The rule of nines out (or proof of nine)

When calculators were not yet popular, accountants used a trick to verify that their accounts were correct: the well-known proof of nines or nines out.

It is a validity test for the manual calculation of sums, subtractions, divisions and multiplications of integers.

Using only the input and output digits of the calculation, many accidental errors can be discovered. Due to its ease of use, this rule can be used even by children in school.

For example, let's look at the result of adding 474 and 853, which should result in 1,327. First we add all the digits of 474 to 15. Then we add the digits of 15 to 6. It can be shown that the number obtained (in this case 6) is the remainder of the division of 474 by 9. Hence the origin from the phrase "474 nines out gives 6". Proceeding in the same way, the "nines out" of 853 will be equal to 7, ie the remainder of the division of 853 by 9 is equal to 7.

When we add the numbers 474 and 853, the "nines out" of the result will always be equal to the "nines out" of the sum of the "nines out" of 474 and the "nines out" of 853. So, to check if the Addition is correct, just find the "nines out" of the result and check if it equals the "nines out" of 13 (13 is the sum of 6 with 7). We know that "13 nines out gives 4". Checking the "nines out" of result 1.327, we found that it also equals 4, which confirms the accuracy of the operation.

It is important to remember that if the result of an add-on account is correct and the proof of nines is done correctly, it will confirm the accuracy of the answer. However, if we get a wrong result in the addition, there are cases where the nine test does not detect the error.

The "nines out" or "nine test" rule takes its name from the fact that the numbers 9 can be ignored in the sums, since they are the same as 0 when calculating the remainder of the division by 9.