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SECTION XI. EVOLUTION AND RADICAL QUANTITIES. (135.) EVOLUTION is the process of finding the root of any quantity.

The square, or second power of a number, is the product arising from multiplying that number by itself once. Thus the square of 8 is 8X8, or 64; the square of 15 is 15x15, or 225.

The square root of a number is that number which, multiplied by itself once, will produce the given num

ber.

Thus the square root of 144 is 12, because 12 mul. tiplied by itself produces 144.

PROBLEM I. (136.) To Extract the Square Root of Numbers.

If a number is a perfect square and is not very large, its root may generally be found by inspection. Thus the first ten numbers are

1, 2, 3, 4, 5, 6, 7, 8, 9, 10; and their squares are

1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Hence the numbers in the first line are the square roots of the corresponding numbers in the second.

Quest.-What is evolution? What is the square root of a number 1 How may the root sometimes be found ?

[graphic]

But if the number is large, the discovery of its root may be attended with some difficulty. The following · principles will, however, assist us in detecting the root.

(137.) I. For every two figures of the square there will be one figure in the root, and also one for any odd figure. Thus the square of 1 is 1; 66 66

9 is 81;
66 66 10 is 100;
66 66

99 is 9801;
100 is 10000;

999 is 998001; 66

1000 is 1000000;

etc., etc., etc. Hence we see that the square root of every number composed of one or two figures will contain one fig. ure; the square root of every number composed of three or four figures will contain two figures ; the square root of a number composed of five or six fig. ures will contain three figures, etc.

Hence, if we divide the number into periods of two figures, proceeding from right to left, the number of figures in the root will be equal to the number of pe. riods.

(138.) II. The first figure of the root will be the square root of the greatest square number contained in the first period on the left.

Every number consisting of more than one figure may be regarded as composed of a certain number of

Quest.-How may we know the number of figures in the root ? What will be the first figure of the root? How may the square of a number of two figures be decomposed ?

tens and a certain number of units. If we represent the tens by a and the units by b, the number may be represented by

a+b, whose square is a®+2ab+b'.

Hence we see that the square of a number com. posed of tens and units contains the square of the tens plus twice the product of the tens by the units, plus the square of the units.

Now the square of tens can give no significant figure in the first right-hand period; the square of hundreds can give no figure in the first two periods on the right. and the square of the highest figure in the root can give no figure except in the first period on the left.

Ex. 1. Let it be required to extract the square root of 2916.

Since this number is composed of four figures, its root will contain two figures; that is, it will consist of a certain number of tens and a certain number of units. Now the square of the tens must be found in the two left-hand figures, 29, which form the first period, and which we will separate from the other two figures by placing a point between them. Now this period contains not only the square of the tens, but also a part of the product of the tens by the units. The greatest square contained in 29 is 25, whose root is 5; hence 5 must be the number of tens whose square is 2500; and if we subtract this from 2916, the remainder, 416, contains twice the product of the tens by the units, plus the square of the units. If then, we divide this number by twice the tens, we

QUEST.-Explain the method of extracting the square root.

shall obtain the units, or possibly a number somewhat too large. This quotient figure can never be too small, but it may be too large, because the remainder, 416, besides twice the product of the tens by the units, contains the square of the units. We therefore complete the divisor by annexing the quotient 4 to the right of the 10, and then, multiplying by 4, we evidently obtain the double product of the tens by the units, plus the square of the units. The entire operation may then be represented as follows:

29.16 154=the root.

25

5x2=104 416

416 In this operation we have, for convenience, written 25 for 2500. The two ciphers are, however, to be regarded as implied. So, also, the divisor is properly 50X2, or 100, which is contained in 416 four times; but we usually omit the last cipher in the divisor, calling it 5x2, or 10, and also omit the last figure of the dividend, leaving 41, which gives the same result as if all the figures were retained, for 10 is contained in 41 fuur times.

(139.) Hence, for the extraction of the square rout of numbers, we derive the following

RULE. 1. Separate the given number into periods of two figures each, beginning at the right hand. The first period on the left will often contain but one figure.

Quest.—Give the rule for extracting the square root of numbers.

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2. Find the greatest square contained in the lefl. hand period: its root will be the first figure of the required root. Subtract the square from the first period, and to the remainder bring down the second period for a dividend. .

3. Double the root already found for a divisor, anu find how many times it is contained in the divideni, exclusive of its right-hand figure; annex the result both to the root and the divisor.

4. 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. If the product should be greater than the dividend, diminish the last figure of the root.

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

Ex. 2. Find the square root of 186624.
The operation is as follows:

18.66.241432

16

83 2 66

2 49 862 17 24

17 24 Consequently, the required root is 432. Ex. 3. Find the square root of 8836.

Ans. 94 Ex. 4. Find the square root of 58564.

Ans. 242

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