Bounds for Integer Expressed in Base k

Theorem

Let $n \in \Z$ be an integer.

Let $k \in \Z$ such that $k \ge 2$.

Let $n$ be expressed in base $k$ notation:

$n = \ds \sum_{j \mathop = 1}^s a_j k^j$

where each of the $a_j$ are such that $a_j \in \set {0, 1, \ldots, k - 1}$.

Then:

$0 \le n < k^{s + 1}$

As none of the coefficients $a_j$ in $\ds \sum_{j \mathop = 1}^s a_j k^j$ is (strictly) negative, the summation itself likewise cannot be negative

Thus:

$0 \le n$

The equality is satisfied when $a_j = 0$ for all $j$.

We then have:

$\ds n$

$=$

$\ds \sum_{j \mathop = 1}^s a_j k^j$

$\ds $

$\le$

$\ds \paren {k - 1} \sum_{j \mathop = 1}^s k^j$

as $a_j \le k - 1$ for all $j$

$\ds $

$=$

$\ds \paren {k - 1} \dfrac {k^{s + 1} - 1} {k - 1}$

$\ds $

$=$

$\ds k^{s + 1} - 1$

$\ds $

$<$

$\ds k^{s + 1}$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-2}$ The Basis Representation Theorem: Exercise $6$