Is the set of the rational numbers equal to the set of periodic numbers?

Asked Feb 19 '21 at 15:24

Active Feb 19 '21 at 15:32

Viewed 30 times

Is $\mathbb Q = \{\text{set of all numbers periodic from some point on}\}$ ?

I am pretty sure that $\{\text{set of all numbers periodic from some point on}\} \subset \mathbb Q$ but i don't know how to prove.

I didn't find any proof about $\mathbb Q \subset \{\text{set of all numbers periodic from some point on}\}$. Maybe it is wrong.

Thanks for the help !

edited Feb 19 '21 at 15:32

asked Feb 19 '21 at 15:24

Jotadiolyne Dicci

Is $0.5$ or $1$ a periodic number? – player3236 Feb 19 '21 at 15:25
@player3236 $.5=.49999\dots$, so I would say, "Yes". – saulspatz Feb 19 '21 at 15:26
@saulspatz The 4 there is destroying the strict sense of periodic/recurring. – user10354138 Feb 19 '21 at 15:27
That is why I inquire whether $0.5$ or $1$ is periodic: I don't know what the definition of "periodic numbers" is. – player3236 Feb 19 '21 at 15:28
Well. I would say no because technicaly. 1 is never being repeated. However 0 is being repeated. I think i will add what i mean by "periodic". – Jotadiolyne Dicci Feb 19 '21 at 15:29
@user10354138 Yes, but I think the OP must mean "periodic from some point on." – saulspatz Feb 19 '21 at 15:29
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Also $.5 = .500000\ldots$ – Hagen von Eitzen Feb 19 '21 at 15:30
@player3236 Well, I admit I'm guessing. – saulspatz Feb 19 '21 at 15:30

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