Digital root

Let ${\displaystyle n}$  be a natural number. For base ${\displaystyle b>1}$ , we define the digit sum ${\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }$  to be the following:

${\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}d_{i}}$ 

where ${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$  is the number of digits in the number in base ${\displaystyle b}$ , and

${\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}$ 

is the value of each digit of the number. A natural number ${\displaystyle n}$  is a digital root if it is a fixed point for ${\displaystyle F_{b}}$ , which occurs if ${\displaystyle F_{b}(n)=n}$ .

All natural numbers ${\displaystyle n}$  are preperiodic points for ${\displaystyle F_{b}}$ , regardless of the base. This is because if ${\displaystyle n\geq b}$ , then

${\displaystyle n=\sum _{i=0}^{k-1}d_{i}b^{i}}$ 

and therefore

${\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}d_{i}<\sum _{i=0}^{k-1}d_{i}b^{i}=n}$ 

because ${\displaystyle b>1}$ . If ${\displaystyle n<b}$ , then trivially

${\displaystyle F_{b}(n)=n}$ 

Therefore, the only possible digital roots are the natural numbers ${\displaystyle 0\leq n<b}$ , and there are no cycles other than the fixed points of ${\displaystyle 0\leq n<b}$ .

In base 12, 8 is the additive digital root of the base 10 number 3110, as for ${\displaystyle n=3110}$ 

${\displaystyle d_{0}={\frac {3110{\bmod {12^{0+1}}}-3110{\bmod {1}}2^{0}}{12^{0}}}={\frac {3110{\bmod {12}}-3110{\bmod {1}}}{1}}={\frac {2-0}{1}}={\frac {2}{1}}=2}$ 

${\displaystyle d_{1}={\frac {3110{\bmod {12^{1+1}}}-3110{\bmod {1}}2^{1}}{12^{1}}}={\frac {3110{\bmod {144}}-3110{\bmod {1}}2}{12}}={\frac {86-2}{12}}={\frac {84}{12}}=7}$ 

${\displaystyle d_{2}={\frac {3110{\bmod {12^{2+1}}}-3110{\bmod {1}}2^{2}}{12^{2}}}={\frac {3110{\bmod {1728}}-3110{\bmod {1}}44}{144}}={\frac {1382-86}{144}}={\frac {1296}{144}}=9}$ 

${\displaystyle d_{3}={\frac {3110{\bmod {12^{3+1}}}-3110{\bmod {1}}2^{3}}{12^{3}}}={\frac {3110{\bmod {20736}}-3110{\bmod {1}}728}{1728}}={\frac {3110-1382}{1728}}={\frac {1728}{1728}}=1}$ 

${\displaystyle F_{12}(3110)=\sum _{i=0}^{4-1}d_{i}=2+7+9+1=19}$ 

This process shows that 3110 is 1972 in base 12. Now for ${\displaystyle F_{12}(3110)=19}$ 

${\displaystyle d_{0}={\frac {19{\bmod {12^{0+1}}}-19{\bmod {1}}2^{0}}{12^{0}}}={\frac {19{\bmod {12}}-19{\bmod {1}}}{1}}={\frac {7-0}{1}}={\frac {7}{1}}=7}$ 

${\displaystyle d_{1}={\frac {19{\bmod {12^{1+1}}}-19{\bmod {1}}2^{1}}{12^{1}}}={\frac {19{\bmod {144}}-19{\bmod {1}}2}{12}}={\frac {19-7}{12}}={\frac {12}{12}}=1}$ 

${\displaystyle F_{12}(19)=\sum _{i=0}^{2-1}d_{i}=1+7=8}$ 

shows that 19 is 17 in base 12. And as 8 is a 1-digit number in base 12,

${\displaystyle F_{12}(8)=8}$ .

