unit one adding & subtracting integers. 1 st ) adding two positive integers find the result then...
TRANSCRIPT
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