The digital root (also repeated digital sum) of a non-negative integer 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, the digital root of 35,79E is 2, because 3 + 5 + 7 + 9 + E = 2E and 2 + E = 11 and 1 + 1 = 2.

Digital roots can be calculated with congruences in modular arithmetic rather than by adding up all the digits, a procedure that can save time in the case of very large numbers.

Digital roots can be used as a sort of checksum, to check that a sum has been performed correctly. If it has, then the digital root of the sum of the given numbers will equal the digital root of the sum of the digital roots of the given numbers. This check, which involves only single-digit calculations, can catch many errors in calculation.

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.

The number of times the digits must be summed to reach the digital root is called a number's additive persistence; in the above example, the additive persistence of 35,79E is 3.

The digital root of a square is 1, 3, 4, 5, 9, or E (thus, we can know that 987,654 is not a square, since its digital root is 6), but the digital root of a cube can be any number.

The digital root of a power of 2 can be any number except E, but the digital root of a power of 3 is 1, 3, 4, 5, or 9.

## Significance and formula of the digital root

It helps to see the digital root of a positive integer as the position it holds with respect to the largest multiple of E less than the number itself. For example, the digital root of 11 is 2, which means that 11 is the second number after E. Likewise, the digital root of 2037 is 1, which means that 2037 − 1 is a multiple of E. If a number produces a digital root of exactly E, then the number is a multiple of E.

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

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

## Abstract multiplication of digital roots

The table below shows the digital roots produced by the familiar multiplication table in the dozenal system.

${\displaystyle \circ}$ 1 2 3 4 5 6 7 8 9 X E
1 1 2 3 4 5 6 7 8 9 X E
2 2 4 6 8 X 1 3 5 7 9 E
3 3 6 9 1 4 7 X 2 5 8 E
4 4 8 1 5 9 2 6 X 3 7 E
5 5 X 4 9 3 8 2 7 1 6 E
6 6 1 7 2 8 3 9 4 X 5 E
7 7 3 X 6 2 9 5 1 8 4 E
8 8 5 2 X 7 4 1 9 6 3 E
9 9 7 5 3 1 X 8 6 4 2 E
X X 9 8 7 6 5 4 3 2 1 E
E E E E E E E E E E E E

The table shows a number of interesting patterns and symmetries and is known as the Vedic square.

## Formal definition

Let

${\displaystyle S(n)}$ denote the sum of the digits of

${\displaystyle n}$ and let the composition of

${\displaystyle S(n)}$ be as follows:

${\displaystyle S^{1}(n)=S(n),\ \ S^{m}(n)=S\left(S^{m-1}(n)\right),\ \text{for}\ m\ge2.}$

Eventually the sequence

${\displaystyle S^{1}(n),S^{2}(n),S^{3}(n),\ldots}$ becomes a one digit number. Let

${\displaystyle S^{*}(n)}$ (the digital root of

${\displaystyle n}$ ) represent this one digit number.

### Example

Let us find the digital root of

${\displaystyle 1853}$ .

${\displaystyle S(1853)=15}$
${\displaystyle S(15)=6}$

Thus,

${\displaystyle S^{2}(1853)=6.}$

For simplicity let us agree simply that

${\displaystyle S^{*}(1853)=\operatorname{dr}(1853)=6.}$

### Proof that a constant value exists

How do we know that the sequence

${\displaystyle S^{1}(n),S^{2}(n),S^{3}(n),\ldots}$ eventually becomes a one digit number? Here's a proof: Let

${\displaystyle n=d_1+10d_2+\cdots+10^{m-1}d_m}$ , for all

${\displaystyle i}$ ,

${\displaystyle d_i}$ is an integer greater than or equal to 0 and less than 10. Then,

${\displaystyle S(n)=d_1+d_2+\cdots+d_m}$ . This means that

${\displaystyle S(n) , unless

${\displaystyle d_2,d_3,\ldots,d_m=0}$ , in which case

${\displaystyle n}$ is a one digit number. Thus, repeatedly using the

${\displaystyle S(n)}$ function would cause

${\displaystyle n}$ to decrease by at least 1, until it becomes a one digit number, at which point it will stay constant, as

${\displaystyle S(d_1)=d_1}$ .

