Digital Root Calculator
Find the digital root of an integer with repeated digit-sum work, additive persistence, digit frequency, and modulo 9 verification.
Last Updated: May 2026
Enter an integer. Commas, spaces, plus signs, and minus signs are accepted.
Digital root
3
Additive persistence
2
Digit count
4
Modulo 9 remainder
3
| Item | Value | Meaning |
|---|---|---|
| Input used | 9,876 | Nonnegative integer input. |
| Absolute value | 9,876 | Digits are summed from this value. |
| First digit sum | 30 | Sum of all digits in the original absolute value. |
| Digit-sum chain | 9876 -> 30 -> 3 | Repeated sums until one digit remains. |
| Modulo 9 check | 9,876 mod 9 = 3 | A positive multiple of 9 has digital root 9. |
Add the digits, then add the digits of the result until one digit remains.
For positive integers, the digital root matches the modulo 9 remainder, using 9 instead of 0 for multiples of 9.
Additive persistence counts how many rounds of digit sums were needed.
Digit-Sum Steps
| Step | Work | Status |
|---|---|---|
| Step 1 | 9 + 8 + 7 + 6 = 30 | Repeat with the new digit sum. |
| Step 2 | 3 + 0 = 3 | Reached one digit. |
Digit Frequency
| Digit | Count | Status |
|---|---|---|
| 0 | 0 | Not present |
| 1 | 0 | Not present |
| 2 | 0 | Not present |
| 3 | 0 | Not present |
| 4 | 0 | Not present |
| 5 | 0 | Not present |
| 6 | 1 | Included in digit sums |
| 7 | 1 | Included in digit sums |
| 8 | 1 | Included in digit sums |
| 9 | 1 | Included in digit sums |
Integer Input Notice
This calculator is for base-10 integer digital roots. Decimal, fraction, and non-decimal base inputs should be converted before using this tool.
Checked by Jitendra Kumar
Digital Root Calculator is checked for formula labels, source links, and result limits.
Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.
How to Use the Digital Root Calculator
Enter an integer. Commas, spaces, plus signs, and minus signs are accepted. The calculator normalizes the input and uses the absolute value for the digit-sum process.
Read the digital root first, then review the additive persistence, repeated digit-sum chain, modulo 9 check, and digit frequency table.
Step 1: Enter an integer
Use any whole number with optional commas, spaces, or sign.
Step 2: Review the digital root
The highlighted result is the final one-digit value.
Step 3: Check the digit-sum chain
Each step shows the digits being added and the next value.
Step 4: Use the modulo 9 check
The remainder confirms the digital root rule for positive integers.
How This Digital Root Calculator Works
The calculator removes formatting characters, keeps the sign for display, and sums the digits of the absolute value. It repeats that process until a one-digit value remains.
Additive persistence is the number of repeated digit-sum rounds. For example, 9876 becomes 30 and then 3, so the persistence is 2.
It also computes the remainder modulo 9. For positive integers, digital root follows dr(n) = 1 + ((n - 1) mod 9), with 9 used for positive multiples of 9.
Digital Root Guide
Digital Root Rules
| Concept | Formula or Example | Meaning |
|---|---|---|
| Digit-sum process | 9876 -> 30 -> 3 | Repeatedly sum digits until one digit remains. |
| Digital root | dr(9876) = 3 | The final one-digit value. |
| Additive persistence | 9876 needs 2 rounds | Counts the digit-sum rounds needed. |
| Modulo 9 formula | dr(n) = 1 + ((n - 1) mod 9) | Works for positive integers. |
| Multiple of 9 | n mod 9 = 0 -> dr(n) = 9 | Positive multiples of 9 use root 9, not 0. |
| Zero | dr(0) = 0 | Zero is already one digit. |
Digital Root Examples
| Number | Digit-Sum Chain | Result |
|---|---|---|
| 9876 | 9876 -> 30 -> 3 | Digital root 3, persistence 2. |
| 123456789 | 123456789 -> 45 -> 9 | Positive multiple of 9. |
| 1000000000000 | 1000000000000 -> 1 | Only one nonzero digit. |
| 4728 | 4728 -> 21 -> 3 | Modulo 9 remainder 3. |
| 999999 | 999999 -> 54 -> 9 | Digital root remains 9 for positive multiples of 9. |
Casting Out Nines
Digital roots are closely tied to casting out nines because summing digits preserves a number's congruence modulo 9. That makes the digital root useful for quick divisibility checks and arithmetic sanity checks.
Keep the research moving with Divisibility Test Calculator, Modulo Calculator, Integer Calculator, and Expanded Form Calculator.
Frequently Asked Questions
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- 1.Wolfram MathWorld - Digital Root(Accessed May 2026)
- 2.Wikipedia - Digital root(Accessed May 2026)