__Introduction__

**Whole Numbers**

Whole numbers include all-natural numbers and zero

i.e., 0, 1, 2, 3, 4, and so on.

__Negative Numbers__

- The numbers lying to the left of the zero with negative sign on the numbers line are called negative numbers.

__The Number Line__

__Integers__

- Set of all positive and negative numbers including zero are called integers.

…, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … are integers.

__Representing Integers on the Number Line__

__Absolute value of an integer__

- When we consider only the numerical value of the integer without considering its sign it is called as the absolute value of an integer.
- Example: Absolute value of -7 is 7 and of +7 is 7.

__Ordering Integers__

- The number increases as we move towards right and decreases as we move towards left on a number line.
- Hence, the order of integers is …, –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5…

__Addition of Integers__

**Positive integer + Negative integer**

- Example: (+5) + (-2) Subtract: 5 – 2 = +3 (Sign of bigger integer)
- Example: (-5) + (2) Subtract: 5-2 = -3 (Sign of bigger integer)

**Positive integer + Positive integer**

- Example: (+5) + (+2) = +7

**Negative integer + Negative integer**- Example: (-5) + (-2) = -7
- Add the two integers and add the negative sign.

__Properties of Addition and Subtraction of Integers__

__Operations on Integers__

- Addition
- Subtraction
- Multiplication
- Division.

__Properties of Addition and Subtraction of Integers__

**Closure under Addition**- a + b and a – b are integers, where a and b are any integers.

**Commutativity Property**- a + b = b + a for all integers a and b.

**Associativity of Addition**- (a + b) + c = a + (b + c) for all integers a, b and c.

**Additive Identity**- Additive Identity is 0, because adding 0 to a number leaves it unchanged.
- a + 0 = 0 + a = a for every integer a.

__Multiplication of Integers__

- Product of a negative integer and a positive integer is always a negative integer.

- Product of two negative integers is a positive integer.

- Product of even number of negative integers is positive.
- Product of an odd number of negative integers is negative.

__Properties of Multiplication of Integers__

**Closure under Multiplication**- Integer Integer = Integer

**Commutativity of Multiplication**- For any two integers a and b,

**Associativity of Multiplication**- For any three integers a, b and c,

**Distributive Property of Integers**- Under addition and multiplication, integers show the distributive property.
- For any integers a, b and c, .

**Multiplication by Zero**- For any integer a,

**Multiplicative Identity**- 1 is the multiplicative identity for integers.

__Dividing Integers__

- (positive integer/negative integer) or (negative integer/positive integer)

⇒ The quotient obtained is a negative integer. - (positive integer/positive integer) or (negative integer/negative integer)

⇒ The quotient obtained is a positive integer.

__Properties of Division of Integers__

For any integer a,

- is not defined

Integers are not closed under division.

Example: result is an integer but

which is not an integer.

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