Talk:Digit sum

Generalise

Would be nice if someone could add a more generalised rule to the mod 9 for base 10. I'm sure it would just be mod (base-1) but the answer could be more tricky to interpret. For example, in binary ds(101) = 10? Syndicate 10:45, 23 October 2005 (UTC)

• May I add some results for base 10?

Let σ(M) be the digit sum of positive integer M. We all know from casting nines that σ(9k+r)=r.

Suppose we want to find σ(314159265358979^{2718281828459045}). Let us state some basic operations and some periodicities:

σ(A+B)=σ(A)+σ(B); σ(A×B)=σ(A)×σ(B); σ(Aa)=[σ(A)]a,
σ(26k+j)=1,2,4,8,7,5 for j=0,1,2,3,4,5, respectively.
σ(3k)=1,3,9 for k=0,1 and k not smaller than 2, respectively.
σ(43k+j)=1,4,7 for j=0,1,2, respectively.
σ(56k+j)=1,5,7,8,4,2 for j=0,1,2,3,4,5, respectively.
σ(6k)=1,6,9 for k=0,1 and k not smaller than 2, respectively.
σ(73k+j)=1,7,4 for j=0,1,2, respectively.
σ(82k+j)=1,8 for j=0,1, respectively.

For the example, 314159265358979=9m+5, and 2718281828459045=6k+5, so

σ(3141592653589792718281828459045)=σ(55)=2.

69.158.119.184 18:41, 16 September 2006 (UTC)

Adding sheree silver

I added info by a doctor in metaphysical philosophy upon advice from another user on the numerology page.Spring12 (talk) 22:37, 23 February 2009 (UTC)

Inequalities

It seems to me if S(n) is a digit sum of n (in any base), then for any a,b:

S(a+b) <= S(a) + S(b)

S(a*b) <= S(a) * S(b)

I cannot find any reference for this on the internet. 94.113.5.84 (talk) 18:05, 7 August 2009 (UTC)

combine with digital root?

I was surprised that they're different pages — Preceding unsigned comment added by 69.76.179.81 (talk) 04:55, 13 November 2011 (UTC)

The digit sum and digital root aren't the same, as you can read in the Applications section of the article:

> The concept of a decimal digit sum is closely related to, but not the same as, the digital root, which is the result of repeatedly applying the digit sum operation until the remaining value is only a single digit. Polish bookworm (talk) 12:33, 25 September 2022 (UTC)

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