Digital Root Explained

Try the digital root calculator.

If you take the digits of any number and add them together, and then add the digits of the resulting number together, and continue doing that until you get a single digit, that single digit is the digital root of the original number.

Digital Root of a Single Digit

The digital root of a single digit is the digit itself.

Examples

  • Digital Root of 9 = 9
  • Digital Root of 4 = 4

Digital Root of a Two Digit Number

With a two digit number, add the two digits together. If your result is a single digit, then that is the digital root of the two digit number.

Examples

  • Digital Root of 18 = 1 + 8 = 9
  • Digital Root of 42 = 4 + 2 = 6

If you get another two digit number after adding the original digits together, then add the digits of the result together to get the digital root.

Examples

  • Digital Root of 97 = 9 + 7 = 16 → 1 + 6 = 7
  • Digital Root of 58 = 5 + 8 = 13 → 1 + 3 = 4

Add the Digits Together to Get the Digital Root

No matter how long the number is, you add all the digits together, then add the digits of the result together, repeating the process until you get a single digit. That digit is the digital root of the original number.

Example

  • Digital Root of 146 = 1 + 4 + 6 = 11 → 1 + 1 = 2
  • Digital Root of 389257 = 3 + 8 + 9 + 2 + 5 + 7 = 34 → 3 + 4 = 7
  • Digital Root of 96871565493528698 = 9 + 6 + 8 + 7 + 1 + 5 + 6 + 5 + 4 + 9 + 3 + 5 + 2 + 8 + 6 + 9 + 8 = 101 → 1 + 0 + 1 = 2.

Shortcut for Digital Root

There is a simple mathematical shortcut for getting the digital root of a number, without having to add each individual digit of each multi-digit result.

digital root of n = 1 + ( (n - 1) % 9 )

The % represents the modulo operation.

For an explanation of why this shortcut works, please read Wolfram MathWorld's Digital Root Page.

Examples

  • Digital Root of 97
    = 1 + ( (97 - 1) % 9 )
    = 1 + ( 96 % 9 )
    = 1 + 6
    = 7
  • Digital Root of 146
    = 1 + ( ( 146 - 1 ) % 9 )
    = 1 + (145 % 9)
    = 1 + 1
    = 2
  • Digital Root of 389257
    = 1 + ( (389257 - 1) % 9 )
    = 1 + (389256 % 9)
    = 1 + 6
    = 7

Creating a Program that can Calculate Digital Roots of Large Numbers

If you are a programmer and want to calculate digital roots of large numbers using the above formula, you will probably need to make use of a code library that is designed to perform mathematical operations on large numbers. For example, PHP has the BCMath arbitrary precision mathematics library and the GNU Multiple Precision library.

Calculating Digital Roots in the Game

The game, Nine Hours Nine Persons Nine Doors, for Nintendo DS, has a built-in calculator that lets you easily determine the digital root of any number. Just press Y to pull up the calculator. You can enter any number and press the Digital Root button to find out the digital root of that number.