* Natural Numbers:* The counting numbers are called natural numbers.

* Whole Numbers:* All natural numbers together along with

*0 (zero)*form the set W of all whole numbers.

* Integers:* All natural numbers, negative natural numbers and

*0 (zero)*together form a set Z of all Integers.

Properties of Addition of Integers

- The sum of two integers is always an integer
- If
*a*and*b*are two integers, then:

*a + b = b + a*

- Associative Law: If
*a, b*, and*c*are integers, then:

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

- Existence of Additive Identity: 0 is the additive identity for integers. Therefore for any integer
*a*we will have:

*a + 0 = 0 + a = a*

- Existence of additive inverse: For each integer
*a*, there exists another integer*(-a)*such that*a + (-a) = 0*

*Note: If a be any integer, then (a+1) will be called the successor of a is (a + 1) and the predecessor of a is (a-1)*

**Properties of Subtraction**

- If
*a*and*b*are two integers, then*(a – b)*is also an integer. - For any integer
*a*, we have*(a – 0) = a*. But*(0 – a) ≠ a* - If
*a, b, c*are integers and*a > b*, then

*(a –c) > (b – c)*

**Properties of Multiplication of Integers**

*Closure Property*: The product of two integer is always an integer*Commutative Law*: For any two integers*a*and*b*we have:

*a × b = b × a*

*Associative Law*: For any three integers*a, b,*and*c*we have

*(a × b) × c = (a × (b × c)*

*Distributive Law of Multiplication over Addition*: For any there integers*a, b,*and*c*we have

*a × (b + c) = a × b + a × c*

*Existence of Multiplicative Identity*: The integer 1 is a multiplicative identity for Integers. So, for any integer a we have:

*a × 1 = 1 × a = a*

*Property of Zero*: For any integer we have:

*a × 0 = 0 × a = 0*

**Properties of Multiplication of Integers**

- If
*a*and*b*are integers, then (a/b ) is not necessarily an integer. - If
*a*is an integer and*a ≠ 0*, then (a/b) = 1 - If
*a*is an integer, then (a/1) = 1 - If
*a*is a non-zero integer, then (0/a) = 0 but (a/0) is not defined. - If
*a, b, c*are integers, then (a/b) ÷ c ≠ a ÷ (b/c) , unless*c*= 1 - If
*a, b, c*, are integers and*a > b*then*(a ÷ c) > (b ÷ c)**,*if*c*is positive*(a ÷ c) < (b ÷ c)**,*if*c*is negative

**Rational Numbers**

The numbers that can be expressed in the form (p/q) , where *p* and *q* are integers and *q ≠ 0* are called rational numbers. Therefore, the set Q of all rational numbers is given by

Examples:

Note:

- Every integer is a rational number, since every integer a can be written as (a/1)
- Every fraction is a rational number
- The square root of every perfect square is a rational number

E.g., √9 , ∛8

- Every terminating decimal is a rational number

E.g., 0.9 = (9/10)

- Every recurring decimal is a rational number

E.g., 0.222222 = (2/9)

* Positive Rational Numbers:* A rational number is said to be positive the numerator or denominator are either both positive and both negative. e.g.

* Negative Rational Numbers: *A rational number is said to be negative is any one of the numerator or denominator is negative (they are of opposite sign) e.g.

__Equivalent Rational Numbers__

If (p/q) is a rational number and *m* is a non-zero integer, then we have . We call these numbers and as equivalent rational numbers.

__Standard form of a Rational Numbers__

A rational number (p/q) is said to be a standard rational number form if *p* and *q* are integers having no common divisors other than 1 and *q* is positive.

*Note: Every rational number can also be represented on a number line**.*

__Addition of Rational Numbers__

- In order to add rational numbers, we must first convert them into rational numbers with positive denominator.

- When the rational numbers have the same denominator, then these numbers can be added as follows:

- If the denominator is not same (or unequal), we find the LCM of the denominator and express each one of the rational numbers having the LCM as its denominator. You could use the following approach and reduce the number once you add them.

__Properties of Addition of Rational Numbers__

- Closure Property: The sum of two rational number is also a rational number. That is, if and are rational numbers than is also a rational number.
- Commutative Law: If and are rational numbers than
- Associative Law: If are rational numbers, then
- Existence of Additive Identity: The rational number 0 is the additive identity for any rational number. Therefore

__Subtraction of Rational Numbers__

- If (p/q) and (m/q) are two rational numbers with the same denominator, then
- If the denominator is not same (or unequal), we find the LCM of the denominator and express each one of the rational numbers having the LCM as its denominator. You could use the following approach and reduce the number once you subtract them.

__Properties of Multiplication__

*Division of Rational Numbers*

__Properties of Division of Rational Numbers__

*Closure Property*: The division of two rational numbers is also a rational number. Therefore if and are two rational numbers, then is also a rational number given .- For any rational number (a/b) , we have
- For any non-zero rational number (a/b) , we have

*How to find a Rational Number between Two Given Rational Numbers?*

Let there be two rational numbers and . Then the rational number in between the two would be .

*If you want to find a large number of rational numbers between two given rational numbers, then first make the denominator equal by taking LCM. After that you can just insert values between the two numerators.*

**Irrational Numbers**

A number that can neither be expressed as a terminating nor as a repeating decimals, is called an irrational number. These are non-terminating and non-repeating numbers. Examples

- Non-terminating and non-repeating numbers such as
- 82485863738485738…. or 1.2456345789560….

- Square root of positive integers that are not perfect squares
- √2 , √5 , √7 …

- Cube root of numbers that are not perfect cubes
- ∛2 , ∛3 , ∛4 …

- Number π is irrational. It has a value that is non-terminating and non-repeating. We only approximate it to (22/7) .

__Properties of Irrational Numbers__

*Sum of two irrational numbers*need not be an irrational number- Take (5+ √2) and (5- √2) as two irrational numbers. The sum is 10 which is a rational number.

*Difference of two irrational*numbers need not be an irrational number- Take (5+ √2) and (7+ √2) as two irrational numbers. The difference is 12 which is a rational number

*Product of two irrational*numbers need not be an irrational number- Take (5+ √2) and (5- √2) as two irrational numbers. The product is 23 which is a rational number.

*Quotient of two irrational*numbers need not be an irrational number- Take (9√2) and (3√2) as two irrational numbers. The quotient is 3 which is a rational number.

8. The *sum or difference of a rational and an irrational* numbers is always irrational.

- Example 2 + √5 is irrational
- Example 2 – √5 is irrational

9. The *product or quotient of a rational and an irrational* numbers is always irrational

- Example 2 × √5 is irrational
- Example 2 ÷ √5 is irrational

**Real Numbers**

All rational and irrational numbers forms the set of all real numbers.

__Real number Line__

- The real number line represents all real numbers.
- On a real number line, each point corresponds to a point and conversely, every point on the real number line corresponds to a real number.
- Between any two real numbers, there exist infinite real numbers.

__Properties of Real Numbers__

*Closure property*- The sum of two real numbers is always a real number.
- The product of two real numbers is always a real number.

*Commutative law*- a + b = b + a
- a × b = b × a

*Associative law*

- (a+b )+c=a+( b+c)
- (a ×b)×c=a ×( b ×c)

*Distributive Laws*of multiplication over addition- a ×(b+c)=a ×b+a ×c

- If a ≠ 0, then 1/a is called the reciprocal of a and a × (1/a)=1

__Comparison of two irrational numbers__

If a and b are two irrational numbers such that* a < b, then √a < √b*