sumofdigits

The sum of digits, often denoted s(n), is a basic arithmetic operation that maps a nonnegative integer to the sum of its decimal digits. For example, s(1234) = 1 + 2 + 3 + 4 = 10. If a number has k digits, the maximum possible value of s(n) is 9k.

Definition and variants: s(n) is defined for nonnegative integers and extends to other bases by summing the digits in that base. For n = 0, s(0) = 0. A related concept is the digital root, obtained by repeatedly applying the digit-sum operation until a single digit remains. The digital root dr(n) satisfies dr(n) ≡ n (mod 9) and, for n > 0, lies in {1, 2, ..., 9}. More precisely, dr(n) = 1 + ((n − 1) mod 9).

Properties: The digit-sum function is invariant under modulo 9, meaning two numbers with the same remainder when divided by 9 have the same digital root. It is not injective; many numbers share the same sum of digits. Additive persistence is the number of times s must be applied to reach a single digit.

Computation: In programming, s(n) is computed by repeatedly taking the last digit (n % 10) and discarding it (n //= 10). For numbers given as strings, one can sum the characters that represent digits. In other bases, the same principle applies with the corresponding base and digit set.

Applications: The digit-sum is used in quick arithmetic checks, such as the casting out nines technique, which uses s(n) modulo 9 to verify computations. The concept also appears in numerical puzzles and is a stepping stone to the study of digital roots and related invariants.