Elementary Algebra and Geometry for Schools

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Positive and Negative Numbers

Positive and negative numbers. In our study of arithmetic and algebra we have used both whole numbers and fractions. We shall continue to use them, but we shall call them positive numbers in order to distinguish Them from numbers of another kind, negative numbers, which we shall find useful.

We live in a world of opposites. Geese fly north in the spring; they fly south in the fall. You may deposit money in your bank; you may withraw money from your bank. You may ride upward in an elevator; you may ride downward in an elevator. We must distinguish between these opposite activities. If we represent a gain in weight by means of a plus oign, we will represent a loss in weight by a minus sign. If we consider flying north as going in a positive direction, then we will consider flying south as going in a negative direction. When both positive and negative  numbers are used, they designate quantities that are in some way "opposite" in character.

Signed numbers. Positive and negative numbers are called signed numbers. They not only indicate size but they also indicate direction. If a football player has lost five yards, we can represent this by -5. How can we represent a gain of five yards? A number without a sign preceding it is assumed to be a positive number.

For example, a thermometer reading of 50 is understood to be 50 above zero, that is + 5°. Only if the minus sign precedes 5° does it mean five degrees below zero. When the plus sign is omitted, the number is still considered to be a signed number.

The number scale. In a football game the line of scrimmage is the position from which we determine whether the ball carrier gains or loses. The number scale is represented in similar fashion. In the figure below a line was drawn in a horizontal position and a point on the line was labeled zero. Units were then laid off to the right and left of this point. The distance (number of units) from the zero point to a point on the right is positive (+), while the distance from the zero point to a point on the left is negative (-). Here again, as in the case of the thermometer, positive numbers are measured in one direction and negative numbers are measured in the opposite direction. If, therefore, we say that the distance from the zero point to another point on the scale is x units, then in some cases x will be a positive number, and in other cases x will be a negative number; that is, the value of x is a signed number. Indicating signed numbers. In the adjacent figures, the first thermometer shows the level of the mercury to be 40 units above the zero point and we can represent this reading by the number + 40, or simply as 40. The second thermometer shows the level of the mercury to be 10 units below the zero point and we can represent this reading by the number -10. • A positive number is written with a plus sign in front of the number, or the plus sign may be omitted; a negative number is written always with a minus sign in front of the number.

If we represented the number of degrees of the temperature by x, then the value of x on the first thermometer is 40, and the value of x on the second thermometer is -10; that is, x can represent either a positive or a negative number.

EXERCISES: Signed numbers

1. If a temperature of 35° above zero is represented by the number +35, what number would represent a temperature of 8° below zero?

2. If an altitude of 800 ft. above sea level is represented by the number +800, what number would represent an altitude of 900 ft. below sea level?

3. If a profit of \$500 is represented by the number + 500, what number would represent a loss of \$700?

4. If a loss in weight of 8 lb. is represented by the number -8, what number would represent a gain in weight of 12 lb.?

5. If a latitude of 50° north of the equator is represented by the number 50, what number would represent a latitude of 40° south of the equator?

6. If a gain of 20 yd. in a football game is represented by the number 20; what number would represent a loss of 10 yd.?

7. In the stock market quotations, if a rise in the price of a stock of \$2 a share is represented by the number +2, what number would represent a drop of \$3 a share in the price of the stock?

8. If a longitude of 70° west of Greenwich meridian is represented by the number 70, what number would represent a longitude of 60° east of the Greenwich meridian?

9. If a canoe trip of 5 miles upstream is represented by the number +5, what number would represent a canoe trip of 9 miles downstream?

10. If a deposit of \$100 is represented by the number +100, what number would represent a withdrawal of \$50?

Answers : 1. -8. 2. -900. 3. -700. 4. + 12. 5. -40. 6. -10. 7. -3. 8. -60 9. -9. 10. -50.

Plus and minus signs. As we explained that the signs +and -are used to indicate the operations of addition and subtraction. That is, 2 +7 indicates. that 7 is to be added to 2, and 8 -5 indicates that 5 is to be subtracted from 8. In these and similar cases, the signs + and - are symbols of operation:

In this chapter we have already seen another use of these + and signs in connection with signed numbers, where +10 means 10 units in one direction and -10 means 10 units in the opposite direction. The signs +and -, when used in these and similar cases, are symbols of quality.

• In algebra each of the signs + and -serves a double purpose and may be used either as a sign of operation or as a sign of quality.

Signs for indicating addition and subtraction. Early algebraists did not use symbols such as we now have, but wrote out their problems, using words to express each operation. This lack of much needed symbols made algebra very difficult. Through hundreds of years mathematicians developed the symbols we use today. The Latin word plus to signify addition was replaced by p which later was simplified to +. The Latin word minus, signifying subtraction, became rn. This was later replaced by our present symbol -. It was not until about 1600 that the signs + and -came into general use. The plus sign signified then, as now, both positive quantities and addition; the minus sign signified both negative quantities and subtraction.

ILLUSTRATIVE EXAMPLES: Addition of signed numbers

As we have already seen, a signed number represents both a quantity and a direction. Suppose we are watching a game of football. In the diagrams which follow, zero represents the original position of the football on the line of scrimmage. The team with the ball is attempting to advance the ball to the right from the zero point. Gains, therefore, will be represented by positive numbers while losses will be represented by negative numbers.

