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

$34716$ 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.


Sources

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