**Chapter Notes and Summary**

• **Integers : **collection of all natural numbers, 0 and negatives of natural numbers are called integers, i.e., we can represent integers on number line.

Thus integers range from {………… – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5,…………}

• **Absolute value of an integer : **absolute value of an integer is numerical value of integer regardless of** **its sign.•

**Successor and predecessor of an integer : Let a be an integer, then :**

(a + 1) is called successor of a.

(a – 1) is called predecessor of a.

•

**Properties of addition on integers :**

(i) Closure property of addition :sum of two integers is always an integer.

(i) Closure property of addition :

**a + b = b + a.**

(ii) Commutative law of addition :

(ii) Commutative law of addition :

**(a + b) + c = a + (b + c).**

(iii) Associative law of addition :

(iii) Associative law of addition :

**•**

**Properties of subtraction of integers :**

(i) Closure property :If a and b are two integers, then (a – b) is also an integer.

(i) Closure property :

**(ii) If a is any integer, then a – 0 = a**

**.**

(iii) If a, b, c are integers and a > b, then (a – c) > (b – c).

**•**

**Properties of multiplication of integers :**

(i) Closure property :product of two integers is always an integer.

(i) Closure property :

**a × b = b × a.**

(ii) Commutative law :

(ii) Commutative law :

**a × (b × c) = (a × b) × c.**

(iii) Associative law :

(iii) Associative law :

**a × (b + c) = a × b + a × c.**

(iv) Distributive law :

(iv) Distributive law :

**•**

**Properties of division on integers :**

(i) If a and b are two integers, then a ÷ b is not necessarily an integer.

**(ii) If a ≠ 0, then a ÷ a = 1.**

**(iii) a ÷ 1 = a.**

**(iv) If a is non-zero integer, then 0 ÷a = 0, but a ÷ 0 is not meaningful.**

**(v) (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) unless c = 1.**

**•**

**Representation of Integer on Number line**To represent integers on a number line, draw a line and mark some points at equal distances on it as shown in

**figure given below. Mark a point as ‘zero‘ on it. Points to right of zero are positive (+ve) integers and are**

**marked as +1, +, 2, +3 etc. or simply 1, 2, 3 etc. Points to left of zero are negative (–ve) integers and are marked**

**as –1, – 2, –3 etc.**

**•**

**Ordering of Integers**On number line, number increases as we move to right and decreases as we move to left.

**Therefore, ……….. < –5 < – 4 < – 3 < – 2 < – 1 < 0 < 1 < 2 < 3 < 4 < 5 and so on.**

**•**

**Addition of Integers**To add two negative integers, we add corresponding positive integers and retain negative sign with sum.

**e.g., find sum of –5 and –3.**

**–5 + (–3) = – (5 + 3) = – 8.**

**To add a positive integer and a negative integer, we ignore signs and subtract integer with smaller numerical**

**value from integer with larger numerical value and take sign of larger one.**

**e.g., (a) Consider –6 and + 4**

**As 6 – 4 = 2, therefore –6 + (+4) = – 2**

**(b) Consider + 5 and – 2**

**5 + (–2) = 5 – 2 = 3**

**Two integers whose sum is zero are called additive inverse of each other.**

**To subtract an integer from a given integer, we add additive inverse of integer to given integer e.g., Subtract :**

(a) 3 from – 4 (b) –3 from – 4

**(a) additive inverse of 3 is –3**

**So, – 4 – 3 = – 4 + (–3) = – (4 + 3) = – 7**

**(b) additive inverse of –3 is 3**

**So, –4 – (–3) = – 4 + 3 = – 1**

**•**

**Addition/Subtraction of Integers on Number Line**Firstly, draw number line and represent first number on it. then, to add/subtract second number in first, we

**move left/right to first number according to second integer (either –ve or + ve)**

**e.g., (i) To add –2 and –3, we move two steps to left of reaching –2. Then, we move three steps to left of –2**

**and reach –5.**

**Thus, –2 + (–3) = – 5**

**(ii) To subtract 2 from –3, we first move three steps to left of 0, reaching –3. Then we move 2 steps to left of**

**3 and reach –5.**

**So, –3 + (–2) = – 3 – 2 = – 5.**