Properties of Integers

The properties of integers are straightforward and assist students in understanding and working with these numbers. These properties are derived from basic mathematical principles. Some of the key properties of integers are mentioned below:

 

• Property 1: Closure Integers are closed under addition, subtraction, and multiplication, meaning the result of these operations on integers is always an integer. Integers are closed under addition, subtraction, and multiplication, and operations with zero. For example, 
  \(0 + 5 = 5\), \(0 × 7 = 7\) division by zero is undefined.
  
   
• Property 2: Commutative Property The sum or product of two integers remains the same regardless of their order: \(a + b = b + a\) and \(a × b = b × a\). The sum or product of integers does not change when their order is switched, including zero. For example, \((0 + 8 = 8 + 0, 0 × 6 = 6 × 0)\).
  
   
• Property 3: Associative Property The way integers are grouped doesn't change their sum or product: \((a + b) + c = a + (b + c)\) and \((a × b) × c = a × (b × c)\). Grouping of integers does not affect the sum or product, including zero. For example, \((0 + 4) + 5 = 0 + (4 + 5)\).
  
   
• Property 4: Distributive Property Multiplication distributes over addition: \(a × (b + c) = a × b + a × c\). Multiplication distributes over addition, and multiplying by zero gives zero. For example, \(0 × (5 + 3) = 0 × 5 + 0 × 3 = 0\).
  
   
• Property 5: Identity Elements The integer 0 is the additive identity \((a + 0 = a)\), and 1 is the multiplicative identity \((a × 1 = a)\). Zero is the additive identity and one is the multiplicative identity, with zero following both rules. For example, \(0 + 0 = 0\).

Students often make mistakes while learning the properties of integers. To avoid confusion, consider these tips and tricks:

 

• Closure Property: Remember that when you add, subtract, or multiply any two integers, the result will always be an integer.
   
• Commutative Property: Keep in mind that changing the order of addition or multiplication doesn't affect the result.
   
• Associative Property: Grouping integers differently in addition or multiplication won't change the outcome.
   
• Distributive Property: Practice distributing multiplication over addition to simplify expressions.
   
• Identity Elements: Use 0 to simplify addition and 1 to simplify multiplication.

Integers are used daily to represent gains, losses, temperatures, elevations, and financial transactions, making calculations and comparisons easier.

• Banking and Finance: Integers are used to keep track of deposits and withdrawals, as well as profits and losses, allowing individuals and businesses to maintain accurate accounts.
   
• Temperature Measurement: Positive and negative integers represent temperatures above and below zero, helping us understand weather conditions clearly.
   
• Elevation and Geography: Heights above sea level are shown as positive integers, while depths below sea level are represented as negative integers, which is essential for mapping and navigation.
   
• Sports Scoring: Integers help record points gained or lost in games and competitions, making it easier to calculate totals and determine winners.
   
• Accounting and Budgeting: Credits and debits in personal or business accounts are represented using positive and negative integers, ensuring proper financial management and planning.