Natural numbers are the set of positive integers, starting from '1' such as 1, 2, 3, and so on.

They have important properties including closure, commutative, associative, identity, distributive, and the zero property of multiplication. The properties of natural numbers are used in mathematical operations like addition, subtraction, multiplication, and division. Let’s learn about these properties of natural numbers.

Natural numbers are the counting numbers we use in everyday life to count objects. They begin from 1 and continue till infinity. Natural numbers do not include 0, decimals, fractions, or negative numbers. The symbol to represent the set of natural numbers is N. 

N = {1, 2, 3, 4, 5, 6, …}

The closure property means that when you do an operation like adding or multiplying numbers in a set, you always get another number from the same set. But when you do math operations like subtraction or division with numbers in a set, you may or may not get a natural number. 
 

• Addition in closure property:  If natural numbers exclude 0, The set of natural numbers (0, 1, 2, 3,...) is closed under addition because if you add any two natural numbers, you always get another natural number.
  
  For example, \(2 + 3 = 5\), which is a natural number.
   
• Subtraction in closure property: The set of natural numbers is not closed under subtraction because sometimes you get a number outside the set.
  
  For example, \(3 – 5 = –2\), which is not a natural number. 
   
• Multiplication in closure property: The set is closed under multiplication. This means that if you multiply any two natural numbers, the result is always a natural number.
  
  For example, \(3 × 5 = 15\), which is also a natural number.
   
• Division in closure property: The set of natural numbers is not closed under division because dividing two natural numbers won’t always give a natural number.
  
  For example, \(5 ÷ 2 = 2.5\), which is not a natural number. 

The commutative property is a mathematical principle that states that the order in which we add or multiply numbers will not affect the result. However, the order in which we subtract or divide will affect the result.
 

• Addition in commutative property: If a and b are any two natural numbers, then the order of adding a and b does not change the result.
  
  For example, \(3 + 5 = 5 + 3\).
   
• Subtraction in Commutative property: If a and b are any two natural numbers, changing their order in subtraction changes the result. That is, \(a – b ≠ b – a\).
  
  For example, \(8 – 2 = 6\),
  
  Which is a natural number,
  
  But \(2 – 8 = – 6\), which is not a natural number.
   
• Multiplication in commutative property: If a and b are any two natural numbers, then changing their order while multiplying will not change the result.
  
  For example, \(9 × 2 = 2 × 9\).
   
• Division in commutative property: If a and b are any two natural numbers, then changing their order while dividing will change the result.
  
  For example, \(18 ÷ 2 ≠ 2 ÷18\).

The associative property states that when you add or multiply numbers, the way you group them (using parentheses) does not change the result. But when you subtract or divide numbers, the way you group them (using parentheses) will change the result.
 

• Addition in associative property: The addition of natural numbers follows the associative property.
  
  For any three natural numbers, let’s say a, b, and c,
  
  Then, \(a + (b + c) = (a + b) + c\).  
  
  For example, \(2 + (5 + 3) = (2 + 5) + 3\).
   
• Subtraction in associative property: The subtraction of natural numbers is not associative.
  
  For any three natural numbers a, b, and c,
  
  Then, \(a - (b - c) ≠ (a - b) - c\).
  
  For example, \(7 - (3 - 5) ≠ (7 - 3) - 5\).
   
• Multiplication in associative property: Multiplication of natural numbers follows the associative property. This means that for any three natural numbers a, b, and c, the grouping does not affect the product.
  
  For any three natural numbers, let’s say a, b, and c,
  
  Then \(a × (b × c) = (a × b) × c\).
  
  For example, \(2 × (5 × 3) = (2 × 5) × 3\).
   
• Division in associative property: Division is not associative, meaning the order of numbers matters.
  
  For any three natural numbers a, b, and c,
  
  Then \(a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c\).
  
  For example, \(7 ÷ (3 ÷ 5) ≠ (7 ÷ 3) ÷ 5\).

An identity is a number that, when added or multiplied by any number, let’s say n, remains n unchanged. This property does not apply to subtraction or division. 
 

