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

* Whole Numbers:* All natural numbers together along with

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

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

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

Properties of Addition of Integers

The sum of two integers is always an integer. If and are two integers, then:

*Associative Law:* If , and are integers, then:

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

*Existence of additive inverse:* For each integer , there exists another integer such that

Note: If be any integer, then will be called the successor of and the predecessor of is

Properties of Subtraction

If and are two integers, then is also an integer.

For any integer , we have . But

If are integers and , then

Properties of Multiplication of Integers

*Closure Property:* The product of two integer is always an integer

*Commutative Law:* For any two integers and we have:

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

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

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

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

Properties of Multiplication of Integers

If and are integers, then is not necessarily an integer.

If is an integer and , then

If is an integer, then

If is a non-zero integer, then but is not defined.

If are integers, then , unless

If , are integers and then

, if is positive

, if is negative

Rational Numbers

The numbers that can be expressed in the form , where and are integers and are called rational numbers. Therefore, the set of all rational numbers is given by

Examples:

Note:

- Every integer is a rational number, since every integer can be written as
- Every fraction is a rational number
- The square root of every perfect square is a rational number E.g.

- Every terminating decimal is a rational number E.g.,

- Every recurring decimal is a rational number E.g

* 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 is a rational number and 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 is said to be a standard rational number form if and are integers having no common divisors other than and 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__

(i) If and are two rational numbers with the same denominator, then

(ii) 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. i.e.

__Properties of Multiplication__

*Closure Property:* The product of two rational numbers is also a rational number. Therefore if and are two rational numbers, then is also a rational number.

*Commutative Law:* If and are two rational numbers, then

*Associative Law: *If , and are rational numbers then

*Existence of Multiplicative Identity:* The rational number is a multiplicative identity for rational numbers. For any rational number we have

*Existence of Multiplicative Inverse:* For every non zero rational number, there exists a multiplicative inverse. For a rational number the multiplicative inverse would be such that

*Distributive Law of Multiplication over Addition:* For any three and we have

*Multiplicative Property of Zero:* For any rational number that we have

Division of Rational Numbers

If and are two rational numbers such that . Then, we define . When is divided by , then is called the dividend, is called the divisor and the result of the division is called the quotient.

__Properties of Division of Rational Numbers__

*Closure Property*: The division of two rational numbers is also a rational number. Therefore if

__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 , we have

For any non-zero rational number , 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
- or

- Square root of positive integers that are not perfect squares
- Cube root of numbers that are not perfect cubes
- Number is irrational. It has a value that is non-terminating and non-repeating. We only approximate it to .

__Properties of Irrational Numbers__

*Sum of two irrational numbers* need not be an irrational number

- Take and as two irrational numbers. The sum is which is a rational number.

*Difference of two irrational* numbers need not be an irrational number

- Take and as two irrational numbers. The difference is which is a rational number

*Product of two irrational *numbers need not be an irrational number

- Take and as two irrational numbers. The product is which is a rational number.

*Quotient of two irrational* numbers need not be an irrational number

- Take and as two irrational numbers. The quotient is which is a rational number.

*Commutative Law: *

- If and are irrational numbers, then
- If and are rational numbers, then

*Associative Law*

- If , and are irrational numbers, then

- If , and are irrational numbers, then

*Distributive Law*

- If , and are irrational numbers, then

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

- Example is irrational
- Example is irrational

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

- Example is irrational
- Example 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*

*Associative law*

*Distributive Laws of multiplication over addition*

- If , then is called the reciprocal of and

__Comparison of two irrational numbers__

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