## WHOLE NUMBERS

Numbers that represent complete (whole) things without pieces (part/fraction) are called whole numbers.

↪ Natural numbers also called Counting numbers begin with 1 i.e., when we start counting something, we start from 1.

Prerequisite / Revise this :

↪ Predecessor of any natural number is the number that comes before it. The predecessor of a number is obtained by subtracting one from the number.

1 has no predecessor in natural numbers because when we subtract one from itself, we get 0 which is not a natural number.

↪ When we subtract any number from itself, we get 0.

↪ 0 (zero) is not a part of counting number as we can’t count 0. It was also not thought as a actual number for many years. But now, we know its significance.

“When we add 0 to the set of natural numbers, we get a new set of numbers called Whole numbers.”
W={0,1,2,3…}

### Time to Think

1) Are all natural number also whole numbers?
2) Are all whole numbers also natural number?
3) Has the natural number 1 no predecessor?
4) Has the whole number 1 no predecessor?
5) Has the whole number 0 no predecessor?
6) Which is smallest whole number?
7) Which is smallest whole number?

1) Yes.
2) No, 0 is not a natural number.
3) Yes.
4) No, predecessor of whole number 1 is 0.
5) Yes.
6) 0.
7) none.

## Properties of Whole numbers

### Closure property

➤ For addition- closed – sum of  any two whole number is whole number,e.g.,

0+2 = 2

5+6 = 11

➤ For multiplication- closed -product of  any two whole number is whole number,e.g.,

0×2 = 0

5×6 = 30

➤ For subtraction- not closed -difference of  any two whole number is not always whole number,e.g.,

0-2 = ?, not a whole number

6-5 = 1

5-6 = ?, not a whole number

➤ For division – not closed – division of  any two whole number is not always whole number,e.g.,

4÷2 = 2

2÷4 = ?, not a whole number

6÷5 = ?, not a whole number

### Commutative property

➤ For addition- commutative – we can add two whole numbers in any order & the result will be same, e.g.,

0+2 = 2 ⇔ 2+0 =2

5+6 =11 ⇔ 6+5 =11

➤ For multiplication- commutative – we can multiply two whole numbers in any order & the result will be same, e.g.,

0×2 = 0 ⇔ 2×0 = 0

5×6 =30 ⇔ 6×5 =30

➤ For subtraction-not commutative – we can not subtract two whole numbers in any order, the result will get changed if we reverse the order, e.g.,

2-0 =2 ⇎ 0-2 =?

6-5 =1 ⇎ 5-6 =?

➤ For division – not commutative – we can not divide two whole numbers in any order, the result will get changed if we reverse the order, e.g.,

4÷2 =2 ⇎ 2÷4 =?

9÷3 =3 ⇎ 3÷9 =?

### Associative property

➤For addition- associative – sum of whole numbers will give same result when the grouping of numbers is changed, e.g.,

(0+2)+5 =2+5 =7 ⇔ 0+(2+5) =2+5 =7

(3+6)+7 =9+7 =16 ⇔ 3+(6+7) =3+13 =16

➤For multiplication- associative – product of whole numbers will give same result when the grouping of numbers is changed, e.g.,

(0×2)×4 =0×4 =0 ⇔ 0×(2×4)= 0×8 =0

(3×4)×7 =12×7 =84 ⇔ 3×(4×7) =3×28 =84

➤For subtraction- not associative – difference of whole numbers may not give same result when the grouping of numbers is changed, e.g.,

(4-2)-1=2-1=1  ⇎  4-(2-1)= 4-1= 3  , results are not same

(6-3)-1=3-1=2  ⇎  6-(3-1)= 6-2 =4  , results are not same

➤For division – not associative – division of whole numbers may not give same result when the grouping of numbers is changed, e.g.,

(6÷3)÷1=2÷1 =2 ⇎ 6÷(3÷1) =6÷3 =2, results are same

(16÷4)÷2=4÷2 =2 ⇎ 16÷(4÷2) =16÷2 =8, results are notsame

### Distributive Property/ Distribution of multiplication over addition :

Multiplying a number with sum of two numbers will give same result when
the number is first multiplied with each addend & then adding the products.

2×(0+5) = 2×5 = 10

2×(0+5) =(2×0)+(2×5) =0+10 =10 (using distributive property)

When we add 0 to any whole number, we get the same number as a result and we can say that the number has kept its identity.

0 is called additive identity for whole numbers (or any number).

### Multiplicative Identity

When we multiply 1 to any whole number, we get the same number as a result and we can say that the number has kept its identity.

1 is called multiplicative identity for whole numbers (or any number).

### Whole Numbers – Problems with Solutions

 ⏪ Natural Numbers Factors & Multiples of Whole Numbers ⏩