The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, in base 10, the digital root of the number 12345 is 6 because the sum of the digits in the number is 1 + 2 + 3 + 4 + 5 = 15, then the addition process is repeated again for the resulting number 15, so that the sum of 1 + 5 equals 6, which is the digital root of that number. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0), which allows it to be used as a divisibility rule.

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