This is important because if you operate a system universally with the same base, the significant number begins to take on an objective importance intra-system.
Stated another way, it's mathematical semantics, the same as arguing an idea is not significant because you can express it in a multitude of different languages, each having an arbitrary words.
Developing via Django (Python) or Rails (Ruby) is similarly predisposed to making one think it's all relative, but when you look closely, the framework doesn't matter at all - there are universal abstractions inherent in web development.
The representation is arbitrary and insignificant, what is significant is what the number or word represents, and what you can do by unraveling it.
Numbers are independent from any base. You may mean the numeral in another base, which would have a different number of digits.
But what is more important, note that not necessarily the two ratios from the quote will have the same number of digits in another base.
I am sure that many of you are familiar with Dr. Feynman's 7-part lecture series entitled The Character of Physical Law, which were part of the "Messenger Lectures" [1], given at Cornell University in 1964.
In the first lecture, Law of Gravitation - An Example of Physical Law, Feynman states the following:
"Question: what is the ratio of the gravitational force to the electrical force? That is illustrated on the next slide.
The ratio of the gravitational attraction to the electrical repulsion is given by a number with 42 digits, and goes off here: all this is written very carefully out, so that's 42 digits.
Now, therein lies a very deep mystery: where could such a tremendous number come from? That means if you ever had a theory from which both of these things are to come, how could they come in such disproportion? From what equation has a solution which has for one, two kinds of forces, an attraction and a repulsion with that fantastic ratio? People have looked for such a large ratio in other places.
They're looking for a large number.
They hope, for example, that there's another large number.
And if you want a large number why not take the diameter of the universe to the diameter of a proton.
Amazingly enough, it also is a number with 42 digits." [2]
[1] https://en.wikipedia.org/wiki/Messenger_Lectures
[2] https://www.youtube.com/watch?v=j3mhkYbznBk#t=48m11s