## Basic Definitions and Formulas- Real Numbers

Contents

**Natural Numbers**

All the **counting numbers** starting from **1** i.e., **1,2,3,.., etc.** are called the natural numbers.

**Whole Numbers**

All the **counting numbers** or the **collection of natural numbers including zero** are called whole numbers, i.e, **0,1,2,3,4,5….etc.**

**Integers**

All the** natural numbers** including **zero (0)** and their** negatives** come under integers.

**Rational Numbers**

The numbers which are of the form p/q, where p and q are integers and q is not equal to zero are called rational numbers.

Ex- 1/2, 3/4, 4 (since 4 can be written in 4/1 form), etc.

** Note**– A rational is said to be in the simplest form if integers p and q do not have a common factor other than 1, and obviously q is not equal to zero.

**Examples-** 1/4, 4/3, 8/7 etc.

Remember **4/8** is a rational number but its simplest form is **1/2.**

### Finding rational numbers between two numbers?

- Finding
**one**rational number between two numbers, let say x and y where x<y

**Formula- (x+y)/2**

- Finding ‘
**n**’ rational numbers between x and y where x<y

1^{st} Step- Find **d= (y-x) / (n+1)**

2^{nd} Step- n numbers are as follows- (x+d), (x+2d),(x+3d),(x+4d)….(x+nd).

**Q- Find a rational number between 1/4 and 1/2?**

**Q- Find 5 rational numbers between 8 and 10?**

**Terminating Decimal**

Every rational number (p/q) can be converted into the decimal form, and if the decimal form comes to an end. For example- 1/2=0.5, 1/4=0.25 etc., then this decimal form is called terminating decimal.

** Note–** Every fraction (p/q) is terminating decimal if the denominator ‘q’ has

**only 2 and 5**as prime factors.

**Repeating or Recurring Decimals**

Decimal forms where a digit or set of digits repeats itself.

For example- 0.3333, 0.999, 0.282828, etc.

** Note-** Here we place a bar over the digit or set of digits which keeps repeating.

Ex- 0.333= 0.¯3 (the bar should be just above 3, its typing error)

**Irrational Numbers**

The numbers which can neither be expressed in terminating nor recurring decimal forms. Ex- 22/7, 0.23540123…, integers which are not perfect squares or perfect cubes.

**Real Numbers**

Set of rational or irrational numbers is called Real Numbers.

**Rationalisation**

The process of converting the irrational denominator into a rational denominator by multiplying its numerator and denominator by a suitable number is called rationalisation.

Example- **3/√5** , Here denominator √5 is an irrational number, So

It can be rationalised by multiplying its denominator and numerator by **√5**.

i.e., 3/√5 = (3/√5)×**( √5/√5) **= (3√5)/5