Digital root

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The digital root (also repeated digital sum) of a number is the number obtained by adding all the digits, then adding the digits of that number, and then continuing 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, somebody 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.

1 Significance and formula of the digital root
2 Abstract multiplication of digital roots
3 Formal definition
- 3.1 Example
- 3.2 Proof that a constant value exists
4 Ramans' formula
5 Some properties of digital roots
6 See also
7 External links

[edit] 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.

[edit] Abstract multiplication of digital roots

The table shows the digital roots produced by the familiar multiplication table in the decimal system. The first column and first row are just the elements of this table that are being multiplied. You can see that for example, 2x5=1; that's because the digital root of 10 is 1 or $S(10)=S(2\times5)=1$

ones	twos	threes	fours	fives	sixes	sevens	eights	nines
1	2	3	4	5	6	7	8	9
2	4	6	8	1	3	5	7	9
3	6	9	3	6	9	3	6	9
4	8	3	7	2	6	1	5	9
5	1	6	2	7	3	8	4	9
6	3	9	6	3	9	6	3	9
7	5	3	1	8	6	4	2	9
8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9

The table shows a number of interesting patterns and symmetries. For example, we can see that all multiples of nine have a digital root of nine too (which can be considered as zeros too). After nine, the table repeats itself infinite times in blocks of multiples of nine.

If we ignore the ninth column and the ninth row, (which are all nines) we are left with the semigroup $\{\mathbb{Z}/(9), \circ\}$ where $\mathbb{Z}/(9)$ means the set of positive integers partitioned by the residue classes modulo nine. Also, the operator $\circ$ means the abstract multiplication between the elements of this semigroup. If ${a, b}$ are elements of $\{\mathbb{Z}/(9), \circ\}$ then $a \circ b$ can be seen as,

$a\times b \equiv c \pmod{9}$

Where $c=S(a\times b)$ is the digital root of $a\times b$

See: W. E. Deskins. Abstract Algebra. Dover, New York, 1964. pp. 162-167.

[edit] 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 $S σ (n)$ (the digital sum of $n$ ) represent this constant value.

[edit] 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

$S(1853)=8\,$

[edit] 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 $S (d 1) = d 1$ .