Digit sum sequence: 9875→29→11→2. The final single digit is the digital root. Shortcut: dr(n) = 1 + (n−1) mod 9 (except dr(0)=0). The number of steps is the additive persistence. Digital roots have deep connections to modular arithmetic (casting out nines).
How to Use
Enter a number:
- Chain: Step-by-step sums
- Digital root: Final value
- Steps: Persistence count
Casting Out Nines
Digital root = n mod 9 (with 0→9). This ancient technique verifies arithmetic: if dr(a×b)≠dr(dr(a)×dr(b)), the multiplication is wrong! Used for centuries as a quick error check.
Patterns
Multiples of 9 have dr=9. Perfect squares have dr∈{1,4,7,9}. Cubes have dr∈{1,8,9}. These 'digital root filters' quickly eliminate candidates in number theory problems.
Step-by-Step Instructions
- 1Enter number.
- 2Compute digit sum.
- 3Iterate until single digit.
- 4Count steps.
- 5Verify formula.