There isn't some state where you are "carrying the one" and the number is different. Carrying is just an algorithm to canonicalize a number w.r.t. to a radix.

There isn't some state where you are "carrying the one" and the number is different. Carrying is just an algorithm to canonicalize a number w.r.t. to a radix. consider the natural number 9. it is always equal to (1+1+1+1+1+1+1+1+1+0) but the radix is what determines allowable digits in the representation. in any radix >9 you simply write 9, now consider radix=7 you will compute the fact 9 = 7^1*1 + 1^1*2, which implies the numeral is 12. because 9 = 9 + 0 = 7 + (9 - 7) = 7 + 2 = 10_7 + 2_7 = 12_7 computing coefficients doesn't change a number or a sum.