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.
is a digital root if it is a fixed point for
, then trivially Therefore, the only possible digital roots are the natural numbers
, and there are no cycles other than the fixed points of
This process shows that 3110 is 1972 in base 12.
in the following ways: The formula in base
The digital root is the value modulo
, which explains why digits can be meaningfully added.
Concretely, for a three-digit number
, To obtain the modular value with respect to other numbers
, 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.
taking the alternating sum of digits yields the value modulo
It helps to see the digital root of a positive integer as the position it holds with respect to the largest multiple of
For example, in base 6 the digital root of 11 is 2, which means that 11 is the second number after
Likewise, in base 10 the digital root of 2035 is 1, which means that
If a number produces a digital root of exactly
With this in mind the digital root of a positive integer
may be defined by using floor function
, as 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.
, the persistence of the number consisting of
repetitions of the digit 1 is 1 higher than that of
The smallest numbers of additive persistence 0, 1, ... in base 10 are: The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×(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.
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.