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digital root (Definition)

Given an integer $ m$ consisting of $ k$ digits $ d_1, \dots, d_k$ in base $ b$, let

$\displaystyle j = \sum_{i = 1}^{k} d_i,$
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$.



"digital root" is owned by PrimeFan. [ full author list (4) | owner history (6) ]
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Other names:  repeated digit sum, repeated digital sum
Also defines:  additive persistence

Attachments:
examples of digital roots in a few selected bases (Example) by PrimeFan
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Cross-references: calculate, multiplicative function, imply, natural number, sum, iterations, number, operation, base, digits, integer
There are 10 references to this entry.

This is version 10 of digital root, born on 2006-06-12, modified 2006-06-15.
Object id is 8017, canonical name is DigitalRoot.
Accessed 2296 times total.

Classification:
AMS MSC: 11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems)
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