Digital Root Calculator
Calculate the digital root of any number by repeatedly summing its digits until a single digit remains.
Enter any non-negative integer (up to 1000 digits) to calculate its digital root
The digital root (also called the repeated digital sum) is found by summing all digits of a number, then summing the digits of the result, and repeating until only a single digit remains.
Algorithm:
- Sum all digits of the number
- If the sum has more than one digit, repeat step 1
- The final single digit is the digital root
Direct Formula:
dr(n) = 1 + ((n - 1) mod 9)
For n > 0, this gives the same result as the iterative method. For n = 0, dr(0) = 0.
Example: For n = 987654321:
- First sum: 9+8+7+6+5+4+3+2+1 = 45
- Second sum: 4+5 = 9
- Digital root = 9
Mathematical Property: The digital root equals n mod 9, except when n mod 9 = 0 (and n ≠ 0), in which case the digital root is 9. This is because 10 ≡ 1 (mod 9), so each digit's positional value doesn't affect the remainder.
Applications: Digital roots are used in divisibility tests (a number is divisible by 9 if its digital root is 9), error checking in arithmetic (casting out nines), and various recreational mathematics puzzles.
What is a digital root?
The digital root of a number is the single-digit value obtained by repeatedly summing the digits until only one digit remains. For example, the digital root of 12345 is 6 (1+2+3+4+5=15, then 1+5=6). It's also known as the repeated digital sum or iterated digit sum.
How is the digital root related to divisibility by 9?
The digital root of any number equals the number modulo 9, with the exception that if the result is 0, the digital root is 9 (unless the original number is 0). This is because 10 ≡ 1 (mod 9), so each digit contributes its face value to the remainder when divided by 9. This property is the basis for 'casting out nines' - a method to check arithmetic.
What is casting out nines?
Casting out nines is an ancient technique for checking arithmetic calculations. Since the digital root is preserved under addition and multiplication, you can verify calculations by comparing digital roots. For example, to check 123 × 456 = 56088: digital root of 123 is 6, digital root of 456 is 6, 6×6=36 has digital root 9, and digital root of 56088 is also 9, confirming the calculation is likely correct.
What is additive persistence?
Additive persistence is the number of times you must sum the digits of a number to reach a single-digit result (the digital root). For example, 9999 has additive persistence 2: 9+9+9+9=36, then 3+6=9. The smallest number with additive persistence 4 is 19999999999999999999999 (23 digits).