CHAPTER XIV INVOLUTION AND EVOLUTION INVOLUTION 107. The index law for involution has been explained in the preceding chapter. As Involution is a form of multiplication in which all the factors are equal to each other, it is evident that, Any required power of a fraction is found by raising the numerator and denominator to the required power. The law of signs in multiplication states that like signs produce a plus product. Hence any pair of equal factors must produce a positive product and any product, consisting of an even number of equal factors, must also be positive. Hence, Any even power of a quantity will be positive. In the case of an odd power, consisting of an odd number of equal factors, if the factors are paired, a single factor will be left over. The product of the pairs of factors will be positive, and this product multiplied by the remaining single factor will yield either a positive or a negative product, according to its sign. Hence, Any odd power of a quantity will have the same sign as the quantity itself. Thus, (xy) = x1y1 and (— xy) = x1y1 but (xy)3 = x3y3 and (— xy)3 = (a b) 4 = a4 - 4a3b6a2b2 4ab3 + b4 From these results, it will be evident that: 1. The number of terms is one greater than the exponent of the binomial. 2. The exponent of the first term is equal to the exponent of the binomial and the exponent of a, the first term of the binomial, decreases by one in each succeeding term while the exponent of b, the second term of the binomial, increases by one in each succeeding term. 3. The coefficient of the first term is 1. 4. The coefficient of the second term is equal to the exponent of the first term. 5. The coefficient of each succeeding term is found by multiplying the coefficient of the preceding term by the exponent of a and dividing that product by one more than the exponent of b. 6. In finding any power of ab, the terms containing the odd powers of b are negative, that is, the terms are alternately plus and minus. Expressions containing 3 or 4 terms may be raised to any required power by the same rules, if the terms of the expression are grouped to form a binomial. 3 (a−b+c-d) 3 = [(a − b) + (c− d)]3 = (a - b)3 + 3(a− b)2 (c− d) + 3(a − b) (c — d)2 + (c — d) 3 = a3-3a2b+3ab2 — b3 +3 (a2 — 2ab + b2) (c− d) + --- = 3(a - b) (c2 2cd + d2) + c3-3c2d + 3cd2 — d3 = a3-3a2b+3ab2 — b3 + 3(a2c - 2abc +t2c - a2d + 2abd — b2d) +3(ac2 2acd+ ad2 — bc2 + 2bcd — bd2) + c3 — 3c2d + 3cd2 — d3 = a3-3a2b+3ab2 - b3+3a2c - 6abc + 3b2c - 3a2d + 6abd - 3b2d+ 3ac2-6acd+3ad2 - 3bc2+ 6bcd - 3bd2+ c3 - 3c2d + 3cd2 — d3. 109. Evolution has been defined as the operation of finding any required root of a quantity. If the quantity is not a perfect power, its root cannot be found exactly. Thus, as 3 is not a perfect square, its square root cannot be found exactly and would be indicated by the use of the radical sign, as √3. Similarly, the cube root of 4 would be indicated, 4. Such roots can be found approximately by methods shown hereafter. From the theory of exponents, we know, that, TO FIND THE ROOT OF A SIMPLE EXPRESSION Take the required root of the numerical coefficient and divide the exponent of each letter by the index of the required root. From the remarks made about the algebraic signs in involution, it is clear that 1. Any even root of a positive number may be either positive or negative and hence is preceded by the sign ±. Thus, as 4 is the square of both + 2 and 2, 2, the square root of 4 will be ± 2. 2. A negative number can have no even root, since every even power is positive. Thus, since - 9 is not the product of two equal factors, it cannot have a square root. An indicated even root of a negative number is called an Imaginary Number, as V-4. 3. Any odd root of a number will have the same sign as the number itself, since every odd power of a number has the same sign as the number. Thus, since the cube of 2 is 8, the cube root of 8 must be 2, and since the cube of +2 is +8, the cube root of +8 must be + 2. To find any required root of a fraction, take the required root of the numerator and of the denominator. Whenever the root of a nuinber expressed in figures is not easily found, resolve the number into its prime factors Thus, to find √11025, 64a6n a3f6m 612 |