We can define the digit root directly for base ${\displaystyle b>1}$  ${\displaystyle \operatorname {dr} _{b}:\mathbb {N} \rightarrow \mathbb {N} }$  in the following ways:

The formula in base ${\displaystyle b}$  is:

${\displaystyle \operatorname {dr} _{b}(n)={\begin{cases}0&{\mbox{if}}\ n=0,\\b-1&{\mbox{if}}\ n\neq 0,\ n\ \equiv 0{\pmod {(b-1)}},\\n{\bmod {(b-1)}}&{\mbox{if}}\ n\not \equiv 0{\pmod {(b-1)}}\end{cases}}}$ 

or,

${\displaystyle \operatorname {dr} _{b}(n)={\begin{cases}0&{\mbox{if}}\ n=0,\\1\ +\ ((n-1){\bmod {(b-1)}})&{\mbox{if}}\ n\neq 0.\end{cases}}}$ 

In base 10, the corresponding sequence is (sequence A010888 in the OEIS).

The digital root is the value modulo ${\displaystyle (b-1)}$  because ${\displaystyle b\equiv 1{\pmod {(b-1)}},}$  and thus ${\displaystyle b^{i}\equiv 1^{i}\equiv 1{\pmod {(b-1)}}.}$  So regardless of the position ${\displaystyle i}$  of digit ${\displaystyle d_{i}}$ , ${\displaystyle d_{i}b^{i}\equiv d_{i}{\pmod {(b-1)}}}$ , which explains why digits can be meaningfully added. Concretely, for a three-digit number ${\displaystyle n=d_{2}b^{2}+d_{1}b^{1}+d_{0}b^{0}}$ ,

${\displaystyle \operatorname {dr} _{b}(n)\equiv d_{2}b^{2}+d_{1}b^{1}+d_{0}b^{0}\equiv d_{2}(1)+d_{1}(1)+d_{0}(1)\equiv d_{2}+d_{1}+d_{0}{\pmod {(b-1)}}.}$ 

To obtain the modular value with respect to other numbers ${\displaystyle m}$ , one can take weighted sums, where the weight on the ${\displaystyle i}$ -th digit corresponds to the value of ${\displaystyle b^{i}{\bmod {m}}}$ . In base 10, this is simplest for ${\displaystyle m=2,5,{\text{ and }}10}$ , where higher digits except for the unit digit vanish (since 2 and 5 divide powers of 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit.

Also of note is the modulus ${\displaystyle m=b+1}$ . Since ${\displaystyle b\equiv -1{\pmod {(b+1)}},}$  and thus ${\displaystyle b^{2}\equiv (-1)^{2}\equiv 1{\pmod {(b+1)}},}$  taking the alternating sum of digits yields the value modulo ${\displaystyle (b+1)}$ .

It helps to see the digital root of a positive integer as the position it holds with respect to the largest multiple of ${\displaystyle b-1}$  less than the number itself. For example, in base 6 the digital root of 11 is 2, which means that 11 is the second number after ${\displaystyle 6-1=5}$ . Likewise, in base 10 the digital root of 2035 is 1, which means that ${\displaystyle 2035-1=2034|9}$ . If a number produces a digital root of exactly ${\displaystyle b-1}$ , then the number is a multiple of ${\displaystyle b-1}$ .

With this in mind the digital root of a positive integer ${\displaystyle n}$  may be defined by using floor function ${\displaystyle \lfloor x\rfloor }$ , as

${\displaystyle \operatorname {dr} _{b}(n)=n-(b-1)\left\lfloor {\frac {n-1}{b-1}}\right\rfloor .}$ 

• The digital root of ${\displaystyle a_{1}+a_{2}}$  in base ${\displaystyle b}$  is the digital root of the sum of the digital root of ${\displaystyle a_{1}}$  and the digital root of ${\displaystyle a_{2}}$ : $${\displaystyle \operatorname {dr} _{b}(a_{1}+a_{2})=\operatorname {dr} _{b}(\operatorname {dr} _{b}(a_{1})+\operatorname {dr} _{b}(a_{2})).}$$  This property can be used as a sort of checksum, to check that a sum has been performed correctly.

