Number of Digits to Represent Integer in Given Number Base

Theorem

Let $b \in \Z$ be an integer such that $b > 1$.

Let $d$ denote the number of digits of $n$ when represented in base $b$.

Then:

$d = \ceiling {\map {\log_b} {n + 1} }$

where $\ceiling {\, \cdot \,}$ denotes the ceiling function.

Let $n$ have $d$ digits.

Then:

$\ds b^{d - 1}$

$\le$

$\, \ds n \, $

$\, \ds < \, $

$\ds b^d$

$\ds \leadsto \ \ $

$\ds b^{d - 1}$

$<$

$\, \ds n + 1 \, $

$\, \ds \le \, $

$\ds b^d$

$\ds \leadsto \ \ $

$\ds d - 1$

$<$

$\, \ds \map {\log_b} {n + 1} \, $

$\, \ds \le \, $

$\ds d$

$\ds \leadsto \ \ $

$\ds $

$\, \ds \ceiling {\map {\log_b} {n + 1} } \, $

$\, \ds = \, $

$\ds d$

$\blacksquare$