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Mastering Integer Addition and Subtraction: Properties and Examples

This resource provides a comprehensive guide to adding and subtracting integers, both positive and negative. It covers various properties of addition such as closure, commutative, associative, and identity. With clear examples, learners can practice adding two positive integers, two negative integers, and combinations of positive and negative integers. The guide also explores the concept of additive inverses and the properties of subtraction, illustrating that subtraction is not commutative or associative. This is an essential tool for mastering integer operations.

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Mastering Integer Addition and Subtraction: Properties and Examples

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  1. Unit one Adding & Subtracting Integers

  2. 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 *

  3. 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*

  4. Find the result:- 6 -5 a)4 + 2 = b) (-4) + (-1) = c) -10 + 3= d) 5 – 9 = e) 0 + (-5) = f) -9 – 8 = g) 0 – 7 = h) 0 – (-3) = -7 -4 -5 -17 -7 3

  5. -6 i) -3 – 3 = j) -7 + 4 = k) (-10) + (-10) = l) (-5) – 0 = m) 33 - -13 = n) -14 - -28 = o) -5 + -10 = p) -4 + 0 = -3 -20 -5 20 -14 5 4

  6. 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) ) = -24th) 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

  7. Write the inverse (0pposite) of the numbers:- -10 12 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) - -25 is k) 0 is l) -(-13) is 0 -45 27 -1 36 30 19 25 0 -13

  8. 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 )

  9. 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................)

  10. 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)

  11. b)113 – 120 + 17 • ( 113 – 120 ) + 17 ( associative) • 113 + 17 – 120 ( commutative) • (113 + 17 ) – 120 ( associative ) • 130 – 120 = 10

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