Digital Root Calculator

What is the Digital Root of a Number?

The digital root (also called the repeated digital sum or sum of digits) of a non-negative integer is the single-digit value obtained by repeatedly summing the digits of the number until only one digit remains.

For instance, to find the digital root of 65,536:

Add the digits: $6 + 5 + 5 + 3 + 6 = 25$
Since 25 is not a single digit, add its digits: $2 + 5 = 7$
7 is a single digit, so the digital root of 65,536 is 7.

The number of steps required to reach a single digit is called the additive persistence of the number. In this example, the additive persistence is 2.

The Digital Root Formula (Modular Arithmetic)

Instead of repeatedly summing digits manually, the digital root of any positive integer in base 10 can be computed directly using modular arithmetic:

$$dr(n) = \begin{cases} 0 & \text{if } n = 0 \\ 9 & \text{if } n \neq 0 \text{ and } n \equiv 0 \pmod 9 \\ n \pmod 9 & \text{if } n \not\equiv 0 \pmod 9 \end{cases}$$

In concise mathematical notation using the floor function:

$$dr(n) = n - 9 \left\lfloor \frac{n - 1}{9} \right\rfloor = 1 + ((n - 1) \pmod 9)$$

Applications and Properties of Digital Roots

1. Casting Out Nines

Casting out nines is a classic sanity check used for centuries to verify arithmetic calculations (addition, subtraction, and multiplication). Because $dr(a + b) = dr(dr(a) + dr(b))$ and $dr(a \times b) = dr(dr(a) \times dr(b))$, you can verify manual calculations by comparing the digital roots of operands and their results.

2. Divisibility Tests

Divisibility by 9: A number is evenly divisible by 9 if and only if its digital root is 9 (or 0 for the number 0).
Divisibility by 3: A number is evenly divisible by 3 if and only if its digital root is 3, 6, 9, or 0.

3. Multiplicative Digital Root & Persistence

While additive digital root uses addition, the multiplicative digital root repeatedly multiplies the digits of a number until a single digit is reached. For example, for 39: $3 \times 9 = 27 \to 2 \times 7 = 14 \to 1 \times 4 = 4$. The multiplicative digital root is 4, and the multiplicative persistence is 3.

Related Digit and Number Tools

Explore related mathematical utilities including our Digit Sum Calculator, Digit Product Calculator, Number of Digits Calculator, and Prime Factors Calculator.

Frequently Asked Questions

What is the digital root of 0?

The digital root of 0 is 0. For all positive integers, the digital root is always an integer between 1 and 9.

Why is the digital root always equal to n mod 9?

In the base 10 positional numeral system, each power of 10 is congruent to 1 modulo 9 ($10^k \equiv 1 \pmod 9$). Consequently, any number $n = \sum d_k 10^k$ is congruent to the sum of its digits modulo 9 ($n \equiv \sum d_k \pmod 9$).

What is additive persistence?

Additive persistence is the number of times you must sum the digits of a number before reaching a single-digit value. For example, 98 has persistence 2 ($9+8=17$, then $1+7=8$).

Can digital roots be calculated in other bases?

Yes. In any radix base $b$, the digital root represents reduction modulo $(b - 1)$. For octal (base 8), it relates to modulo 7, and for hexadecimal (base 16), it relates to modulo 15.

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