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### Operations on Whole Numbers:

We can perform four basic operations on whole numbers. Addition, Subtraction, Multiplication, Division.

Let us see the properties of these operations on whole numbers.

**Addition:**

**Example:** Add

1) 59, 21

2) 118, 122

**Solution:**

##### Properties of Addition on Whole Numbers:

**Property 1: (Closure Property)**

The sum of two whole numbers is always a whole number.

That is, for any two whole numbers a and b, a + b is always a whole number.

**Verification:** We have,

1) 18 + 7 = 25 is a whole number

2) 23 + 12 = 35 and 35 is a whole number.

**Property 2:** **(Commutative law)**

The addition of whole numbers is commutative, i.e, for any two whole numbers a and b, we have **a + b = b + a**

**Verification:** We have,

15 + 26 = 41 and 26 + 15 = 41

Therefore, 15 + 26 = 26 + 15.

**Property 3:** **(Associative Law of Addition)**

If a, b, c are any three whole numbers then **(a + b) + c = a + (b + c)**.

**Verification:** Consider the Integers 6, 1 and 3.

We have,

(6 + 1) + 3 = (7) + 3 = 10.

And, 6 + (1 + 3) = 6 + (4) = 10.

Therefore, (6 + 1) + 3 = 6 + (1 + 3)

**Property 4:** **(Additive Property of Zero)**

If a is any whole number then **a + 0 = a** and **0 + a = a**.

**Verification:** We have,

(i) 8 + 0 = 8

(ii) 235 + 0 = 235

While adding 3 or more whole numbers, we group them in such a way that the calculation becomes easier. We arrange them suitably and add.

**Example: ** Add

1) 62 + 70 + 8

2) 43 + 62 + 17 + 38

**Solution:**

**Subtraction:**

Subtraction and Addition are inverse operations. For every addition there are two **subtraction facts**.

For example,

2 + 8 = 10 is an addition fact. From this, we can state two related subtraction facts. They are,

10 – 8 = 2 and 10 – 2 = 8.

##### Properties of Subtraction on Whole Numbers:

**Property 1:**

If a and b are 2 whole numbers such that a > b or a = b then a – b is a whole number;otherwise, subtraction is not possible in whole numbers.

**Verification:**

1) If we subtract 2 equal whole numbers, we get the whole number 0.

(9 – 9 ) = 0, (3 – 3) = 0

2) If we subtract a smaller whole number from a larger one, we always get a whole number.

(23 – 5) = 18 is a whole number

(54 – 20) = 34 is a whole number

3)Clearly, we cannot subtract larger whole number from a smaller whole number.

(20 – 54) is not defined in whole numbers.

**Property 2:**

For any 2 whole numbers a and b, (a – b) is not equal to (b – a)

**Verification:**

1) (9 – 4) = 5 but (4 – 9) is not defined in whole numbers.

2) (23 – 2) = 21 but (2 – 23) is not defined in whole numbers.

**Property 3:**

For any whole number a, we have: (a – 0) = a but (0 – a) is not defined in whole numbers.

**Verification:**

1) (23 – 0) = 23 but (0 – 23) is not defined in whole numbers.

1) (27 – 0) = 23 but (0 – 27) is not defined in whole numbers.

**Property 4:**

If a, b, c are any 3 whole numbers, then in general (a – b) – c is not equal to a – (b – c).

**Verification:**

Consider the numbers 14, 11 and 2

(14 – 11) – 2 = (3) – 2 = 1

14 – (11 – 2) = 14 – (9) = 5

(14 – 11) – 2 is not equal to 14 – (11 – 2)

**Property 5:**

If a, b, c are any 3 whole numbers such that a – b = c, then b + c = a.

**Verification:**

16 – 7 = 9, 7 + 9 = 16

**Multiplication:**

Multiplication is a short form of addition of the same number several times.

For example,

7 + 7 + 7 + 7 = four times 7

= 4 x 7

= 28

4 x 7 means four times seven or seven times four

In multiplication, if we use some simple and easy techniques, we can get the products easily and quickly.

**Example:**

Multiply 283 by 101

**Solution:**

##### Properties of Multiplication on Whole Numbers:

**Property 1: (Closure Property)**

The product of two whole numbers is always a whole number.

**Verification:** We have,

1) 16 x 5 = 80, 80 is a whole number

2) 12 x 7 = 84, 84 is a whole number

**Property 2: (Commutative Law for Multiplication)**

For any two whole numbers a and b, we have **a x b = b x a**.

**Verification:** We have,

40 x 5 = 200 and 5 x 40 = 200

Therefore, 40 x 5 = 5 x 40.

**Property 3:** **(Associative Law for Multiplication)**

If a, b, c are any three whole numbers then **(a x b) x c = a x (b x c)**.

**Verification:** Consider the whole numbers 7, 12 and 4.

We have,

(7 x 12) x 4 = (84) x 4 = 336.

And, 7 x (12 x 4) = 7 x (48) = 336.

Therefore, (7 x 12) x 4 = 7 x (12 x 4)

**Property 4:** **(Distributive Law for Multiplication over Addition)**

If a, b, c are any three whole numbers then **a x (b + c) = (a x b) + (a x c)**.

**Verification:** Consider the whole numbers 2, 12 and 19.

We have,

2 x (12 + 19) = 2 x (31) = 62.

And, 2 x 12 + 2 x 19 = 24 + 38 = 62.

Therefore, 2 x (12 + 19) = 2 x 12 + 2 x 19

**Property 5:** **(Distributive Law for Multiplication over Subtraction)**

If a, b, c are any three whole numbers then **a x (b – c) = (a x b) – (a x c)**.

**Verification:** Consider the whole numbers 7, 10 and 4.

We have,

7 x (10 – 4) = 7 x 6 = 42.

And, 7 x 10 – 7 x 4 = 70 – 28 = 42.

Therefore, 7 x (10 – 4) = 7 x 10 – 7 x 4 = 70 – 28

**Property 6:** **(Multiplicative Property of 1)**

For any whole number a, we have a x 1 = 1 x a = a.

**Verification:** We have,

1) 11 x 1 = 1 x 11 = 11

2) 5 x 1 = 1 x 5 = 5

**Property 7:** **(Multiplicative Property of 0)**

For any whole number a, we have a x 0 = 0 x a = 0.

**Verification:** We have,

1) 7 x 0 = 0

2) 5 x 0 = 0

**Division:**

Division is the **inverse operation of multiplication**.

Let a and b be 2 whole numbers. Dividing a by b means finding a whole number c such that b x c = a and we write \(\frac{a}{b}\) = c.

Thus, a/b = c that implies a = b x c.

**Example:**

Dividing 35 by 7 is the same as finding a whole number which when multiplied by 7 gives 35.

Clearly, such a number is 5, as 7 x 5 = 35.

Let a and b be two given whole numbers such that a > b. On dividing a by b, let q be the quotient and r be the remainder.

Then we have: a = bq + r, where 0 <= r < b

This result is known as division algorithm.

Thus, **dividend = (divisor x quotient) + remainder**.

**Properties of Division on Whole Numbers:**

**Property 1:**

If a and b are nonzero whole numbers, then \(\frac{a}{b}\) is not always a whole number.

**Verification:** We have,

9 and 2 are two whole numbers, but \(\frac{9}{2}\) is not a whole number.

**Property 2: (Division by Zero)**

If a is a whole number, then \(\frac{a}{0}\) is meaningless.

**Property 3:**

If a is a nonzero whole number, then \(\frac{0}{a}\) = 0.