Digital Root Calculator

Background.

Add up the digits of a number, then add up the digits of the answer, and keep going until one digit is left. That last digit is the number's digital root, and it is more useful than the party trick suggests. Before machines did arithmetic, bookkeepers used it to check long multiplications by casting out nines, and it is still the quickest test of whether a large number divides by 3 or by 9.

It works because 10 is one more than 9. Every power of ten — 10, 100, 1,000 — leaves a remainder of 1 when divided by 9, so a digit in any position contributes exactly its face value to that remainder. Adding the digits therefore never changes a number's remainder on division by 9, and repeating the sum just shrinks the number until only the remainder is left. The one twist is that a multiple of 9 ends at 9 rather than 0.

So the repeated adding can be skipped. For any positive whole number n, the digital root is 1 + ((n − 1) mod 9): subtract one, take the remainder on division by 9, add the one back. The answer here comes from that single step, and it always agrees with summing the digits by hand, however long the number.

Whole numbers from 0 up to 9,000,000,000,000,000 are accepted, written in ordinary base ten; zero is the only number whose digital root is 0. Bigger numbers are still within reach: add their digits once yourself and enter the total, which has the same digital root. Other bases follow the same idea with a different divisor, and negative numbers have no single agreed convention, so both are left out.

What is digital root calculator?

The digital root of a non-negative integer is the single digit obtained by repeatedly summing its decimal digits. For positive integers it always lies between 1 and 9 and equals the remainder on division by 9, except that multiples of 9 give 9 instead of 0; the digital root of 0 is 0. Because it depends only on that remainder, it behaves predictably under arithmetic: the digital root of a sum or a product equals the digital root of the sum or product of the separate digital roots.

How to use this calculator.

1. 01Type the whole number without commas, spaces or a decimal point. Decimals and negative numbers are refused, since the digital root is defined here only for non-negative integers.
2. 02For a number larger than 9,000,000,000,000,000, add its digits once yourself and enter that total instead. The result is identical, because summing digits never changes the remainder on division by 9.
3. 03Read the digital root. A 9 means the number divides exactly by 9; a 3, 6 or 9 means it divides exactly by 3.
4. 04To check a multiplication, find the digital roots of both factors, multiply them, reduce that product to one digit, and compare it with the digital root of your answer.

The formula.

dr(n) = 0 if n = 0, and otherwise dr(n) = 1 + ((n − 1) mod 9). The mod 9 operation gives the remainder after dividing by 9, a whole number from 0 to 8. Shifting n down by one before taking the remainder, and back up by one afterwards, maps those remainders onto 1 to 9 instead of 0 to 8, so multiples of 9 come out as 9. The formula matches repeated digit summing because 10 ≡ 1 (mod 9): a number with digits dₖ…d₁d₀ equals the sum of dᵢ × 10ⁱ, and each 10ⁱ leaves remainder 1, so the number and the sum of its digits share a remainder. The digits' order plays no part, which is why swapping two of them, or inserting or deleting zeros, leaves the digital root unchanged.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.