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Math Tools

Digit Sum Calculator

Calculate digit sums, digital roots, and explore number properties

Digit Sum & Digital Root

Calculate the sum of digits and find the digital root

Supports very large numbers (no limit)

Quick Examples

Step-by-Step Breakdown

See each step in calculating the digital root

Batch Calculator

Calculate digit sums for multiple numbers at once

Number Properties

Explore divisibility and other properties using digit sums

Digit Sum Reference

Formulas, applications, and interesting facts

Key Formulas

Digital Root Formula
dr(n) = 1 + ((n - 1) mod 9)

For n > 0; dr(0) = 0

Alternative Formula
dr(n) = n mod 9, if n mod 9 ≠ 0; else 9
Divisibility Rules Using Digit Sum
Divisible by 3 If digit sum is divisible by 3
Divisible by 9 If digit sum is divisible by 9
Digital Root = 9 Number is divisible by 9
Digital Root = 3, 6, 9 Number is divisible by 3
Interesting Numbers
Harshad Numbers
Numbers divisible by their digit sum

Examples: 18 (1+8=9, 18÷9=2), 27, 36, 45...

Self Numbers
Numbers not expressible as n + digit_sum(n)

Examples: 1, 3, 5, 7, 9, 20, 31, 42...

Keith Numbers
14, 19, 28, 47, 61, 75, 197, 742...

Numbers that appear in their own digit-based sequence

Persistence Records

Additive persistence counts how many times you sum digits to reach a single digit.

Persistence 1 10, 11, ... 99
Persistence 2 19, 28, 37, 46...
Persistence 3 199, 299, 399...
Persistence 4 19999999999999999999999 (23 nines)

Multiplicative persistence: The smallest number with persistence 11 is 277777788888899.

Understanding Digit Sums

What Is a Digit Sum?

The digit sum is simply the sum of all digits in a number.

  • 12345 → 1+2+3+4+5 = 15
  • 99999 → 9+9+9+9+9 = 45
  • Also called "digit addition"
What Is a Digital Root?

The digital root is the single-digit value obtained by repeatedly summing digits.

  • 12345 → 15 → 6
  • Always a single digit (1-9)
  • Also called "repeated digit sum"
  • Related to modulo 9
Additive Persistence

The number of times you must sum digits to reach a single digit.

  • 5 → persistence 0 (already single)
  • 19 → 10 → 1 (persistence 2)
  • Useful in number theory