Digital root, in full
Because 10 leaves remainder 1 when divided by 9, every power of ten does too, so a number and its digit sum always share the same remainder modulo 9. Repeating the digit sum preserves that remainder while shrinking the number, and it stops once a single digit is left. The closed form is 1 + ((n - 1) mod 9) for any n of 1 or more, which returns 9 rather than 0 for multiples of nine. Only 0 itself has digital root 0.
That mod-9 invariance is why casting out nines works as an arithmetic check. Multiply the digital roots of two numbers, take the digital root of the result, and compare it with the digital root of the claimed product. A mismatch proves an error. A match proves nothing, since the test cannot see a transposed pair of digits. The digital root of 9 times anything is 9, and that fixed point is the source of a good deal of nine-related folklore.
Numerological reduction is built on the same repeated digit sum but changes the stopping rule. It halts at 11, 22 and 33 rather than pushing on to a single digit, so those master numbers survive. The two procedures agree on most inputs and part company exactly where the tradition treats a number as significant. core/numerology.ts keeps both: reduceNum stops at the masters, singleDigitRoot applies 1 + ((n - 1) mod 9) and does not. The digital root is arithmetic, the reduction is tradition.
- Probability
- Repeated digit sum, Digit root
A worked example
Take 4729. The digits sum to 4 + 7 + 2 + 9 = 22, and 2 + 2 = 4, so the digital root is 4. The formula agrees: 4729 - 1 = 4728, 4728 divided by 9 leaves remainder 3, and 1 + 3 = 4. Numerological reduction on the same number stops a step earlier, at 22, because 22 counts as a master number and is left unreduced.
Sources
The digital root of n has the closed form dr(n) = 1 + ((n - 1) mod 9).
Wolfram MathWorld — Digital Root
See also
Where it comes up on this site
Digital root: frequently asked
Add the digits, then add the digits of the result, and keep going until one digit is left. For 4729 the chain runs 22, then 4. The shortcut 1 + ((n - 1) mod 9) reaches the same place in one step.
Because the digital root tracks the remainder on division by nine, and multiples of nine leave nothing. The convention maps that case to 9 rather than 0 so the answer is always a digit from 1 to 9.
Almost. Both repeat the same digit sum, and they part company on when to stop. Numerology halts at 11, 22 or 33 and keeps them whole. The mathematical digital root carries on, turning 11 into 2 and 22 into 4. Same arithmetic underneath, different halting rule, and the difference shows up on every Life Path that lands on a master number.