# Definition:Digital Root

## Definition

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

Let $n$ be expressed in base $b$ notation.

Let $n_1 = \map {s_b} n$ be the digit sum of $n$ to base $b$.

Then let $n_2 = \map {s_b} {n_1}$ be the digit sum of $n_1$ to base $b$.

Repeat the process, until $n_m$ has only one digit, that is, that $1 \le n_m < b$.

Then $n_m$ is the digital root of $n$ to the base $b$.

## Examples

### $34716$ Base $10$

In conventional base $10$ notation, we have:

$\map {s_{10} } {34716} = 3 + 4 + 7 + 1 + 6 = 21$

and then:

$\map {s_{10} } {21} = 2 + 1 = 3$.

So the digital root of $34716$ (base $10$) is $3$.

### $10010111101$ Base $2$

In binary notation, we have:

 $\ds \map {s_2} {10010111101_2}$ $=$ $\ds 1 + 0 + 0 + 1 + 0 + 1 + 1 + 1 + 1 + 0 + 1$ $\ds = 7 = 111_2$ $\ds \map {s_2} {111_2}$ $=$ $\ds 1 + 1 + 1$ $\ds = 3 = 11_2$ $\ds \map {s_2} {11_2}$ $=$ $\ds 1 + 1$ $\ds = 2 = 10_2$ $\ds \map {s_2} {10_2}$ $=$ $\ds 1 + 0$ $\ds = 1 = 1_2$

It is pretty obvious that the digital root of any non-zero number in base $2$ is always $1$.

## Also see

• Results about digital roots can be found here.