Digital Root — Definition, Formula & Examples

The digital root of a number is the single digit you get by repeatedly adding up all its digits until only one digit remains.

The digital root of a positive integer $n$ is the value obtained by iterating the digit-sum function until a single-digit result is produced. Equivalently, for $n \geq 1$ , the digital root equals $1 + ((n - 1) \mod 9)$ , which always yields a value from 1 to 9.

Key Formula

\text{dr}(n) = 1 + ((n - 1) \bmod 9)

Where:

$n$ = A positive integer
$\text{dr}(n)$ = The digital root of n, always a value from 1 to 9

How It Works

Start with any positive integer. Add all of its digits together. If the result has more than one digit, add those digits together again. Repeat until you reach a single digit — that single digit is the digital root. This process always terminates because each step produces a smaller number. The digital root is closely tied to modular arithmetic: it tells you the remainder when the original number is divided by 9, except that a remainder of 0 corresponds to a digital root of 9.

Worked Example

Problem: Find the digital root of 9473.

Step 1: Add the digits of 9473.

9 + 4 + 7 + 3 = 23

Step 2: The result 23 has more than one digit, so add its digits.

2 + 3 = 5

Step 3: Verify with the formula: compute (9473 − 1) mod 9, then add 1.

1 + ((9473 - 1) \bmod 9) = 1 + (9472 \bmod 9) = 1 + 4 = 5

Answer: The digital root of 9473 is 5.

Why It Matters

Digital roots provide a quick way to check arithmetic — if the digital root of your answer doesn't match what you expect, you've likely made an error. This technique, sometimes called "casting out nines," has been used for centuries to verify multiplication and addition. It also serves as a natural first encounter with modular arithmetic before studying it more formally.

Common Mistakes

Mistake: Assuming the digital root of a multiple of 9 is 0.

Correction: The digital root is never 0 for a positive integer. Multiples of 9 (like 27 or 81) have a digital root of 9, not 0.