I have seen in wikipedia that irrational numbers have infinite continued fraction but I also found $$1=\frac{2}{3-\frac{2}{3-\ddots}}$$ so my question is that does that mean $1$ is irrational because it can be written as an infinite continued fraction?
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Irrational numbers have infinite decimal expansion. But so does $\frac13$. – MJD Oct 17 '21 at 11:40
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No it does not, since the logic behind the statement is
$x$ irrational $\implies$ $x$ can be written as an infinite continued fraction.
However, this does not necessarily mean that rationals cannot have an infinite continued fraction.
L0Ludde0
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2But in fact the simple continued fraction is infinite if and only if the number is irrational. – Gerry Myerson Oct 17 '21 at 11:54
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