Question on continued fraction?

Question

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?

Answer 1

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.

Answer 2

The theorem about irrationals and and infinite continued fractions is for simple continued fractions. See here