The "magical" number 9

Written by Ketil Hansen & Kristian Klunderud

Sum of the digits

There are a lot of "magical" qualities with the number 9. Let us take a number where 9 is a factor. E.g. 69399987889889883648255731946. If we take the sum of the digits of this number 6+9+3+9+9+9+8+7+8+8+9+8+8+9+8+8 +3+6+4+8+2+5+5+7+3+1+9+4+6 we end up with 189. The sum of the digits of this number is 1+8+9 = 18, and the sum of the digits is 9. Another number with 9 as a factor is 5886. The sum of the digits is 27, and the sum of the digits of this number is 9. This is a unique quality to a number. The reason why only 9 of the cardinal numbers (0-9) has this quality, is that only 2, 3 and 9 as a factor will be divisible with its sum of its digits. The cardinal numbers 2 and 3 may however be a factor of a higher cardinal number, and therefor we can omit the theory.

Let us take a closer look on the number 9. If you start with the number 9, and add 9 we get 18. Here we see that we subtract the number 1 from the 1-position, and add the number 1 to the 10-position(78-1, 77+10, 87). The absolute sum of the digits will not be changed.

Opposite numbers

If we take a number ab where a¹ b and the opposite number ba, and subtract the smallest from the largest we always end up with a number where the sum of the digits is 9, and the number is therefor divisible with 9. E.g. 75-57 = 18, where the sum of the digits is 9. We can do this with larger numbers as well, and we will then get a sum of the digits which is divisible with 9. E.g. 763-367 = 396 , where the sum of the digits is 18. We can do this with all "opposite" numbers, except for palindromic numbers, a number which is read the same forward and backward. (5665, 3553).

The explanation is similar to the one above:

If we take the smallest combination of numbers (10 - 01), the 10-position turns to a 1-position, and the sum is 9. If we change one of the digits with the number 1, the opposite number will be multiplied with 10, or divided with 10. We will get a change of 9 to the original difference. We used 75-57 in the example. This may be split into (70 - 07) + (05 - 50). Again this may be split into 7(10 - 01) + 5(01 - 10). We will end up with the following situation: (7*9 + 5*-9) = 9(7-5).