Repeatedly sum the digits of a nonnegative integer until only one digit remains. Each submitted value produces digital root plus the intermediate reasoning.
Digit checks, recreational number theory, and casting-out-nines verification use this invariant. It can detect many arithmetic errors but cannot prove a calculation correct. A related application of Digital Root is casting out nines.
The digital root compresses a base-ten integer to one digit while preserving its remainder modulo 9, with positive multiples of 9 represented by 9 rather than 0.
This definition sets the boundary between digital root and a neighboring calculation with similar inputs.
The digits of 987654321987654321 sum to 90; 9 + 0 gives digital root 9.
Digit order does not matter to the result, so transposed digits often escape this check. Zero is assigned digital root zero.
An impossible digital root sign or magnitude should prompt a Digital Root input review before rounding.
Add every digit. If the total has more than one digit, repeat until a single digit remains.
Start the digital root setup by pairing every source number with nonnegative integer. Convert unlike units before typing, and postpone rounding until the displayed digital root is ready to report.
Adding 9 to a positive number leaves its digital root unchanged; adding 1 normally advances it, wrapping 9 back to 1. Use that direction of change to check the displayed digital root before copying it elsewhere.
A digital root summarizes one number, while casting out nines compares an entire claimed operation. Keep that boundary in mind when interpreting the numerical result.
When copying the result elsewhere, include its label and any squared, linear, angular, or percentage unit implied by the inputs. That record distinguishes a calculated digital root from an unlabeled number and makes later checking substantially easier.
For later verification, record nonnegative integer before rounding the digital root. The unrounded working value can feed subsequent steps while the rounded value serves presentation.
Digital roots also reveal divisibility by three: roots 3, 6, or 9 indicate a multiple of three. This works because ten and every power of ten leave remainder one when divided by three, so replacing place values with their digits preserves that remainder.
Yes, many numbers share each residue class modulo 9.
Yes; a positive integer is divisible by 9 when its digital root is 9.
Each decimal place is congruent to 1 modulo 9.
Zero.