Dividing 22 by 4 (2t) gives 5 (u) for a quotient. This unit's figure (5) is to be added to the double of the tens (40), and the sum multiplied by the unit's figure.

Multiplying 40+5=45(2t+u), by 5 (u), the product is 225, which is double the tens plus the units, multiplied by the units. As there is no remainder, we conclude that 25 is the exact square root of 625.

In squaring and doubling the tens, it is customary to omit the ciphers, and to add the unit's figure to the double of the tens, by merely writing it in the unit's place. The actual operation is usually performed as in the margin.

625 25

400

45 225

2. Required to extract the square root of 59049.

Since this number consists of five places of figures, its square root will consist of three places. (Art. 174.) We, therefore, separate it into three periods.

In performing this operation, we find the square root of the number 590, on the same principle as in the preceding example.

225

59049 243

4

44/190

176

483 1449

1449

We next consider 24 as so many tens, and proceed to find the unit's figure (3) as in the preceding example.

From these illustrations, we derive the following

Rule for the Extraction of the Square Root of Numbers.-1st. Separate the given number into periods of two places each, beginning at the unit's place. (The left period. will often contain but one figure.)

2d. Find the greatest square in the left period, and place its root on the right, after the manner of a quotient in division. Subtract the square of the root from the left period, and to the remainder bring down the next period for a dividend.

3d. Double the root already found, and place it on the left for a divisor. Find how many times the divisor is contained in the dividend, exclusive of the right hand figure, and place the figure in the root and also on the right of the divisor.

4th. Multiply the divisor thus increased by the last figure of the root; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.

5th. Double the whole root already found for a new divisor, and continue the operation as before, until all the periods are brought down.

NOTE.-If, in any case, the division can not be effected, place a cipher in the root and divisor, and bring down the next period

176. In extracting the square root of numbers, the remainder may sometimes be greater than the divisor, while the last figure of the root can not be increased. To explain this,

Let a and a+1, be two consecutive numbers.

Then, (a+1)2=a2+2a+1, the square of the greater.

Their difference is 2a+1. Hence,

The difference of the squares of two consecutive numbers is equal to twice the less number, increased by unity.

Therefore, when any remainder is less than twice the root already found, plus one, the last figure can not be increased.

177. Since X, the square root of is; that is, 14 V

=

V25. Hence, we have the following

Rule for Extracting the Square Root of a Fraction.Extract the square root of both terms.

When the terms of a fraction are not perfect squares, they may sometimes be made so by reducing. Thus,

Find the square root of 29.

Here,

20 4X5

45 9X5

By canceling the common factor 5, the fraction

becomes the square root of which is .

When both terms are perfect squares, and contain a common factor, the reduction may be made either before or after the square root is extracted.

178. A Perfect Square is a number whose square root can be exactly ascertained; as, 4, 9, 16, etc.

An Imperfect Square is a number whose square root can not be exactly ascertained; as, 2, 3, 5, 6, etc.

Since the difference of two consecutive square numbers, a2 and a2+2a+1, is 2a+1; therefore, there are always 2a imperfect squares between them.

Thus, between the square of 5 (25) and the square of 6 (36), there are 10 (2a=2×5) imperfect squares.

A quantity, affected by a radical sign, whose root can not be exactly found, is called a radical, or surd, or irrational root; as, √/2, †/5, etc.

The root of such a quantity, expressed with more or less accuracy in decimals, is called the approximate value, or approximate root. Thus, 1.414+ is the approximate value of 2.

179. It might be supposed, that when the square root of a whole number can not be expressed by a whole number, it might be exactly equal to some fraction. That it can not, will now be shown.

Let c be an imperfect square, as 2, and, if possible, let its square

Now, by supposition, a and b have no common factor; therefore, their squares, a2 and b2, can have no common factor, since to square a number, we merely repeat its factors. Consequently,

a2

b2

must be in its lowest terms, and can not be equal to a whole a2 number. Hence, the equation c= is not true, and the supposi

α

tion on which it is founded, that is, that √c=, is false; therefore, the square root of an imperfect square can not be a fraction.

APPROXIMATE SQUARE ROOTS.

180. To explain the method of finding the approximate square root of an imperfect square, let it be required to find the square root of 5 to within.

If we reduce 5 to a fraction whose denominator is 9 (the square of 3, the denominator of the fraction), we have 5=45.

Now, the square root of 45 is greater than §, and less than ; hence, §, or 2, is the square root of 5 to within .

To generalize this explanation, let it be required to ex

tract the square root of a to within a fraction

an2 N2

1

n

Write a (Art. 127) under the form and denote the entire part of the square root of an2 by r. Then, an2 will be comprised between 2 and (r+1)2, and the square root of will be comprised

+1

n

an 2

n2

Rule for Extracting the Square Root of a Whole Number to within a Given Fraction.-1. Multiply the given number by the square of the denominator of the fraction, which determines the degree of approximation.

2. Extract the square root of this product to the nearest unit, and divide the result by the denominator of the fraction.

1. Find the square root of 3 to within 3.

Since the square of 10 is 100, the square of 100, 10000, and so on, the number of ciphers in the denominator of a decimal fraction is doubled by squaring it. Therefore,

When the fraction which determines the degree of approximation is a decimal, add two ciphers for each decimal place required; and, after extracting the square root, point off from the right one place of decimals for each two ciphers added.

6. Find the square root of 3 to five places of decimals Ans. 1.73205.

7. Find the square root of 7 to five places of decimals Ans. 2.64575. 8. Find the square root of 500. Ans. 22.360679+.

181. To find the approximate square root of a fraction, 1. Required to find the square root of to within 4.

4-4×7-38

The square root 28 is greater than and less than ; therefore, is the square root of to within less than 4. Hence, to find the square root of a fraction to within one of its equal parts,

the

Rule.-Multiply the numerator by its denominator, extract. square root of the product to the nearest unit, and divide the result by the denominator.