Digital root

The digital root (also repeated digital sum) of a number 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 $65,536$ is $7$, because $6+5+5+3+6 = 25$ and $2+5 = 7.$

Digital roots can be calculated with congruences 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. For example, since the digital root of a sum is always equal to the digital root of the sum of the summands' digital roots. A person adding long columns of large numbers will often find it reassuring to apply casting out nines to his or her result—knowing that this technique will catch the majority of errors.

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 sum is called a number's additive persistence; in the above example, the additive persistence of 65,536 is 2.

Significance and formula of the digital root

It helps to see the digital root of any positive integer $n$ as the position $n$ holds with respect to the last multiple of nine to the left of $n$. For example, the digital root of 11 is 2, which means that 11 is the second number after 9. The digital root of 23 is 5, this means that 23 is the fifth number after a multiple of nine to the left of 23; in this case, 18. The digital root of 2035 is 1 which means that 2035-1, that is 2034, is a multiple of nine.

The digital roots of {1,2,3,4,5,6,7,8} which are the same digits themselves, reveal their position with respect to 0. The digital roots of nine and all of its multiples are nine, however, they all play the same role that zero plays for the integers from 1 to 8. It helps to think of the number nine and all its multiples as a kind of zero or zeros, so that the other integers be able to reveal their position or digital roots with respect to them. This is in part the nature of the decimal system.

With this in mind we may think of the digital root of the positive integer $n$ as $S(n)$, defined by:

$S(n)=n-\max_{0< x\le 9x}(9x<n)$

which precisely says that,

$S(n)=n-9\left\lfloor\frac{n}{9}\right\rfloor.$

This formula will give the digital root of $n$ and will assign the value 0 to all $n$ which are multiples of nine.

Abstract multiplication of digital roots

The table below shows the digital roots produced by the familiar multiplication table in the decimal system. You can see that for example, 2x5=1; that's because the digital root of 10 is 1 or

$S(10)=S(2\times5)=1.\$

	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	1	3	5	7	9
3	3	6	9	3	6	9	3	6	9
4	4	8	3	7	2	6	1	5	9
5	5	1	6	2	7	3	8	4	9
6	6	3	9	6	3	9	6	3	9
7	7	5	3	1	8	6	4	2	9
8	8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9	9

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

Formal definition

Let $S(n)$ denote the sum of the digits of $n$. Eventually the sequence $S(n),S(S(n)),S(S(S(n))),\dotsb$ becomes constant. Let $S_{\sigma}(n)$ (the digital sum of $n$) represent this constant value.

Example

Let us find the digital sum of $1853$.

$S(1853)=17\,$

$S(17)=8\,$

Thus,

$S_{\sigma}(1853)=8.\,$

For simplicity let us agree simply that

$S(1853)=8.\,$

Proof that a constant value exists

How do we know that the sequence $S(n),S(S(n)),S(S(S(n))),\dotsb$ eventually becomes constant? Here's a proof:

Let $x=d_1+10d_2+\dotsb+10^{n-1}d_n$, with $0\le d_i\in\mathbb{Z}<10$ (For all $i$, $d_i$ is an integer greater than or equal to $0$ and less than $10$). Then, $S(x)=d_1+d_2+\dotsb+d_n$. This means that $S(x)<x$, unless $d_2,d_3,\dotsb,d_n=0$, in which case $x$ is a one-digit number. Thus, repeatedly using the $S(x)$ function would cause $x$ to decrease by at least 1, until it becomes a one-digit number, at which point it will stay constant, as $S(d_1)=d_1$.

Congruence formula

The formula is:

$\mbox{dr}(n) = \begin{cases} n\ ({\rm mod}\ 9)\ n\ \ne 0\ ({\rm mod}\ 9) \\ 9\ \ \ \ \ \ \ \ \ \ \ \ \ n\ \equiv 0\ ({\rm mod}\ 9) \end{cases}$

or,

$\mbox{dr}(n) = 1\ +\ [n-1 ({\rm mod}\ 9)].\$

To generalize the concept of digital roots to other bases b, one can simply change the 9 in the formula to b - 1.

The digital root is the value modulo 9 because $10 \equiv 1 \mod 9,$ and thus $10^k \equiv 1^k \equiv 1 \mod 9,$ so regardless of position, the value mod 9 is the same – $a00 \equiv a0 \equiv a \mod 9$ – which is why digits can be meaningfully added. Concretely, for a three-digit number,

$\mbox{dr}(abc) = a\cdot 10^2 + b\cdot 10 + c \cdot 1 \mod 9 = a\cdot 1 + b\cdot 1 + c \cdot 1 \mod 9 = a + b + c \mod 9$

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 $10^k$ modulo n, or analogously for $b^k$ for different bases. This is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 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 (even numbers end in 0, 2, 4, 6, or 8).

Also of note is $n=11:$ since $10 \equiv -1 \mod 11,$ and thus $10^2 \equiv (-1)^2 \equiv 1 \mod 11,$ taking the alternating sum of digits yields the value modulo 11.

Some properties of digital roots

• Digital root of a square is 1, 4, 7, or 9.
• Digital root of a perfect cube is 1, 8 or 9.
• Digital root of a prime number (except 3) is 1, 2, 4, 5, 7, or 8.
• Digital root of a power of 2 is 1, 2, 4, 5, 7, or 8. Digital roots of the powers of 2 progress in the sequence 1, 2, 4, 8, 7, 5, 1. This even applies to powers less than 1; for example, 2 to the power of 0 is 1; 2 to the power of -1 (minus one) is .5, with a digital root of 5; 2 to the power of -2 is .25, with a digital root of 7; and so on, ad infinitum in both directions.
• Digital root of an even perfect number (except 6) is 1.
• Digital root of a star number is 1 or 4.
• Digital root of a nonzero multiple of 9 is 9.
• Digital root of a nonzero multiple of 3 is 3, 6 or 9.
• Digital root of a triangular number is 1, 3, 6 or 9.
• Digital root of a factorial ≥ 6! is 9.
• Digital root of Fibonacci Series is a repeating pattern of 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9.
• Digital root of the product of twin primes, other than 3 and 5, is 8. The digital root of the product of 3 and 5 (twin primes) is 6.
• Digital root of a non-zero number is 9 if and only if the number is itself a multiple of 9

See also

• digit sum
• Vedic square

References

• F. M. Hall: An Introduction into Abstract Algebra. 2nd edition, CUP ARchive 1980, ISBN 9780521298612, p. 101 (online copy at Google Books)
• Bonnie Averbach, Orin Chein: Problem Solving Through Recreational Mathematics. Courier Dover Publications 2000, ISBN 0486409171, pp. 125-127 (online copy at Google Books)
• T. H. O'Beirne: Puzzles and Paradoxes. In: New Scientist, No. 230, 1961-4-13, pp. 53-54 (online copy at Google Books)¨
• The video game 999: Nine Hours, Nine Persons, Nine Doors uses the digital roots as part of the plot.

External links

• pattern of digital root using MS Excel
• Weisstein, Eric W., "Digital Root" from MathWorld.