# More irrational than the Golden Ratio?

According to this video, $$\varphi$$ is the most irrational number, due to its continued fraction form having $$1$$, the smallest natural number, in the denominators.

Is it not possible to construct a "more irrational" number by using $$0$$?

For example,

$$\iota = 1 + \cfrac{1}{0 + \cfrac{1}{1 + \cfrac{1}{0 + \cfrac{1}{1+\cdots} } } }$$

Based on the argument in the video, this would appear to be more irrational.

What am I missing?

• This number is $1+1+1+1+...$ because the recipricals cancel. Sep 25 '18 at 20:11

No because $$\iota = 1 + \cfrac{1}{0 + \cfrac{1}{1 + \cfrac{1}{0 + \cfrac{1}{1+\cdots} } } } = 1 + \cfrac{1}{ \cfrac{1}{1 + \cfrac{1}{ \cfrac{1}{1+\cdots} } } }$$\ $$=1 + 1+\cfrac{1} {\cfrac{1}{1+\cdots}}=1+1+1+\ldots$$
• I don't understand how you go from form two to three, if anything, it looks like the $1/1$s would cancel, and it would be identical to the continued fraction form of $\varphi$ - which I guess also answers my question. Sep 25 '18 at 20:30
• No, when two of the $1$’s cancel the rest is in the numerator, not the denominator Sep 25 '18 at 20:59