Properties of Integers

Integers have various properties, the major properties of integers are:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property
• Identity Property
• Additive Inverse
• Multiplicative Inverse

Closure Property

Closure property of integers states that if two integers are added or multiplied together their result is always an integer. For integers p and q

• p + q = integer
• p × q = integer

Example:

(-8) + 11 = 3 (An integer)
(-8) × 11 = -88 (An integer)

Commutative Property

Commutative property of integers states that for two integers p and q

• p + q = q + p
• p × q = q × p

Example:

(-8) + 11 = 11 + (-8) = 3
(-8) × 11 = 11 × (-8) = -88

But the commutative property is not applicable to the subtraction and division of integers.

Associative Property

Associative property of integers states that for integers p, q, and r

• p + (q + r) = (p + q) + r
• p × (q × r) = (p × q) × r

Example:

5 + (4 + 3) = (5 + 4) + 3 = 12
5 × (4 × 3) = (5 × 4) × 3 = 60

Distributive Property

Distributive property of integers states that for integers p, q, and r

• p × (q + r) = p × q + p × r

For Example, Prove: 5 × (9 + 6) = 5 × 9 + 5 × 6

Solution:

LHS = 5 × (9 + 6) 
        = 5 × 15
        = 75

RHS = 5 × 9 + 5 × 6 
        = 45 + 30
        = 75

Thus, LHS = RHS Proved

Identity Property

Integers hold Identity elements both for addition and multiplication. Operation with the Identity element yields the same integers, such that

• p + 0 = p
• p × 1 = p

Here, 0 is Additive Identity, and 1 is Multiplicative Identity.

Additive Inverse

Every integer has its additive inverse. An additive inverse is a number that in addition to the integer gives the additive identity. For integers, Additive Identity is 0. For example, take an integer p then its additive inverse is (-p) such that

• p + (-p) = 0

Multiplicative Inverse

Every integer has its multiplicative inverse. A multiplicative inverse is a number that when multiplied to the integer gives the multiplicative identity. For integers, Multiplicative Identity is 1. For example, take an integer p then its multiplicative inverse is (1/p) such that

• p × (1/p) = 1

Integers are any number including 0, positive numbers, and negative numbers. Examples of integers are 3, 70, -92, 234, -3567 etc. Examples of numbers that are not integers are -1.3, 3/4, 2.78, and 345.97

In this article, we have covered everything about what are integers in maths, integers definition, types of integers, etc. to Integers classes 6 and 7.