Casting out nines

Casting out nines is any of three arithmetical procedures:

To "cast out nines" from a single number, its decimal digits can be simply added together to obtain its so-called digit sum. The digit sum of 2946, for example is 2 + 9 + 4 + 6 = 21. Since 21 = 2946 − 325 × 9, the effect of taking the digit sum of 2946 is to "cast out" 325 lots of 9 from it. If the digit 9 is ignored when summing the digits, the effect is to "cast out" one more 9 to give the result 12.

More generally, when casting out nines by summing digits, any set of digits which add up to 9, or a multiple of 9, can be ignored. In the number 3264, for example, the digits 3 and 6 sum to 9. Ignoring these two digits, therefore, and summing the other two, we get 2 + 4 = 6. Since 6 = 3264 − 362 × 9, this computation has resulted in casting out 362 lots of 9 from 3264.

For an arbitrary number, ${\displaystyle 10^{n}d_{n}+10^{n-1}d_{n-1}+\cdots +d_{0}}$, normally represented by the sequence of decimal digits, ${\displaystyle d_{n}d_{n-1}\dots d_{0}}$, the digit sum is ${\displaystyle d_{n}+d_{n-1}+\cdots +d_{0}}$. The difference between the original number and its digit sum is

Because numbers of the form ${\displaystyle 10^{i}-1}$ are always divisible by 9 (since ${\displaystyle 10^{i}-1=9\times \left(10^{i-1}+10^{i-2}+\cdots +1\right)}$), replacing the original number by its digit sum has the effect of casting out

lots of 9.

If the procedure described in the preceding paragraph is repeatedly applied to the result of each previous application, the eventual result will be a single-digit number from which all 9s, with the possible exception of one, have been "cast out". The resulting single-digit number is called the digital root of the original. The exception occurs when the original number has a digital root of 9, whose digit sum is itself, and therefore will not be cast out by taking further digit sums.

The number 12,565, for instance, has digit sum 1 + 2 + 5 + 6 + 5 = 19, which, in turn, has digit sum 1 + 9 = 10, which, in its turn has digit sum 1 + 0 = 1, a single-digit number. The digital root of 12,565 is therefore 1, and its computation has the effect of casting out (12,565 − 1) ÷ 9 = 1,396 lots of 9 from 12,565.