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Digit Sum Calculator: Sum of Digits and Digital Root

Add up the digits of one integer or a whole list, of any length, and see each digital root. Results one per line, ready to copy or download. Nothing is uploaded.

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About the digit sum calculator

This calculator adds up the digits of an integer and can show its digital root as well. Enter one integer or a whole list, one per line or separated by commas or spaces, and each result appears on its own line, ready to copy or download as a text file. Integers may be up to 10,000 digits long and the sum is always exact, so the page also works for very large numbers that an ordinary calculator would round. Nothing is uploaded: the sums are calculated in your browser.

How to find a digit sum

Steps

  1. The Digit sum operation is already selected.
  2. Type or paste your integers. A minus sign is ignored for the sum, because it is not a digit.
  3. Turn on Show digital root to see the single-digit root next to each sum.
  4. Copy the results, or press Download .txt to save them.

Worked examples

A digit sum

The digit sum of 98765 is 9 + 8 + 7 + 6 + 5 = 35. The digit sum of 1,000,000 is 1, and the digit sum of a number written with a thousand nines is 9,000.

The digital root

Summing the digits again until one digit is left gives the digital root. For 98765 the sum is 35, then 3 + 5 = 8, so the digital root is 8. The same value comes straight from the formula below, without repeated summing.

dr⁡(n)=1+((n−1) mod 9),n>0\operatorname{dr}(n) = 1 + ((n - 1) \bmod 9), \quad n > 0

Why digit sums are useful

A number is divisible by 3 exactly when its digit sum is, and divisible by 9 exactly when its digit sum is. This is the basis of casting out nines, the old check for hand arithmetic: the digital root of a product equals the digital root of the product of the digital roots, so a mismatch reveals a mistake. Digit sums also appear in check digits, in puzzles, in numerology and in number theory, where sequences such as the digit sums of the integers are studied for their own sake.

Other digit operations

This page is one operation of the Digit Tools, which can also multiply digits, reverse or sort them, build palindromes and add thousands separators. To add up a list of whole numbers rather than their digits, use the Number List Calculator, and to generate integers to test with, use the Random Number Generator.

Open-source credits

  • The calculator is implemented in-house for this site and uses no third-party library.
  • The digital root follows the formula 1 + (n − 1) mod 9, and the tests check it against the published sequence OEIS A010888; digit sums correspond to OEIS A007953.

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