Is there any proper way through which we can check whether a number is rational or not because finding the decimal expansion at times is not viable. For example the number $48/121$ shows no sign of termination or repetition of its digits well to around 12 to 13 decimal places but is actually rational.
2026-03-25 13:55:37.1774446937
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Is there any proper way through which we can check whether a number is rational or not
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In order to decide if a decimal representation is that of a rational number or not, you need all (infinitely many) digits, and that's never going to happen. You cannot determine if it rational or not from finitely many digits from its decimal.
The task you apparently want to do is not possible to carry out by human or computer. You could, though, pick a rational that is very very close to the decimal expansion in question.
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If it is a fraction of two integers, say $m/n$ with $n\ne 0$ then it is rational.
If it is presented by its decimal expansion then you check to see if it has eventualy repeating or terminating decimals.
If none of those happen then it is not rational.
Some numbers such as $\pi$ or $e$ or $\sqrt n$ if $n$ is not a perfect square, have been proved to be irrational.
Some very interesting questions are out there regarding irrational numbers.
For example finding two irrational numbers $\alpha$ and $\beta$ such that $ \alpha ^{\beta}$ is rational.
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Depends entirely how expressed. One would have doubts about $$ \sqrt[3]{-2 + \sqrt 5} + \sqrt[3]{-2 - \sqrt 5} $$ but be more certain about the real-valued root of $$ t^3 + 3 t + 4 $$
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Here's a summary of several comments, that may help.
A number is rational if it can be written as the quotient of two integers. That's the definition, so there's no need to ask how to show that $48/121$ is rational. There it is as a quotient of integers.
There is a theorem that says that a number is rational if and only if it has a decimal representation that is eventually periodic (it might have a whole bunch of zeroes at the end; that counts). So you know that if you work out the decimal expansion of $48/121$ it will eventually have a repeating block - even if you have to look further than you are willing to calculate to see it.
Looking for a repeating block is rarely a good way to see whether a number is rational. Usually you have to try a different kind of argument. There are well known ones that show $\sqrt{2}$ is irrational, but they don't work by proving there is no repeating block in $1.4146\ldots$.
You can sometimes argue directly from the decimal expansion. The number $$ 0.1010010000001000000000000000000000000100\ldots $$ where the number of zeroes between the ones follows the factorial pattern $$ 1, 2, 6, 24, 120, \ldots $$ is not rational. (It's even transcendental, but that's a longer story.)
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All numbers in the form
$$\frac p q \quad p,q\in \mathbb{Z}\quad q\neq 0\iff \frac p q\in \mathbb{Q} $$
are defined rational. Usually to guarantee an unique representation we set $q$ positive and $(p,q)=1$ that is relatively primes.
In such cases we can't have doubts.
When we face with decimal expression it can be shown that when the decimal part becomes eventually periodic the number has a rational expression (there are simple algorithms to reconstruct the fraction from the decimal expression).
Rational numbers are written as ratios, i.e., fractions.
If you can write it as a fraction, it's rational. Otherwise, no. If it repeats or terminates, it's rational. Otherwise, irrational.