The accepted answer to this question regarding the continuum hypothesis contains the sentence:
"I am assuming you know that |ℝ|=2^(ℵ0), which can be proven by looking at binary expansions of numbers in [0,1] (discounting countably many numbers with non-unique expansions)."
Does the "binary" in binary expansion pertain to why two occupies the base of the exponential expression on the right side of the equation? Why two and not another quantity? My understanding is that GCH allows for the subscript of the aleph number exponent to change while the base remains equal to two.
The fact that the $2$ in $2^{\aleph_0}$ is the same as the $2$ in binary just means that the correspondence is easier to prove. If you wanted to prove that $|\Bbb R|=3^{\aleph_0}$ (which is also true), you could use ternary expansions to prove it. And in fact $|\Bbb R|=n^{\aleph_0}$ for any integer $n\ge 2$.
By the way, you don't need GCH (or indeed CH) for any of this. The subscript of $\aleph$ has to be zero, independently of GCH or CH.