• Addition in identity property: When zero is added to any natural number, the result remains the same natural number.
  
  If n is any natural number, then \(n + 0 = 0 + n = n\).
  
  Thus, zero is the additive identity for natural numbers.
  
  For example, \(8 + 0 = 0 + 8 = 8\).
   
• Multiplication in identity property: When any natural number is multiplied by 1, the result remains the same natural number.
  
  If n is any natural number, then \(n × 1 = 1 × n = n\).
  
  Thus, 1 is the multiplicative identity for natural numbers.
  
  For example, \(5 × 1 = 1 × 5\).

The distributive property helps simplify expressions by multiplying a number by each term inside the brackets. It is an easy math rule that helps make calculations simpler by spreading (distributing) a number to each term inside brackets. 

• Distributive property of multiplication over addition: Multiplication of natural numbers follows the distributive property over addition.
  
  This means that for any three natural numbers a, b, and c,
  
  That is, \(a × (b + c) = (a × b) + (a × c)\).
  
  For example, \(2 × (3 + 2) = 2 × 5 = 10\). 
  
  Using the distributive property, \((2 × 3) + (2 × 2) = 6 + 4 = 10\). 
   
• Distributive property of multiplication over subtraction: Multiplication of natural numbers also follows the distributive property over subtraction.
  
  This means that for any natural numbers a, b, and c.  
  
  That is, \(a × (b – c) = (a × b) – (a × c)\).
  
  For example, \(3 × (5 – 2) = 3 × 3 = 9\) 
  
  Using the distributive property, \((3 × 5) – (3 × 2) = 15 – 6 = 9\)

Operations with natural numbers help us perform basic calculations such as addition, subtraction, multiplication, and division of whole numbers. Each operation follows specific rules, known as properties, that tell us how numbers behave when we perform operations on them. Understanding these properties makes it easier for students to solve problems quickly and correctly.

Understanding the properties of natural numbers becomes much easier when students learn simple strategies, use visual tools, and connect the ideas to real-life situations. Here are a few tips to help students master the properties of natural numbers. 

• Always remember that natural numbers are the counting numbers starting with one and continuing to infinity. Knowing this helps you easily identify when properties apply.
   
• Use mnemonics to remember properties, such as CCAD (“Cats Can Always Dance.”)
   
• Understand which operations follow which properties. Only addition and multiplication follow all significant properties. 
   
• Start with simple numbers for calculations, then gradually increase their difficulty.
  Parents can ask children to count fruits, steps, toys, or books. This builds familiarity with natural numbers. 
   
• Teachers can use properties such as blocks, counters, or interactive charts to help students visualize each property.

The properties of natural numbers are used in many real-life situations. Here are some examples: 
 

1. The commutative property applies when counting money, as the order of adding bills or coins doesn’t change the total.
   
   For example, if a store sells 3 shirts at  $10 each, the total price is the same whether you calculate \(3 × 10\) or \(10 × 3\).
    
2. While grocery shopping, if you group items differently, the total price remains the same.
   
   For example, if apples cost $2, bananas cost $3, and oranges cost $4, you can add them as \((2 + 3) + 4\) or \(2 + (3 + 4)\), and the total remains $9. 
    
3. When calculating the discounts in shopping, The distributive property applies when calculating discounts.
   
   For example, a 5% discount on a $20 shirt and $30 jeans is \(5\% \times (\$20 + \$30) = 5\% \times \$50 = \$2.50 \), or \((5\% \times \$20) + (5\% \times \$30) = \$1 + \$1.50 = \$2.50 \).
    
4. When setting the time, adding 0 minutes to the current time does not change it (additive identity). If you multiply a recipe’s ingredient amounts by 1, the quantity stays the same (multiplicative identity). 
    
5. Multiplying a number by a sum is the same as multiplying it by each added and then adding the products.
   
   For example, we can apply this when we buy 3 gift boxes, each containing 2 toys and 4 candies. \(3 × (2 + 4) = (3 × 2) + (3 × 4) = 6 + 12 = 18\).