What this calculator does
Summing the digits of a number, then summing the digits of that result, and repeating, always ends at a single digit. That endpoint is the digital root, and it is reached surprisingly quickly even for large numbers.
It is really modular arithmetic in disguise. The digital root is the remainder on division by 9, with 9 itself standing in for a remainder of zero, so 99 reaches 9 while 12345 reaches 6.
The formula
The digits are summed, and the process repeats on each result until a single digit remains. Every intermediate step is shown.
| Term | Meaning |
|---|---|
| Digital root | The single digit reached by repeated digit summing. |
| Casting out nines | The old arithmetic check that uses this property. |
| Modulo 9 | The operation the digital root is equivalent to, with 9 replacing 0. |
The inputs explained
| Field | What to enter |
|---|---|
| Whole number | The whole number to reduce. The sign is ignored. |
When to use it
Checking arithmetic
Casting out nines catches many addition and multiplication errors.
Testing divisibility by 9
A digital root of 9 means the number divides by 9.
Recreational mathematics
Digital roots underpin a great many number tricks.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How quickly does each number reduce?
The full reduction chain for each.
Questions
Why is the digital root the remainder modulo 9?
Because summing the digits preserves the remainder on division by 9, for the same reason the divisibility rule for 9 works. Repeating until one digit remains therefore lands on that remainder, with 9 standing in for zero.
What is casting out nines?
An old check for arithmetic. Take the digital roots of the operands, perform the same operation on them, and compare against the digital root of the answer. A mismatch proves an error, though a match does not prove correctness.
Why does casting out nines miss some errors?
Because it only checks the remainder modulo 9. Any error that happens to be a multiple of 9, including transposing two digits, passes undetected. It catches most errors but not all.
Does the digital root of zero exist?
It is zero by convention, which is the one case that breaks the modulo 9 pattern. Every other multiple of 9 has a digital root of 9 rather than 0.
For divisibility rules generally, see the divisibility test calculator. For modular arithmetic, see the modular exponentiation calculator.