Digital Root Calculator

The digit sum of a number is the result of adding together its individual digits repeatedly until only one digit remains. Although it may seem like a little trick at first, it provides a clear introduction to understanding modular arithmetic and forms the basis for a classic test method called the "casting out nines". This calculator will find the digit sum of any integer, show the sum of its individual digits, output how many reduction steps were required, and optionally display a multiplication version.

What is a digit sum?

You take a number and add its individual digits together. If the result has more than one digit, you continue adding those digits together. This process is repeated until only one digit remains. That final digit is the digital sum.

Take for example the number 34758. If we sum up its digits, we get the sum of 3, 4, 7, 5 and 8, which leads to 27. As it is still a two-digit number, we continue adding. The addition of 2 and 7 results in 9. Therefore, the digital root of 34758 is 9.

The digit sums of positive integers are all between 1 and 9, with the exception of zero which has a digit sum of 0. No matter how large the original number is, this process always terminates.

A simple formula.

It is possible to find the solution without processing each digit individually. For the base-10 system there is a direct formula:

Simply put, the digital root is what remains after a number is divided by nine. If the remainder is zero, it is treated as nine. This is why the digital root is at the core of the method of nines. The same principle can be applied to any base b by replacing nine with the number that results from subtracting one from b.

How to use this calculator:

When a whole number is entered the tool instantly displays the digital root, which is the sum of its individual digits, the additive persistence and the number of digits. The sum of the individual digits only shows the result from the first processing while the digital root is the result after a series of steps have been fully completed.

When the multiplication option is enabled, the tool multiplies rather than adds the individual digits. This allows for calculating the product of the individual digits, the multiplicative digital root and the multiplicative persistence. As the overall product becomes zero when any single digit is zero, an option can be enabled to ignore zeros. When the base option is enabled, numbers can be read in binary, hexadecimal or any base from 2 to 36.

Persistence and product of individual digits.

The additive persistence is the number of times you need to sum the digits until a single-digit number is obtained. Most numbers require one or two steps, but the smallest number that requires three additions is 19999999999999999999999.

In the multiplicative version each step consists of multiplying the digits together. The number of steps is the multiplicative persistence. The smallest number with a multiplicative persistence of 11 is 277777788888899, and no numbers have been found with a multiplicative persistence greater than 11. As the product of the digits will always be less than the original number, and if any digit is 0 it immediately becomes 0, this method is used less often to test for persistence than the additive variant.

Use of the digital root

The most common use is to check arithmetic operations. When two numbers are added or multiplied, the digital root of the result is always the same as the digital root of the sum or product of the digital roots of the original numbers. A mismatch indicates an error. This forms the basis for a checking method called "casting out nines".

The digital root can also be used to determine divisibility. If the digital root is 9 then the number is divisible by 9. If the digital root is 3, 6 or 9 it is divisible by 3. Besides being used for checks, the digital root is a simple starting point for learning congruence arithmetic and number theory.

The table below shows the significance of each output.