52. Subtraction of signed numbers. By subtracting 7 from 12 we have always meant finding the number which added to 7 gives 12. This meaning is easily applied to signed numbers. Since +7 added to +5 gives +12, then +7 subtracted from +12 gives +5.

What number added to -3 gives -7?
What number added to -4 gives -2?
What number added to +6 gives +2?

In the following exercises subtract the lower number from the upper by finding the number which added to the subtrahend gives the minuend.

Check the preceding results by adding the subtrahend and remainder.

In each of the preceding examples think the sign of the subtrahend changed and add the resulting number to the minuend. Compare the result with that already found. Can you make a rule for subtraction?

It is interesting to see how the rule applies in finding the difference between two readings on the thermometer scale. EXAMPLE. Find the difference between a reading of +29° and a reading of +35°.

SOLUTION. Starting at +29° and going to +35° we pass over 6° and go in the positive direction. Therefore the result of subtracting +29° from +35° is +6°. This result will also be obtained by adding -29° and +35° on the scale, that is, by going +35°, then —29° and reading from 0 to the stopping point.

Find the remainder when -6 is subtracted from +2 by starting at 6 on the scale and counting the spaces to +2. Then find it by adding +6 to +2 on the scale. This illustrates the

Rule for subtraction. Change the sign of the subtrahend and add.

53. Applications of addition and subtraction of signed numbers.

We have seen that numbers above and below zero on the thermometer are represented by + and numbers, respectively.

In the same way $15 of debt may be represented by -$15 and $15 of credit by +$15.

East longitude is usually called - and west longitude +. Ten years B.c. may be called -10 years and 10 years A.D. may be called +10 years.

In many other cases opposite kinds of numbers may be indicated by the plus and the minus signs.

Exercise 58

-$15. What is

1. A boy's account shows +$25 and -$15. his balance?

2. To how much does a debit of $40 and a credit of $25 amount?

3. A vessel is in 15° east longitude and sails 25° westward. What is then its longitude?

4. A vessel is in 12° south latitude and sails 20° northward. What is its latitude then?

5. A vessel is in latitude -12° and sails -15°. What is its latitude then?

6. A man was born in the year 18 B.C. and lived 52 years. In what year did he die?

7. How old was a man at his death who was born 4 B.C. and died 29 A.D.?

How fast

8. A train runs south 15 mi. an hour and a brakeman walks forward on it at the rate of 3 mi. an hour. and in what direction is the brakeman moving?

9. How fast is the brakeman of the preceding problem moving if he walks toward the rear of the train, the rates being the same as before?

10. A pupil takes three tests to determine his standing in a subject. In the first his mark is 8 points below passing, in the second it is 17 points above passing, and in the third it is 6 points below. Does he pass? By what margin?

11. In a football game the ball moves 12 yards toward the north goal, then 8 yards toward the south, then 5 yards north and stops. How far and in what direction is the stopping from the starting point?

12. In a certain election there are 38 votes for and 27 against a certain candidate. What is the candidate's plurality?

13. Julius Cæsar was born 100 B.C. and died 44 B.C. How old was he when he died?

14. Octavius Cæsar was born 63 B.C. and died 14 a.d. How old was he when he died?

15. A boy has debts of 10, 25¢, $2.25, and 40%; he has credits of 854, 60¢, $1.25, and 50¢. Represent these debts and credits by signed numbers and find their sum.

16. A merchant has debts of $100, $500, and $750; he has credits of $50, $600, $10, and $2000. Represent these debts and credits by signed numbers and find their sum.

54. Multiplication of signed numbers. The temperature

fell 3° each hour for 4 hours.

What was the total fall?
-2° each hour for 5 hours.
If it changed +6° each hour

The temperature changed What was the total change? for 3 hours, what was the total change? If it changed -7° each hour for 6 hours?

The pupil has known that 3×7 means that 7 is to be added 3 times, beginning with zero. We may think +3x+7 to mean the same as 3×7 and do to the +7 what the sign before the 3 tells us to do, that is to add +7 three times. Thus +3· +7=0+(+7)+(+7)+(+7)=+21.

Then +3 · −7=0+(−7)+(−7)+(−7) = −21.

.

Similarly -3 +7 means that +7 is to be subtracted three times beginning with zero. Thus,

Also,
That is

-3 +7=0-(+7)~(+7)-(+7)=-21.
-37-0-(-7)-(-7)-(-7)=+21.

(+3)(+7)=+21.
(+3)(−7) = −21.

(−3)(+7)= −21.

(-3)(−7)=+21.

From this the pupil should get the following

Rule. To find the product of two factors multiply their absolute values. If the two factors have like signs the product is plus, if unlike signs the product is minus.

55. Division of signed numbers. When given the product of two numbers and one of the numbers, the process by which the other number is found is called division.

Supply the missing number:

(a)()(+3) = +12.
(b) (-3)()=+12.
(c) (+4)()=-12.
(d) (-4)(
(-4)( )=-12.

From (a) we see that (+12)÷(+3)=+4.
From (b) we see that (+12)÷(−3) = −4.
From (c) we see that (-12)+(+4)=-3.
From (d) we see that (-12)÷(-4)=+3.

12÷3=4

As in multiplication, so also in division, the quotient of two numbers with like signs is positive and with unlike signs is negative. This statement is sometimes shortened to like signs give plus and unlike signs give minus.