Master digital roots with precision, handling massive integers and custom bases effortlessly. View iterative summation steps, additive persistence, and visual insights instantly online anytime. Customize numeral systems, validate patterns, and study number theory behavior deeply. Download structured outputs as CSV logs and PDF summaries.
Tip: Include a semicolon and base per line to mix numeral systems.
Large ranges are truncated for performance. Export for offline analysis.
For positive integers in base b, the digital root obeys:
drb(n) = 0 if n = 0; else 1 + ((n − 1) mod (b − 1)).
This congruence mirrors the classic mod‑9 rule in base ten.
| Input Number | Base | Digit Sums Sequence | Additive Root | Additive Persistence |
|---|---|---|---|---|
| 98765 | 10 | 9+8+7+6+5=35 → 3+5=8 | 8 | 2 |
| 1A3F | 16 | 1+10+3+15=29 → 2+9=11 → 1+1=2 | 2 | 3 |
| 999999 | 10 | 9+9+9+9+9+9=54 → 5+4=9 | 9 | 2 |
| 101101 | 2 | 1+0+1+1+0+1=4 → 1+0+0=1 (base 2) | 1 | 2 |
For standard positional systems, additive root and digital root are equivalent concepts.
| Term | Description |
|---|---|
| Additive root | Single digit obtained from repeated digit sums in a base. |
| Digital root | Common name for the same operation in many texts. |
| Dependence | Both depend on base; formula uses modulus base−1. |
| Number | Base | Root | Persistence |
|---|---|---|---|
| 38 | 10 | 2 | 2 |
| 99 | 10 | 9 | 2 |
| 199 | 10 | 1 | 3 |
| 1A3F | 16 | 2 | 3 |
| User | How Additive Root Helps |
|---|---|
| Students | Understand modular arithmetic and digit behavior interactively. |
| Teachers | Create demonstrations of iterative numeric processes. |
| Researchers | Scan ranges for unusual persistence characteristics. |
| Developers | Add quick digit-sum checks to internal tools. |
| n mod 9 | Digital Root |
|---|---|
| 0 and n>0 | 9 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
Digits map to values 0–35 using 0–9 then A–Z.
| Symbol | Value | Symbol | Value | Symbol | Value |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 2 | 2 |
| 3 | 3 | 4 | 4 | 5 | 5 |
| 6 | 6 | 7 | 7 | 8 | 8 |
| 9 | 9 | A | 10 | B | 11 |
| C | 12 | D | 13 | E | 14 |
| F | 15 | G | 16 | H | 17 |
| I | 18 | J | 19 | K | 20 |
| L | 21 | M | 22 | N | 23 |
| O | 24 | P | 25 | Q | 26 |
| R | 27 | S | 28 | T | 29 |
| U | 30 | V | 31 | W | 32 |
| X | 33 | Y | 34 | Z | 35 |
Illustrative examples known to require multiple summations.
The additive root is obtained by repeatedly summing digits (in a chosen base) until a single digit remains.
In base ten for positive integers: dr(n) = 0 if n = 0, otherwise dr(n) = 1 + ((n - 1) mod 9).
For other bases, the calculator iterates numeric digit sums, providing transparent stepwise traces.
Yes. It uses arbitrary‑precision math when available to convert bases, and performs digit‑sum iterations as strings, so extremely long inputs are handled gracefully.
Letters map to values A=10 through Z=35. Ensure your selected base is greater than the largest digit value present in the input.
It is the number of iterations required to reach the final single‑digit root by repeatedly summing a number’s digits in the chosen base.
Digit sums preserve value modulo b−1 due to positional weights. That invariant yields the 1 + ((n−1) mod (b−1)) expression for positive integers, generalizing the classic base‑ten mod‑9 trick.
The summation uses absolute digits. The final single‑digit value is reported, and you may conceptually attach the sign if your use case expects it.
Yes. Provide a default base, or specify a per‑line base using “number;base”. Invalid lines are reported without halting other calculations.
Use the CSV and PDF buttons near each table. The PDF includes a compact table render suitable for attachments or records.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.