« ΠροηγούμενηΣυνέχεια »
5th. Bring down the next two figures as before, to form a new dividend, and double the root already found, for a divisor, and proceed as before. The root will be doubled, if the right hand figure of the last divisor be doubled.
If it happens that the divisor is not contained in the dividend when the right hand figure is rejected, a zero must be written in the root, and also at the right of the divisor; and the next figures must be brought down, and then a new trial made.
If it happens that the figure annexed to the root is too small, it may be discovered as follows.
The second power of a + 1 is a' + 2a + 1.
That is, if we have the second power of any number, the second power of a number larger by 1, is found by multiplying the first number by 2, increasing the product by i, and adding it to the power. For example, the second power of 10 is 100; the second power of 11 is 100 + 2 x 10 + 1 = 121. The second power of 12 is 121 + 2 X 11 +1 = 144, &c.
If then the remainder, after subtraction, is equal to twice the root already found plus 1, or greater, the last figure of the root must be increased by 1.
In the last example, the first dividend was 43,8 and the divisor 14; the figure put in the root was 3, and the remainder was 9. If 2 instead of 3 had been put in the root, the remainder would have been 154, which is considerably larger than twice 72, and would have shown, that the figure should be 3 instead of 2.
There are many numbers, of which the root cannot be exactly assigned in whole or mixed numbers. Thus 2, 3, 5, 6, 7, have no assignable roots. That is, no number can be found, which, multiplied into itself, shall produce either of these numbers. This is the case with all whole numbers, which have not an exact root in whole numbers.
This may be proved, but the demonstration is so difficult, that few learners would comprehend it at this stage of their progress. The proof may be found in Lacroix's Algebra. The learner, however, may easily satisfy himself by trial. We shall soon find a method of approximating the roots of these numbers, sufficiently near for all purposes.
XXIX. Extraction of the second Root of Fractions. Fractions are multiplied together by multiplying their numerators together, and their denominators together. Hence the second power of a fraction is found by multiplying the numerator into itself, and the denominator into itself; thus the second power of is = The second power of
Hence the root of a fraction is found b
72 by extracting the root of the numerator, and of the denominator ; thus the root of is 7.
If either the numerator or denominator has no exact root, the root of the fraction cannot be found exactly. Thus the root of 'I is between, and / or 1. It is nearest to .
The denominator of a fraction may always be rendered a perfect second power, so that its root may be found ; and for the numerator, the number which is nearest to the root must be taken. Suppose it is required to find the root of. If both terms of the fraction be multiplied by 5, the value of the fraction will not be altered, and the denominator will be a perfect second power,
= The root is nearest f. This is exact, within less than .
If it is necessary to have the root more exactly ; after the fraction has been prepared by multiplying both its terms by the denominator, we may again multiply both its terms by some number that is a perfect second power. The larger this number, the more exact the result will generally be.
If both terms be multiplied by 144, which is the second power of 12, it becomes 18, the root of which is nearest to 14. This is the true root within less than o'ó.
We may approximate in this way the roots of whole numbers, whose roots cannot be exactly assigned.
If it is required to find the root of 2, we may change it to a fraction, whose denominator is a perfect second power.
2 = 1
The root of is nearest to ji 11gThis differs from the true root by a quantity less than i'sIf greater exactness is required, a number larger than 144 may be used.
1. What is the root of 8" ? 2. What is the root of 3. What is the root of 1312. = 5370? 4. What is the root of 28317? 5. What is the approximate root of ? 6. What is the approximate root of is? 7. What is the approximate root of 34 ? 8. What is the approximate root of 1725 ? 9. What is the approximate root of 3 ? 10. What is the approximate root of 7? 11. What is the approximate root of 417 ?
The most convenient numbers to multiply by, in order to approximate the root more nearly, are the second powers of 10, 100, 1000, &c., which are 100, 10000, 1000000, &c. By this means, the results will be in decimals.
To find the root of 2 for instance, first reduce it to hundredths.
2=its, the approximate root of which is 14 = 1.4.
Again 2 = 1:885, the approximate root of which is 141 = 1.41.
Again, 2 = 4:66:66, the approximate root of which is 1414 = 1.414.
In this way we may approximate the root with sufficient accuracy for every purpose. But we may observe, that at every approximation, two more zeros are annexed to the number. In fact, if one zero is annexed to the root, there must be two annexed to its power ; for the second power of 10 is 100, that of 100 is 10000, &c.
This enables us to approximate the root by decimals, and we may annex the zeros as we proceed in the work, always annexing two zeros for each new figure to be found in the root, in the same manner as two figures are brought down in whole numbers.
The root of 2 then may be found as follows.
2 (1.41421, &c. root.
11 90,0 (2824
60 40,0 (28282
3 83 60,0 (282841
1 00 75 9 12. What is the approximate root of 28 ? 13. What is the approximate root of 243 ? 14. What is the approximate ræot of 27068? 15. What is the approximate root of 243 ? 2431 = 243 7 = 18375 = '1837668', &c. The approximate root of which is 10= 15.6, &c.
But it is plain that this may be performed in the same manner as the above. For if the number 243375000 be prepared in the usual way, it stands thus ; 2,43,37,50,00. Now
243375020 = 243.375000. If we take this number and begin at the units and point towards the left, and then towards the right in the same manner, the number will be separated into the same parts, viz. 2,43.37,50,00. The root of this number may be extracted in the usual way, and continued to any number of decimal places by annexing zeros.
N. B. The decimal point must be placed in the root, before the first two decimals are used. Or the root must contain one half as many decimal places as the power, counting the zeros which are annexed.
16. What is the approximate root of 213.53 ?
Questions producing pure Equations of the Second
1. A mercer bought a piece of silk for £16. 4s.; and the number of shillings which he paid per yard, was to the number of yards, as 4 to 9. How many yards did he buy, and what was the price of a yard ?
Let x = the number of shillings he paid per yard.
= 27. 4
Ans. 27 yards, at 12s. per yard. 2. A detachment of an army was marching in regular column, with 5 men more in depth than in front; but
the enemy coming in sight, the front was increased by 845 men; and by this movement the detachment was drawn up in 5 lines, Required the number of men.