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2. Same sign …..ADD….ADD…..ADD Different sign …..SUBTRACT…SUBTRACT…. SUBTRACT All the time you have to pick….THE GREATER NUMBER SIGN No sign means….positive sign

3. Adding Positive Numbers : 2 + 5 = 7  ( + 2 ) + ( + 5 ) = 7 Adding Negative numbers: - 2 – 2 = - 4  ( - 2 ) + ( - 2 ) = - 4

4. Subtracting numbers: • 8 + 5 = -3 As 8 is negative and 3 is positive Different sign …..SUBTRACT…SUBTRACT…. SUBTRACT 8 – 5 = 3 8 > 5 All the time you have to pick….THE GREATER NUMBER SIGN -3

5. 5 – 2 = 3 As 2 is negative and 5 is positive Different sign …..SUBTRACT…SUBTRACT…. SUBTRACT 5 – 2 = 3 5 > 2 All the time you have to pick….THE GREATER NUMBER SIGN + 3

6. Critical thinking: In 1996, divers discovered a ship wreck believed to be the pirate Blackbeard's flagship "queen Anne's Revenge", the ship's cannons were found 21 feet below the water's surface, and its bell was found 20 feet below the surface. Which was closer to the surface, the cannons or the bell?

7. Answer: • Name the integers that represent each situationscannons 21 feet below the surface -21 • Bell 20 feet below the surface - 20 • Water surface 0 • The bell is closer to the surface than the cannons • -20> -21

8. Properties of Addition operation Z

9. Closure property : Z is closed under the addition operation . Means that the sum of two integers is an integer, therefore addition operation is possible in Z . Example : 3 + 2 = 5 -4 + 6 = 2

10. Commutative property : Means that sum of any two integers does not change if we change their places . Example : 9 + 5 = 5 + 9 = 14 - 6 + (-10 ) = - 10 + ( - 6 ) = - 1

11. Associative property of addition : Means that we can group numbers in brackets in different ways and still we get the same answer . Example : ( 3 + 7 ) + 4 = 3 + ( 7 + 4 ) = 7 + ( 3 + 4 ) = 14 ( - 7 + 2 ) + 9 = -7 + ( 2 + 9 ) = 4

12. The Additive –identity : Zero is the additive identity ( neutral ) in Z as in N Example : 0 + 5 = 5 + 0 = 5

13. The Additive Inverse: For each positive integer ( 5 ) on the number line there is an opposite negative integer ( - 5 ) where their sum is zero. • example: 5 + ( -5 ) = 0

14. Properties of subtraction operation in Z

15. Closure property: Z is closed under the subtraction operation, means that the difference between any two integers is an integer, therefore subtraction operation is always possible in Z Examples: 7 – 5 = 2 -9 – 8= -17 5 – 9 = - 4 9 – 3 = 6

16. Commutative property: The subtraction operation isn’t commutative in Z Example: 5 – 8 = -3 8 – 5 = 3 8 – 5 5 – 8 Subtraction is not commutative

17. Associative property: Subtraction operation isn’t associative in Z Example: -9 – (3 – 8) = -9 – (- 5) = -9 + 5 = -4 (-9 – 3) – 8 = - 12 – 8 = - 20 subtraction operation isn’t associative

18. Properties of multiplication operation in Z :

19. Closure property : Z is closed under multiplication, means that the product of any two integers is an integer . Example : 7 x 8 = 56 2 x- 7 = -14

20. Commutative Property : The multiplication operation is commutative in Z, If a  Z , b  Z then a x b = b x a Example : -2 x 4 = - 8 4 x -2 = -8

21. Associative Property in Multiplication : Example : (-2 x 4) x 3 = -24 -2 x ( 4 x 3 ) = -24 Multiplication is associative.

22. The multiplicative identity : Example : -7 x 1 = -7 4 x 1 = 4

23. The distributive property : It means distributing multiplication operation over addition operation Example : 5 x ( 3 + 7 ) =( 5 x 3 ) + ( 5 x 7 ) = 15 + 35 = 50

24. Properties of Division operation in Z

25. Closure property : Z is not closed under division operation. The division operation is not always possible in Z . - 36 ÷ 9 = -4 • 34 ÷ 4 = 8.5 not belong to Z

26. Commutative Property : The division operation is not commutative in Z Division by zero is impossible in Z as it was in N 14 ÷ 2 2 ÷ 14

27. Associative Property : Division operation is not associative in Z ( 24 ÷ 4 ) ÷ 2 24 ÷ ( 4 ÷ 2) 6 ÷ 2 = 3 24 ÷ 2 = 12

28. Example: 116 + 190 + ( -116 ) = 116 + ( -116 ) + 190 = Comm.prop. (116 + ( -116 ) ) + 190 = Ass.prop. 0 + 190 = add.inverse 190 add.identity

29. Evaluation Solve: -9 - 3 = 2) 5 – 8 = 3) -9 + 2 = Solve using the distributive property: 3 x ( 5 + 7 ) Solve using the properties: 115 + 120 + ( -115 ) =