Properties of Whole Numbers in Detail

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Properties of Whole Numbers

Closure Property

  • Two whole numbers are said to be closed if their operation is also the whole number.
Closure property on Addition for Whole Number
  • Whole numbers are closed under addition as their sum is also a whole number.
0+2=2
1+3=4
5+6=11

So Whole number are closed on Addition

Closure property on Multiplication for Whole Number
  • Whole numbers are not closed under subtraction as their difference is not always a whole number.
0×2=0
1×4=4
5×1=5

So Whole number are closed on Multiplication

Closure property on subtraction of Whole number
  • Whole numbers are not closed under subtraction as their difference is not always a whole number.
5−0=5
0−5=?
1−3=?
3−1=2

So Whole number are not closed on Subtraction

Closure property on Division of Whole number
  • Whole numbers are not closed under division as their result is not always a whole number.
2/1=2

1/2=? ( not a whole number.)
0/2=0
2/0=? ( not defined )

( Division by Zero is undefined)

So Whole Number are not closed on Division

In short

Closure Property- If a and b are any two whole numbers, then a+b, axb are also whole numbers.

Commutative property

  • Two whole numbers are said to be commutative if their result remains the same even if we swap the positions of the numbers.
Commutativity property on Addition for Whole Number

  • The addition is commutative for whole numbers as their sum remains the same even if we interchange the position of the numbers.
2 + 5 = 7

5 + 2 = 7

So Whole number are Commutative on Addition

Commutativity property on Multiplication for Whole Number

  • Multiplication is commutative for whole numbers as their product remains the same even if we interchange the position of the numbers.
9 × 5 = 45

5 × 9 = 45

So Whole number are Commutative on Multiplication

Commutativity property on subtraction of Whole number
  • Subtraction is not commutative for whole numbers as their difference may be different if we interchange the position of the numbers.
9 – 2 = 7

2 – 9 = (-7) which is not a whole number.

So Whole number are not Commutative on Subtraction

Commutativity property on Division of Whole number
  • The division is not commutative for whole numbers as their result may be different if we interchange the position of the numbers.
5 ÷ 1 = 5

1 ÷ 5 =, not a whole number.

So Whole Number are not Commutative on Division

In short

You can add two whole numbers in any order. You can multiply two whole numbers in
any order.

Commutative property- If a and b are any two whole numbers, then
  • a+b=b+a 
  • a×b=b×a

Associative property

  • The two whole numbers are said to be associative if the result remains the same even if we change the grouping of the numbers.
Associativity property on Addition for Whole Number
  • The addition is associative for whole numbers as their sum remains the same even if we change the grouping of the numbers.
3 + (2 + 5) = (3 + 2) + 5

3 + 7 = 5 + 5

10 = 10

So Whole number are Associative on Addition

Associativity property on Multiplication for Whole Number
  • Multiplication is associative for whole numbers as their product remains the same even if we change the grouping of the numbers.
3 × (5 × 2) = (3 × 5) × 2

3 × (10) = (15) × 2

30 = 30
  
So Whole number are Associative on Multiplication

Associativity property on subtraction of Whole number
  • Subtraction is not associative for whole numbers as their difference may change if we change the grouping of the numbers.
8 – (10 – 2) ? (8 – 10) – 2

8 - (8) ? (-2) – 2

0 ? (-4)
  
So Whole number are not Associative on Subtraction

Associativity property on Division of Whole number
  • The division is not associative for whole numbers as their result may change if we change the grouping of the numbers.
24 ÷ 3 ? 4 ÷ 2

8 ? 2

So Whole Number are not Associative on Division

So in Short
If a, b and c are any two whole numbers, then
  • (a+b)+c = a+(b+c)
  • (a×b)×c = a×(b×c).

Distributive property

Distributive Property of Multiplication over Addition

If a, b and c are any three whole numbers, then
a (b+c) = a×b + a×c

Evaluate using distributive property 15 × 45

Solution

15 × 45 = 15 × (40 + 5)

= 15 × 40 + 15 × 5

= 600 + 75

= 675

Additive Identity
  • If we add zero to any whole number the result will the same number only. So zero is the additive identity of whole numbers.
  • If a is any whole number, then  a+0=a=0+a.
Example

2+0=2
0+3=3
5+0=5

Multiplicative Identity
  • If we multiply one to any whole number the result will be the same whole number. So one is the multiplicative identity of whole numbers.
  • If a is any whole number, then a×1=a=1×a
Example

1×1=1
5×1=5
6×1=6

Multiplication by zero

If a is any whole number, then a×0=0=0×a.

Example

1×0=0
5×0=0
0×0=0

Division by zero

If a is any whole number, then a ÷ 0 is not defined

Fun Time

Question 1 : Which of the following is not defined?
A)10+0
B)10−0
C)10×0
D)10÷0

Question 2 : Find the value of 6536 � 91 + 9 � 6536?
A)588240
B)594776
C)58824
D)653600

Question 3 : Which of the following is true
A)Every whole number has predecessor
B)The product of two whole numbers need not to be whole number
C)1 is the identity for multiplication of whole numbers.
D)1 is the identity for addition of whole numbers.

Question 4 : Which of the following is not true
A) Whole number are closed on addition
B) Whole number are Commutative on Multiplication
C) Whole number are Commutative on Subtraction
D) Whole number are Commutative on addition

Question 5: The product of a non-zero whole number and its successor is always?
A)even number
B)odd number
C)prime number
D)divisible by 5