OBS.-Mixed numbers may be reduced to improper fractions, before involving: Thus, 22-12; or they may be reduced to decimal: Thus, 23-2.4. The powers of the nine digits, from the first to the ninth, may be seen by the following 5th power..1 32 243 1024 1296 2401 4096 6561 32768 59049 6th power.. 1 64 729 4096 15625 7th power.. 1128 2187 16384 46656 117649 262144 531441 78125 279936 823543 2097152 4782969 8th power.. 1256 6561 65536 390625 1679616 5764801 16777216 43046721 9th power.. 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 EVOLUTION. Art. 225.-EVOLUTION, the opposite of Involution, is the extracting of the root of any number, or the finding of such a number as, when multiplied into itself a certain number of times, will produce a given number. Thus, 3 is the square root of 9, because 3 × 3=9; also, 3 is the cube root of 27, because 3x3x3=27. Any given power may be found by a continued multiplication of the number into itself; yet there are numbers whose precise root can never be found; but, by the use of decimals, we can arrive sufficiently near for all practical purposes. A number whose precise root cannot be found, is called a surd, or irrational number, and its root a surd root. The square root may be denoted by this character, /, called the radical sign, placed before the power; and the other roots by the same sign, with the index of the root placed over it, or by the fractional indices placed on the right hand. Thus, the square root of 9 is expressed, √/9, or 94, and the cube root of 27 thus: 27, or 273. QUESTIONS.-1. What is Evolution? 2. How may any given power be found? 3. Can the precise roots of all powers be found? 4. How can we approximate sufficiently near for practical purposes? 5. What is a number called whose precise root cannot be found? 6. What is the advantage of denoting roots by the fractional indices? The method of denoting roots by the fractional indices is preferable, as, by it, not the root only is denoted, but the power. The numerator of the index denotes the power, and the denominator the root of the number over which it is placed. 1 If the power is expressed by several numbers, with the sign + or between them, a line, or vinculum, is drawn from the top of the sign over all the numbers. Thus, the square root of 12+4 is √12+4=4, and the cube root of 357–14 is √357-14=7. EXTRACTION OF THE SQUARE ROOT. Formation of the Square, and Extraction of the Square Root. Art. 226.-It has been shown, that to obtain the square of any number, whether entire or fractional, we have only to multiply that number into itself. Therefore, To extract the square root, is to find a number, which, multiplied into itself once, will produce a given number. The principle applied in the extraction of the square root, will be better understood by attending, first, to the formation of the square. The square of any number expressed by a single figure, will contain no figure of a higher denomination than tens. (See Table of Powers.) Numbers which are produced by the multiplication of a number into itself, are called perfect squares. There are but nine perfect squares among all the numbers, which can be expressed by one or two figures. The square roots of all other numbers, expressed by one or two figures, will be found between two whole numbers differing from each other by unity. Thus, 37, which is comprised between 36 and 49, has for its square root a number between 6 and 7; and 95, which is comprised between 81 and 100, has for its square root a number between 9 and 10. What is the square of 32 ? tens. units. 32=3+2 3+2 6+4 9+ 6 9+12+4=1024 Thus, it appears, that the square of a number made up of tens and units, contains the square of the tens, plus twice the products of the tens into the units, plus the square of the units. What is the square root of 1024 ? It is evident, that the root will contain more than one figure, since the number is composed of more than two places; and it will contain no more than two, for 1024 is less than 10,000, the square of 100. It will also be perceived, from the foregoing process, that the square of the tens, the first figure of the root, must be found in the two left-hand figures, which we will separate from the others by a point; thus, 1024. The two parts, of two figures each, are called periods. The period 10 is comprised between the squares, 9 and 16, whose roots are 3 and 4; hence, 3 is the tens, or the first figure of the root sought. 