**1. RECALL****Natural Numbers**

The numbers 1, 2, 3…… are used for counting objects is called ‘Natural Numbers’. Set of natural

numbers are represented by ‘N’.

Set of natural numbers N = { 1, 2, 3 …… }**Whole Numbers**

Including ‘0’ in the collection of natural numbers is called whole numbers.

Set of whole numbers are represented by ‘W’.

Set of whole numbers W = {0, 1, 2, 3 ……}

Numbers greater than 0 are called positive numbers. Extending the number line to the left of 0 allow us to picture negative numbers, numbers that are less than 0.

When a single + sign or no sign is in front of a number, the number is a positive number. When a single – sign is in front of a number, the number is a negative number.

– 5 indicates “negative five”.

5 and + 5 indicate “positive five”.**The number 0 is neither positive nor negative.****Reading and writing integers**

• The sign of an integer is read first before the number.**Example** : – 5 is read as ‘negative five’.

+ 9 is read as ‘positive 9’ or simply ‘nine’.

• 0 is an integer but it is nothing positive nor negative.**Representing Integers on Number Lines**

Integers can be represented on a number line.**The number line shows that every integer has an opposite number except ‘0’.**

**Comparing the Values of Two Integers**

Number line can be used to compare the values of two integers.**1. Horizontal number line**

(A) On a horizontal number line, an integer is greater than the integer on its left.

(B) On a horizontal number line, an integer is less than the integer on its right.

**2. Vertical number line**

(A) On a vertical number line, an integer is greater than the integer below it.

(B) On a vertical number line, an integer is less than the integer above it.**Arranging Integers in Order**

1. Number lines can be used to arrange order, integers in increasing or decreasing order.

2. The value of integers on a horizontal number line increases from left to right and decreases from right to left.

Writing Positive and Negative Integers to Represent Word Descriptions**A positive or negative number is used to denote:**

(A) An increase or a decrease in value**Example**:

(i) Rs. 70 withdrawn is denoted by –Rs. 70.

(ii) Rs.70 deposited is denoted by + Rs.70.

(B) Values more than zero values less than zero

**Example** :

(i) – ${18}^{0}C$ denotes a temperature that is ${18}^{0}C$below ${0}^{0}C$.

(ii) +${18}^{0}C$ denotes a temperature that is ${18}^{0}C$ above ${0}^{0}C$.

(C) A positive direction or a negative direction (opposite direction)**Example**:

(i) –$20$ $C_{0}$denotes an anticlockwise rotation of ${20}^{0}$.

(ii) +${20}^{0}$ denotes a clockwise rotation of ${20}^{0}$.

(iii) +5 m denotes a direction 5 m to the right.

(iv) –5 m denotes a direction 5 m to the left.

(D) Position above or below sea level

(i) The sea level is taken as 0 m.

(ii) Anything above sea level is taken as positive. For example if a bird is flying at 50m above, then we say +50m.

(iii) Similarly if a submarine lies 150 m below sea level we write it as –150m.**Comparison of Integers: **If we represent two integers on the number line, then the integer

occurring on the right is greater than that occurring on the left.**Note**:

• 0 is less than every positive integer

• Every negative integer is less than every positive integer.

• The greater is the integer, the lesser is its negative.

$\text{i.e.,}ab\Rightarrow -a-b$

**2. PROPERTIES OF INTEGERS ON ADDITION AND SUBTRACTION**

(A) If a and b are two integers then a + b = c; where c is also an integer.

(B) For any two integers a and b

a + b = b + a

Which means that if we change the order of the integers, even then their sum does not change.

(C) For any three integers a, b and c

(a + b) + c = a + (b + c)

This means that even if we rearrange the integers their sum does not change.

(D) If a is any integer then a + 0 = a

This means that the sum of any integer and zero is the integer itself.**Example**: – 10 + 0 = – 10

6 + 0 = 6

– 15 + 0 = – 15

(e) For every integer a (which is not zero) there is another integer – a such that

a + (– a) = 0**Example**: 3 + (– 3) = 0

5 + (– 5) = 0

6 + (– 6) = 0

(f) The difference of any two integers is an integer i.e.

If a and b are two integers then a – b = c, where c is also an integer.

(g) In the whole numbers, 0 has no predecessor. But in integers – 1 is the predecessor of 0, –2 is the predecessor – 1 and so on.

Thus if a is any integer, then a – 1 is its predecessor.

(h) If a is any integer then a – 0 = a

**Like signs Unlike signs**

+ (+y) = +y + (– y) = – y

– (–y) = + y – (+ y) = – y**3. MULTIPLICATION OF INTEGERS**

1. The multiplication of an integer with a positive integer is the repeated addition of the integer.

2. Rules for multiplication of integers:

(A) The product of two integers is positive when both integers have like signs [as in (i) and (iv)].

(B) The product of two integers is negative when both integers have unlike signs [as in (ii) and (iii)].

(C) The product of an integer and zero is always zero [as in (v) and (vi)].**Property 1**: If a and b are integers than a × b is also an integer**Property 2:** If a and b are two integers then a × b = b × a**Property 3:** If a, b and c are any three integers then a × (b × c) = (a × b) × c**Property 4:** The product of an integer and zero (0) is always zero (0), i.e., for any integer a, a × 0 = 0 × a = 0.**Property 5 :** The product of an integer and 1 is the integer itself.

i.e., for any integer a, a × 1 = 1 × a = a.**Property 6:** For any integers a, b and c,

a × ( b + c) = (a × b) + (a × c)

a × ( b – c) = (a × b) – (a × c)

(A) When the number of negative integers in a product is ODD, the product is negative.

(B) When the number of negative integers in a product is EVEN, the product is positive.