Properties of Whole Numbers says that the addition and multiplication of the whole numbers give the same result as a whole number. If you consider subtraction of whole numbers, then it doesn’t give the result as a whole number. Let us see all the whole number properties along with examples in the entire article.

## Whole Number Properties

Check out the below whole number properties and their proofs with the help of examples here. The different types of Properties of Whole Numbers are

- Closure Property
- Commutative Property
- Associative Property
- Additive Identity
- Multiplicative Identity
- Distributive Property
- Multiplication by zero
- Division by zero

### 1. Closure Property

The closure property of whole numbers addition says that by adding the two whole numbers we will get the same output as a whole number. If a and b are two whole numbers, then the addition of a and b (a + b) is a whole number i.e, c.

a + b = c

Also, the closure property of whole numbers multiplication says that by multiplying the two whole numbers we will get the same output as a whole number. If a and b are two whole numbers, then the multiplication of a and b (a * b) is a whole number i.e, c.

a * b = c

**Example:** a = 4, b = 6.

c = a + b = 4 + 6 = 10.

c = a * b = 4 * 6 = 24

Therefore, 10, 24 are whole numbers.

### 2. Commutative Property

The commutative property of whole numbers addition says that we get the whole number as output whatever the order the input whole numbers are added. If a and b are two whole numbers, then the addition of a and b (a + b) is equal to the addition of b and a (b + a).

a + b = b + a

The commutative property of whole numbers multiplication says that we get the whole number as output whatever the order the input whole numbers are multiplied. If a and b are two whole numbers, then the multiplication of a and b (a * b) is equal to the multiplication of a and b (b * a).

a * b = b * a

**Example:** a = 2, b = 3.

a + b = 2 + 3 = 5.

b + a = 3 + 2 = 5

Therefore, a + b = b + a

a * b = 2 * 3 = 6

b * a = 3 * 2 = 6

Therefore, a * b = b * a

### 3. Associative Property

The associative property of whole numbers addition says that when you add whole numbers by grouping them in any order the result will be the same. If a, b and c are three whole numbers, then a + (b + c) = (a + b) + c.

The associative property of whole numbers multiplication says that when you multiply whole numbers by grouping them in any order the result will be the same. If a, b and c are three whole numbers, then a * (b * c) = (a * b) * c.

**Example:** a = 8, b = 4, and c = 10.

a + (b + c) = 8 + (4 + 10) = 8 + 14 = 22

(a + b) + c = (8 + 4) + 10 = 12 + 10 = 22.

Therefore, a + (b + c) = (a + b) + c

a * (b * c) = 8 * (4 * 10) = 8 * 40 = 320

(a * b) * c.= (8 * 4) * 10 = 32 * 10 = 320

Therefore, a * (b * c) = (a * b) * c.

### 4. Additive Identity

When we add a whole number to zero, then the value of a whole number remains the same. If a is a whole number, then a + 0 = a = 0 + a.

**Example:
**a = 3

3 + 0 = 3

0 + 3 = 3

Therefore, a + 0 = a = 0 + a

### 5. Multiplicative Identity

When we multiply a whole number with one, then the value of a whole number remains the same. If a is a whole number, then a * 1 = a = 1 * a.

**Example:**

a = 3

3 * 1 = 3

1 * 3 = 3

Therefore, a * 1 = a = 1 * a

### 6. Distributive Property

If a, b and c three are whole numbers, then the distributive property of multiplication over addition becomes a * (b + c) = (a * b) + (a * c). Also, the distributive property of multiplication over subtraction is a * (b – c) = (a * b) – (a * c)

**Example: **a = 4, b = 3, and c = 2

a * (b + c) = 4 * (3 + 2) = 4 * 5 = 20

(a * b) + (a * c) = (4 * 3) + (4 * 2) = 12 + 8 = 20

Therefore, a * (b + c) = (a * b) + (a * c)

a * (b – c) = 4 * (3 – 2) = 4 * 1 = 4

(a * b) – (a * c) = (4 * 3) – (4 * 2) = 12 – 8 = 4

Therefore, a * (b – c) = (a * b) – (a * c)

### 7. Multiplication by zero

When we multiply a whole number with zero, then the value of a whole number becomes zero. If a is a whole number, then a * 0 = 0 = 0 * a.

**Example:**

a = 3

3 * 0 = 0

0 * 3 = 3

Therefore, a * 0 = 0 = 0 * a

### 8. Division by zero

We can’t define the division of a whole number by zero. If a is a whole number, then a/0 is not defined.

## Some More Properties of Whole Numbers

Check out some important Whole Number Properties below.

- The number zero is the first and smallest whole number.
- We can’t define the last or greatest whole number.
- All natural numbers along with zero are called whole numbers.
- There are uncountable or infinitely many whole numbers available.
- Each number is 1 more than its previous number.
- All natural numbers are whole numbers.
- All whole numbers are not natural numbers.
- Even Whole Numbers (E): The whole numbers divisible by 2 or the multiples of 2 are called Even Whole Numbers. It represents with a letter E. E = {2, 4, 6, …..}
- Odd Whole Numbers (O): The whole numbers which are not divisible by 2 or not the multiples of 2 are called Odd Whole Numbers. It represents with a letter O. O = {1, 3, 5, 7, …..}