CHAPTER VII.

INVOLUTION AND EVOLUTION,

WITH THE ARITHMETIC OF SURDS.

INVOLUTION.

165. DEF. A Power of a number is the number which arises from successive multiplications by itself: the operation by which it is obtained is termed Involution ; and the Degree or Order of the power is denoted by the number of factors employed.

Thus, taking the number 2, we shall have the powers of it as follows:

22, the first power of 2:

2 × 2 =

4, the second power of 2:

2 × 2 × 2

=

8, the third power of 2:

2 × 2 × 2 × 2 = 16, the fourth power of 2:
2 × 2 × 2 × 2 × 2 = 32, the fifth power of 2:

2 × 2 × 2 × 2 × 2 × 2 = 64, the sixth power of 2:

and so on, as far as we please :

but instead of expressing these multiplications at length, which would soon become inconvenient, we denote the same operations by means of Indices, or small figures placed a little above the line to the right of the quantities whose powers are intended to be exhibited: thus, what is put down above may be denoted by

21 = 2:
22 = 4:

23 = 8:

25 = 32:

21 = 16:

26 = 64, &c.:

where the Index sometimes called the Exponent is equal to the number of factors and is greater by one than the number of operations.

In the same manner the second powers of the nine digits are expressed: thus,

The second and third powers of numbers are styled their Squares and Cubes in reference to their application to Geometry, as will be seen hereafter: and the operations by which all powers are obtained are merely those of Multiplication.

166. To find the powers of a vulgar fraction or of a quantity expressed decimally, a similar process is used: thus,

and exactly in the same manner the powers of a quantity

expressed by factors are found:

thus, the square of 2 × 7 = (2 × 7) × (2 × 7)

= 2 × 2 × 7 × 7 = 22 × 72 = 4 × 49 = 196.

=

Hence it appears that a power of a fraction is equal to the fraction formed by raising both its numerator and denominator to the power, and that the power of a quantity formed by factors is found by raising each factor to the power.

A mixed quantity is represented as a simple fraction or as a decimal, before the process is applied.

167. This notation furnishes important conclusions with respect to powers,

from which we infer that the Multiplication and Division of powers of the same quantity are expressed by the Addition and Subtraction of their indices.

Similarly, we have the fourth power of 3o expressed by 32 × 3 × 32 × 3 = 3 = 3***; or, the Involution of powers is expressed by the Multiplication of their indices: and conversely.

Ex. Let it be required to find the 6th power of 13.

and the same result will be obtained by effecting any of the operations indicated below:

13® = 13% x 13* = 133 x 133 = 135 x 13.

168. When one power of a quantity is divided by a higher power of the same quantity, the quotient may be expressed by the power of a fraction: thus,

7° ÷ 7a = (7 × 7) ÷ (7 × 7 × 7 × 7)

where the difference of the indices is employed in the numerator or denominator according as the dividend or divisor is the higher power.

If the indices of the dividend and divisor be the same, this notation extended will give us the representation of unity or 1 in the form of the power of any number or quantity whatever, as 7 for instance, whose index is 0,

since 17 ÷ 7a = 74-4 = 7o.

EVOLUTION.

169. DEF. A Root of a number is such a number as being multiplied into itself one or more times produces it; and the operation by which this root is obtained is called Evolution.

Thus, the second or square root of 16 is 4, because the square of 4 is 16, or 42 = 4 × 4 = 16.

The third or cube root of 512 is 8, since the cube of 8 is 512, or 838 x 8 x8 = 512:

and similarly of vulgar fractions and decimals.

This operation is expressed by the sign which is called the Radical Sign, with a small figure placed on its left to particularize the root intended: thus,

/16-4 and 512 = 8 :

but the square root is denoted by the sign only, without the small figure, as being of most frequent occurrence.

These operations are also indicated by means of the primitive fractions,, &c., used as indices, so that the indices,, &c., denote operations exactly the reverse of those expressed by the indices 2, 3, &c., respectively: thus,

EXTRACTION OF THE SQUARE ROOT.

170. In this operation, having only one magnitude to work with, we cannot avail ourselves of any of the fundamental operations of Arithmetic: we shall therefore merely put down such instructions as will enable the student to extract the square root, without entering into the reasons upon which they are founded, these reasons admitting of a much clearer exposition by means of Algebraical Symbols than any that could be given in particular numbers. See the Appendix.

171. Repeating what was said in Article (165), we

have

Digits:

1, 2, 3, 4, 5, 6, 7, 8, 9:

Squares:

1, 4, 9, 16, 25, 36, 49, 64, 81:

whence, by mere inspection, we are enabled to find the square roots of all numbers that can be produced by the squaring of a single figure: but it is evident that this statement will not be sufficient for finding the square roots of quantities consisting of more than two figures; and recourse must therefore be had to other expedients.

172. From the number of figures in any proposed quantity, to find the number of figures in its square root. Since, the square root of 1 is 1:

the square root of 100 is 10:

the square root of 10000 is 100:

the square root of 1000000 is 1000: &c.: we see immediately that the square root of a number of fewer than three figures must consist of only one figure: that of a number of more than two figures and fewer than five, of two figures: that of a number of more than four figures and fewer than seven, of three figures, and so on: whence it follows, that if a dot or full point be placed over every alternate figure, beginning at the units' place, the number of such points will be the same as the number of figures in the square root.

This is called the Rule for Pointing, and may easily be extended to decimals: thus,

since, the square root of .01 is .1:

the square root of .0001 is .01:

the square root of .000001 is .001: &c.:

we infer that the quantity proposed must first be made to have an even number of decimal places, and then the pointing must proceed from the place of units towards the right hand over every alternate figure as before: and the number of such points will be the same as the number of decimal places in the square root.

Rule for the Extraction of the Square Root.

Point the alternate figures of the number proposed, beginning at the place of units, so as to form as many periods of two figures each as possible: find the greatest square number contained in the first period on the left hand, put down its root on the right as in division, and subtract it from that period. To the remainder bring

H. A.

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