Properties of Operations on Integers
OBJECTIVES
a. State and illustrate the different properties of the operations on integers:
closure; commutative; associative; distributive; identity; inverse
b. Rewrite given expressions according to the given property.
c. Appreciate the concept of properties on operation on integers in real-life
situation.
PROPERTIES OF OPERATIONS ON INTEGERS
1. Closure Property
Two integers that are added and multiplied remain as integers.
The set of integers is closed under addition and multiplication.
2. Commutative Property
Changing the order of two numbers that are either being added
or multiplied does not change the value.
3. Associative Property
Changing the grouping of numbers that are either being added
or multiplied does not change its value.
4. Distributive Property
When two numbers have been added / subtracted and then
multiplied by a factor, the result will be the same when each number is
multiplied by the factor and the products are then added / subtracted.
5. Identity Property
Additive Identity
- states that the sum of any number and 0 is the given number. Zero,
“0” is the additive identity.
Multiplicative Identity
- states that the product of any number and 1 is the given number,
a • 1= a. One, “1” is the multiplicative identity.
6. Inverse Property
In Addition
- states that the sum of any number and its additive inverse, is zero.
The additive inverse of the number a is –a.
In Multiplication
- states that the product of any number and its multiplicative inverse or
1
reciprocal, is 1.The multiplicative inverse of the number a is
𝑎
Supply missing terms using the given property.
1. 5 + (-8) = (-8) + Commutative Property
2. (-4) x (2 + 4) = (-4) x + (-4) x Distributive Property
3. (8 + 3 )+7 = + ( 3+7) Associative Property
4. (-10) = (-10) + Identity Property
5. 2 + =0 Inverse Property
Rewrite the following expressions using the given property.
1. 12a – 5a _________________ Distributive Property
2. (7a)b _________________ Associative Property
3. 8 + 5 _________________ Commutative Property
4. -4(1) _________________ Identity Property
5. 25 + (-25) _________________ Inverse Property
6. ([-2] + [-1]) +3 _________________ Associative Property
7. (2 x 5) x 7 _________________ Distributive Property
8. 5 x ( 8 + 5) _________________ Distributive Property
9. -9(9) _________________ Closure Property
1
10. 3 x 3 _________________ Inverse Property
Complete the Table: Which property of real number justifies each statement?
Given Property
1. 0 + (3)=-3
2. 2 (3-5)=2(3)-2(5)
3. (-6) +(-7)=(-7) + (-6)
4. 1 x (-9)= -9
1
5. -4 x 4 =1
6. 2 x (3x7)=( 2x3) x 7
7. 10 + (-10)=0
8. 2 (5)= 5(2)
1 1
9. 1 x (4 )= 4
10. (-3) (4+9)=(-3)(4) + (-3) (9)
Fill in the blanks and determine what properties were used to solve the equations.
1. 5 x ( ______ + 2) = 0
2. -4 + 4 = _____
3. -6 + 0 = _____
4. (-14 +14) + 7 = ______
5. 7 x ( _____ + 7) = 49
