Basic Definitions- Number System
All the counting numbers starting from 1 i.e., 1,2,3,…, etc. are called the natural 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.
All the natural numbers including zero (0) and their negatives come under integers.
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
- 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?
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 that keeps repeating.
Ex- 0.333= 0.¯3 (the bar should be just above 3, its typing error)
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.
Set of rational or irrational numbers is called Real Numbers.
The process of converting the irrational denominator into a rational denominator by multiplying its numerator and denominator by a suitable number is called rationalization.
Example- 3/√5, Here denominator √5 is an irrational number, So
It can be rationalized by multiplying its denominator and numerator by √5.
i.e., 3/√5 = (3/√5)×( √5/√5) = (3√5)/5