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
= 75RHS = 5 × 9 + 5 × 6
= 45 + 30
= 75Thus, 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 – Definition, Properties and Worksheet
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.
Table of Content
- What are Integers?
- Types of Integers
- Integers on a Number Line
- Rules of Integers
- Arithmetic Operations on Integers
- Properties of Integers
- Applications of Integers
- Examples on Integers