BOOK V. INVOLUTION-EVOLUTION-EXTRACTION OF THE SQUARE ROOT-EXTRACTION OF THE CUBE ROOT-EXTRACTION OF THE ROOTS OF ALL POWERS-OF AN EQUATION-GEOMETRICAL PROPORTION-COMPOUND PROPORTION-POSI TION-ALLIGATION PERMUTATION ARRANGEMENT COMBINATION-NEWTON'S BINOMIAL FORMULA-ARITH METICAL PROPORTION AND PROGRESSION-GEOMETRICAL PROGRESSION-LOGARITHMS-USE OF THE TABLE. SECTION XIX. INVOLUTION-EVOLUTION-EXTRACTION OF THE SQUARE ROOT. Involution, or Raising of Powers. 354. This consists in the continual multiplication of a quantity into itself. The successive products are called the powers of the given quantity, and each shows, by its appellation, the number of times that it contains the given quantity as factor, the quantity itself being the first power or root. Thus, let a be any quantity; then, a is the first power or root of all the powers of a; a Xa, or aa, is the second power or square of、 a; a XaXa, or aaa, is the third power or cube of a; a X axaxa, or aaaa, is the fourth power of a. This being the product of two squares, is also called the biquadrate, from bis and quadrans. Other names have been given to several of the higher powers, indicative of their formation from the square and cube; but these have generally fallen into disuse, and a ×a×a×a×a, or aaaaa, is simply called the fifth power of a; aaaaaa, the sixth, and so on. For the sake of convenience, these products are written successively a, a2, a3, a*, a3, a, &c., and are read, a, a squared, a cubed, a fourth, a fifth, a sixth, &c. The figure on the right is called the index or exponent, because it shows the number of times that a is factor in each product. Hence, it is evident, that if we add the indices of two or more powers of a quantity, their sum will be the index of the power to which the quantity would be raised by multiplying those powers together. The scholar must remember, that the exponent of the root, or quantity itself, though seldom written, is a unit, and must be so considered in adding or subtracting the exponents. Thus, a × a2, or aa2 a3; a3a2 = a1; a3a3 =a5; aa2a3, a2a3a2 or a3a3 a; a2a2a2a3, aa2a3, a1a2a*, a2a3a3 or a*a* — a3. 355. From the above it is plain, that a quantity may be raised to the higher powers in various ways, and the higher the power, the greater the variety. Thus, to find the twelfth power of 2, as we know that the cube of 2 is 8, we say 8×8 X8X8=4096, which is easier than to perform the continual multiplication 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 X 24096. Because either of two numbers taken from their sum leaves the other, the index of either of two factors taken from the index of their product will leave the index of the other factor. But the product divided by either factor gives the other; therefore, to divide any higher power of a quantity by a lower, we have only to subtract the index of the divisor from the index of the dividend, and the remainder will be the index of the a3 quotient. Thus, = a; = 39 a a3 as a3; a; also, a3 a5 a2; a 33 4; 36 356. A vulgar fraction is each of its terms to that power. a a3a3 raised to any power by raising For example, the fifth power 32 25 25 or may be expressed thus: 35' and is equal to 243 This is obvious from the rule for multiplying fractions. and then involved as a fraction. Thus, (24) = (5)⋅ = 3125 329731. 55 - 25 357. A decimal fraction, whether finite or infinite, is involved by the rules already given for the multiplication of decimals. We may, however, remark, that a finite decimal may be involved as a whole number, after which, the number of decimal places, of which the result must consist, is determined by multiplying the number of decimals in the given number by the index of the power to which it is raised. Thus, (,04)3=,000064 Having found that the cube of 4 is 64, as the given number contains 2 decimals, we multiply 2 by 3, the index, and have 6 for the number of decimal places in the result. Then, as 64 contains only two figures, we subtract 2 from 6, and the remainder 4 is the number of ciphers which we must prefix to 64. The cube of ,04 is therefore 64 millionths, as above. The reason of the above operation will be seen by reflecting, that as the factors are alike, and as the product must contain as many as are contained in both, the square will contain twice as many as the given number, the cube, 3 times as many, &c. The scholar may verify the following results: Powers of the Nine Digits, to the Tenth Power, inclusive. 9th. 10th. 11 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1024 59049 1048576|9765625 60466176 282475249 1073741824 3486784401 Evolution, or Extraction of Roots. 358. Evolution is the method of finding a quantity from any given power of that quantity. The quantity so found is called the root of the power from which it is extracted, and assumes its name accordingly: that is, if found from the square, it is called the square root, if from the cube, the cube root, &c. Extraction of the Square Root. 359. If the square of a number consists of only two figures, its root cannot consist of more than one, and must, in whole numbers, be one of the nine digits. This is evident, because the product of two factors (121) contains as many figures as both of them or one figure less; and therefore, if the root or quantity contained two figures, the product, or square, would contain at least three. Again, if the square contains three figures, the root must contain at least two, because (121) the product cannot contain more figures than both factors, which would be the case if the root contained only one figure. It is evident, therefore, that if the square contains four figures, the root will contain two; if five figures, the root will contain three, &c.; and, consequently, that the number of figures of which the root will consist, may be determined by separating the figures of the square into periods of two figures each. 360. When the root contains more than one figure and has its unit figure significant, it may always be considered as a number of tens and units. Now a quantity, consisting of two parts, is called a binomial, from bis and nomen, a name, and the square of a binomial contains the square of each of its terms, together with twice their product. Let the root consist of tens and units, and let a represent the tens, and b, the units; then, to find the square of the binomial a + b, we involve it thus, (See art. 138 :) The binomial a + b, being multiplied first by a, and then by b, we have for the sum of the products a2+2ab+b2; that is to say, the square of the tens, the square of the units, and twice the product of the tens and units, and this is the formula by means of which the square root is extracted. This may be exemplified by figures, thus: in squaring 12, the number of inches in a foot, we have 144, the number of square inches in a square foot: but 12 consists of 10+2, and in squaring this binomial, as above, we have 10+2 10 + 2 100+ 20 20+ 4 100+(20) 2+4 or 100 + 40 + 4=144; that is to say, the square of the tens, twice the product of the tens and units, and the square of the units. Or we may represent the square foot geometrically, thus: a b In which the same parts are clearly developed, their sum being ་ ་ ! as before. |