What is a digital root?
The digital root of a number is the single digit you reach by adding its digits, then adding the digits of that result, and so on until only one digit remains. For 12345:
- 1 + 2 + 3 + 4 + 5 = 15
- 1 + 5 = 6
So the digital root of 12345 is 6. Every positive integer has a digital root between 1 and 9, and zero's digital root is zero.
If you only need to add the digits once, use our Digit Sum Calculator.
The divide-by-nine shortcut
You never actually have to iterate. For a positive integer, the digital root is the remainder when the number is divided by 9, except that a remainder of 0 corresponds to a digital root of 9.
Digital root = 1 + ((n − 1) mod 9), for n > 0 12345 divided by 9 leaves a remainder of 6, which is the digital root. This works because 10 leaves a remainder of 1 when divided by 9, so every place value in a number contributes just its own digit to the remainder — which is the same thing as adding up the digits. The calculator shows both the iterated rounds and the shortcut, so you can check one against the other.
Why multiples of nine give 9, not 0
The digital root of 18 is 9, not 0, even though 18 divides by 9 exactly. Adding the digits gives 1 + 8 = 9 and the process stops there, since 9 is already a single digit. A digital root of 0 belongs to the number 0 alone. This is the result people most often expect to come out wrong, so it's worth checking against the iterated rounds the calculator shows.
What digital roots are used for
- Casting out nines: a classic method for checking hand arithmetic. If the digital roots of the operands do not agree with the digital root of the answer, an error has been detected. Matching digital roots do not guarantee that the answer is correct.
- Divisibility by 9: a positive number is divisible by 9 when its digital root is 9, and divisible by 3 when its digital root is 3, 6, or 9.
- Number theory and puzzles: digital roots appear in competition maths, recreational number problems, and divisibility exercises.
Frequently asked questions
What's the difference between a digital root and a digit sum?
A digit sum adds the digits once and stops. A digital root repeats that addition until a single digit is left. For 12345 the digit sum is 15 and the digital root is 6. When a number's digit sum already has one digit, the two answers are identical.
Can a digital root be 0?
Only for the number 0 itself. Every positive integer has a digital root from 1 to 9. Multiples of nine give 9, not 0.
How many rounds of adding does it take?
Very few, however large the number. Even a 1000-digit number has a digit sum of at most 9000, which reduces to a single digit in two more rounds. The calculator reports the number of rounds it took.