• The digital root of ${\displaystyle a_{1}-a_{2}}$  in base ${\displaystyle b}$  is congruent to the difference of the digital root of ${\displaystyle a_{1}}$  and the digital root of ${\displaystyle a_{2}}$  modulo ${\displaystyle (b-1)}$ : $${\displaystyle \operatorname {dr} _{b}(a_{1}-a_{2})\equiv (\operatorname {dr} _{b}(a_{1})-\operatorname {dr} _{b}(a_{2})){\pmod {(b-1)}}.}$$ 

• The digital root of ${\displaystyle -n}$  in base ${\displaystyle b}$  is $${\displaystyle \operatorname {dr} _{b}(-n)\equiv -\operatorname {dr} _{b}(n){\bmod {b-1}}.}$$ 

• The digital root of the product of nonzero single digit numbers ${\displaystyle a_{1}\cdot a_{2}}$  in base ${\displaystyle b}$  is given by the Vedic Square in base ${\displaystyle b}$ .

• The digital root of ${\displaystyle a_{1}\cdot a_{2}}$  in base ${\displaystyle b}$  is the digital root of the product of the digital root of ${\displaystyle a_{1}}$  and the digital root of ${\displaystyle a_{2}}$ : $${\displaystyle \operatorname {dr} _{b}(a_{1}a_{2})=\operatorname {dr} _{b}(\operatorname {dr} _{b}(a_{1})\cdot \operatorname {dr} _{b}(a_{2})).}$$ 

The additive persistence counts how many times we must sum its digits to arrive at its digital root.

For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9.

There is no limit to the additive persistence of a number in a number base ${\displaystyle b}$ . Proof: For a given number ${\displaystyle n}$ , the persistence of the number consisting of ${\displaystyle n}$  repetitions of the digit 1 is 1 higher than that of ${\displaystyle n}$ . The smallest numbers of additive persistence 0, 1, ... in base 10 are:

0, 10, 19, 199, 19 999 999 999 999 999 999 999, ... (sequence A006050 in the OEIS)

The next number in the sequence (the smallest number of additive persistence 5) is 2 × 10^{2×(1022 − 1)/9} − 1 (that is, 1 followed by 2 222 222 222 222 222 222 222 nines). For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm.^[1]

The example below implements the digit sum described in the definition above to search for digital roots and additive persistences in Python.

def digit_sum(x: int, b: int) -> int:
    total = 0
    while x > 0:
        total = total + (x % b)
        x = x // b
    return total

def digital_root(x: int, b: int) -> int:
    seen = set()
    while x not in seen:
        seen.add(x)
        x = digit_sum(x, b)
    return x

def additive_persistence(x: int, b: int) -> int:
    seen = set()
    while x not in seen:
        seen.add(x)
        x = digit_sum(x, b)
    return len(seen) - 1

Digital roots are used in Western numerology, but certain numbers deemed to have occult significance (such as 11 and 22) are not always completely reduced to a single digit.

Digital roots form an important mechanic in the visual novel adventure game Nine Hours, Nine Persons, Nine Doors.

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• Ghannam, Talal (4 January 2011), The Mystery of Numbers: Revealed Through Their Digital Root, CreateSpace Publications, pp. 68–73, ISBN 978-1-4776-7841-1, archived from the original on 29 March 2016, retrieved 11 February 2016 (online copy, p. 68, at Google Books)
• Hall, F. M. (1980), An Introduction into Abstract Algebra, vol. 1 (2nd ed.), Cambridge, U.K.: CUP Archive, p. 101, ISBN 978-0-521-29861-2 (online copy, p. 101, at Google Books)
• O'Beirne, T. H. (13 March 1961), "Puzzles and Paradoxes", New Scientist, 10 (230), Reed Business Information: 53–54, ISSN 0262-4079 (online copy, p. 53, at Google Books)
• Rouse Ball, W. W.; Coxeter, H. S. M. (6 May 2010), Mathematical Recreations and Essays, Dover Recreational Mathematics (13th ed.), NY: Dover Publications, ISBN 978-0-486-25357-2 (online copy at Google Books)

• Patterns of digital roots using MS Excel
• Weisstein, Eric W. "Digital Root". MathWorld.