Talk:Digital root

Proposed mergers
"Digital root" is the term given by Mathworld. See also. I for one think that repeated digital sum and digital sum should be merged into this article. They can be listed as synonyms if anyone can cite professional mathematics papers using them. Anton Mravcek 22:38, 9 February 2006 (UTC)

0 or 9
Does anyone have a strong opinion on whether 0 or 9 makes more sense as a digital root? 0 is better for calculations, and more elegant, but you can't reduce any further than 9 with the standard "add up the digits" process. Is there a convention? Should this be mentioned in the article? ~ CZeke 23:15, 21 July 2006 (UTC)

This article is inconsistent for n=0 (and doesn't even mention negative numbers). There are three formulas and definitions: Assuming that dr(0)=0 is the expected/desired definition, two improvements seem necessary: remove or fix the floor-based formula; and make explicit that a non-standard (not floor-based) definition of mod is used. Marnix.klooster (talk) 11:55, 22 December 2017 (UTC)
 * The first (using the floor function) results in dr(0)=9.
 * The third (using mod) is ambiguous, and depends on the definition of mod and how it works for a negative left argument. Using the mathematical / Python definition, this formula is equivalent to the first, and results in dr(0)=9. But apparently C's definition of mod is being used, making dr(0)=0.
 * The second just says "Let {\displaystyle S(n)} S(n) denote the sum of the digits of {\displaystyle n} n", which presumably means S(0)=0, which implies dr(0)=S*(0)=0.

Some information
I have some information about digital roots, but I'm not sure exactly how to put it in an encyclopedic format. I hope I won't be too unclear: The digital roots of squares follow the cycle 1,4,9,7,7,9,4,1,9, and then repeat. So you can take a number, n, and if it is a perfect square, its digital root is n mod 9 of the series if you replace the zero with a nine. Also, if you find the digital root of a number taken to the power of a number like four, you can generate a similar series for four by taking the number of the first series and then taking the number you get of the second series to get third series. i.e. : for the series of six, where 1,4,9,7,7,9,4,1,9... is the series of two, the fourth number would be seventh number of the three series, 1,8,9,1,8,9,1,8,9, because seven is the fourth number of the two series. By this find out that the fourth series is 1,7,9,4,4,9,7,1,9, the six series is 1,1,9,1,1,9,1,1,9, and the even exponents repeat in that pattern after that. The series of multiples of threes, however starts with the three series repeating 1,8,9,1,8,9,1,8,9..., the six series being the one defined above, and repeating form there. The next two prime numbers, 5 and 7, have series 1,5,9,7,2,9,4,8,9..., and 1,2,9,4,5,9,7,8,9..., respectively. Then the prime numbers alternate after this; i.e. the 11 series is the same as the five series, the thirteen series is the same as the seven series, the nineteen series is the same as the five series, etc. Both the seven and five series' multiples repeat after six rounds, but if this made any sense to you, then you can calculate how they do using the method I described above. I hope that made sense to some people. Ask me about what I meant for the more unclear parts, Nat2 (talk) 22:44, 15 December 2008 (UTC) Also, the series for the first power is 1,2,3,4,5,6,7,8,9... . Kind of obvious, but worth pointing out. Nat2 (talk) 22:50, 15 December 2008 (UTC)
 * I'm proposing (see below) a table describing these patterns. However, they actually repeat after just six patterns, and are independent of whether or not prime numbers are involved. So actually, the 17 series is the same as that for 5 and 11, and the 19 series is the same as for 7 and 13. Although 25 is not a prime number, it too follows 7, 13, and 19. The other series follow the same pattern, whether prime or not. — Glenn L (talk) 06:47, 4 November 2012 (UTC)

Patterns
I teach math to elementary school children and this is one of the things we have them do to practice their multiplication. To make it interesting for them, we have them draw patterns dictated by the repeating pattern of the digital sum for continuing multiples. The work is done on grid paper and the instructions are to move up, right, down, left, counting off each value along the grid paper, eventually ending up back where the student started. Each number 1 - 9 has a distinct pattern, and all other numbers are duplicates based on their own digital root (ex. 34's pattern is the same as 7's). If anyone is familiar with this, I would love help in adding this to the article, I can provide images of the patterns which I have generated myself in Mathematica. Wutchamacallit27 (talk) 22:34, 26 July 2010 (UTC)

Some properties of digital roots
I propose splitting this section on the main page, with one subsection for powers, which form just six patterns, from squares to seventh powers, as partially described above:


 * Digital roots of square numbers progress in the sequence 1, 4, 9, 7, 7, 9, 4, 1, 9. This pattern also applies to 8th, 14th, 20th, etc. powers.
 * Digital roots of perfect cubes progress in the sequence 1, 8, 9. This pattern also applies to 9th, 15th, 21st, etc. powers.
 * Digital roots of fourth powers progress in the sequence 1, 7, 9, 4, 4, 9, 7, 1, 9. This pattern also applies to 10th, 16th, 22nd, etc. powers.
 * Digital roots of fifth powers progress in the sequence 1, 5, 9, 7, 2, 9, 4, 8, 9. This pattern also applies to 11th, 17th, 23rd, etc. powers.
 * Digital roots of sixth powers progress in the sequence 1, 1, 9. This pattern also applies to 12th, 18th, 24th, etc. powers.
 * Digital roots of seventh powers progress in the sequence 1, 2, 9, 4, 5, 9, 7, 8, 9. This pattern also applies to 13th, 19th, 25th, etc. powers.

It might be instructive to make a table showing these patterns. — Glenn L (talk) 06:38, 4 November 2012 (UTC)

May I note that 3^32, or 1853020188851840, has a digital root of 8, and is yet a multiple of 3. — Preceding unsigned comment added by 129.234.252.66 (talk) 09:26, 28 January 2013 (UTC)
 * You miscalculated. It should have been obvious to you that you miscalculated, because 332 should not have come out as an even number. The correct value is 1853020188851841. —David Eppstein (talk) 15:24, 28 January 2013 (UTC)

Further to the information above on the properties of digital roots with various powers, I'm also wondering how to show the property that the Digital Root of n^x (for simplicity, call this function DR(n^x) ) is equal to DR(DR(n)^x)) Such that, if x=2 you take a number and square it, the digital root of the number is equal to the square of the digital root of the square. Still difficult to explain, a couple of examples should make it more obvious:

For 22: DR(22) = 2 + 2 = 4

Of the square:

22^2 = 484 DR(484): 4 + 8 + 4 = 16 1 + 6 = 7

And: DR(22)^2 = 16 DR(16) = 7

So DR(22^2) is the same as the digital root of DR(22)^2 The fact that we get the nice Digital Sum in this case (4 + 8 + 4 = 16 = 4^2) is not always held, as in the following example:

For 35: DR(35) = 8

Of the square: DR(35^2) = DR(1225): 1 + 2 + 2 + 5 = 10 1 + 0 = 1

And:

DR(35)^2 = 8^2 = 64 6 + 4 = 10 1 + 0 = 1

This holds for all positive integers of n and x Potkettle (talk) 15:25, 5 August 2013 (UTC)

So far, no-one has mentioned the noteworthy fact that many sequences of digital roots consist of palindromic groups that repeat endlessly. This seems to apply mostly with the plane figurate numbers. Does anyone else agree that this is worth pointing out in the main article? Surely, this cannot fall foul of the Original Research pitfall. --DStanB (talk) 15:06, 25 January 2015 (UTC)
 * ...although it probably would be (OR, that is) if I suggested they should be known as Characteristic Signatures of their respective series. --DStanB (talk) 15:13, 25 January 2015 (UTC)