# Definition:Digital Root

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## 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

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 $3$.

In binary notation, we have:

- $\map {s_2} {10010111101_2} = 1 + 0 + 0 + 1 + 0 + 1 + 1 + 1 + 1 + 0 + 1 = 7 = 111_2$

- $\map {s_2} {111_2} = 1 + 1 + 1 = 3 = 11_2$

- $\map {s_2} {11_2} = 1 + 1 = 2 = 10_2$

- $\map {s_2} {10_2} = 1 + 0 = 1 = 1_2$

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

## Also see

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**casting out nines** - Weisstein, Eric W. "Digital Root." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/DigitalRoot.html