## Congruence formula

The formula is:

${\displaystyle \operatorname{dr}(n) = \begin{cases}0 & \mbox{if}\ n = 0, \\ \mathcal{E} & \mbox{if}\ n \neq 0,\ n\ \equiv 0\pmod{\mathcal{E}},\\ n\ {\rm mod}\ \mathcal{E} & \mbox{if}\ n \not\equiv 0\pmod{\mathcal{E}}\end{cases}}$

or,

${\displaystyle \operatorname{dr}(n) = 1\ +\ ((n-1)\ {\rm mod}\ \mathcal{E}).}$

The digital root is the value modulo E because

${\displaystyle 10 \equiv 1\pmod{\mathcal{E}},}$ and thus

${\displaystyle 10^k \equiv 1^k \equiv 1\pmod{\mathcal{E}},}$ so regardless of position, the value mod E is the same –

${\displaystyle a\cdot 100 \equiv a\cdot 10 \equiv a\pmod{\mathcal{E}}}$ – which is why digits can be meaningfully added. Concretely, for a three-digit number,

${\displaystyle \operatorname{dr}(abc) \equiv a\cdot 10^2 + b\cdot 10 + c \cdot 1 \equiv a\cdot 1 + b\cdot 1 + c \cdot 1 \equiv a + b + c \pmod{\mathcal{E}}}$.

To obtain the modular value with respect to other numbers n, one can take weighted sums, where the weight on the kth digit corresponds to the value of

${\displaystyle 10^k}$ modulo n, or analogously for

${\displaystyle b^k}$ for different bases. This is simplest for 2, 3, 4, 6, and 10, where higher digits vanish (since 2, 3, 4, and 6 divide 10), which corresponds to the familiar fact that the divisibility of a dozenal number with respect to 2, 3, 4, 6, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, 8, or X). Also of note is the modulus 11: since

${\displaystyle 10 \equiv -1\pmod{11},}$ and thus

${\displaystyle 10^2 \equiv (-1)^2 \equiv 1\pmod{11},}$

taking the alternating sum of digits yields the value modulo 11.

## Some properties of digital roots

The digital root of a number is zero if and only if the number is itself zero.

${\displaystyle \operatorname{dr}(n)=0 \Leftrightarrow n=0.}$

The digital root of a number is a positive integer if and only if the number is itself a positive integer.

${\displaystyle \operatorname{dr}(n)>0 \Leftrightarrow n>0.}$

The digital root of

${\displaystyle n}$ is

${\displaystyle n}$ itself if and only if the number has exactly one digit.

${\displaystyle \operatorname{dr}(n)=n \Leftrightarrow n \in \{0,1,2,3,4,5,6,7,8,9,\mathcal{X},\mathcal{E}\}.}$

The digital root of

${\displaystyle n}$ is less than

${\displaystyle n}$ if and only if the number is greater than or equal to 10.

${\displaystyle \operatorname{dr}(n)

The digital root of

${\displaystyle a}$ +

${\displaystyle b}$ is digital root of the sum of the digital root of

${\displaystyle a}$ and the digital root of

${\displaystyle b}$ .

${\displaystyle \operatorname{dr}(a+b) = \operatorname{dr}(\operatorname{dr}(a)+\operatorname{dr}(b) ).}$

The digital root of

${\displaystyle a}$ -

${\displaystyle b}$ is congruent with the difference of the digital root of

${\displaystyle a}$ and the digital root of

${\displaystyle b}$ modulo E.

${\displaystyle \operatorname{dr}(a-b) \equiv \operatorname{dr}(a)-\operatorname{dr}(b) \pmod{\mathcal{E}}.}$

Especially, we can define the digital root of minus

${\displaystyle n}$ as follows:

${\displaystyle \operatorname{dr}(-n) \equiv -\operatorname{dr}(n) \pmod{\mathcal{E}}.}$

The digital root of

${\displaystyle a}$ ×

${\displaystyle b}$ is digital root of the product of the digital root of

${\displaystyle a}$ and the digital root of

${\displaystyle b}$ .

