Digital roots

Reducing a numeral to a single digit is older than the decimal point reached Europe. Hippolytus, the Roman bishop who died around 235, described digit-summing of Greek numerals in The Refutation of all Heresies, and the Syrian philosopher Iamblichus (c. 245 – c. 325) covered it in his commentary on Nicomachus of Gerasa's Introduction to Arithmetic. Both used it to reduce a numeral to a "root" between 1 and 9. Neither connected it to checking a calculation.

That step appears first in India. The earliest surviving work to use digit sums as a check on arithmetic is the Mahāsiddhānta, written around 950 by Aryabhata II (c. 920 – c. 1000). Around 1020 the Persian polymath Ibn Sina — Avicenna in Latin — set out full details of what he called the Hindu method of checking calculations by casting out nines, and the technique travelled west alongside the Hindu–Arabic numerals themselves: Leonardo of Pisa describes the procedure in Liber Abaci in 1202. European arithmetic texts carried it for centuries afterwards as a quick proof of a long multiplication.

It works because 10 ≡ 1 (mod 9): every power of ten leaves remainder 1, so a number and its digit sum always have the same remainder on division by nine. The same reasoning is why it fails as an error check. Reordering digits does not change their sum, so writing 1324 for 1234 passes the test. Modern check digits — the Luhn algorithm on a bank card, the mod-11 digit on an ISBN-10 — weight each position differently precisely to catch that transposition.

The digit-sum check also circulates today under the banner of "Vedic mathematics", after Bharati Krishna Tirtha's book of that name, published posthumously in 1965. Tirtha said his sixteen sutras came from a pariśiṣṭa, an appendix, to the Atharvaveda. K. S. Shukla asked him to point them out in a standard edition and was told they appeared only in a version Tirtha alone had seen. S. G. Dani, Kim Plofker and others have since noted that the Vedas contain none of the sutras, and that the methods rely on decimal notation, which reached India far later. The techniques work; the attribution does not.

Key properties

• For n ≥ 1 the digital root in base b is 1 + ((n − 1) mod (b − 1)), and the digital root of 0 is 0 — so it is just the residue modulo b − 1, with b − 1 standing in for 0.
• In base 10 it works because 10 ≡ 1 (mod 9): every power of ten leaves remainder 1, so a number and its digit sum are congruent modulo 9.
• dr(a + b) = dr(dr(a) + dr(b)) and dr(a × b) = dr(dr(a) × dr(b)), which is exactly what makes casting out nines a valid check on addition and multiplication.
• A positive integer is divisible by 9 precisely when its digital root is 9, and by 3 precisely when its digital root is 3, 6 or 9.
• The digital root of a perfect square is always 1, 4, 7 or 9, because the squares leave only the remainders 0, 1, 4 and 7 modulo 9.
• Casting out nines cannot detect a transposition of digits: reordering them leaves the digit sum, and therefore the root, unchanged.
• The smallest numbers with additive persistence 1, 2 and 3 are 10, 19 and 199, and additive persistence has no upper bound.
• Multiplicative persistence behaves quite differently: 277777788888899 needs 11 steps of multiplying its digits, the most of any known number, and it is conjectured that nothing does better.

Where they turn up

• The divisibility tests for 3 and 9 taught in primary arithmetic are digital-root tests in disguise.
• Casting out nines was the standard hand check on long multiplication in European arithmetic from Liber Abaci onwards; modern check digits such as Luhn and ISBN-10 displaced it because they also catch transposed digits.
• Western numerology reduces a birth date or a name to a single digit — the "life path number". That is a belief system, not a mathematical result.
• "Vortex based mathematics", promoted by Marko Rodin, is built on the 1-2-4-8-7-5 cycle formed by the digital roots of the powers of two. It is pseudomathematics with no standing in number theory, although the cycle itself is real.
• The parlour trick where you scramble a number’s digits, subtract the smaller from the larger, and always get a multiple of 9 — because a number and any rearrangement of its digits are congruent modulo 9.
• Recreational number theory keeps returning to digit iteration: Harshad (Niven) numbers, happy numbers and the persistence problems all come from the same family of operations.