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, …..}