${\displaystyle \operatorname{dr}(ab) = \operatorname{dr}(\operatorname{dr}(a)\cdot\operatorname{dr}(b) ).}$
• The digital root of a nonzero number is E if and only if the number is itself a multiple of E.
${\displaystyle \operatorname{dr}(n)=\mathcal{E} \Leftrightarrow n=\mathcal{E}m \quad \text{for } m=1,2,3,\ldots.}$
${\displaystyle \operatorname{dr}(n!)=\mathcal{E} \Leftrightarrow n \ge \mathcal{E}.}$
• The digital root of a square is 1, 3, 4, 5, 9 or E. Digital roots of square numbers progress in the sequence 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, E.
• The digital root of a cube can be any number. Digital roots of perfect cubes progress in the sequence 1, 8, 5, 9, 4, 7, 2, 6, 3, X, E.
• The digital root of a prime number (except E) can be any number except E.
• The digital root of a power of 2 can be any number except E. Digital roots of the powers of 2 progress in the sequence 1, 2, 4, 8, 5, X, 9, 7, 3, 6. This even applies to negative powers of 2; for example, 2 to the power of 0 is 1; 2 to the power of -1 (minus one) is .6, with a digital root of 6; 2 to the power of -2 is .3, with a digital root of 3; 2 to the power of -3 is .16, with a digital root of 7; and so on, ad infinitum in both directions. This is because negative powers of 2 share the same digits (after removing leading zeroes) as corresponding positive powers of 6, whose digital roots progress in the sequence 1, 6, 3, 7, 9, X, 5, 8, 4, 2.
• The digital root of a power of 3 is 1, 3, 4, 5, or 9. Digital roots of the powers of 3 progress in the sequence 1, 3, 9, 5, 4. This even applies to negative powers of 3; for example, 3 to the power of 0 is 1; 3 to the power of -1 (minus one) is .4, with a digital root of 4; 3 to the power of -2 is .14, with a digital root of 5; and so on, ad infinitum in both directions. This is because the negative powers of 3 share the same digits (after removing leading zeroes) as corresponding positive powers of 4, whose digital roots progress in sequence 1, 4, 5, 9, 3.
• The digital roots of powered numbers progress in sequence (only certain for positive powers, although in for some exceptions it also may occur for negative powers), and this is because of one of the previously shown properties. As the digital root of a b is congruent with the multiple of the digital root of a and the digital root of b modulo E, the digital root of a a will also do it. So, for example, as shown above, powers of 2 will follows the sequence 1, 2, 4, 8, 5, X, 9, 7, 3, 6; Powers of 49 (whose digital root is 2) will also follow this sequence. The very sequence follows this rule, and is appliable to any other number.
${\displaystyle \operatorname{dr}(a^n) \equiv \operatorname{dr}(a)^n \pmod{\mathcal{E}}.}$
• The digital root of an even perfect number is 1, 4, 6 or X.
• The digital root of a centered dozagonal number (centered 10-gonal number), a centered hexagram number (centered 6-gram number), or a star number is 1, 2, 4, 5, 7 or E, their digital roots progressing in the sequence 1, 2, 4, 7, E, 5, E, 7, 4, 2, 1.
• The digital root of a centered hexagonal number (centered 6-gonal number, or hex number) is 1, 3, 4, 6, 7 or 8, their digital roots progressing in the sequence 1, 7, 8, 4, 6, 3, 6, 4, 8, 7, 1.
• The digital root of a triangular number is 1, 3, 4, 6, X or E. Digital roots of triangular numbers progress in the sequence 1, 3, 6, X, 4, X, 6, 3, 1, E, E.
• The digital root of a Fibonacci number is 1, 2, 3, 5, 8, X or E. Digital roots of Fibonacci numbers progress in the sequence 1, 1, 2, 3, 5, 8, 2, X, 1, E.
• The digital root of a Lucas number is 1, 2, 3, 4, 7, X or E. Digital roots of Lucas numbers progress in the sequence 1, 3, 4, 7, E, 7, 7, 3, X, 2.
• The digital root of the product of twin primes, other than E and 11, is 2, 3, 4, 8 or X. The digital root of the product of E and 11 (twin primes) is E.