1. Integer Subtraction Subtraction with positive and negative integers

2. Objectives Know that subtracting an integer is done by adding its opposite

3. Over view of how the objectives will be achieved The context of finding the new temperature after a given fall in temperature will be used to demonstrate how to subtract positive integers. The context of finding the change between an initial temperature and a final temperature will be used to demonstrate how to be able to subtract negative integers.

4. Introductory Ideas Introductory Problem Mrs. Johnson has her refrigerator set to operate at five degrees above zero. The thermostat develops a fault and causes the temperature to fall by three degrees. The word ‘falls’ suggests that we need to subtract two degrees from the setting of five degrees. The temperature will therefore become; 5 - 3 = 2 that is, two degrees above zero. What will the temperature be inside the refrigerator?

5. Introductory Ideas After several days the fault worsens, and the temperature falls by a further seven degrees. The temperature will become 2 – 7 = ? What will the temperature become now? Following the pattern of the previous slide, we need to subtract seven from the two degrees above zero.

6. Introductory Ideas To find the new temperature we now need to be able to subtract seven from two. We can not do this with ordinary counting numbers but we can with integers. Recall from the presentation on Negative Numbers, a fall of seven degrees is a change of - 7 degrees. Also, from the presentation on Integer Addition, the new temperature is found by combining 2 with - 7: The new temperature will become 2 + - 7

7. Introductory Ideas The two expressions for the temperature after a fall of seven degrees are: 2 – 7 2 + - 7 These two expressions must be equivalent, and because they are equivalent we conclude that: 2 – 7 = 2 + - 7 and,

8. Introductory Ideas To summarize the results of the previous slide: the subtraction of positive seven can be replaced with the addition of negative seven We can hypothesize that the subtraction of any positive number can be replaced with the addition of its opposite.

9. Subtraction of a positive integer is the addition of its opposite We will now prove the hypothesis that subtraction of any positive integer can be replaced by adding its opposite. This proof relies on two facts: the combination of a rise in temperature of any amount with a fall of the same amount is no change, that is; + T + - T = 0 the subtraction of a number from itself is zero, that is: + T - + T = 0

10. Subtraction of a positive integer is the addition of its opposite Let N be any integer, positive or negative and + T be a positive integer and write: N - + T If we add – T + + T to N we still have N, but: Now, + T - - T is zero, so: N + - T + T - T N + - T + + T - + TN - + T N + - T

11. Subtraction of a positive integer is the addition of its opposite Putting all of this together gives: Adding – T + + T to N - + T N + - T + + T - + TN - + T N + - T 0

12. Subtracting a negative integer Introductory problem Mrs. Johnson has her refrigerator set to operate at two degrees above zero. She changes the thermostat setting to five degrees above zero. What is the change in temperature in Mrs. Johnson’s refrigerator? The temperature will increase from two degrees above zero to five degrees above zero, that is, will increase by three degrees. This is a change in temperature of +3o+3o

13. Subtracting a negative integer The change in temperature is the difference between the temperature the thermostat was set to (5 o ), and the temperature setting it was changed from (2 o ): Change in temperature = 5 o – 2 o This problem demonstrates that the change is found by subtracting from the final value (in this case 5 o ) the initial value (in this case 2 o )

14. Subtracting a negative integer Development Problem What would the change in temperature be had Mrs. Johnson been operating her refrigerator at - 2 o, and then changed the thermostat setting to 5 o ? From the previous slide we know that change = final - initial In this problem the final is + 5 o and the initial value is - 2 o The change will be + 5 - - 2 = ?

15. Subtracting a negative integer A temperature change from - 2 o to + 5 o is a rise of seven degrees, + 7 o that is, a change of; + 5 o - - 2 o = + 7 o So the change, given by, final – initial must be + 7 Therefore, This shows that the subtraction - 2 is achieved by adding + 2

16. Subtracting a negative integer To summarize the results of the previous slide: the subtraction of negative two can be replaced with the addition of positive two We can hypothesize that the subtraction of any negative number can be replaced with the addition of its opposite.

17. Subtraction of a negative integer is the addition of its opposite The following proves the hypothesis: Let N be any integer, positive or negative and - T be a negative integer and write; N - - T Adding + T + - T to N N - - T N + + T + - T - - T 0 N + T