The Set of integers

1. The Set of integers Remember: The set of natural number N = {0 , 1 , 2 , 3 , ……… } The set of Counting number C = { 1 , 2 , 3 , ……… } The set of integers Z = { … , -3 , -2 , -1 , 0 , 1 , 2 , 3 … } The set of positive integers Z+ = { 1 , 2 , 3 , …… } The set of negative integers Z-= { 1 , 2 , 3 , …… }

2. Positive number : a number greater than zero. 0 1 2 3 4 5 6

3. Negative number : a number less than zero. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4. Negative Numbers Are Used to Measure Temperature

5. Negative Numbers Are Used to Measure Under Sea Level 30 20 10 0 -10 -20 -30 -40 -50

7. Definition • Opposite Numbers – numbers that are the same distance from zero in the opposite direction -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

8. Remarks: • Z = Z+ { 0 } Z- • N Z • Z+ Z • Z- Z Z {0} Z+ Z- N

9. We can use a Venn Diagram to represent the set of integers Z Z+ {0} Z-

10. In Summary Natural

11. The absolute value of an integer a = = The positive value of a For example: = 3 = 7 = 0

12. Express each of the following sets using the listing method: • The set of integers less than 4 = { 3 , 2 , 1 , 0 , -1 , ……… } b) The set of integers greater than -3 = { -2 , -1 , 0 , 1 , 2 , ……… } c) The set of integers between -5 and 2 { -4 , -3 , -2 , -1 , 0 , 1 }

13. d) The set of non-negative integers = { 0 , 1 , 2 , 3 , 4 , ………….. } e) The set of non-positive integers = { 0 , -1 , -2 , -3 , -4 , ………….. } f) The set of even positive integers = { 2 , 4 , 6 , 8 , ………… }

14. Exercise Put a) l -3l Z- b) -2 N c) {-5} Z d) N Z e) – l-7l Z- f) Z- Z g) Z+ Nh) Z N

15. Find the value of x in each of the following: • l x l = 8 x = -8 or 8 • l -2 l = x x = 2 • -3 {-1 , 3 , x , -2 } x = -3 • X { 4 , 3, -2 } { -4 , -3 , -2 } x = -2 • { 5 , x } { -2 , 0 , 2} = { 5 , -2 , -5 , 0 , 2 } x = -5

16. Complete: 1 • … is the smallest positive integer. • Sea level represented by the number … • … is the greatest negative integer. • The complement of Z- with respect to Z = …. • The complement of N with respect to Z = … • The complement of Z- with respect to Z- = …. • The absolute value of -567 is …. 0 -1 N Z- { } 567

17. Put > , < or = • 2 -8 • -7 3 • l -4 l 9 • 0 -6 • - l -15 l -8 • -2 -11

18. Complete: 1 • is the smallest positive integer. • Sea level represented by the number • is the greatest negative integer. • The complement of Z- with respect to Z = • The complement of N with respect to Z = • The complement of Z- with respect to Z- = • The absolute value of -567 is 0 -1 N Z- { } 567

19. Positive numbers move to the right • Negative numbers move to the left • + 3 + (-8) = -5

20. Ordering and Comparing Integers • Arrange the following integers ascendingly: 3 , -12 , 5 , -7 , 1 The order: -12 , -7 , 1 , 3 , 5

21. (2) Arrange the following integers descendingly: 8 , - 54 , 9 , - 22 , 0 , - 35 The order: , -22 , -35 , -54 9 , 8 , 0

22. > (3) put > , < or = • 5 -7 b) -8 20 • I -14 l 4 • I -2 l 2 • 5 + I -5 l 0 • -30 -13 < > = > <

23. (4) Write the previous integer and the next integer for each of the following integers: a) 19 b) -15 c) o Solution • 18 , 20 • - 16 , -14 c) -1 , 1

24. (5) Write the integers between each two integers of the following: • -4 , 3 • -5 , 1 Solution • -3 , -2 , -1 , 0 , 1 , 2 b) -4 , -3 , -2 , -1 , 0