## Basic Definitions and Formulas- Real Numbers

### 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

1st Step- Find d= (y-x) / (n+1)

2nd 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

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