1. Suppose the fullback gains 3 yd. on his first attempt and then gains 4 yd. on his second try. What is the result (sum) of his two efforts?

Solution. He obviously gained 7 yd. We represent the sum of +3 and +4 on a number scale as follows: 2. Starting again from the zero point, suppose the left halfback loses 2 yd. on his first try. He fumbles on his second try, losing 6 yd. What is the result (sum) of his two efforts?

Solution. He, of course, lost 8 yd. We represent the sum of -2 and -6 on a number scale as follows: 3. Suppose that Bill makes two attempts to advance the ball. He is successful in his first try, gaining 3 yd. On his next try, however, he loses 5 yd. What is the result (sum) of his two efforts?

Solution. He loses 2 yd. We represent the sum of +3 and -5 on a number scale as follows: 4. Suppose, in the above example, that Bill had first lost 5 yd. and then gained 3 yd. Would the result have been the same?

Solution. Yes, the result would still be a 2-yd. loss. Observe following diagram: Referring to the number scale given below, find the sum of each of the following: Answers : 1. 7. 2. -3. 3. 3. 4. -7 .5. -10. 6. -1. 7. -2. 8. -1. 9. 0. 10. 8. 11. -10. 12. 7. 13. -9. 14. -10. 15. -5. 16. 5. 17. 9. 18. 1. 19. 5. 20. -8. 21. 0. 22. 0. 23. 3. 24. -6. 25. 1. 26. -7. 27. 0. 28. -2. 29. -7. 30. -3. 31. -6.

Absolute value. The absolute value of a signed number is the magnitude of the number regardless of its sign. Thus the absolute value of -7 is 7, just as the absolute value of + 7 is 7; the absolute value of -23 is 23; the absolute value of + 23 is 23; etc.

Rules for the addition of signed numbers. In the preceding exercises you found the following results for the addition of signed numbers: We observe that the following rules apply to these problems:

• To add two or more numbers with like signs: add the absolute values of the numbers and prefix their common sign. To add two numbers of unlike signs: find the difference of their absolute values and prefix the sign of the number whose absolute value is the larger.

ILLUSTRATIVE EXAMPLES: Applying the rules for addition of signed numbers

 1. Find the sum of 27 and -34. Solution. Since the numbers 27 and -34 have unlike signs, we take the difference of their absolute values (the difference between 27 and 34, which is 7) and prefix the sign of the number whose absolute value is larger. The absolute value of -34, which is 34, is larger than the absolute value of 27; hence we prefix the minus sign. 2. Find the sum of - 18 and -14. Solution. The numbers -18 and -14 have like signs; therefore we add -14 their absolute values (we add 18 and 14, which is 32) and prefix their common sign (which is minus), 3. Find the sum of -8a and 3a. Solution. We learned before to add like (+)  terms by adding their numerical coefficients and multiplying this number by their common literal part. To add the numerical coefficients, we apply the rules for the addition of signed numbers. EXERCISES: Applying the rules for addition of signed numbers

Find the sum of each of the following: Answers : 1. 29. 2. 30. 3. -8. 4. -2. 5. -24. 6. + 26. 7. -3. 8. -3. 9. 0. 10. 28. 11. -50. 12. 6. 13. -29. 14. -58. 15. 18. 16. 0. 17. 23x. 18. 39a. 19. 9b. 20. -13x. 21. -29y. 22. -8xy. 23. -29a2. 24. -37p. 25. 9ab. 26. 14s2. 27. -102y. 28. 0. 29. -ax. 30. 0. 31. 34xy. 32. -x. 33. 88. 34. -7. 35. 7. 36. -28. 37. 0. 38. 0. 39. 20. 40. -51. 41. 1. 42. -50. 43. 0. 44. 2. 45. -71. 46. -6. 47. 0. Answers : 48. 11x. 49. 3y. 50. -4a2. 51. -17b. 52. -3x2. 53. -2p. 54. 21y. 55. -6ab. 56. 6x3. 57. -6p2. 58. 11y. 59. 25pq. 60. -14ab. 61. 0. 62. -38x.

ILLUSTRATIVE EXAMPLES: Adding more than two signed numbers

The addition of more than two signed numbers can be performed in either of the following ways:

1. Find the sum of the following: Solution. We can add the numbers in order, as follows: 16 plus - 22 is - 6; -6 plus - 8 is -14; -14 plus 11 is -3; -3 plus 13 is 10; hence the sum is 10. That is An alternate solution is to add together all the positive numbers, then add together all the negative numbers, and finally add these two sums together. Thus: 2. Find the sum of the following: Solution. To add like terms, we add the numerical coefficients of the terms and indicate that this sum is multiplied by their common literal part (their common literal part is x2). Adding the positive coefficients, then adding the negative coefficients, and finally, adding these sums : Find the sum of each of the following: Answers : 1. 6. 2. 6. 3. 1. 4. 9. 5. 1. 6. 0. 7. 33. 8. -1. 9. 12. 10. -22. 11. -51. 12. x. 13. -8a. 14. -6y. 15. 0. 16. -6x. 17. -13y. 18. -5p. 19. 58y. 20. 32x. 21. 20y. 22. -53xy. 23. -65t. 24. 66y

Subtraction of signed numbers. In arithmetic, when we were asked to subtract 15 from 22 we wrote: In arriving at this answer, we found the difference between 15 and 22; that is, we found the number which, when added to 15, would produce 22. Let us apply the same procedure to algebraic subtraction.