Unit one Adding & Subtracting Integers

1 st ) Adding two positive integers Find the result then represent it on the number line = * 0* 1* 2* 3* 4* 5* 6* 7* 8* 9* = * 0 * 1* 2 * 3* 4* 5* 6* 7* 8* 2nd) Adding two negative integers (-5) + (-2) = * -7* -6* -5* -4* -3* -2* -1* 0* 1* 2* (-3) + (-4) = * -7 * -6* -5 * -4 * -3 * -2 * -1 * 0 * 1 * 2 *

3 rd ) Adding (ve+) & (ve-) integers 6 + ( -4 ) = * 0* 1* 2* 3* 4 * 5* 6* 7* 8* 7 + ( -8 ) = * -1* 0* 1* 2* 3* 4* 5* 6* 7* 8* (-4 ) + 5 = * -4* -3* -2* -1* 0* 1* 2 * 3* 4* 5*

Find the result:- a)4 + 2 = b) (-4) + (-1) = c) = d) 5 – 9 = e) 0 + (-5) = f) -9 – 8 = g) 0 – 7 = h) 0 – (-3) =

i) -3 – 3 = j) = k) (-10) + (-10) = l) (-5) – 0 = m) = n) = o) = p) =

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 + (-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 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 is c) 0is d) 45 is e) -27 is f) 1 is g)- 36 is h) -30 is i) – 19 is j) is k) 0 is l) -(-13) is

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: = 5 + ( -7 ) (...commutative ) 9 + ( -9 ) = 0 (...additive inverse ) 0 + ( -11) = -11 (... Additive identity ) ( ) + 2 = -8 + ( ) (... Associative ) ( ) + 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 – ( 113 – 120 ) + 17 ( associative) – 120 ( commutative) ( ) – 120 ( associative ) 130 – 120 = 10