Rational Numbers Between Two Rational Numbers

Rational Numbers:

A number is a mathematical object which is used to count, measure, and label the mathematical concepts. Numbers are the basic units of Mathematics. All types of numbers used in Mathematics are grouped under the banner of a Number system. Various kinds of numbers include prime numbers, composite numbers, odd numbers, even numbers, natural numbers, whole numbers, integers, decimal numbers, fractions, rational numbers, irrational numbers, real numbers, and imaginary numbers. All the numbers that exist in reality are called the real numbers. All the numbers that do not exist and are assumed to explain a few mathematical concepts are called imaginary numbers. Real numbers are broadly classified into rational numbers and irrational numbers. Rational numbers are the numbers that can be expressed in the form of a fraction whose denominator is not equal to zero. Irrational numbers are the numbers that cannot be expressed in the form of a fraction such that the denominator is not equal to zero.(image will be updated soon) 


Rational Numbers Between Two Rational Numbers:

Rational numbers are the numbers which can be expressed in the form of p and q where q ≠ 0. Examples for rational numbers are prime and composite numbers, odd and even numbers, decimals and fractions. A number of rational numbers between two rational numbers can be located. Between any two rational numbers, countless rational numbers can be found. 


How to Find Rational Numbers Between two Rational Numbers With the Same Denominator Value?

  • The first step in determining the rational numbers between two rational numbers is to check the value of the denominators.

  • If the denominator values are the same, check the value of the numerators.  

  • If the numerators differ by a large value, then the rational numbers between the two rational numbers can be written in the increments of one for the numerator without altering the value of the denominator.

  • Example: The 5 rational numbers between 1/9 and 7/9 are 2/9,3/9,4/9,5/9,6/9

  • If the values of the numerators differ by a lesser value than the number of rational numbers to be found, then the numerators and denominators of both the rational numbers are multiplied by multiples of 10. 

Example: If 10 rational numbers are to be found between 2/7and 5/7, both the rational numbers are to be multiplied with 10/10.


2/7 x 10/10=20/70

5/7 x 10/10=50/70


The 10 rational numbers between 2/7 and 5/7 can be written as the rational numbers between 20/70 and 50/70. The 10 rational numbers are 21/70, 22/70,23/70, 24/70, 25/70, 26/70, 27/70, 28/70, 29/70 and 30/70.

How to find Rational Numbers Between two Rational Numbers with Different Values of Denominators?

  • To find the rational numbers between two rational numbers with different denominators, the denominators should be equated. 

  • Equating the denominators can be done either by finding their LCM or by multiplying the denominators of one to both the numerator and denominator of the other.


Example:

If the rational numbers between ⅔ and ¾ is to be found. 

LCM of 3 and 4 is 12. When the denominators are equated by LCM method, the equivalent rational numbers are 8/12 and 9/12.

The same rational numbers will be obtained when the denominator of one rational number is multiplied to the numerator and denominator of the other. 

2/3 x 4/4 = 8/12 and 3/4 x 3/3 = 9/12.

  • Once the denominators are equated, the same rules of finding the rational numbers between two rational numbers having the same denominator is used.


Rational Numbers Between two Rational Numbers Example:

1. Find rational numbers between ¼ and ½ (at least 5).

Solution:

The rational numbers ¼ and ½ have different denominators.

Equate the denominator. 

1/4 x 2/2 = 2/8 and 1/2 x 4/4 = 4/8

So the rational numbers are 2/8 and 4/8.

5 rational numbers between these two rational numbers cannot be written. So, both the numerator and denominator of the two rational numbers is multiplied by 10.

2/8 x 10/10 = 20/80 and 4/8 x 10/10 = 40/80

The 5 rational numbers between ¼ and ½ therefore are 21/80, 22/80, 23/80, 24/80 and 25/80.


2. How to find the rational numbers between two rational numbers ¼ and ¾. (At Least 10)

Solution:

The rational numbers ¼ and ¾ have the same denominators. 

It is not possible to write 10 rational numbers between ¼ and ¾ .

So, the numerator and denominator of both the rational numbers are multiplied by 10.

1/4 x 10/10 = 10/40 and 3/4 x 10/10 = 30/40

Therefore, the 10 rational numbers between ¼ and ¾ are:

11/40, 12/40, 13/40, 14/40, 15/40, 16/40, 17/40, 18/40, 19/40 and 20/40


Fun Facts: 

  • Rational numbers can always be expressed in the form of ratios.

  • The denominator of a rational number cannot be an irrational number because any rational number divided by an irrational number is an irrational number.

FAQ (Frequently Asked Questions)

1. What are the Properties of Rational Numbers?

A. Rational numbers are the numbers which can be represented in the form of fractions with non zero denominators. 


Properties of Rational Numbers:

  • Rational numbers are closed under addition, subtraction, multiplication, and division. i.e. the results obtained by performing any of these four operations are rational numbers. 

  • Rational numbers obey the commutative and associative law of addition and multiplication.

  • Rational numbers also satisfy the distributive law of addition over multiplication. 

  • The additive identity of a rational number is 0 and its additive inverse is the number itself. 

  • Multiplicative identity of a rational number is 1 and its multiplicative inverse is the reciprocal of the number.

  • Between any two rational numbers, an infinite number of rational numbers can be positioned.

2. How to Find Rational Numbers Between two Rational Numbers?

A. The steps to be followed to determine the rational numbers between any two rational numbers varies with the value of their denominators and the number of rational numbers to be found. If the denominators are not equal, the denominators are equated by LCM method. After equating the denominators, the rational numbers between them are written with rational numbers with numerators incremented by 1. If the number of rational numbers to be determined is more, then the numerator and denominator of both the rational numbers are multiplied by the multiples of 10 depending on the number of rational numbers to be found.