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.)
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.