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I was observing a high school algebra class and they were discussing irrational versus rational numbers. Irrational go on forever (the digits after the decimal) and don’t ever repeat. Rational repeats. Such as $1.000000...$ or $3.333333...$ or $4.482482482482...$

The teacher wrote a number like this on the board:

$7.36183648747382...$

(I don’t remember the exact number, except that it was seemingly random digits. The dots just indicate it continues in a random fashion, forever.)

He then asked the students if it was irrational or rational. He concluded irrational because there’s no repetition and it goes on forever.

My question is, how can you truly know it doesn’t ever repeat? What if the first 25 digits repeat forever? What if the first million?

Theoretically, the first trillion could repeat forever. How in the world do you truly confirm a number is irrational?

(I can only guess at the StackExchange tags. Please fix as necessary.)

Peter
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aliteralmind
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Sep 24 '21 at 12:14
  • I don’t understand. How can it truly be determined a number is irrational? How can it truly be determined that seemingly random digits following the decimal don’t, in fact, repeat were you take more digits into account? In other words, how can you determine a number is truly irrational unless you manually traverse the decimal digits infinitely, which is impossible? – aliteralmind Sep 24 '21 at 12:18
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    You are right that one cannot determine if a number is irrational simply by looking at its first $n$ digits. You have to prove it using a different method. For example $\sqrt{2}$ is shown to be irrational using a proof by contradiction. – Alphie Sep 24 '21 at 12:22
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    I don't understand what the number $7.36183648\dots$ is. It is a meaningless sequence of digits that came out of thin air as far as I am concerned. Certainly, if the digits repeat then the repetition begins after what is pictured or the period of repetition is longer than what is currently pictured... but it is not clear if this is supposed to be a specific number like $\sqrt{54}$ or if it just happens to be a rational number near that. How were the digits decided upon? This is crucial. – JMoravitz Sep 24 '21 at 12:23
  • @JMoravitz Fair enough. I don’t remember what the number was. It was just random as far as I could tell. I clarified the post. – aliteralmind Sep 24 '21 at 12:27
  • @JMoravitz I would think that any sequence of digits could be a valid decimal, therefore implying that any number of digits could repeat forever. I’m obviously missing something. – aliteralmind Sep 24 '21 at 12:29
  • As alluded to already, those numbers which are proven to be rational are proven to be so by showing that they satisfy the properties of being rational, either by showing that they must repeat or by showing the number is equal to a ratio of integers. A number which is irrational is proven to be so by proving it never repeats or by showing no such ratio of integers exists. See for instance Square root of 2 is irrational or consider numbers like $0.1011011101111011111\dots 0\underbrace{11\dots 1}_{more~1's}\dots$ – JMoravitz Sep 24 '21 at 12:30
  • As for any finite sequence of digits could be the start of a rational number, of course they could. If every digit after the ellipses is equal to zero... for instance $1.2341234783\dots$ was actually equal to identically $1.2341234783$ then this is obviously a ratio of integers $\frac{12341234783}{100000000000}$ (possibly more or less zeroes, I didn't bother counting). Without being told how the digits continue... we don't know how the digits continue and in particular we can't rule out the possibility that every digit following is zero or follows some other rational pattern. – JMoravitz Sep 24 '21 at 12:35
  • Instead of "doesn't ever repeat" you should use the formulation "is non-periodic" (or "is periodic" in the case that there is a period). Note that $e$ seems to be rational, if we only look at the first few digits : $2.718281828$ , but the repition of "$1828$" is a coincidence. What we actually need is that some block is repeating forever (or of course a terminating decimal expansion) – Peter Sep 24 '21 at 12:58
  • Also note that irrationality proofs are usually extremely difficult. For example, we do still NOT know whether $\ e+\pi\ $ is rational. – Peter Sep 24 '21 at 13:10

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The teacher wrote this number on the board:

7.36183648747382...

He then asked the students if it was irrational or rational. He concluded irrational because there’s no repetition and it goes on forever.

Without having any restriction on what the "$\dots$" are supposed to mean, one cannot decide whether the number is rational or not.

Just showing finitely many digits (no matter whether they have some pattern or not) does not tell anything about the semantics of "$\dots$".

In particular, if there is no restriction on the "$\dots$", then make them all 0's and the number is rational.

Or just plug in the digits of $\pi$ and the outcome is irrational.

To determine whether a number is irrational, you need enough information on the number. Knowing all digits and knowing whether the sequence is finally periodic or not is fine, but for many numbers that is not the case.

For example, $\pi$ or Euler's Number are irrational, but we don't know this because we knew all digits (we don't). The proof of irrationality comes from deep results of algebra, and only from there you can infer that no representation in a number system to some base is periodic.

emacs drives me nuts
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  • Thanks for your response. It’s my BS, not his. I clarified my question soon before you posted your answer. Also see the comments on the question. – aliteralmind Sep 24 '21 at 12:39
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    Why the abbreviation "BS" ? Does it stand for "bull-shit" ? Anyway an excellent answer (+1) showing that the exercise is ill-posed. – Peter Sep 24 '21 at 13:04
  • @Peter: I am not a native speaker. If you find it too rude, I'll remove it. I don't want to offend anyone. – emacs drives me nuts Sep 24 '21 at 13:34