Properties of Natural Numbers

Natural numbers are the set of positive integers used for counting and ordering.

Here are the key properties of natural numbers with detailed explanations and examples:

The four operations, addition, subtraction, multiplication and division on natural numbers.

1. Closure Property

The set of natural numbers is closed under addition and multiplication. This means that if you add or multiply any two natural numbers, the result is always a natural number.

This shows that the sum of two natural numbers is also a natural number.

(1) Closure property:

Addition of Natural Numbers

Closure property of addition

a + b = c

Addition Example: 3+5=8

Example: 3 + 4 = 7

Example: 15 + 9 = 24

Subtraction of Natural Numbers

The Subtraction of two natural numbers is not always a natural number.

Subtraction Example:

Example: 6 – 2 = 4 is a natural number, but 2 – 6 = – 4 is not a natural number.

5−7=−2 (not a natural number)

When we subtract a larger natural number from a smaller natural number, we do not get a natural number.

However, natural numbers are not closed under subtraction and division because subtracting or dividing two natural numbers does not always yield a natural number.

Multiplication of Natural Numbers

Closure property of Multiplication

a x b = c

Closure property of a + b = c

Multiplication Example: 4×6=24

Example: 3 x 4 = 12

Example: 12 x 6 = 72

This shows that the set of natural numbers are always closed under addition and multiplication.

The product of two natural numbers is also a natural number.

Division of Natural Numbers:

When we divide natural numbers the resultant number may, or may not be a natural number.

The Division of two natural numbers is not always a natural number.

Division Example: 7÷2=3.5 (not a natural number)

Example: 8/2 = 4 is a natural number, but 

                 2/8 = 1/4 is not a natural number.

The addition and multiplication of two or more natural numbers will always a natural number, but this is not closed under subtraction and division. Subtracting or division of two natural numbers will not always give a natural number.

2. Associative Property

The associative property states that the way in which numbers are grouped in addition or multiplication does not change the result.

(2) Associative Property: The sum or product of any three natural numbers remains the same even if the grouping of numbers is changed.

Associative property of Addition:

a + (b + c) = (a + b) + c

Addition Example: (2+3)+4=2+(3+4)=9

3 + (4 + 2) = (3 + 4) + 2

3 + 6 = 9

7 + 2 = 9

Associative property of Multiplication:

a x (b x c) = (a x b) x c

Multiplication Example: (2×3)×4=2×(3×4)=24

3 x (4 x 2) = (3 x 4) x 2

3 x 8 = 12 x 2 = 24

This shows that the set of natural numbers N is assocoative under addition and multiplicaion but this is not closed under subtraction and division.

Subtraction: a – (b – c) ≠ (a – b) – c

= 4 – (8 – 5) = 4 – 3 = 1

and (4 – 8) – 5 = – 4 -5 = -9

Subtraction: a – (b – c) ≠ (a – b) – c

4 – (8 – 5) = 4 – 3 = 1 and

(4 – 8) – 5 = – 4 – 5 = -9

Division: a ÷(b ÷ c) (a ÷ b) ÷ c

2 (3 ÷ 9)= 6 and (2 ÷ 3) 9 = 2/27

This shows that the set of natural numbers N is associative under addition and multiplication but this is not closed under subtraction and division.

3. Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not change the result.

Commutative Property: Addition and multiplication of natural numbers show the commutative property.

Example: a + b = b + a and a x b = b x a

Example: 3 + 4 = 4 + 3 and 3 x 5 = 5 x 3

Subtraction and division of natural numbers do not show the commutative property.

Division: a ÷ (b ÷ c) ≠ (a ÷ b)÷ c

Example: a – b ≠ b – a and

a ÷ b ≠ b ÷ a

4. Distributive Property

The distributive property relates addition and multiplication, stating that multiplying a number by a sum is the same as doing each multiplication separately.

Example: 2×(3+4)=(2×3)+(2×4)=6+8=14

Distributive Property:

Multiplication of natural numbers is always distributive over addition.

Example : a x (b + c) = ab + ac

Example : 5 x (3 + 4) = 5 x 3 + 5 x 4 = 12 + 20 = 35 = 5 x 7 = 35

Multiplication of natural numbers is also distributive over subtraction.

Example : a x (b – c) = ab – ac

Example : 6 x (9 – 4) = 6 x 9 – 6 x 4 = 54 – 24 = 30 = 6 x 5 = 30

Multiplication of natural numbers is also distributive over subtraction.

Example : a x (b – c) = ab – ac

5. Identity Property

The identity property includes the additive identity and the multiplicative identity.

• Additive Identity: The number 0 is the additive identity because adding 0 to any natural number does not change the number.
  • Example: 5+0=5
• Multiplicative Identity: The number 1 is the multiplicative identity because multiplying any natural number by 1 does not change the number.
  • Example: 7×1=7

6. Well-Ordering Property

Every non-empty subset of natural numbers has a least element. This property is foundational in proofs and mathematical induction.

• Example: In the subset {4, 7, 1, 3}, the least element is 1.

7. Infinite Set

The set of natural numbers is infinite. There is no largest natural number because you can always add 1 to any natural number to get another natural number.

• Example: 1, 2, 3, 4, 5, …, n, n+1, …

8. No Negative Numbers or Fractions

Natural numbers do not include negative numbers, zero (in some definitions), or fractions.

• Example: Natural numbers are 1, 2, 3, 4, …, but not -1, 0, 1.5, etc.

Examples of Natural Numbers

• Counting objects: 1, 2, 3, 4, …
• Ordinal numbers: 1st, 2nd, 3rd, 4th, …
• Set of natural numbers: {1, 2, 3, 4, …}

In summary, natural numbers have several important properties that make them fundamental in mathematics. They are closed under addition and multiplication, associative, commutative, distributive, and have identity elements for addition and multiplication. They also have the well-ordering property and are an infinite set without negative numbers or fractions.