We will learn to insert rational numbers between two rational numbers. Let us recall integers and properties of various operations on them. We know between two nonconsecutive integers x and y there are (x  y  1) integers. However, there is no integer between two consecutive integers.
For example, between 7 and 7 there are 7  (7)  1 = 7 + 7  1 = 14 – 1 = 13 integers. The integers are 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5 and 6 but there is no integer between 2 and 3 since they are consecutive integers.
Thus, we find that between two given integers there may or may not lie any integer.
How to insert many rational numbers between two rational numbers?
We can insert infinitely many rational numbers between any two rational numbers. This property of rational numbers is known as the dense property.
How to find out some rational numbers lying between two given rational numbers, say between 4/7 and 2/7. The four rational numbers 3/7, 2/7, 1/7, 0/7 and 1/7 lies between 4/7 and 2/7.
We can apply the same procedure to insert more rational numbers between 4/7 and 2/7.
The rational numbers 4/7 and 2/7 can also be written as 40/70 and 20/70 respectively.
Clearly, 39/70, 38/70, 37/70, 36/70, 35/70, …….., 0/70, 1/70, 2/70, 3/70, 4/70, …….., 18/70, 19/70 are rational numbers between 4/7 and 2/7.
The total number of these rational numbers is same as the number of integers between 40 and 70, i.e., 70  (40)  1 = 70 + 40  1 = 110  1 = 109.
Similarly, by rewriting 4/7 and 2/7 as 400/700 and 200/700, we can insert 700  (400)  1 = 700 + 400  1 = 1100  1 = 1099 rational numbers between 4/7 and 2/7.
Therefore, we can apply the same procedure to insert as many rational numbers between 4/7 and 2/7.
Solved examples on rational numbers between two rational numbers:
Find out 100 rational numbers lying between 9/19 and 5/19.
Solution:
We have,
9/19 = 9 × 10/19 × 10 = 90/190 and,
5/19 = 5 × 10/19 × 10 = 50/190
We know that
90 < 89 < 88 < 87 < 86 < 85 < …….. < 25 < 24 < 23 < 22 < …….. < 1 < 0 < 1 < 2 < …….. < 9 < 10
⇒ 90/190 < 89/190 < 88/190 < 87/190 < 86/190 < 85/190 < …….. < 25/190 < 24/190 < 23/190 < 22/190 < …….. < 1/190 < 0/190 < 1/190 < 2/190 < …….. < 9/190 < 10/190
Hence, < 89/190 < 88/190 < 87/190 < 86/190 < 85/190 < …….. < 25/190 < 24/190 < 23/190 < 22/190 < …….. < 1/190 < 0/190 < 1/190 < 2/190 < …….. < 9/190 < 10/190 are the 100 rational numbers between 9/19 = 90/190 and 5/19 = 50/190.
● Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
Standard form of a Rational Number
Equality of Rational Numbers using Standard Form
Equality of Rational Numbers with Common Denominator
Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
Rational Numbers in Descending Order
Representation of Rational Numbers on the Number Line
Rational Numbers on the Number Line
Addition of Rational Number with Same Denominator
Addition of Rational Number with Different Denominator
Properties of Addition of Rational Numbers
Subtraction of Rational Number with Same Denominator
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Rational Expressions Involving Addition and Subtraction
Simplify Rational Expressions Involving the Sum or Difference
Multiplication of Rational Numbers
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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● Rational Numbers  Worksheets
Worksheet on Equivalent Rational Numbers
Worksheet on Lowest form of a Rational Number
Worksheet on Standard form of a Rational Number
Worksheet on Equality of Rational Numbers
Worksheet on Comparison of Rational Numbers
Worksheet on Representation of Rational Number on a Number Line
Worksheet on Adding Rational Numbers
Worksheet on Properties of Addition of Rational Numbers
Worksheet on Subtracting Rational Numbers
Worksheet on Addition and
Subtraction of Rational Number
Worksheet on Rational Expressions Involving Sum and Difference
Worksheet on Multiplication of Rational Number
Worksheet on Properties of Multiplication of Rational Numbers
Worksheet on Division of Rational Numbers
Worksheet on Properties of Division of Rational Numbers
Worksheet on Finding Rational Numbers between Two Rational Numbers
Worksheet on Word Problems on Rational Numbers