Is there any proper way through which we can check whether a number is rational or not

Question

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.

Not following. The number $\frac {48}{121}$ is obviously rational. What are you asking? — lulu, Aug 04 '18 at 19:23
If you have a number like $e$ or $\pi$ for which you can use analytic methods to write out the decimal, it can be extremely difficult to determine rationality. Usually methods specific to the number or the functions that produce it are needed. — lulu, Aug 04 '18 at 19:25
If you divide a/b where a and b are integers, you get a rational number. "Rational" comes from "ratio", and a division is just that -- a ratio! — Max, Aug 04 '18 at 19:28
Just to say, if you take an irrational number like $e$ and divide it by a huge power of $10$, say $10^{10^9}$, you'll get an irrational number for which the decimal starts with a colossal string of $0's$...about a billion $0's$ in this case. If that's all you look at, it sure looks rational for a long time, but it isn't. — lulu, Aug 04 '18 at 19:31
@CaptainAmerica16 has posted a wildly inaccurate (terminating!) decimal expansion of the number in question. The version posted by lulu is what you want. — Cameron Buie, Aug 04 '18 at 20:31
@CameronBuie I'm so sorry! I wasn't even paying attention to my notation! If anyone is willing, could you show me how to represent non-terminating decimals through mathjax? — CaptainAmerica16, Aug 04 '18 at 20:45
@MohdKhan Please recall that if the OP is solved you can evaluate to accept an answer among the given, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work — user, Sep 06 '18 at 23:05

score 3 · Answer 1 · answered Aug 04 '18 at 19:40

3

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 $$ t^3 + 3 t + 4 $$

answered Aug 04 '18 at 19:40

Will Jagy

139,541

score 1 · Answer 2 · answered Aug 04 '18 at 19:29

1

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.

answered Aug 04 '18 at 19:29

Sean Roberson

9,110

score 1 · Answer 3 · answered Aug 04 '18 at 19:32

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.

score 1 · Answer 4 · answered Aug 04 '18 at 19:34

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.

score 1 · Answer 5 · answered Aug 04 '18 at 20:00

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.)

I think it might be good to note that "[arguing] directly from the decimal expansion" might work to nudge you towards a hypothesis but is by no means a proof. — Jam, Aug 04 '18 at 20:39

score 0 · Answer 6 · answered Aug 04 '18 at 20:13

0

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).

answered Aug 04 '18 at 20:13

user

154,566

Any repeated decimal is rational. – David G. Stork Aug 04 '18 at 21:07
What do you mean with repeated? isn't it included in "periodic"? – user Aug 04 '18 at 21:09

Is there any proper way through which we can check whether a number is rational or not

6 Answers6