366. The Radical Sign, √, is used to indicate a root. Thus,

V 16 means the second, or square, root of 16.

25 means the third, or cube, root of 25.

The number in the opening of the sign, called the Index of the root, denotes the name of the root.

The index of the square root may be understood. Thus, √ 81, or 81, may indicate the second, or square, root of 81. 367. Evolution is the process of finding roots. It is the reverse of involution.

SQUARE ROOT.

368. To extract the square root of a number is to find one of the two equal factors which produce it.

369. The square of a number contains twice as many figures as the root, or twice as many less one.

12: = 1.

Thus,

92 = 81. 99298'01.

102 = 1'00.

1002 = 1'00'00.

999299'80'01.

370. If a power is separated into periods of two figures each, beginning at the decimal point, the number of periods will show the number of figures in the root. Thus,

The square root of 2'35.'92'96 contains two integral and two decimal figures.

371. The square of the highest order of units in the root is found in the highest period of the power. Thus,

9 tens2, or 902

9 hundreds2, or 9002
9 thousands2, or 90002

= 81 hundreds, or 81'00.

= 81 ten-thousands, or 81'00'00. 81 millions, or 81'00'00'00.

=

372. The parts which make up a second power may be learned by a careful inspection of the process of multiplication by which the power is produced. For example, let us square 36, keeping its tens and ones and their products distinct and separate. Thus,

1296 = 302 + 2 (30 × 6) + 6o = tens2 + 2 tens × ones + ones2.

That is, the square of any number composed of tens and one equals

373. The square of the tens, plus two times the tens times the ones, plus the square of the ones, or t2 + 2txo + o2.

highest period (Art. 371). Taking out of the 12 hundreds the

largest "tens 2" possible, 900, and placing its root, 3 tens, or 30, at the right, there remains 396, which must be the "2 tens X ones + ones2" (Art. 372). The "ones" being but a small part of the 396, we may treat this number as the approximate product of the "2 tens X ones." Dividing this product, 396, by one of its factors, "2 tens," or 60, we have 6 as the other factor, the " ones." Taking from 396 the "2 tens X ones," or 60 × 6, or 360, there remains 36, which contains the "ones?" Taking the ones2, or 62, from 36, nothing remains. Hence we conclude that 30+ 6, or 36, is the root required.

Geometrical Explanation of Square Root.

374. As the length of a square is the square root of its area, the method of extracting the square root of a number may be illustrated by the process of finding the length of a square, its area being given.

It is required to find the length of a square containing 1296 square inches.

Solution.

A square containing 1296 sq. in. cannot be 40 in. long, for a 40-inch square contains 1600 sq. in. It must be more than 30 in. long, for a 30-inch square contains but 900 sq. in. The length of the given square must therefore be between 30 and 40 inches.

C

Removing from the given square. A, a 30-inch square, B, containing 900 sq. in., there remains a surface containing 396 sq. in., largely made up of the rectangles C and D, whose length is evidently that of the square, B. It is obvious that the width of these rectangles added to the length of the square, B, will give the required length of the given square, A. Now the width of a rectangle is found by dividing its area by its length (Art. 218). The length of each of these rectangles, C and D, is

30

D

30 in., and their united length 2 x 30 in., or 60 inches. Dividing their approximate area, 396 sq. in., by their length, 60 in., we have as their probable width 6 inches.

Removing the rectangles C and D, there remains the little square, E, whose length is evidently the width of the rectangles removed. Combining the two rectangles and the little square, we find their united length to be 60+ 6, or 66 in. Multiplying their length and width together, we find their area to be 66 × 6, or 396 sq. in., the exact area of that portion of the given square, 4, remaining after the removal of the square, B.

180

+ 6

We therefore conclude that the length of the given square is 36 inches.

The process may be shortened by the omission of the ciphers, and proved by involution.

375. Rule for finding the Square Root of a Number.

Beginning at the decimal point, separate the given number into periods of two figures each.

Find the greatest square in the left period, and place its root at the right; subtract the square of this root from the first period, and to the remainder annex the next period for a dividend.

Divide this dividend, omitting the last figure, by double the root already found, and annex the quotient to the root and also to the divisor.

Multiply the divisor as it now stands by the last root figure, and subtract the product from the dividend.

If there are more periods to be brought down, proceed in the same manner as before.

NOTE 1.

If 0 occurs in the root, annex 0 to the divisor and another period to the dividend, and proceed as before.

NOTE 2. - If there is a remainder after using all the periods, we can only approximate to the root. But nearer and nearer approximations can be obtained by annexing and using successive periods of decimal ciphers.