∞, just as the number of sides of a triangle is called three and represented by 3. Also, if we write the smallest decimal fraction possible by putting down a row of one or more numerals with a decimal point to the left of them, and then writing forever between the point and the numerals a row of ciphers (or any other character expressing simply the absence of the figures 1, 2, 3, 4, 5, 6, 7, 8, 9), we have a decimal fraction called zero and often expressed by the character 0. With this explanation of ∞ and 0 we see that for all values of a greater than - and less than co, =0. α Also for values of a less than 0 and greater than +0, 168. and are said to be in indeterminate form because their values will depend on those of the numerator and denominator when written out, or otherwise fully expressed. 2x Thus 2= becomes when x is supposed to be infinity, and 2= 2(x-1) x-1 0' becomes when x is supposed to be 1, so that we 0 which we cannot do must know just what the 's and O's are before we can tell the value of the fractions. In the same way we might be required to find the value of the fraction satisfactorily without knowing what the next decimal figures should be. 0 On the other hand ∞ 0' .0001 and other related forms may not only be in indeterminate form, but the value may be really indeterminate. Thus, solving in the regular way the equations any value whatever if we give y a value to suit, so that the value of x is indeterminate. Inasmuch as 1 0 =0, what is true of CHAPTER XXII. INVOLUTION. 169. DEFINITION. Involution is the general name for multiplying an expression by itself so as to find its second, third, fourth, or any other power. Involution may always be effected by actual multiplication. Here, however, we shall give some rules for writing down at once (1) any power of a simple expression; (2) the square and cube of any binomial; (3) the square and cube of any multinomial. 170. It is evident from the Rule of Signs that (1) no even power of any quantity can be negative; (2) any odd power of a quantity will have the same sign as the quantity itself. NOTE. It is especially worthy of notice that the square of every expression, whether positive or negative, is positive. 171. From definition we have, by the rules of multiplication, (a2)3 = a2. a2. a2= a2+2+2 = a¤ ̧ (-3)=(-3)(− x3)=x3+3=x6. (-a)=(-a)(-a) (— a3)= — ab+6+5 — — a15 ( − 3a3) 4 = ( − 3)1 (a3)4=81a12. Hence we obtain the following rule: RULE. To raise a simple expression to any proposed power: (1) Raise the coefficient to the required power by Arithmetic, and prefix the proper sign found by Art. 35. (2) Multiply the index of every factor of the expression by the exponent of the power required. (2) (-3ab3)6=729a6b18 2ab3 4 16a4b12 (3) (ty) = 812*y** It will be seen that in the last case the numerator and the denominator are operated upon separately. EXAMPLES XXII. a. Write down the square of each of the following expressions: Write down the cube of each of the following expressions: 3y3 1 15. 16. - 2xy 1 19. 20. 4a4. Write down the value of each of the following expressions : 31. 7x3y1. 32. 22 a3. 33. (3a2b3). 34. (-a2x). 35. (-2xy)5. 36. The square of the sum of two quantities is equal to the sum of their squares increased by twice their product. RULE 2. The square of the difference of two quantities is equal to the sum of their squares diminished by twice their product. Example 1. (x+2y)2= x2+2.x. 2y+(2y)2 = x2+4xy+4y2. Example 2. (2a3 – 3b2)2 = (2a3)2 – 2. 2a3. 3b2 +(3b2)? 173. These rules may sometimes be conveniently applied to find the squares of numerical quantities. Example 1. The square of 1012 (1000+12)2 TO SQUARE A MULTINOMIAL. 174. We may now extend the rules of Art. 172 thus: (a+b+c)2={(a+b)+c}2 =(a+b)2+2(a+b)c+c2 =a2+b2+c2+2ab+2ac+2bc. In the same way we may prove (a−b+c)2=a2+b2+c2 −2ab+2ac - 2bc [Art. 172, Rule 1.] (a+b+c+d)2=a2+b2+c2+d2+2ab+2ac+2ad+2bc+2bd+2cd. In each of these instances we observe that the square consists of (1) the sum of the squares of the several terms of the given expression; (2) twice the sum of the products two and two of the several terms, taken with their proper signs; that is, in each product the sign is + or according as the quantities composing it have like or unlike signs. NOTE. The square terms are always positive. The same laws hold whatever be the number of terms in the expression to be squared. RULE. To find the square of any multinomial: to the sum of the squares of the several terms add twice the product (with the proper sign) of each term into each of the terms that follow it. Ex. 1. (x-2y-3z)2=x2+4y2+9z2-2.x.2y-2.x.3z+2.2y.3z = x2+4y2+9z2-4xy-6xz+12yz. Ex. 2. (1+2x-3x2)2=1+4x2+9x1+2.1.2x−2.1.3x2-2.2x.3x2 =1+4x2+9x+4x-6x2-12x3 =1+4x-2x2—12x3+9x1. EXAMPLES XXII. b. Write down the square of each of the following expressions: TO CUBE A BINOMIAL. 173. By actual multiplication, we have (a+b)3=(a+b)(a+b)(a+b) =a3+3a2b+3ab2+b3 (a - b)3=a3-3a2b+3ab2-b3 From these results we obtain the following rule: ..(1), (2). RULE. To find the cube of any binomial: take the cube of the first term, three times the square of the first by the second, three times the first by the square of the second, and the cube of the last. If the binomial be the sum of the quantities, all signs will be +; if the difference of two quantities, the signs will be alternately + and -, commencing with the first. EXAMPLES XXII. c. Write down the cube of each of the following expressions: |