Again, if you have $10 but owe $11, what you really have is a debt of $1. This may be expressed by writing (+10)+(-11) = −1. Finally, if you owe $10 and for some reason are obliged to owe $6 more, what you really have is a debt of $16. This may be expressed by writing (-10)+(-6)=-16. These four illustrations when examined carefully show that the rule for adding any two numbers (positive or negative) is as follows: RULE FOR ADDING TWO NUMBERS. To find the sum of two numbers whose signs are opposite, take their difference, regarding each as positive, and prefix the sign of the larger number to the result. To find the sum of two numbers whose signs are the same, take their sum, regarding each as positive, and prefix the common (same) sign to the result. NOTE. Just as it is the stronger boy who wins in the tug of war (see Fig. 16, § 16), so, in adding two numbers whose signs are opposite, the stronger, or larger, number is the one that leaves its sign upon the answer. For example, (+16)+(-8)=+8, but (−16)+(+8)=-8. ORAL EXERCISES State the sum in each of the following exercises, and explain how your answer comes from the Rule above. 1. A gain of $8 and a gain of $4. 2. A gain of $10 and a loss of $3. 3. A loss of $5 and a gain of $15. 4. A loss of $10 and a loss of $15. 5. A debt of $60 and an asset of $80. 6. A rise in temperature of 10° and a fall of 20°. Perform the additions in each of the following exercises: 19. Two boys start from the same place and walk in opposite directions, the one going east at the rate of 3 miles an hour and the other going west at the rate of 2 miles an hour. Taking the east direction as positive, state (by a number) the position of each at the end of two hours. 20. A boat was rowed upstream 2 miles and then allowed to float 5 miles downstream. Taking upstream as positive, what number represents the position of the boat at the end of the trip? 18. Addition of Several Numbers. When we add several numbers instead of merely two, we first add the + numbers by themselves, then the numbers by themselves, then we add (by the rule in § 17) the two sums thus obtained. For example, in finding (+8)+(−7)+(−6)+(+4) the steps are as follows: 19. Note. Instead of +5 it is customary to write simply 5, and in the same way +3 is written simply 3, and so on. In other words, whenever a number occurs without any sign, the + sign is to be understood. The sign is never omitted. ORAL EXERCISES State the sum in each of the following exercises. · 1. (+3)+(-4)+(+5). 2. 2+(-1)+4. [HINT. See § 19.] 3. (-4)+6+(−1). 4. (-3)+(-4)+3+2. 5. 6+7+3+(-6). 6. 2+(-1)+3+1+(−2). 7. (-2)+(-3)+(−1)+2. 8. (-2)+3+(-4)+7. 9. (-1)+2+(-3)+1. 10. 16+(-15)+4. 11. (-21)+3+7+(−1). WRITTEN EXERCISES Find the sum in the following exercises. 1. 15, 12, 32, 8, and -4. 5. 24, 6, 5, 10, and -7. 9. At the beginning of the year a class in algebra had 30 members; during the year 4 entered and 6 withdrew. How many were in the class at the end of the year? Show how to obtain your answer by using a negative number to represent those who withdrew. 10. By use of negative numbers, solve the following example: If a carrier pigeon can fly 60 miles an hour, at what rate will it go when flying directly against a wind blowing at the rate of 40 miles an hour? [HINT. Give each rate its proper sign and add.] 11. Solve Ex. 10 when it is supposed that the pigeon can fly only 30 miles an hour. What does the negative sign of your answer indicate? 12. A steamer which can go 10 miles an hour in still water is running against a current flowing 8 miles an hour. How fast and in what direction is the steamer moving. by negative numbers. Work 13. Solve Ex. 12 when the steamer is not only going at 10 miles an hour against a current of 8 miles an hour, but a wind is blowing against it at the rate of 1 mile an hour. 14. An elevator starts from a certain floor, goes up 50 feet, then down 30 feet, up 35 feet, up 45 feet, and there stops. What number represents its final position with reference to the starting place? 15. Alexander the Great founded the city of Alexandria in Egypt in the year -322, and started there a great university. This university lasted for 292 years, when it fell into the hands of the Romans, who continued it for 671 years more. In what year did it close? 16. Add 3 a, 2 a, and −4 a. [HINT. See § 11.] 18. Find the value of each of the following when x=1 19. Tell (by inspection) what value x must have in each 20. The Size of Numbers. In arithmetic we think of 3 as greater than 2, of 5 as greater than 3, of 4 as less than 7, etc. Also, we regard 0 (called zero) as the least of all numbers. However, when we come to use negative numbers as well as positive, as we have been doing in this chapter, we must regard -1 as even less than 0, for if a man has a debt of $1 (that is, if he has $1) he really has that much less than no money at all. Likewise, -2 we must regard as less than 0, and indeed it must be less than 1. In the same way, - 5 is less than -3, while - 10 is less than -8, etc. The whole situation in this matter is vividly brought out to the eye in the figure below: Here the + numbers are arranged in their order (as on a yardstick) running to the right of the point marked 0, while the numbers are similarly arranged to the left of that point. Observe that if you start anywhere on the line and go to the right, the numbers you meet are constantly increasing (in the sense explained above), while if you go to the left, they are constantly decreasing. Thus the figure shows all numbers (positive and negative) arranged in their increasing order as one reads from left to right. In the figure above only the integers, 1, 2, 3, etc. and their negatives, -1, -2, -3, etc. are printed in, but a complete figure would give markings also for fractions such as 1, 2, -3-57. Thus, goes at the point situated halfway between 0 and 1; again, 21 goes at the point one third the way from 2 to 3; -3 goes at the point three fourths the way from 0 to 1, and -5 goes at the point which is seven eighths the way from -5 to -6. In this way, every fraction has a definite position on the line. 5 |