures in the root.
Find the greatest number the square of which is contained in or does not exceed the number in the first period at the left. As no period consists of more than two figures, it will be seen in a moment which of the numbers, from 1 to 9 inclusive, when multiplied by itself, approaches most nearly to the period in question. This is the first figure of the root, which place in the form of a quotient to the right of the given number.
Subtract the square of the first root-figure from the first period, and to the right of the remainder, if any, annex the second period.
Divide the number thus formed, exclusive of its units' figure, by twice the part of the root already found ; annex the quotient to the root-figure and also to the divisor. Next, multiply the divisor as it now stands by the part of the root last obtained, and subtract the product from the number above mentioned, consisting of the first remainder and the second period.
Should there be more periods to be brought down, proceed in a similar manner step after step until all the periods are exhausted, taking as the new divisor in each case the double of the root-figures previously ascertained, and otherwise repeating the operation directed with regard to the extraction of the second figure of the root.
When a remainder is left after the last period has been used, the number is thus proved to have no exact square root ; but by setting a decimal point after the number and annexing pairs of ciphers to represent additional periods, the work may be continued to any extent, and a result consisting of a whole number and decimal fraction obtained, which if multiplied into itself shall differ from the given number by as small an amount as we choose. Example (1.) Extract the square root of 393,129.
39 31 29 (627
Here we begin by dividing the number into three periods of two figures each
The greatest square* number less than 39, the first period, is 36, the square root of which, 6, is placed in the quotient, and the square, 36, subtracted from 39, leaving the remainder 3.
We then bring down the 2nd period, 31, annexing it to the remainder 3; double 6, the figure in the root, for a trial divisor, and see how often 12 goes into 33, that is, 331 wanting its units' figure. The quotient 2 is placed as the second figure of the root, and is also written after the divisor 12, making the latter 122. Twice 122 is now taken from 331, and to the remainder, 87, is brought down the third period 29.
Lastly, we double the root-figures 62, for a new trial divisor, or what is the same thing, add the second figure of the root to the preceding divisor, and see how often the result, 124, is contained in 872, the units’ figure of the dividend being omitted as before ; the quotient 7 is annexed to the root-figures and also to the divisor 124 ; 1247 is multiplied by 7, and the product deducted from 8729. As there is no remainder, we conclude that 627 is the exact square root of the given number. To prove this, we multiply 627 by itself, and find the square to be 393,129.
If at any point of an operation the number to be divided by the trial divisor should be less than it, we must annex a cipher to the root-figures and also to the trial divisor, bring down the next period, and proceed according to the rule. Should the new divisor prove too large, another cipher must be placed both in the root and the divisor, and the next period brought down, and so on. Example (2.) Find the square roots of 18,550,249 ; 77,841 and 10. (1) (2)
Approx. sq. root
* By a "square number" is meant one which has an exact square root, thus 4, 9, 16, 25, &c. are square numbers.
+ It is obvious that the actual annexing of ciphers to the given number m be dispensed with at the pleasure of the operator, as in ordinary division of decimals, and the supposed periods brought down as often as is necessary.
In (1) there is an example of what should be done when the dividend, minus its last figure, will not contain the trial divisor.
In (2) it will be observed that the second remainder, 49, is greater than the divisor 47; but if the second figure of the latter had been made 8 instead of 7, the product of 8 x 48 or 384 could not be subtracted from 378.
In (3) five pairs of ciphers are annexed to 10, which evidently is not a square number, and an approximate square root is then extracted to five places of decimals. By adding more ciphers and continuing the process, a result may be arrived at, the square of which shall be within any assigned degree of nearness to 10. In order to make the last decimal figure required as correct as possible, an additional figure should be obtained, and if that amounts to or exceeds 5, the preceding figure should be increased by a unit. Thus, in the present instance, if four places of decimals were considered sufficient, the root extracted to one place further, and then cut down, should be written 3.1623, as the fifth figure proves to be 7.
When there are decimals in the root, there will be twice the number of them in its square.
Accordingly, the number of decimal places in the square must always be even ; from which it follows that in dividing into periods a number consisting of an integer and a fraction, pairs of figures should be marked off to the right as well as to the left of the decimal point. As a cipher may be put at the end of a series of decimals without altering its value, the number of places can thus be rendered even in all instances.
Contracted Process.—The operation of extracting a root to many figures of decimals may be greatly shortened, and but little accuracy sacrificed, by employing contracted division, as soon as one more than half the required number of decimals have been found. If, for example, the extraction of a root extending altogether to 12 figures were undertaken, the last five figures might be determined by simple division with nearly as much correctness as if the regular process were continued.
Example (3.) Extract the square root of 115.29735 to 10 figures, including decimals.
1|15.29 7350(100737660362 207)1529
2143)8073 21467)164450 214746)1418100
4 Answer. 10.73766036. Had the result been worked out all through as the rule directs, the figures would have been 10.737660359, which, after cutting off and allowing for the last place of decimals, are the same as those otherwise obtained.
The two kinds of abbreviation used in the process above shown, are described on pages 63 and 90 respectively. It is plain that the method of contracted division may be safely employed in the latter part of the calculation, as the difference between the successive trial divisors amounts only to the double of one of the root figures, and this addition cannot affect the contracted divisor, from the fact that its final figures are successively cut off.
Square Root of a Decimal.-In computing the square root of a number which consists entirely of decimals, we consider, that as the square root of 100 is 10, that of 10,000 is 100, &c., 80 the square root of .86, for instance, must be the tenth of that of 100 x 86, or 86; that the square root of •0441 must be the one hundredth part of that of 10,000 x .0441 = 441, and so on. Accordingly the rule for the extraction of the square root of a decimal is–Make the number of decimal places even, if necessary, by annexing a cipher, proceed as in the case of a whole number,
and point off from the root as many decimal places as there are periods of two figures each in the given number together with the cipher that may have been added. Example (4.) What is the square root of •0002704 to 6 decimal places ?
.0002704 = .00027040
17 Here a cipher is written after the given decimal, so that the number of places may be even. The preliminary ciphers are then thrown away as being of no effect in the operation, and the remaining figures treated as a whole number. When all the periods have been used, four decimal places are made in the root by prefixing a cipher to the three places which have been so far obtained, since there are four periods or eight figures in the proposed decimal, counting the added cipher. The process is then continued until one more than the required number of places has been found, contracted division being employed in the calculation of the last three figures.
To extract the square root of a vulgar fraction.–1°, Convert the fraction into a decimal and ascertain its square root by the preceding rule ; or 29. Reduce the fraction to its lowest terms, and extract the roots of the numerator and denominator separately, forming a fraction with the results ; or 3o. Multiply the numerator and denominator together, extract the root of the product, and divide by the denominator. The last process is especially applicable where the terms of the given fraction have not exact square roots.
Example (5.) Find the square roots of 1: and 1. respectively, by each of the foregoing methods.
•7647 = .8745
(1.) % = 5625 & V5625 = •75 : 4 = -7647 &
✓ (2.) to - 1 Vio 용
1' = yit= $1931 (3.V ==1 ®) Vis=\*o**_-1-:V ##*H_
806 . =
4 16 =
It is difficult for persons who have no knowledge of algebra to obtain a clear perception of the principle on which the extraction of the square root depends, but the subject may be brought to a certain extent within the scope of arithmetical