**Chapter Notes and Summary**

• **Natural Numbers : **Counting numbers are called natural numbers. It is denoted by N.i.e., N = {1, 2, 3, 4, 5, …}

**(i) first and smallest natural number is 1.**

**(ii) There is no least or greatest natural number.**

**•**

**Whole numbers :**All natural numbers together with ‘0’ are called whole numbers. It is denoted by W i.e., W =

**{0, 1, 2, 3, 4, 5, …}**

**(i) number 0 is first and smallest whole number.**

**(ii) All natural numbers are whole numbers. There are infinitely many or uncountable number of whole numbers.**

**•**

**Successor :**successor of a whole number is number obtained by adding 1 to it.

**•**

**Predecessor :**predecessor of a whole number is one less than given number.

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**Operation on whole numbers :**

•

**Properties of addition :**

(i) Closure property :If a and b are any two whole numbers, then (a + b) is also a whole number.

(i) Closure property :

**If a and b are any two whole numbers, then a + b = b + a.**

(ii) Commutative law :

(ii) Commutative law :

**If a is any whole number, then a + 0 = 0 + a = a. So, zero is identity for**

(iii) Additive property of zero :

(iii) Additive property of zero :

**addition of whole numbers.**

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

(iv) Associative law :

(iv) Associative law :

**•**

**Properties of subtraction :**

(i) If a and b are two whole numbers such that a > b > 0, then a – b is a whole number, otherwise, subtraction is

**not possible in whole numbers.**

**(ii) For any two whole numbers a and b, (a – b) ≠ (b – a).**

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

**(iv) If a, b and c are any three whole numbers, then (a – b) – c ≠ a – (b – c).**

**(v) If a, b and c are whole numbers such that a – b = c, b + c = a.**

**•**

**Properties of multiplication :**If a, b and c are three whole numbers, then

**(a × b) is also a whole number.**

(i) Closure property :

(i) Closure property :

**a × b = b × a.**

(ii) Commutative law :

(ii) Commutative law :

**a × 0 = 0 × a = 0.**

(iii) Multiplicative property of zero :

(iii) Multiplicative property of zero :

**a × 1 = 1 × a = a. So, one is identity for multiplication of whole numbers.**

(iv) Multiplicative property of 1 :

(iv) Multiplicative property of 1 :

**a × (b × c) = (a × b) × c.**

(v) Associative law :

(v) Associative law :

**a × (b + c) = (a × b) + (a × c)**

(vi) Distributive law of multiplication over addition :

(vi) Distributive law of multiplication over addition :

**a × (b – c) = (a × b) – (a × c) l Properties of division : If a and b are two non-zero whole numbers, then :**

(vii) Distribution law of multiplication over subtraction :

(vii) Distribution law of multiplication over subtraction :

(i) a ÷ b is not always a whole number.

**a ÷ 0 is meaningless.**

(ii) Division by 0 :

(ii) Division by 0 :

**(iii) 0 ÷ a = 0.**

**•**

**Number line :**In mathematics, a number line is a picture of a gradual straight line that serves as abstraction for

**whole numbers denoted by W**

**.**

To represent whole number on number line, draw a straight line and mark a point on it and label it ‘0’ (zero).

**Starting from ‘0’ (zero) on line mark equal intervals (of unit length) to right to 0 and label them as 1, 2, 3, ….**

**The distance between these points labelled as 0, 1, 2, …is called as unit distance.**

**The number line for whole numbers is shown as below :**

**Note : Important and common facts of natural numbers.**

• 1 is first and smallest natural number.

**• This is not possible that any natural number has no successor.**

**• There is no predecessor of number 1.**

**• (Except 1) Every natural number can be found by adding 1 to previous natural number.**

**• There are infinitely many natural numbers.**

**• There is no last or greatest natural number.**