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 , …… }

• Positive number : a number greater than zero.

0 1 2 3 4 5 6

• Negative number : a number less than zero.

0 1 2 3 4 5 6-1-2-3-4-5-6

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level

0102030

-10-20-30

-40-50

Definition• Opposite Numbers – numbers that are the same distance from

zero in the opposite direction

0 1 2 3 4 5 6-1-2-3-4-5-6

Remarks:

• Z = Z+ { 0 } Z- • N Z• Z+ Z• Z- Z

Z

Z- Z+{0}

N

• We can use a Venn Diagram to represent the set of integers

Z- Z+{0}

Z

10

In Summary

Natural

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

Express each of the following sets using the listing method:

a) 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 }

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 , ………… }

ExercisePut

a) l -3l Z- b) -2 N c) {-5} Z d) N Z

e) – l-7l Z-

f) Z- Z g) Z+ N

h) Z N

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

Complete:

• … 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 ….

10

-1NZ- {}567

Put > , < or =

• 2 -8• -7 3• l -4 l 9• 0 -6• - l -15 l -8• -2 -11

Complete:

• 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

10

-1NZ- {}567

19

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

Ordering and Comparing Integers

(1) Arrange the following integers ascendingly: 3 , -12 , 5 , -7 , 1

The order: -

12 - , 7 , 1 , 3 , 5

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

The order:

9 , 8 , 0- , 22 - , 35 - , 54

(3) put > , < or =a) 5 -7b) -8 20c) I -14 l 4d) I -2 l 2e) 5 + I -5 l 0f) -30 -13

><

>

><

=

(4) Write the previous integer and the next integer for each of the following integers:

a) 19 b) -15 c) oSolution

a) 18 , 20

b) - 16 , -14

c) -1 , 1

(5) Write the integers between each two integers of the following:

a) -4 , 3b) -5 , 1

Solution c) -3 , -2 , -1 , 0 , 1 , 2

b) -4 , -3 , -2 , -1 , 0