19 AXIOMS. 27. Ax. I. If equals be added to equals, the sums are equal; but if equals be added to unequals, the sums are unequal: thus, if a=x, then a+b=x+b, or if a⇒x and by, then a+b=x+y; but if a>x, then a+b> x+b; and if a<x, then a+b<x+b. 28. II. If equals be subtracted from equals, the differences are equal; but if equals be subtracted from unequals, or unequals from equals, the differences are unequal: thus, if a=x, then a-b-x-b; or if a=x and b=y, then a-b-x-y: but if a>, then a-b>x-b; and if ax, then a-b<x--b, &c. 29. Corollary 1. From Ax. I. it follows, that if equals be multiplied by equals, the products are equal; but if equals be multiplied by unequals, the products are unequal: for multiplication is nothing else than a number of successive additions. 30. Cor. 11. Hence, if equals be divided by equals, the quotients are equal: for, by the nature of division, each dividend is a product, of which the divisor and the quotient are the factors; and since the divisors are equal, the remaining factors, the quotients, must be equal, else (Cor. 1.) the products, the dividends, could not be equal. For a similar reason, if equals be divided by unequals, or unequals by equals, the quotients are unequal. 31. Cor. III. Again, equal quantities raised to the same power are equal, and equal quantities raised to different powers are unequal: thus, if a=x, then a”=a”; but if a>, then a">x"; and if a<x, then a"<x"; for involution is only a succession of multiplications. Also, the same roots of equal quantities are equal; for if it were otherwise, the given quantities, being obtained by raising those roots to the same power, could not be equal. 20 CHAP. II.-FUNDAMENTAL OPERATIONS. ADDITION. ART. 32. Connect the several quantities to be added by their respective signs, incorporating like quantities. (Art. 23). First, Let the quantity to be added be simple. If it be required to add a to x; this, from the nature of algebraic notation, can only be done by placing the sign + between them, which gives x+a for the sum. And if it be required to add is done (Art. 6) by writing -a tox; this Secondly, Let the quantity to be added be compound. It is plain that to add a compound quantity is the same as to add its separate terms; that is, and x+(a+b)=x+a+b; x+(ab)=x+a+(-b); that is, (Art. 6) +a-b. Thirdly, Let some of the quantities to be added be like. Thus, to x+3a add y+5a: this would be done by writing x+3a+y+5a, which (Art. 7 and 23) is the same as x+y+8a. It is often convenient to place the quantities to be added over one another: thus the last example may be written as under. x+3a y+5a x+y+8a 1. 4x2y2—5xy 5ax+x2y2 7x2y2+3xy -2ax-5x2y2 7x2y2-2xy+3ax 3. 2ax-362y 3ax-2b2y ax-b2y 5ax-4b2y ax-b2y 12ax-11b2y EXERCISES. 2. 4a-2x+56 3b-5a+7x -4a+2x-66 -3a+2b+4x-2y 4. 4a+2x -3a-3x a+7x 2a-4x -5a+x 5. a+b -d d-e -C a+3b-2e За-6х 2a-3x 6. 4x(a+b)-1—3√√x 7. ax+by+5x2 bx-3y+2x2 —3x2—dx+dy 4√x+2x(a+b)—7 8-x-5x(a+b) 3x(a+b)-5x-5 4x(a+b)-5x-5 ax+bx dx+by+dy-3y+4x2 or, (a+b—d)x+(b+d—3)y+4x2* * This answer, in connexion with Art. 23, shows that to multiply the sum of any number of quantities by another quantity is the same as to multiply those quantities separately, and then to add the products. 14. Add together x+a-c, 3b+2d, a + 4x, -3x+5a. 15. Add together (a+b)—d√(x2-y2)—mnp, 2ax+8mnp + 3 (a + b) √x, p−3d √(x2-y2)+2bx, -5mnp+3p-2(a+b)√x. 16. Add together 3/x-(a+b), −2√x+√(a+b), √x-3√(a+b), 4√x+2√(a+b). 17. Add together 3a/x-2y-5+ax2, ax+11 —a√x+33/y, —2ax2-43/y+4ax, -8 +2a√x —3ax +23/y, m√x—bx+6+4ax2 28. Add together —a2 and 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. -a. a2x3 and ax3. Lab and a+b. x+p-y and y-x. 6+7x and -3x+x2. 1+ and 1-x. -11 and -5x2. 2a+b and 5a2—b. a2 and a3. b3 and 3b. 6a"+5x-3y and -4x+zy -2a". 5a and -11a; also -5a and 11a. 5axy and 7ax. -9 and 5. 2x(a+b) and 3x(a+b). x+a and -a; also a-1 and 1. 5x+36--a and a-2b. 4b4-1+2 and -3b4+5—3√x. 3a"-b"+3x2, 2am—3b"—xo, and a"+4b"-2. 3a2-5a+1, 7a2+2a-4, and 13a2-4a. a+b—c, a+c—b, and c+b—a. (x+y) and (x—y). 6axy and -4xay. 2a+1 and 3a+1. a+b+c―d, a+b+d—c, a+c+d -b, and b+c+d-a. |