Given an integer $m$ consisting of $k$ digits $d_{1},\dots,d_{k}$ in base $b$, let

then repeat this operation on the digits of $j$ until $j<b$. This stores in $j$ the digital root of $m$. The number of iterations of the sum operation is called the additive persistence of $m$.

The digital root of $b^{x}$ is always 1 for any natural $x$, while the digital root of $yb^{n}$ (where $y$ is another natural number) is the same as the digital root of $y$. This should not be taken to imply that the digital root is necessarily a multiplicative function.

The digital root of an integer of the form $n(b-1)$ is always $b-1$.

Another way to calculate the digital root for $m>b$ is with the formula $m-(b-1)\lfloor{{m-1}\over{b-1}}\rfloor$.