Digital Root CalculatorFree online math calculator — instant results, no sign-up
Reduce an integer's decimal digits to its digital root.
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About the Digital Root Calculator
The Digital Root Calculator reduces an integer's decimal digits to a single digit by repeatedly adding them together. For the default input of 98765, the digit sums are 35 and then 8, so the digital root is 8.
- Integer
- The whole number whose base-10 digits you want to reduce to a digital root; it has no units and defaults to 98765.
Formula and method
For a nonnegative integer n: dr(n) = 0 if n = 0; otherwise, dr(n) = 1 + ((n − 1) mod 9). For the default input: dr(98765) = dr(9 + 8 + 7 + 6 + 5) = dr(35) = 3 + 5 = 8.
- nThe nonnegative integer being reduced.
- dr(n)The digital root of n, a single digit from 0 through 9.
- modThe remainder operation; mod 9 returns the remainder after division by 9.
Add all the decimal digits of the input. If the sum has more than one digit, add its digits again, repeating until a single digit remains. With 98765, the first sum is 9 + 8 + 7 + 6 + 5 = 35, and the next sum is 3 + 5 = 8. The modulo formula gives the same result without repeated summation.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Integer98765
Results
- Result8
How to use the Digital Root Calculator
- 1Enter the whole number in the Integer field, or leave the default value of 98765.
- 2Run the calculation to reduce the number's decimal digits.
- 3For the default input, verify that the first digit sum is 9 + 8 + 7 + 6 + 5 = 35.
- 4Reduce the remaining two-digit sum: 3 + 5 = 8.
- 5Read the Result row as the final digital root.
Reading your result
The Result row displays the single digit left after repeated digit summation. For Integer = 98765, Result = 8. This value is unitless; it is not the number of summation steps or the first digit sum.
Limitations
The definition and formula here use nonnegative integers written in base 10. Fractions and decimal values are outside this definition. If negative inputs are supported, the usual convention is to use the digits of the absolute value, ignoring the minus sign. Leading zeros do not change the result. Extremely long inputs require exact integer or digit-string handling; rounding an input before calculation can change its digital root.
Common questions
Why is the digital root of 98765 equal to 8 rather than 35?+
35 is only the first digit sum. Because it still has two digits, the process continues with 3 + 5 = 8.
Is a digital root always the remainder after division by 9?+
Not quite. A positive multiple of 9 has digital root 9, even though its remainder is 0. The input 0 has digital root 0. The formula 1 + ((n − 1) mod 9) handles positive integers correctly.
Does rearranging the digits change the digital root?+
No. Rearranging the digits leaves their sum unchanged, so it also leaves the final digital root unchanged.
Can a digital root tell me whether the original integer is divisible by 3 or 9?+
Yes. For positive integers, a digital root of 3, 6, or 9 indicates divisibility by 3, and a digital root of 9 indicates divisibility by 9. The default input's digital root is 8, so 98765 is divisible by neither.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.