natural numbers identified with binary strings

It is convenient to identify a natural number $n$ with the $n$th binarystring in lexicographic order:

The more common binary notation for numbers fails to be a bijectionbecause of leading zeroes. Yet, there is a close relation: the $n$thbinary string is the result of stripping the leading 1 from the binarynotation of $n+1$.

With this correspondence in place, we can talk about such things asthe length $l(n)$ of a number $n$, which can be seen to equal$\lfloor \log (n+1)\rfloor$.