# Bounds for Integer Expressed in Base k

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## 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:

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

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

Then:

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

## Proof

As none of the coefficients $a_j$ in $\displaystyle \sum_{j \mathop = 1}^s a_j k^j$ 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:

\(\displaystyle n\) | \(=\) | \(\displaystyle \sum_{j \mathop = 1}^s a_j k^j\) | |||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \paren {k - 1} \sum_{j \mathop = 1}^s k^j\) | as $a_j \le k - 1$ for all $j$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {k - 1} \dfrac {k^{s + 1} - 1} {k - 1}\) | Sum of Geometric Sequence | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle k^{s + 1} - 1\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle k^{s + 1}\) |

$\blacksquare$

## Sources

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