1024(32 9 3x2=6)124 62 × 2= 124 We write 3, the first figure of the root, on the right of the given number, and its square, 9, we subtract from 10, the left-hand period, and to the remainder we bring down the next period. Having subtracted the square of the tens from the given number, the remainder, 124, contains twice the product of the tens into the units, plus the square of the units; but since tens into units cannot give a product of less name than tens, it follows that the right-hand figure, 4, can form no part of the double product of the tens into the units; therefore, if we divide 12, twice the product of tens into the units, by twice 3, the tens of the quotient, we shall obtain the unit figure of the root. We will now write this quotient figure on the right of the other, and multiply 62 by 2, the last quotient figure. We thus obtain, 1st, the square of the units; 2d, twice the product of the tens into the units; hence 32 is the required root. What is the square root of 572 ? Operation. 4 43)172 129 43 In this example the remainder, 43, shows that 572 is not a perfect square; but 23 is the greatest square contained in 572; that is, it is the entire part of the root. This may be shown, thus: The difference between the squares of two consecutive numbers, is equal to twice the less number, plus 1. The difference between the squares of 8 and 9 is 17=8×2+1, and 23×2+1=47, which is greater than 43, the remainder, which shows that 23 is the entire part of the root. The foregoing rule may now be applied to finding the length of one side of a square surface, whose area is expressed by the given number. EXAMPLES. Art. 227.-1. What is the length of one side of a square garden, containing 576 square rods, or what is the square root of 576? We first distinguish* the number whose root is to be found, into periods of two figures each, denoted by the index of the root. By the number of periods, we perceive that the root will consist of two figures, a unit and ten. As the second power of ten cannot be less than a hundred, we look for the square of tens in the second, or left-hand period, which is 5. We find the nearest square in 5 to be 4, and its root 2, or 2 tens, which we place in the quotient as the first figure of the root; 576(2 4 Fig. 1. and its square 4, or 400, under the period, and subtracting it, we have a remainder of 1, or 100, to which we add 76, the next period. Had the garden contained but 400 square rods, we should now have obtained the length of one side, 2 tens=20, and 20×20=400; consequently, 400 rods would be disposed of in the form of a square. (See Fig. 1.) But we have a remainder of 176 rods, to be added to the square, and in such a manner that its form shall not be altered. We must, therefore, make an equal addition on two sides. Then 20+20=40, the length of the whole addition. find the width of the addition, we place the double of the root 20 rods. 20 20 400 20 rods. To * It is distinguished into periods of two figures each, because the second power can never have more than twice as many figures as its root, and never but one less than twice as many. The third power can never have more than three times as many figures as its root, and never but two less than three times as many. Distinguish, therefore, any number into periods of as many figures as are denoted by the index of the root. 576(24 already found, on the left hand of the dividend, for a divisor. If we divide 176, the number of rods to be added, by 40, the length of the addition, (or 17 by 4, rejecting the unit figure of the dividend and divisor,) we have 4 rods, the width of the addition. Then 40 × 4=160, the 44)176 Proof: 24 x 24 576 24 rods. Fig. 2. 20 × 4 80 20 × 20 400 16 20×4=80 number of rods added on the two sides; still there is a remainder of 16 rods. As the additions made are no longer than the sides of the square, there will be a deficiency in the corner, (see Fig. 2,) of a square whose sides are equal to the width of the addition, 4×4 16 rods. We therefore place 4, the last quotient figure, on the right of the divisor, because its square is necessary to supply this deficiency. The whole divisor now multiplied by the last quotient figure, equals 176, the number of rods which were to be added to the square. We have now obtained 24, the root of 576, or the length of one side of a square garden containing 576 square rods. Proof by Involution: 24 X 24=576. 24 rods. From the preceding example and illustration we derive the following RULE. I. Distinguish the given number into periods of two figures each, by putting a dot over the units, and another over the hundreds, and so on. The dots show the number of figures of which the root will consist. II. Find the root of the greatest square number in the lefthand period, and place it as a quotient in division. Place the square of the root found, under said period, and subtract it therefrom, and to the remainder bring down the next period, for a dividend. III. Double the root already found, for a divisor; see how often the divisor is contained in the dividend, (excepting the right-hand figure,) and place the result for the next figure in the root, and also on the right hand of the divisor. |