Persistence of a number assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest
Numerical digit digits in describing numbers, they are relatively unimportant to modern mathematics. Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits. The digital root is the single-digit number obtained by summing the digits of a given number, then summing the digits of the result, and so on until a single-digit number is obtained. Casting out nines is a procedure for checking arithmetic done by hand. To describe it, let formula_5 represent the digital root of formula_6, as described above. Casting out nines makes use of the fact that