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1st dividend 37,0 (14 = 1st divisor. 7.2 × 72 = 51,84
There is, however, a method, which will save considerable labour in multiplying.
In the last example, for instance, having found the second figure of the root 2, instead of raising the whole 72 to the second power, we may abridge it very much by observing, that the second power of the 70, answering to a” in the formula, has already been found and subtracted ; therefore it only remains to find 2 a b + bo, and subtract it also. But the 140 is 2 a, and the figure 2 found for the root answers to b; therefore if we add 2 to 140, it becomes 142 = 2 a -i- b. If this be now multiplied by 2 or b, it becomes
This completes the second power of 72, which, subtracted from 370, leaves 86 as before. t
Prepare as before, and find the third figure of the root. Observe that the 2d power of 720 or a* + 2 a b + b has already been found and subtracted ; it only remains to find the other parts, viz. 2 (a + b) c + co. The divisor 1440 answers to 2 (a + b). Add 6, the figure of the root just found, to this, and it becomes 1446, answering to 2 (a + b) + c. If this be multiplied by 6, it becomes 1446 × 6 = 8676 = 2 (a + b) c + co. This completes the second power of 726, which, subtracted from 8676, the number remaining in the work, leaves nothing. Operation. 52,70,76, (726 root. 49
}st dividend 370 14 1st divisor.
2d dividend 8676 144 2d divisor
The same principle will apply when the root consists of any number of figures whatever.
What is the root of 533837732164 P
In the first place I observe that the second power of the tens can have no significant figure below hundreds, therefore the two right hand figures may be rejected for the present. Also the second power of the hundreds can have no significant figure below tens of thousands, therefore the next two may be rejected. For a similar reason the next two may be rejected. In this manner they may all be rejected two by two until only one or two remain. Begin by finding the root of these, and proceed as above.
After separating the figures two by two, as explained above, I find the greatest second power in the left hand division. It is 49, the root of which is 7. I subtract 49 from 53, and bring down the next two figures, which makes 438. Now considering the 7 as tens, I proceed as if I were finding the root of 5338: that is, I double the 7, which makes 14 for a divisor, and see how many times it is contained in 43, rejecting the 8 on the right. I find 3 times. I write 3 in the root at the right of 7, and also at the right of 14. I multiply 143 by 3, and subtract the product from 438. I then bring down the next two figures, which make 937. I double 73, or, which is the same thing, I double the 3 in 143; for the 7 was doubled to find 14. This gives 146 for a divisor. I seek how many times 146 is contained in 93, rejecting the 7 on the right, as before. I find it is not contained at all. I write zero in the root, and also at the right of 146. I then bring down the next two figures. I seek how many times 1460 is contained in 9377, rejecting the 3 on the right. I find 6 times. I write 6 in the root, and at the right of 1460, and multiply 14606 by 6, and subtract the product from 93773. I then bring down the next two figures, and double the right hand figure of the last multiplicand, and proceed as before; and so on, till all the figures are brought down. The doubling of the right hand figure of the last multiplicand, is always equivalent to doubling the root as far as it is found.
From the above examples, we derive the following rule for extracting the second root.
Ist. Beginning at the right, separate the number into parts of two figures each. The left hand part may consist of one or two
2nd. Find the greatest second power in the left hand part, and write its root as a quotient in division. Subtract the second power from the left hand part.
3d. Bring down the two next figures at the right of the remainder. Double the root already found for a divisor. See how many times the divisor is contained in the dividendrejecting the right hand figure. . Write the result in the root, at the right of the figure previously found, and also at the right of the divisor.
4th. Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the whole dividend.
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 + 2 a + 1.
That is, if we have the second power of any number, the se– cond power of a number larger by 1, is found by multiplying the first number by 2, increasing the product by 1, and adding it to the power. For example, the second power of 10 is 100; the second power of 11 is 100 + 2 × 10 + 1 = 121. The second power of 12 is 121 + 2 × 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 3 is 3 × 3 = 24. The second power of
a ; , a a a” - 5 IS W X T} – j." Hence the root of a fraction is found by extracting the root of the numerator, and of the denominator; thus the root of #} is 4. If either the numerator or denominator has no exact root, the root of the fraction cannot be found exactly. Thus the root of #} 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 . This is exact, within less than 4.
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 ####, the root of which is nearest to #4. This is the true root within less than or.
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.