The digital root of a number is what you get when you add up its digits, then add up the digits of that answer, and keep going until only one digit is left. It sounds like a curiosity, but it is one of the most useful tools in Vedic Maths: it lets you check an addition, subtraction or multiplication in a few seconds, without redoing it. This guide explains how to find a digital root quickly, why it works, how to use it as a check, and where it can let you down.
Add the digits. If the answer has more than one digit, add those digits too, and repeat until one digit is left.
The digital root is also the remainder when you divide the number by 9. For 764, 9 × 84 = 756, which leaves 8, the same as the digit sum. For 3856, 9 × 428 = 3852, which leaves 4. The one exception is a multiple of 9: it leaves no remainder, but its digital root is 9. In digit-sum work, 9 and 0 count as the same thing.
Because 9 and 0 are the same in digit sums, you can skip any 9 in the number, and any group of digits that adds up to 9. This is called casting out nines, and it makes long numbers quick.
Adding 9 to a number never changes its digital root, which is why 1, 10, 19 and 28 all have a digital root of 1.
Every power of 10 is one more than a multiple of 9: 10 is 9 + 1, 100 is 99 + 1, 1,000 is 999 + 1. So a number like 764 is 7 hundreds, 6 tens and 4 ones, and dividing by 9 leaves the same remainder as 7 + 6 + 4. The digits are all that matter, and the place values drop out. Our article on the Vedic number system covers place value in more depth.
Digital roots follow the same rules as the numbers they come from. The digital root of a sum is the digital root of the digital roots added together, and the same goes for differences and products. So you can check a calculation by doing it again with single digits.
Is 487 + 356 = 843 right? The digital root of 487 is 1 (4 + 8 + 7 = 19, then 1 + 9 = 10, then 1). The digital root of 356 is 5. Add the roots: 1 + 5 = 6. The digital root of 843 is 8 + 4 + 3 = 15, then 6. They match, so the answer passes.
Is 912 − 457 = 455 right? The roots are 3 and 7. Since 7 is bigger than 3, add 9 first: 3 + 9 − 7 = 5. The digital root of 455 is 4 + 5 + 5 = 14, then 5. They match.
Is 47 × 23 = 1081 right? The roots are 2 (from 4 + 7 = 11) and 5. Multiply them: 2 × 5 = 10, whose digital root is 1. The digital root of 1081 is also 1, so the answer passes. This check is especially useful with the long Vedic methods, such as Urdhva Tiryagbhyam multiplication, where a single carry slip changes the answer.
Turn it into a multiplication. If 1081 ÷ 23 is 47, then 47 × 23 must be 1081, which you can check as above. With a remainder, add its root at the end.
A digital root check proves an answer is wrong when the roots don't match. When they do match, the answer is very likely right, but not certain. Two kinds of mistake slip through:
So use the digital root alongside a quick size estimate. 47 × 23 should be close to 50 × 20, about 1,000, so an answer like 10,081 is wrong even if its root matches.
Answers: (1) 3: 4 + 8 + 7 + 2 = 21, then 3. (2) 9: cast out the three 9s, and 1 + 8 is 9. (3) It passes: the roots are 8 and 7, which add up to 15, so 6, and 843 also has a root of 6. (4) It passes: the roots are 2 and 7, which multiply to 14, so 5, and 1,904 also has a root of 5. (5) No: its digital root is 2. (6) It is wrong: 38 has a root of 2 and 27 a root of 9, which multiply to 18, so 9, but 1,036 has a root of 1. The right answer is 1,026.
For a shorter introduction with more examples, read our guide to the digit sum in Vedic Maths. To get quicker at adding digits in your head, play the arithmetic memory game or practise addition in practice mode. More checking and calculation methods are in our Vedic Maths guides.
Almost. The digit sum is what you get after adding the digits once, which can have more than one digit. The digital root keeps going until one digit is left. For 764, the digit sum is 17 and the digital root is 8.
Because you can throw away any 9, or any digits that add up to 9, without changing the result. Removing them is the same as subtracting multiples of 9, which doesn't change the remainder when dividing by 9.
No, only that it is probably right. It always catches an answer whose root doesn't match, but it misses swapped digits and answers that are out by a multiple of 9. Pair it with a rough estimate.