Natural number fomula

Question

There is natural number formula

$ a=a_na_{n-1}....a_0a_1$

For example: how to choose n for 1001, ?

It means i need to use 3 places value or how to do that ?

$1001=a_0a_1a_2a_3 $

n=3 ?

Answer 1

I had to read three or four times to get what you mean. but I finally figured out that you meant: how do we know how many digits a number has. And if the one's digit counts as the $0$th term what is the index of the final term.

If $100000...... 000 = 10^k \le a \le 99999......9 = 10^{k+1} -1< 10^{k+1}$ then $a$ has $k+1$ digits and the index of the final term is $k$.

$\log_{10} (10^k) \le \log_{10} a < \log_{10} 10^{k+1}$ so

$k \le \log_{10} a < k+1$.

The the index of the final term is $k =\lfloor \log_{10} a\rfloor$, that is the largest integer equal to or less than $\log_{10} a$.

But the actual number of digits is $k+1$.