Digital Root Calculator
The digital root of a positive integer is the single digit you get by repeatedly adding the number's digits together until only one digit remains. Enter any positive integer and this calculator shows you every iteration from the original number down to the final single digit. For example, start with 9,875: add 9+8+7+5 to get 29, then 2+9 to get 11, then 1+1 to get 2. The digital root is 2, reached in three iterations. There is also a direct formula: for any n greater than 0, the digital root equals 1 + (n minus 1) mod 9. This means every number with the same digital root is in the same residue class modulo 9. Digital roots are used in arithmetic checking (casting out nines), divisibility rules, and recreational mathematics. The digital root of a sum equals the digital root of the sum of the individual digital roots, making it a quick sanity check on large calculations.
Digital root formula
Iterative method: sum all digits of n. If result has more than one digit, repeat.
Direct formula: dr(n) = 1 + (n - 1) mod 9, for n > 0.
Special case: dr(0) = 0.
The iterative and formula methods always agree. The formula uses modular arithmetic: since summing digits is equivalent to reducing modulo 9, the digital root is simply the residue of n modulo 9, with residue 0 replaced by 9 (except for n = 0).
Digital roots 1 to 18
| n | Digit sum | Digital root | n | Digit sum | Digital root |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 10 | 1 | 1 |
| 2 | 2 | 2 | 11 | 2 | 2 |
| 3 | 3 | 3 | 12 | 3 | 3 |
| 4 | 4 | 4 | 13 | 4 | 4 |
| 5 | 5 | 5 | 14 | 5 | 5 |
| 6 | 6 | 6 | 15 | 6 | 6 |
| 7 | 7 | 7 | 16 | 7 | 7 |
| 8 | 8 | 8 | 17 | 8 | 8 |
| 9 | 9 | 9 | 18 | 9 | 9 |
Digital root: frequently asked questions
What is a digital root?
A digital root is the single digit obtained by repeatedly summing the digits of a positive integer until only one digit remains. For example, the digital root of 493 is 4+9+3 = 16, then 1+6 = 7. So the digital root is 7.
Is there a shortcut formula for digital roots?
Yes. For any positive integer n, the digital root equals 1 + (n - 1) mod 9, except that the digital root of 0 is defined as 0. This is because the digital root is the remainder when n is divided by 9, with remainder 0 replaced by 9 (for multiples of 9 other than 0).
What is the digital root of a multiple of 9?
The digital root of any positive multiple of 9 is always 9. For example, 18 gives 1+8=9, and 9999 gives 9+9+9+9=36, then 3+6=9. The only exception is 0 itself, whose digital root is 0.
What are digital roots used for?
Digital roots appear in divisibility tests, modular arithmetic, and recreational mathematics. They can quickly check arithmetic: the digital root of a product equals the digital root of the product of the digital roots. They also arise in casting out nines, a method for checking multiplication.
How many iterations does it take to reach the digital root?
For most numbers, only one or two iterations are needed. The maximum number of iterations grows very slowly with the size of the number. A 10-digit number with all 9s (9,999,999,999) requires at most 3 iterations to reach a single digit.
Sources
- Modular arithmetic and digital roots: Wolfram MathWorld, Digital Root.
- Number theory reference: NIST Special Publication 330.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.