The machine uses 1 and 0 because it's easier to do binary arithmetic (in electronics) than denary arithmetic. 1 and 0 map very well to the intrinsic states 'on' and 'off'.
You can make bigger numbers in base 2 (binary) using only 0 and 1, just the same as you can make bigger numbers in base 10 (denary) with only 0,1,2,3,4,5,6,7,8 and 9. You can represent any number in base 10 in base 2, so it doesn't really matter.
Base 10 is quite arbitrary anyway. Why do you consider 0,1,2,3,4,5,6,7,8, and 9, as 'enough' unit numbers... why not 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F? Or base 200 with some new unique symbol for each unit?