Of +3? Of -8? 41. What is the absolute value of +7? 42. What is the absolute value of x when x = 6? 43. What is the value of a when a - +4 = 0 ? 45. What is the value of a when a + a = 0 ? - 48. What is the value of a +7, when a 10? a= =7? When a 3? 49. What is the value of a -+5, when a = +9? a=+5? When a =-5? When When 50. Show that if a = : 7 and b = 5, (a + b) (a − b) = a2 — b2. 51. The reading of a thermometer was +20° and its mercury falls 30°. What is the reading now? 52. If I say a man is worth $1000, what does the expression mean to you? 53. Indicate by symbols the difference between 100 years B.C. and 100 years A.D. 54. What is the combined weight, under water, of a piece of metal weighing 5 lb. and a piece of cork weighing −1 lb., that is, pulls up with a force of 1 lb. ? Express in algebraic language: 55. The sum of any two numbers diminished by either one of them is equal to the other. 56. The sum of any three numbers diminished by any one of them is equal to the sum of the other two. 57. The minuend diminished by the remainder is equal to the subtrahend. 58. The product divided by the multiplier is equal to the multiplicand. 59. The area of a rectangle is equal to the product of its base by its altitude. 60. The area of a circle is equal to the square of its radius multiplied by π. 61. The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. 62. If s is the side and a the area of a square, what principle is expressed by a = 63. Ifb is the base, s the area, and a the altitude of a triangle, what principle is expressed by s = (a × b)? 1 64. If z is the hypotenuse, a the base, and y the altitude of a right-angled triangle, what principle is expressed thus, z2x2 = y2? ADDITION 60. 1. If a man is worth $300 and earns $100 more, what is he then worth? What is the sum of +$300 and +$ 100 ? · + $300++$100: = 2. If a man begins the year $300 in debt, and during the year incurs a debt of $100, how much does he then owe? What is the sum of -$300 and $100? -$300 +-$100 = 3. If a person has $300 and owes $100, what is his net capital? What is the sum of $300 and -$100? + $300 + $100 == 4. If I have $500 and incur a debt of $1000, what is my net capital? What is the sum of $500 and $1000 ? +$500-$1000 = 5. To find the sum in the last example, do we add or subtract the absolute values? In the first example? In the second? In the third? 6. Which of the four sums are positive? Which negative? 7. In which of the examples do the addends have like signs? Unlike signs? 61. Addition is the process of uniting one algebraic number with another into one aggregate. In this process the one number is said to be added to the other; the expression a+b means that b is to be added to a. In algebra the word sum is extended to include the result of adding negative numbers, as well as that of adding positive and negative numbers. 62. Numbers with Like Signs. In algebra the process of adding two or more positive numbers, or two or more negative numbers, is the same as that of adding in arithmetic, except that the sign of quality is to be prefixed to the sum. Thus, the sum of 2 positive units and 3 positive units is (2+3) positive units, or 5 positive units; that is, +2 + +3 = +(2 + 3), or +5. In like manner, −2+ −3 = −(2+3), or -5; that is, to add -2 and -3 means to add 2 and 3 and to prefix the sign to the sum. Add the following: 63. Numbers with Unlike Signs. The following equalities were considered in Art. 52 : In (1) we see that +20 and 20 cancel each other, that is, their sum is 0. In (2) we see that 15 negative units cancel 15 of the 20 positive units, leaving 5 positive units (5), which is the sum. That is, In (3) we see that 20 positive units cancel 20 of the 30 negative units, leaving 10 negative units (-10), which is the sum. That is, +20 +30 = ̄(30 — 20), or −10. Why is In (2) how is the absolute value of the sum found-by adding or by subtracting? Why has it the + sign? How is the absolute value of the sum found in (3)? it negative? Which number has the greater absolute value? Upon what does the quality of the sum depend? 64. We may illustrate the adding of positive and negative numbers by representing them as standing on opposite sides of zero on a line laid off into units of length, thus: 5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5.... 1. The sum of +2 and +3 may be found by counting from +2 (whose distance from 0 is +2) 3 units to the right, or in the positive direction. The result is +5, the number of units from 0 to the right. found? How may the sum of 2 and 3 be 2. The sum of 2 and 3 may be found by counting 3 units to the left, or in the negative direction, from +2. It is 1. How may the sum of 2 and 3 be found? 65. A consideration of the preceding exercises and examples gives rise to the following method of finding the sum of two algebraic numbers: 1. If the two numbers have like signs, add arithmetically their absolute values, and prefix to the result their common sign. 2. If they do not have the same sign, take the difference between their absolute values, and prefix to the result the sign of the one having the greater absolute value. |