Digital root · Steps shown
Digital Root Calculator
The digital root of a number is the single digit left after you add its digits, then add the digits of that answer, until one digit remains. It always equals the remainder when the number is divided by 9, with 9 standing in for a remainder of 0, so 07101856 becomes 28, then 10, then 1.
- Digits counted
- Remainder ÷ 9
- Steps
- Doubling path
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How it works
Three ways to the same digit
1. Add the digits until one is left
0 + 7 + 1 + 0 + 1 + 8 + 5 + 6 = 28. Then 2 + 8 = 10. Then 1 + 0 = 1. Each round is one step, and the number of rounds is called the additive persistence. Most numbers take one to three steps.
2. Divide by 9
The remainder after dividing by 9 is the digital root, with a remainder of 0 meaning 9. This works because 10 is one more than 9, so every place value (10, 100, 1000...) leaves a remainder of 1, and a number leaves the same remainder as the sum of its digits. The calculator does both and shows that they agree.
3. Cast out nines
By hand, cross out every 9 and every group of digits that adds to 9, because they do not change the remainder. In 3748562, cross out 3 and 6, 7 and 2, 4 and 5. Only 8 is left, and 8 is the digital root (3 + 7 + 4 + 8 + 5 + 6 + 2 = 35, then 3 + 5 = 8). Bookkeepers used this trick for centuries to catch mistakes in long sums: the digital root of an answer must match the digital root worked out from its parts.
Where the 3 6 9 pattern comes from
Keep doubling a digit and collapsing the answer. 1, 2, 4, 8, 16 → 7, 14 → 5, 10 → 1, and the loop starts again. 3 and 6 only swap with each other, and 9 doubles into 9. Every multiple of 3 lands on 3, 6 or 9. The circle on this page draws that path for whatever number you type.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| The digital root equals the remainder after dividing by 9, with 0 shown as 9. | Real |
| Every multiple of 9 has a digital root of 9. Every multiple of 3 lands on 3, 6 or 9. | Real |
| The digital root of a product equals the digital root of the product of the roots, which is why casting out nines catches errors. | Real |
| Every perfect square has a digital root of 1, 4, 7 or 9. | Real |
| Doubling cycles 1, 2, 4, 8, 7, 5 and never reaches 3, 6 or 9. | Real |
| Your digit describes your personality, your life path or your destiny. | Legend |
| Tesla called 3, 6 and 9 the key to the universe. | No source |
Questions
Digital root FAQ
What is a digital root?
It is the single digit you end up with when you add the digits of a number, then add the digits of that answer, and keep going until one digit is left. The digital root of 07101856 is 1, because 0+7+1+0+1+8+5+6 = 28, 2+8 = 10 and 1+0 = 1.
What is the fastest way to find a digital root?
Divide the number by 9 and take the remainder. That remainder is the digital root, except that a remainder of 0 means the root is 9 (for any number other than 0). By hand you can also cross out every 9 and every group of digits that adds to 9, then add what is left.
What is the digital root of 0?
0. It is the only number whose digital root is 0. Every other whole number has a root from 1 to 9.
What is additive persistence?
It is how many rounds of adding it takes to reach one digit. 199 takes 3 rounds (19, 10, 1). The smallest number that takes 4 rounds is 19999999999999999999999, a 1 followed by 22 nines. The smallest that takes 5 rounds has more than 2 sextillion digits.
Why do 3, 6 and 9 come up so often?
Every multiple of 3 has a digital root of 3, 6 or 9, and every multiple of 9 has a digital root of 9. When you keep doubling a number and taking its digital root, the roots 1, 2, 4, 8, 7 and 5 loop forever and never reach 3, 6 or 9. That is plain arithmetic in base 10, and it is where the 3 6 9 legend comes from.
Does a digital root have a spiritual meaning?
No evidence says so. The calculation is exact math, but numerology meanings attached to each digit are tradition and story, not findings.
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