Properties of Integers :

1. Closure property:

For any two integers a and b, a + b is also an integer (a, b ∈ Z then a + b ∈ Z)

i.−5 and 3 are integers and −5 + 3= −2 is also an integer.
ii.−3 and −2 are integers and −3 + (−2) = −5 is also an integer.

For any two integers a and b, a × b is also an integer (a, b ∈ Z then a × b ∈ Z)

i.−5 and 3 are integers and (−5) × 3= −15 is also an integer.
ii.−3 and −2 are integers and (−3) × (−2) = 6 is also an integer.

For any two integers a and b, a − b is also an integer (a, b ∈ Z then a − b ∈ Z)

i.−5 and 3 are integers and −5 − 3= −8 is also an integer.
ii.−3 and −2 are integers and −3 − (−2) = −1 is also an integer.

For any two integers a and b (b ≠ 0), a ÷ b need not an integer (a, b ∈ Z then a ÷ b ∉ Z)

i.9 and 10 are integers and 9 ÷ 10 = 0.9 is not an integer.
ii.−6 and −2 are integers and −6 ÷ (−2) = 3 is also an integer.

∴ Integers are closed under addition, subtraction and multiplication. But need not be closed under division.

Do you know?
∈ and ∉ are used to indicate whether the given element belongs to the given collection or not.
Example:
0 ∈ W (0 belongs to whole numbers)
0 ∉ N (0 does not belong to natural numbers)
−3 ∈ Z (−3 belongs to integers)

2. Commutative Law :

For any two integers a and b, a + b = b + a

i.(−6) + 3 = −3 and 3 + (−6) = −3.
 So, (−6) + 3=3 + (−6)
ii.(−6) + (−5) = −11 and (−5) + (−6) = −11.
  So, (−6) + (−5) =(−5) + (−6)

For any two integers a and b, a × b = b × a

i.(−6) × 3 = −18 and 3 × (−6) = −18.
 So, (−6) × 3=3 × (−6)
ii.(−6) × (−5) = 30 and (−5) × (−6) = 30.
  So, (−6) × (−5) =(−5) × (−6)

In general, for any two integers a and b, a − b ≠ b − a

i.−6 − 3 = −9 and 3 − (−6) = 3 + 6 = 9.
 So, −6 − 3 ≠ 3 − (−6)
ii.(−6) − (−5) = −1 and (−5) − (−6) = 1.
  So, (−6) − (−5) ≠ (−5) − (−6)

In general, for any two integers a,b (b ≠ 0), a ÷ b ≠ b ÷ a

2 ÷ 10 = 0.2 and 10 ÷ 2 = 5
So, 2 ÷ 10 ≠ 10 ÷ 2

∴ Integers are commutative under addition and multiplication. But not commutative under subtraction and division.

3. Associative Law :

For any three integers a, b and c, (a + b) +c = a + (b + c)

(−6 + 3) + 2 = −3 + 2 = −1
−6 + (3 +2)=−6 + 5= −1
 So,(−6 + 3) + 2= −6 + (3 + 2)

For any three integers a, b and c, (a × b) × c = a × (b × c)

(−6 × 3) × 2= −18 × 2= − 36
−6 × (3 × 2)=−6 × 6 = −36
So, (−6 × 3) × 2= −6 × (3 × 2)

In general, for any three integers a, b and c, (a – b) – c ≠ a – (b – c)

(−6 − 3) −2= −9 −2 = −11
−6 − (3 − 2) = −6 −1 = −7
So, (−6 − 3) −2 ≠ −6 − (3 −2)

In general, for any three integers a, b and c, (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

(−18 ÷ 6) ÷ 3 = −3 ÷ 3 = −1
−18 ÷ (6 ÷ 3)= −18 ÷ 2 = −9
So, (−6 ÷ 3) ÷ 2 ≠ −6 ÷ (3÷2)

∴ Integers are associative under addition and multiplication. But not associative under subtraction and division.

4.Identity property :

For any integer a, a + 0 = 0 + a = a

Zero is the additive identity.

For any integer a, a × 1 = 1 × a = a

1 is the multiplicative identity.

5.Additive inverse property :

For any integer a, there exists an integer (–a) such that a + (–a) = (–a) + a = 0.

Both the integers a and −a are called additive inverse of each other.

What should be multiplied by 6 to get multiplicative identity 1? Is it exist in integers?

Multiply 6 by`1/6` we get 1. But `1/6`is not in integers.
So integers do not have the inverse identity.

6. Distributive law :

For any integers a, b and c, a × (b + c) = a × b + a × c

i)Let us take three integers −2, 1 and 3,
−2 × (1 + 3) = [(−2) × 1] + [(−2) × 3]
−2 × 4 = −2 + (−6)
−8 = −8
ii)Let us take three integers −1, 3 and −5,
−1 × [3 + (−5)] = [(−1) × 3] + [(−1) × (−5)]
−1 × (−2) = −3 + (+5)
2 = 2

Multiplication distribute over the addition of integers

Verify −3 × [ (−4) − 2] = [(−3) × (−4)] − [(−3) × 2]. Is multiplication distribute over subtraction of integers? Write your conclusion.

L.H.S=−3 × [ (−4) − 2]
=−3 × (−6) =18.
R.H.S=[(−3) × (−4)] − [(−3) × 2]
=(12)−(−6)=12+6=18
L.H.S=R.H.S
Multiplication distributes over subtraction of integers.

If a negative integer multiplied even number of times, the product is a positive integer. If a negative integer is multiplied odd number of times, the product is a negative integer.

(−1) × (−1) = + 1
(−1) × (−1) × (−1) = −1
(−1) × (−1) × (−1) × (−1) = + 1
(−1) × (−1) × (−1) × (−1) × (−1) = −1

EXAMPLES

Example 7 : Find the additive inverses of (+2) , (−3),5, −8, 1 and 0.

Additive inverse of +2 = − (+2) = −2.
Additive inverse of −3 = − (−3) = + 3.
Additive inverse of 5 = − 5.
Additive inverse of −8 = − (−8) = + 8.
Additive inverse of 1 = −1.
Additive inverse of 0 = 0

Example 8: Multiply the following using associative law. i) −25 × (−4) × 2 × (− 8) ii) (−20) × (−2) × (−5) × 7

i) −25 × (−4) × 2 × (−8) = [ −25 × (−4)] × 2 × (−8)
= [100 × 2] × (−8)= 200 × (−8)= −1600
ii) (−20) × (−2) × (−5) × 7 = (−20) × [(−2) × (−5)] × 7
= [(−20) × 10] × 7 = −200 × 7 = −1400

Example 9 : Are (−42) × (−7) and (−7) × (−42) equal? Write the law.

(−42) × (−7) = + 294 (−7) × (−42) = +294
∴ (−42) × (−7) = (−7) × (−42)
It is multiplicative commutative law.

Example 10 :Simplify 26 × (−48) + (−48) × (−36) using suitable laws.

26 × (−48) + (−48) × (−36)
= (−48) × 26+ (−48) × (−36) (Commutative law)
= (−48) × [26+ (−36)] (Distributive law)
= (−48) × (−10) = 480

EXERCISE-3