Unit oneAdding & Subtracting

Integers

1st ) Adding two positive integers

• Find the result then represent it on the number line• 3 + 5 = ..8...... • -1* 0* 1* 2* 3* 4* 5* 6* 7* 8* 9*

• 4 + 3 = ...7..... • -1* 0 * 1* 2 * 3* 4* 5* 6* 7* 8*• 2nd) Adding two negative integers• (-5) + (-2) =..-7.....• -8* -7* -6* -5* -4* -3* -2* -1* 0* 1* 2* • (-3) + (-4) = ....-7.. • -8* -7 * -6* -5 * -4 * -3 * -2 * -1 * 0 * 1 * 2 *

3rd ) Adding (ve+) & (ve-) integers

• 6 + ( -4 ) = ....2.....

• -1* 0* 1* 2* 3* 4 * 5* 6* 7* 8* • 7 + ( -8 ) = .....-1.....

• -2* -1* 0* 1* 2* 3* 4* 5* 6* 7* 8*• (-4 ) + 5 = ...1......

• -5* -4* -3* -2* -1* 0* 1* 2 * 3* 4* 5*

Find the result:-a)4 + 2 =

b) (-4) + (-1) =

c) -10 + 3= d) 5 – 9 = e) 0 + (-5) = f) -9 – 8 = g) 0 – 7 =

h) 0 – (-3) =

6

-5

-7

-4

-5

-17

-7

3

i) -3 – 3 = j) -7 + 4 = k) (-10) + (-10) = l) (-5) – 0 =

m) 33 - -13 = n) -14 - -28 = o) -5 + -10 = p) -4 + 0 =

-6

-3

-20

-5

20

-14

5

4

Properties of addition in ( Z )

• 1st) Closure property: addition is closed in ( Z )• Example : 5 ϵ Z & -2 ϵ Z , then 5 + ( -2 ) = 3 ϵ Z• 2nd) Commutative property : if a , b ϵ Z , then a + b = b + a• Example : 9 + (-4 ) = 5 & (-4) + 9 =5 then 9 + (-4) = (-4) + 9 =5• 3rd) Associative property : if a , b , c ϵ Z then a + b + c = ( a + b )+ c = a + (b +c)• Example : 5 + (-4) + (-3) = ( 5 + (-4) ) +( -3 ) = -2• = 5 + ( (-4) + (-3) ) = -2

4th) Additive identity ( neutral) element in (Z) is ( zero )• Example : * 6 + 0 = 0 + 6 = 6 * -4 + 0 = 0 + (-4) = -4• 5th) Additive inverse ( opposite ) property: the additive inverse of a is ( -a )• Where : a + (-a) = 0 example : additive inverse of (3 is -3) for 3 + (-3) = 0• Note that : 1) the additive inverse of zero is zero because 0 + 0 = 0• The additive inverse of a is (-a) & the additive inverse of (-a) = a• The additive inverse of (-a) is -(-a) = a

Write the inverse (0pposite) of the numbers:-

a)10 is b) -12 isc) 0isd) 45 is e) -27 isf) 1 isg)- 36 ish) -30 isi) – 19 is j) - -25 is k) 0 is l) -(-13) is

-1012

0-4527

-1363019

250

-13

Possibility of Subtraction in (Z)

• Subtraction is closed in Z : * 10 – 6 =4 ϵ Z * -5 – 3 = - 8 ϵ Z

• Subtraction is not commutative in Z : 4 – 3 = 1 but 3 – 4 = -1

Then 4 – 3 ≠ 3 – 4 • Subtraction is not associative in Z : where the result

of 5 – 3 – 1

• ( 5 – 3 ) - 1 =1 but 5 - (3 - 1) = 3 then ( 5 – 3 ) – 1 ≠ 5 – ( 3 – 1 )

Write the Property of each of the following:-

• -7 + 5 = 5 + ( -7 ) ( ...commutative......................)

• 9 + ( -9 ) = 0 (...additive inverse...................)

• 0 + ( -11) = -11 ( ... Additive identity.................)

• (-8 + 5 ) + 2 = -8 + ( 5 + 2 ) ( ... Associative .....................)

• (14 + 6 ) + 10 = (14 + 10 ) + 6 (..... commutative.................)

• –b + b = 0 ( ... additive inverse................)

Use the Property of Addition in (Z) to find the result :-

• a)-5 + (-8) + 5

• (-5 + (-8) ) + 5 ( associative )

• (-5 + 5) + (-8) ( commutative& assoc.)

• 0 + (-8) (additive inverse)

• = -8 ( additive identity)

• b)113 – 120 + 17

• ( 113 – 120 ) + 17 ( associative)

• 113 + 17 – 120 ( commutative)

• (113 + 17 ) – 120 ( associative )

• 130 – 120 = 10