Digital Root Calculator (repeated digit sum, additive persistence)

Understanding the digital root calculator — concept and principle

This calculator finds the digital root of a whole number, the single digit left after adding its digits again and again. It shows the chain of digit sums, the additive persistence (how many rounds were needed), n mod 9 and the shortcut formula as a check. Use it for number puzzles, divisibility checks and quick arithmetic verification.

The digital root equals the remainder on division by 9 (with 9 instead of 0).

How it is calculated

Repeat: replace the number by the sum of its digits, until one digit is left.

Additive persistence = number of rounds needed

Shortcut for n > 0: digital root = 1 + (n − 1) mod 9

That is n mod 9, except that multiples of 9 give 9. The digital root of 0 is 0.

The principle

Each digit sum keeps the remainder after division by 9, because every power of 10 leaves remainder 1. Repeating the sum shrinks the number while keeping that remainder, until only one digit from 1 to 9 is left. So the digital root is simply the remainder mod 9, with 9 in place of 0 for positive multiples of 9. This is the basis of the old check called casting out nines: the digital root of a product equals the digital root of the product of the digital roots.

Worked example

Default input: 9,876.

First round: 9 + 8 + 7 + 6 = 30.
Second round: 3 + 0 = 3.
Chain: 9,876 → 30 → 3; additive persistence 2.
Shortcut: 9,876 = 9 × 1,097 + 3, so n mod 9 = 3, and 1 + (9,875 mod 9) = 3.
Digital root: 3.

How to read the result

A digital root of 9 means the number is divisible by 9; 3, 6 or 9 means it is divisible by 3. To check a calculation, compare digital roots: if 123 × 456 = 56,088, then the roots 6 × 6 = 36 → 9 must match the root of 56,088, which is 5 + 6 + 0 + 8 + 8 = 27 → 9. They match, which is consistent with a correct answer.

Common mistakes

Stopping after one round when the result still has two digits.
Giving 0 as the digital root of a positive multiple of 9.
Treating a matching digital root as proof that a calculation is right.
Including signs or decimal points.

Keep in mind: The check misses errors that change the answer by a multiple of 9, such as swapped digits.

Frequently asked questions

What is the digital root of 9,876?

Is there a formula for the digital root?

Yes: for n > 0 it is 1 + (n − 1) mod 9.

What is additive persistence?

The number of digit-sum rounds needed to reach a single digit.

Sources: E. W. Weisstein, "Digital Root", Wolfram MathWorld

Checked 9 Nov 2026. Weisstein, E. W., “Digital Root”, Wolfram MathWorld (digit sums repeated until one digit remains; the number of steps is the additive persistence; for positive n the root is 1 + [(n − 1) mod 9]; the digital root of 0 is 0).

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