## Natural Numbers

Numbers which start from one (1) and continue up to infinitive are known as natural numbers. Generally, The collection of all natural numbers is denoted by N.

N = {1, 2, 3, 4, …….n}

## Whole Numbers

zero (0) and all Natural Numbers are known as whole numbers. Generally, The collection of all whole numbers are denoted by W. All Natural Numbers are whole numbers but all Whole numbers are not Natural Numbers, because zero is a Whole number but not Natural Number.

W = {0, 1, 2, 3, …n}

## Integers

The collection of all negative natural numbers and whole numbers are called integers. Generally, The collection of all Integers is denoted by Z or I. All Natural numbers and Whole numbers can be an Integer but all Integers are not Whole numbers or Natural Numbers.

Z = {-n …, -3, -2, -1, 0, 1, 2, 3, … n }

## Rational Number

A number can be called a rational number, if it can be presented in the form , where p and q are integers and q ≠ 0. Generally, The collection of rational numbers is denoted by Q. All Integers also comes under this group.

e.g.,

### Equivalent Rational Numbers

If the denominator and numerator of two or more rational numbers are equal on their stand form or reducible form, Then they all are called Equivalent Rational Numbers.

e.g.,

### How many rational numbers are there between two rational numbers ?

There can be infinitely many rational numbers between any two given rational numbers.

We can adopt two processes to find a rational number between any two rational numbers.

Process 1. Let a and b are two rational numbers then a third rational number in between these two

Process 2. Let a and b are two rational numbers and we have to find n rational numbers between a and b. Then first we have to multiply the numerator and denominator of a and b with and then increase the numerator of first number by 1 to get the next number.

Example : Let us find 5 rational numbers between 1 and 3. in this example, n=5, a=1 and b=3 so by multiplying the numerator and denominator of a with 3 we get by multiplying the numerator and denominator of b with 3 we get Five rational numbers in between a and b are means

## Irrational Number

A number can be called an irrational number, if it can not be written in the form , where p and q are integers and q ≠ 0, and its decimal representation is non-terminating and non-repeating.

e.g., √2, √5, π, … Proof of irrationality of these numbers will be covered in class 10th.

Is Square root of all non perfect square numbers are irrational numbers ?

Yes. Square root of all non perfect square numbers are irrational, because they can not be represented in the form of where p and q are integers and q ≠ 0, and their decimal representation is non-terminating and non-repeating.

## Real Numbers

The collection of all rational numbers and irrational numbers together can be called as real numbers, which is denoted by R and can be presented on number line.

## Decimal Expansions of Rational Number

If we find the decimal expansion of rational numbers by dividing p with q, we may get :

- terminating decimal if remainder becomes zero. Example
- non-terminating recurring decimal if the remainder never becomes zero. Example

In other words we can say any number whose decimal expansion is terminating decimal or non-terminating recurring decimal is a rational number.

So, If the decimal expansion of any number is non-terminating non-recurring then it is an irrational number. For example √2 and , as √2 = 1.4142135623730950488016887242096…

= 3.14159265358979323846264338327950…

Operations on Real Numbers

- The sum or difference of a rational number and an irrational number is always an irrational number.
- The product or division of a non-zero rational number and an irrational number is always an irrational number.
- If we add, subtract, multiply or divide two irrational numbers, the result may be rational or irrational.

## Radicand

The number under the radical symbol is known as radicand. So in 3 is index and 5 is radicand.

## Laws of Radicals

Let a and b be positive real numbers. Then,

## Rationalising the Denominator

To rationalise the denominator of, it is multiplied by, where a and b are integers.

## Laws of Exponents

If a is any real number greater then zero and p,q are rational numbers then,

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