DigitalRoot 2006-06-12 16:39:58 2006-06-15 18:41:45 Definition additive persistence % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given an integer $m$ consisting of $k$ digits $d_1, \dots, d_k$ in base $b$, let $$j = \sum_{i = 1}^{k} d_i,$$ then repeat this operation on the digits of $j$ until $j < b$. This stores in $j$ the {\em digital root} of $m$. The number of iterations of the sum operation is called the {\em 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 $m - (b - 1)\lfloor {{m - 1} \over {b - 1}} \rfloor$.