Digital Root Calculator
Calculate the digital root of any integer by repeatedly summing its digits. Shows step-by-step process and additive persistence.
Additive digital root: Sum the digits repeatedly until a single digit remains.
Multiplicative digital root: Multiply the digits repeatedly until a single digit remains.
About this tool
A digital root is the single digit you get by repeatedly adding up the digits of a number until only one digit remains. This process is fundamental in numerology, modular arithmetic, and mathematical puzzles, and it reveals the underlying digital fingerprint of any integer. The Digital Root Calculator automates this iterative process, showing you each step and the total number of reductions required, known as additive persistence.
To use it, simply enter any positive integer and the tool displays the complete reduction sequence—how the digits combine at each stage—along with the final single-digit result and a count of how many iterations it took. It is useful for students learning about digit sums, educators demonstrating number properties, and anyone curious about the recursive structure hidden within large numbers.
One useful insight is that digital roots connect to division by 9: a number's digital root is congruent to the number modulo 9 (with special handling for multiples of 9). All calculations run locally in your browser, so you can explore the digital roots of any numbers you like without leaving your device.
Frequently Asked Questions
Code Implementation
def digit_sum(n: int) -> int:
return sum(int(d) for d in str(abs(n)))
def digital_root(n: int) -> int:
"""Returns the additive digital root (1-9, or 0 for 0)."""
if n == 0:
return 0
remainder = n % 9
return remainder if remainder != 0 else 9
def additive_persistence(n: int) -> tuple[int, list[int]]:
steps = [n]
count = 0
while n >= 10:
n = digit_sum(n)
steps.append(n)
count += 1
return count, steps
def multiplicative_digital_root(n: int) -> tuple[int, list[int]]:
steps = [n]
count = 0
while n >= 10:
product = 1
for d in str(n):
product *= int(d)
n = product
steps.append(n)
count += 1
return count, steps
n = 493
print(f"Digital root of {n}: {digital_root(n)}")
persistence, steps = additive_persistence(n)
print(f"Steps: {' -> '.join(map(str, steps))}")
print(f"Additive persistence: {persistence}")Comments & Feedback
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