Question

Find four rational numbers between \[\dfrac{3}{7}\] and \[\dfrac{5}{7}\] . Find two irrational numbers between \[4.5\overline 6 \] and \[5.\overline 1 \] .

Answer 1

Hint: Here, in the given question, we are asked to find four rational numbers between two given numbers and two irrational numbers between two different numbers in the later part of the question. Before solving this question, we will first understand the both types of numbers and then find the solution according to the definitions and their properties.

Complete step-by-step solution:
Rational Numbers: The numbers which can be expressed in the form \[\dfrac{p}{q}\] are rational numbers, where \[p\] is the numerator and \[q\] is a non-zero denominator. Rational numbers are either terminating or non-terminating repeating. For example: \[0.2525,0.8,0.6777...\] etc.
Irrational numbers: All the real numbers which are not rational numbers are the irrational numbers. The decimal form of irrational numbers is non-terminating and non-repeating. For example: \[\sqrt 2 ,1.2345 \ldots ,0.7070070007 \ldots \] etc.
Now, to find four rational numbers between \[\dfrac{3}{7}\] and \[\dfrac{5}{7}\] , we will multiply and divide both the numbers by \[4\] ,n
 \[
  \dfrac{3}{7} \times \dfrac{4}{4},\dfrac{5}{7} \times \dfrac{4}{4} \\
   \Rightarrow \dfrac{{12}}{{28}},\dfrac{{20}}{{28}} \\
 \]
Hence, the four rational numbers between \[\dfrac{3}{7}\] and \[\dfrac{5}{7}\] are: \[\dfrac{{13}}{{28}},\dfrac{{15}}{{28}},\dfrac{{17}}{{28}},\dfrac{{19}}{{28}}\] .
And, the two irrational numbers between \[4.5\overline 6 \] and \[5.\overline 1 \] are: \[4.712711271127.....\] and \[5.010010001....\] .

Note: There are infinite rational numbers as well as irrational numbers present between any two rational numbers. So, there can be infinite solutions for this question. Also, note that the given two numbers \[4.5\overline 6 \] and \[5.\overline 1 \] are rational numbers, because they are non-terminating but repeating. There is an alternate method to find the rational numbers between two given rational numbers. Firstly, the average of two given numbers will be a rational number between those two numbers. And then finding the required number of times the average of any one of the given numbers and the answer obtained from the previous step will give us the right answer.