# Definition:Digit Sum

## Definition

Let $n \in \Z: n \ge 0$.

The digit sum of $n$ to base $b$ is the sum of all the digits of $n$ when expressed in base $b$.

That is, if:

$\displaystyle n = \sum_{k \ge 0} r_k b^k$

where $0 \le r_k < b$, then:

$\displaystyle s_b \left({n}\right) = \sum_{k \ge 0} r_k$

## Examples

In conventional base 10 notation, we have:

$s_{10} \left({34716}\right) = 3 + 4 + 7 + 1 + 6 = 21$

In binary notation, we have:

$s_{2} \left({10010111101_2}\right) = 1 + 0 + 0 + 1 + 0 + 1 + 1 + 1 + 1 + 0 + 1 = 7$

## Also see

Compare with digital root.