a a 00. co, 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 co and 0 we see that for all values of a greater than -00 and less than oo, =0. Also for values of a less than –0 and greater than +0, 0 168. and are said to be in indeterminate form because 0 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(x - 1) O' 2= becomes when x is supposed to be 1, so that we 0 must know just what the co's and O's are before we can tell the value of the fractions. In the same way we might be required .0001 to find the value of the fraction which we cannot do .0003' satisfactorily without knowing what the next decimal figures should be. 0 On the other hand and other related forms may o O not only be in indeterminate form, but the value may be really indeterminate. Thus, solving in the regular way the equations x + y +2=0, 2x+2y+4=0, 4-4 0 we get x= and we can easily see that u can have 2-2 any value whatever if we give y a value to suit, so that the value 1 0 of x indeterminate. Inasmuch as =0, what is true of is equally so of o 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; any multinomial. (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, (42)3=a2. ar. a-=a2+2+2 = ab. (-3a2)=(-3)*(a3)4=81a12. 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. Examples. (1) (– 2x2)5 - 32x10. 16a4812 81x8y4 It will be seen that in the last case the numcrator and the denomi. nator are operated upon separately. (3) (3x+y) z ab3. 11. EXAMPLES XXII. a. Write down the square of each of the following expressions : 1. 3ab3. 2. a'c. 3. 7ab2. 4. 1162c3. 5. 4a4box2. 6. 5x2yó. 7. - 2abc. 8. - 30%) 2 4 9. 4xyzu. 10. 12. 3.c%y 1 16. - 2xy2. 2xy 5ab3 1 3a5 17. 18. 13.06.203. 19. 20. 2xy 44. 5.73 • Write down the cube of each of the following expressions : 21. 2ab2. 22. 3.x3. 23. 4x4. 24. - 3a3b. 25. - 5ab%. 26. - 63c2x. 27. - 6a6. 23. - 2a1c?. 1 3.25 2 29. 31. 720@y4. 32. 3y2 5a3 Write down the value of each of the following expressions : 33. (3a263)4 34. (-a-x)" 35. (- 2.xoy)5. 36. 30. 2.2010 2y3 3y 3a4 To SQUARE A BINOMIAL. 172. By multiplication we have (a+b)=(a+b)(a+b) .(1), (2). RULE 1. 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. =à2—2ab + b2...... Example 1. (x+2y)'=x+2.1.2y+ (27) =x2 + 4xy +4ya. =4a6 - 12a3b2 +964. 173. These rules may sometimes be conveniently applied to find the squares of numerical quantities. Example 1. The square of 1012=(1000+12)2 =(1000)2 + 2.1000. 12+(12) =1024144. = (100)2 – 2.100.2+(2) =9604. =(a+b)2+2(a+b)c+c2 [Art. 172, Rule 1.] =a? +62 +c2 + 2ab +2ac+2bc. In the same way we may prove (a - b+c)2=a2 +62 +02 – 2ab + 2ac – 2bc (a+b+c+d)2=a2 +62 +c2 + 2 + 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—32)2=X2 +4y2 + 9z2–2.4.24-2.x.3z +2.27.32 = 22 +4y2 +922 — 4xy – 6xz+12yz. Ex. 2. (1+2% — 3x2)2=1+4x2 +924 +2.1.2x — 2.1.3x2 — 2.22.3x2 =1+4x2+9x4 + 4x — 6x2 — 1233 EXAMPLES XXII. b. Write down the square of each of the following expressions : 1. a +36. 2. a-36. 3. 3—5y. 4. 2x+3y. 5. 33 — y. 6. 3x+5y. 7. 9x-2y. 8. 5ab-C. 9. pq-r. 10. 3 - abc. 11. Qx+2by. 12. 22–1. 13. a-b-C. 14. a+b-c. 15. a+26+c. 16. 2a-36+4c. 17. 22—42–22. 18. xy + yz+zx. 19. 3p-2q+4r. 20. 22 — 3+1. 21. 2x2 + 3x – 1. 22. 3-y+a-b. 23. 2x+3y+a26. 24. m-n-p-9. 1 3 2 3 a —26+Ă 26. -3629 27. 3. 22–2+ 2 3 2 с 25. a To CUBE A BINOMIAL. 173. By actual multiplication, we have (a+b)3=(a+b)(a+b)(a+b) (1), (a —8)8=a8 — 3a2b +3ab2 — 38 (2). From these results we obtain the following rule: 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 : 1. 3+a. 2. 3 - a. 3. 2-2y. 4. 2ab-3c. 5. 22 +4y2. 6. 4.c2— 5y2 7. 293-362. 8. 5x5 – 4y4. 26 22 9. a 10. +2. 11. - 3x. 12. @ +24. 3. 3 |