Rational number

What is a Rational number?

• A rational number is a number that can be in the form of p / q
• where p and q are integers and q is not equal to zero.
• Rational numbers are denoted by Q

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Rational number Examples

• Please note that π is not a Rational number
• *π = 22/ 7, = 3.1428571428571… is an approximation, is close but not accurate.
• Hence π is not a Rational number.

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Rational number Properties

• All natural numbers & Integers are rational numbers.
• Representation is not unique. The same rational number can be written in many ways;

• The above property is useful to add, subtract and compare rational numbers

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Rational number Addition

Add : 

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Rational number Subtraction

Subtract : 

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Rational number comparison

Compare : is 

>> Statment is False as 10 is greater than 9

Answer : 

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Rational number Reduce form

• Reduce form Rational number reduce form is p/q
• if p & q have no common factors (or 1 as a common factor) OR
• Reduced form, gcd(p,q) = 1

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Example: get reduced form of prime number

• • Factor of 18 : 1, 2, 3, 6, 9, 18
  • Factor of 60 : 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • The greatest common factor/divisible is 6, it is known as gcd,
  • gcd(18, 60) = 6

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To get a reduced form divide the numerator & denominator by 6

Reduce form

-as only 1 is the common factor in 3 & 10

• • • The factor of 3   =  1, 3
    • The factor of 10 = 1, 2, 5, 10

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Prime factorization :

Every integer can be written in the form of primary factors

• Example : 18  = 2 x 3 x 3 = 2 x 3²
• Example: 60 = 2 x 2 x 3 x 5 = 2² x 3 x 5

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Rational number Density

• Between any two rational numbers, we can find another rational number.
• These Rational numbers are called dense.

Example:

• • Between rational no 4 & 5 or we can say between and

= (4+5) /2 = (rational number)

• • Between 4 &   we can find out another prime number;

= (4 +  ) / 2 =   (rational number)

• • Between    and    we can find out another prime number;

= (      +     ) / 2  =   (rational number)

1. • in the same way, you will continuously get rational numbers in between any two rational numbers  ……………….

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Rational number Density vs Integer number

• There will not be any integer between two consecutive integers
• Exp. between 4 & 5 there is no integer, therefore integers are called discrete (discontinuous).
• Cannot talk of the next rational number
• Cannot talk of the previous rational number
• Example 1 – we can not say, what is the previous rational number of [latex]\frac{1}{2}[/latex]
• Example 2 – we can not say,  what is the next rational number of [latex]\frac{1}{2}[/latex]