ORAL EXERCISES In each of the following exercises, state which of the two numbers is the larger. To do this, locate the numbers at their proper places on the line shown in the figure on page 34 ; or better, on a similar line which you can now draw for yourself. 1. 6, 9. 2. 6, -9. 3. -6, 9. 4. -6, -9. 5. . 6. }, 7. – Ź, 1. 21. Subtraction of Positive and Negative Numbers. Suppose you owe $2, but your father, without your knowing it, pays the debt. The result is that you are $2 better off than before. In other words, the result of taking away, or subtracting, a $2 debt is to increase what you have by $2. If, for example, you had $10 in cash before the debt was paid, all you really had was $8, but now the whole $10 is yours. Stated in terms of negative numbers, this means that subtracting –2 from 8 is the same as adding 2 to 8; that is, 8-(-2)=8+2=10. Now, let us suppose another case. Suppose you are $2 in debt when you accidentally break a window valued at $5. Even though you have no money to pay for it at the moment, you really have had $5 taken away, or subtracted from you. It is just the same as adding a new $5 debt to the old $2 one, and the result is a $7 debt. In terms of negative numbers, this says that subtracting 5 from –2 iş the same as adding - 5 to – 2; that is, -2-5=-2+(-5)=–7. Next, suppose you have $10, but wish to buy a picture costing $15. This is like taking away, or subtracting, $15 from yourself when you have only $10, and the result is that you go $5 into debt. It is the same as adding a $15 debt to the $10 that is in your pocket; that is, . 10-15=10+(-15)= -5. Finally, suppose you owe $10, but your father pays $8 of this amount. You then owe but $2. This is really subtracting an $8 debt from a $10 debt, and its effect is the same as though your father had made you a present of $8, simply adding it to whatever you had in the first place. Stated in terms of negative numbers, this says that -10-(-8)=-10+8=-2. A careful study of the four cases above shows us two important things : (1) When both positive and negative numbers are being used, we may always subtract one number from another, no matter whether the number subtracted is smaller than the one from which it is taken (as in arithmetic) or not. For example, in the third case we subtracted 15 from 10, obtaining the result – 5. (2) The rule for subtracting any number (positive or negative) from another is as follows: RULE FOR SUBTRACTING ONE NUMBER FROM ANOTHER. Change the sign of the subtrahend (or number subtracted) and proceed as in addition. (See § 17.) For example, to subtract –6 from 3 we simply change the – 6 to 6 and add the result to 3; that is, we have 3 - (- 6) = 3 + 6 = 9. Ans. ORAL EXERCISES 1. State the result of each of the following subtractions : (a) 10-(-6). (c) 3-6. (e) 5a-6 a. (b) – 10-(-6). (d) -4-(-6). (f) –4 x-(-5x). 2. In each of the following exercises subtract the smaller number from the larger one. (a) 2, 3. (e) 2, – 3. () – , 1. (6) –1, 2. (1) 1, (j) – 1, (c) 6, -7. (g) 1, 1: (k) -1, – ģ. (d) -5, -6. (h) 1, -1. (1) -1, -2. 3. In each of the following exercises subtract the larger number from the smaller one. (a) 6, 8. (e) -2, -3. (i) – 1, 1. (b) –1, 2. (f) 2, – 3. (j) - }, -1. (c) 4, 1. (g) 1, 1. (k) -3, -1 (d) 4, -1. (h) , -1 (1) – Ž, 4. 4. On a certain day the thermometer stood at 75o. The next day it stood at 45°. What was the drop in temperature? 5. On a certain day the thermometer stood at 10°. The next day it stood at -6°. What was the drop in temperature? [Hint. As in Ex. 4, subtract the lower temperature from the higher.) 6. How long was it from the year +60 to the year +100? 7. Augustus Cæsar lived from the year – 63 to the year +14. How old was he when he died ? [Hint. As in Ex. 6, subtract the earlier date from the later one.] 8. One ship had a latitude of +25°, while another one had a latitude of – 18°. What was their difference of latitude ? WRITTEN EXERCISES 1. Find the result of each of the following subtractions : (a) 192— 261. (d) 80-(-45). (g) .6–.7 (6) – 150-270. (e) 1-1. (h) 2.5-4.9 (c) 80–45. (f) 0-1. (i) 2.5 –4.09 2. When ready to ascend, a balloon, including its basket, weighed – 1500 lb. If the basket alone weighed 100 lb., how much did the balloon alone weigh? 3. The weather map for February 1, 1916, gave the following as the maximum (highest) and minimum (lowest) temperatures for that day : From this table it appears that the range of temperature at Chicago was 30° — 24°, or 6o. What was the range at each of the other cities mentioned in the table? 4. From the table in Ex. 3, calculate how far below the freezing point (which is 32°) the temperature fell in Montreal. 5. Referring again to the table in Ex. 3, how much colder did it become in Duluth than in Chicago? in Montreal than in New York? in Helena than in New Orleans ? 6. If x=1 and y=2, find the value of each of the following expressions : (a) 2 x– 3 y. (d) V2 y=x. (g) V4y—x?. (b) x2 – y? (e) V2 y-(-x). (h) Voy – 4 x. (c) 23 — 43. (1) -V2y-(-x). (i) Y6y - (- 15). 7. Tell (by inspection) what must be the value of x in each of the following equations : (a) x-1=2. (c) 2+x=1. (e) -3+x=-1. (6) 2-2=-3. (d) 2+x=-1. (f) ~—4= -5. 8. What is the number which added to 2 gives – 4? * [Hint. Let x represent the unknown number, form an equation, and then solve it by inspection. Compare § 6.] For further exercises on this topic, see review exercises, p. 47, and Appendix, p. 292. 22. Addition and Subtraction of Several Numbers. In adding and subtracting several numbers we may proceed from left to right, performing each addition and subtraction as we come to it. For example, 5+3–7 – 10+1=8–7–10+1=1-10+1 =-9+1= -8. Ans. NOTE. Another way is to add the + terms by themselves and the – terms by themselves, then take the sum of the two results. Thus, in the example above we have +5 - 7 WRITTEN EXERCISES Simplify each of the following expressions. 1. 2–6+4-3-2. 2. – 2+4-6+7+3. 3. 1-2+3–4+5-6+7. 4. 1+(-2)-(-3) +4. 5. 2-(-3)+(-4)-(-6).–7. 6. 25 – 30+17–21–45. 7. 101 – 75+36-175 — 256. 8. 1-}+i. 9. }-{+1-1. 10. .2-.6+.08–.004 11. .001 –.01+.1-1+10. 12. 2-1-31+.2-.01 |