Properties of Integers

The set of numbers that includes the whole numbers and negative numbers are the integers. For example, -3, -2, -1, 0, 1, 2, 3. Integers are a subset of rational numbers, but they don't include fractions or decimals. The only integer that is neither positive nor negative is 0, and it is denoted by the letter “Z”.


The integers can be classified as: positive integers, negative integers, and zero. Let’s learn about them in detail:

 

• Positive integers: Whole numbers that are greater than 0, that is 1, 2, 3, 4, 5, …. 
  
   
• Negative integers: Numbers that are less than zero, that is -1, -2, -3, -4, … 
  
   
• Zero: Zero is the only integer that is neither positive nor negative. It is represented as 0
   

Now that we know what integers are, let's learn about the properties that integers follow. Integers follow a few mathematical properties that help make calculations easier. These properties include: 

 

• Closure property
   
• Associative property
   
• Commutative property
   
• Distributive property
   
• Identity property
  
   

  Property	Definition	Example
  Closure	The closure property of integers states that the sum, difference, and product of integers will always result in another integer. It does not apply to division.  That is, a + b ∈ Z, a - b ∈ Z, a × b ∈ Z.	(-3) + 5 = 2, where -3, 2, 5 are integers 7 × (-2) = -14, where -14, -2, 7 are integers  9 - (-5) = 14, where -5, 9, 14 are integers
  Commutative	The commutative property of integers states that changing the order of numbers does not affect the result in addition and multiplication. It is not applicable for subtraction and division.  That is a + b = b + a, a × b = b × a, a - b ≠ b - a,  a ÷ b ≠ b ÷ a.	(-6) + 5 = 5 + (-6) = -1 (-8) × 6 = 6 × (-8) = -48
  Associative	The associative property states that the way of grouping the integers does not affect the result in addition and multiplication. That is, a + (b + c) = (a + b) + c = (a + c) + b, a × (b × c) = (a × b) × c = (a × c) × b.	2 + (3 + 5) = (2 + 3) + 5 = (2 + 5) + 3 = 10 5 × (4 × 6) = (5 × 4) × 6 = (5 × 6) × 4 = 120
  Distributive	The distributive property of integers states that the product of multiplying a number with the sum or difference of two numbers is equal to the sum or difference of the product of each addend separately. It applies to addition and subtraction. That is  a × (b + c) = ab + ac, a × (b - c) = ab - ac.	3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27 2 (6 - 3) = 2 × 6 - 2 × 3 = 12 - 6 = 6
  Identity	The identity property is applicable for addition and multiplication.  The identity property of addition states that the sum of any integer with 0 is the same integer.    The identity property of multiplication: In multiplication, the product of any integer with 1 is the integer itself.  That is a + 0 = a, (-a) + 0 = -a, a × 1 = a, (-a) ×1 = -a.	2 + 0 = 2 -5 + 0 = -5 6 × 1 = 6 -6 × 1 = -6

   

In the real world, integers are used to represent quantities such as temperature, financial transactions, engineering calculations, and more. In this section, let’s learn a few real-world applications of the properties of integers. 

 

• The commutative property of integers is used when splitting a bill among friends or groups. For example, if three friends need to pay a total of $10, and they contribute $2, $3, and $5, the total remains the same no matter who contributes which amount.  
  
   
• To combine the expenses of a group we use associative property For example, when adding the expense, the order of grouping will not affect the result. 
  
   
• Integers are used in sports to express the scores, points, players statistics, and ranking. 
  
   
• In transportation, integers are used to calculate the fuel consumption and plan the scheduling and logistic controls.