Properties on
Operation on Integers
Closure Property
Addition and Subtraction:
Closure property under addition and
subtraction states that the sum or
difference of any two integers is also an
integer.
If a and b are integers (Z) then:
a + b = Z or a – b = Z
Closure Property
Multiplication:
Closure property under multiplication
states that the product of any two
integers is also an integer.
If a and b are integers (ℤ) then:
a·b=ℤ
Closure Property
Division:
Division of integers does not hold the
closure property.
If a and b are integers, then a ÷ b may or
may not be an integer, which is why
closure property is not applicable for
division of integers.
Commutative Property
Addition:
Commutative property of addition states
that, two integers can be added in any
order.
If a and b are integers then,
a + b = b + a.
Commutative Property
Multiplication:
Commutative property of multiplication
states that, two integers can be
multiplied in any order.
If a and b are integers then,
a · b = b · a.
Commutative Property
Subtraction and Division:
Subtraction and division do not hold
commutative property for integers.
If a and b are integers, then:
a–b≠b–a
a÷b≠b÷a
Associative Property
Addition:
Associative property of addition states
that if three integers are added, it makes
no difference whether which two are
added first.
If a, b, or c are integers then,
(a+b) + c = a + (b+c).
Associative Property
Multiplication:
Associative property of multiplication
states that if three integers are
multiplied, it makes no difference
whether which two are multiplied first.
If a, b, or c are integers then,
(a · b) · c = a · (b · c).
Associative Property
Subtraction and Division:
Subtraction and division do not hold
associative property for integers.
If a, b, and c are integers then,
a – (b – c) ≠ (a – b) – c
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c
Identity Property
Addition:
Identity property of addition states that any
integer added to the identity element zero
(0) will remain unchanged. Zero (0) is the
identity element of addition.
If a is an integer, then,
a + 0 = a or 0 + a = a.
Identity Property
Multiplication:
Identity property of multiplication states that
any number multiplied to the identity
element 1 will remain unchanged. 1 is the
identity element for multiplication.
If a is an integer, then,
a · 1 = a.
Inverse Property
Addition:
Inverse property of addition states that the
sum of an integer and its additive
inverse(opposite) is the identity element 0.
a and (-a) are additive inverses.
If a is an integer, then,
a + (-a) = 0
(-a) + a = 0.
Inverse Property
Multiplication:
Inverse property of multiplication states that
the product of an integer and its
multiplicative inverse (reciprocal) is the
identity element 1.
If a is an integer, then.
1
𝑎
1
𝑎
a · = 1 and · a = 1
provided that a ≠ 0.
Distributive Property
Distributive property of multiplication over
addition
or
subtraction
states
that
multiplication distributes over addition or
subtraction.
If a, b and c are integers then,
a · (b + c) = (a · b) + (a · c)
or
a · (b – c) = (a · b) – (a · c)
Which property of real number justifies
each statement?
1.
0 + (3)= – 3
_______
2.
2 (3 – 5)=2(3) – 2(5)
_______
3.
(– 6) +(– 7)=(– 7) + (– 6) _______
4.
(1)(– 9)= – 9
_______
5.
10 + (– 10)=0
_______
Rewrite the following expressions using
the given property.
1.
12a – 5a=
_______ (Distributive)
2.
(7a)b =
_______ (Associative)
3.
8 + 5=
_______ (Commutative)
4.
– 4(1)=
_______ (Identity)
5.
25 + (– 25)= _______ (Inverse)