I saw a YouTube video recently that mentioned the golden number, phi, being a viable base for a number system, given $\Phi^2 = \Phi + 1$. The result would be the units place representing multiples of $\Phi^0$, the next value left be multiples of $\Phi^1$, and so on up, with values right being decrimenting integers from $-1$ onward.
The example in the video shows the value $6$ in decimal being represented as $1010.0001$ in base phi. The video also seemed to use only $0$ and $1$ to indicate absence and presence of that place's value, which I would call binary except that it isn't base $2$.
In any case, I've found that every integer could be represented two different ways if only 0 and 1 are allowed: {1 = 1 & 0.11, 2 = 1.11 & 10.01, 3 = 11.01 & 100.01, 4 = 101.01 & 100.1111, 5 = 101.1111 & 1000.0111, 6 = 1001.0111 & 1010.0001, etc.} Beyond these, I calculated 1/2, 1/3, 1/4, 1/5, and 1/6. All are repeating non-terminating decimal numbers. ($1/2$ is $0.01001001001\dots$)
My question is simply: Are there any benefits to a system such as this?
Writing numbers with phi base, at minimum, requires more characters than binary. As log_2(10) indicates how many more characters need to be used in binary to display a given number in decimal, $\log_\Phi(10)$ does the same, and shows that even more values are required. Yet, the results above indicate that even more characters are required, given the necessary decimal values for integer numbers.
I wonder if due to the base being irrational, it would allow for easier mathematics in some areas than a rational base would allow. But as irrationals tend to be unique, not necessarily easily (or possibly) converted to other irrationals, I can't see how it'd have wide applications. Hence why I'm asking those more knowledgeable than I.
EDIT: to clarify "benefit", I know that a highly composite base makes fractions much easier to represent as decimals, given there are more terminating fractions. Base 2 has the benefit of being simple (large benefit for computers), base 10 has the benefit of being how many fingers we have, etc. So benefits could be directly mathematical, or otherwise makes things simpler. Since the golden ratio occurs naturally, such as in a nautilus shell and flower petals/seeds, perhaps some benefit(s) to the base would exist.