Question

Between two rational numbers, there exists :-
A) No rational number
B) Only one rational number
C) Infinite numbers of rational numbers
D) No irrational number

Answer 1

Hint: A rational number is a number of the form \[\dfrac{p}{q}\] where p and q can be any integer such that \[q \ne 0\] and also q cannot be any number in a root or we have to rationalise it. Try to do this by taking any 2 rational numbers and see what they have in between.

Complete Step by Step Solution:
Any number of the form \[\dfrac{p}{q}\] is a rational number so let us take two rational numbers for example and see what are the correct options for that so let us take 3 and 4 here in the first number \[p = 3\& q = 1\] and in the second number \[p = 4\& q = 1\] . Now, If we take any decimal between 3 and 4 that can also be transformed into an rational number for example let us take 3.24 that can be written as \[\dfrac{{324}}{{100}}\] similarly there may exist hundreds of thousands of decimals that can be transformed into rational numbers so clearly the first and second options are incorrect there exists a rational number between 2 rational number and that too multiple of them exists. Also option C is correct there exist infinite numbers if rational numbers between 2 rational numbers. Now let us check for the last options which is given as no irrational number exists between 2 rational number which is also a false statement because \[\pi \] exists between 3 and 4 as \[\pi \approx 3.14159\] which indeed in between 3 and 4 and \[\pi \] is a irrational number because the true value is still unknown.
Therefore option C is the only correct option.

Note: There exist multiple and infinite numbers of irrational and rational numbers between two rational numbers; it is also said that the number line for an irrational number is even bigger than for a rational number so the numbers we are learning are a fraction of what exist in the real number line.