Fatality of Numbers
(1867)

The Number 9

The laws governing numbers are perplexing to the uncultivated mind, and the results arrived at by calculation are so astonishing, that it cannot be matter of surprise if superstition has attached itself to numbers.

But even to those who are instructed in numeration, there is much that is mysterious and unaccountable, much that only an advanced mathematician can explain to his own satisfaction. The neophyte sees the numbers obedient to certain laws; but why they obey these laws he cannot understand; and the fact of his not being able to do so, tends to give numbers an atmosphere of mystery which impresses him with awe.

For instance, the property of the number 9, discovered, I believe, by W. Green, who died in 1794, is inexplicable to anyone but a mathematician. The property to which I allude is this, that when 9 is multiplied by 2, by 3, by 4, by 5, by 6, &c., it will be found that the digits composing the product, when added together, give 9. Thus:

 2 x 9 = 18 3 x 9 = 27 4 x 9 = 36 5 x 9 = 45 6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81 10 x 9 = 90 and " " " " " " " " 1 + 8 = 9 2 + 7 = 9 3 + 6 = 9 4 + 5 = 9 5 + 4 = 9 6 + 3 = 9 7 + 2 = 9 8 + 1 = 9 9 + 0 = 9

It will be noticed that 9 x 11 makes 99, the sum of the digits of which is 18 and not 9, but the sum of the digits 1 + 8 equals 9.

 9 x 12 = 108 9 x 13 = 117 9 x 14 = 126 and " " 1 + 0 + 8 = 9 1 + 1 + 7 = 9 1 + 2 + 6 = 9

And so on to any extent.

M. de Maivan discovered another singular property of the same number. If the order of the digits expressing a number be changed, and this number be subtracted from the former, the remainder will be 9 or a multiple of 9, and, being a multiple, the sum of its digits will be 9.

For instance, take the number 21, reverse the digits and you have 12; subtract 12 from 21 and the remainder is 9. Take 63, reverse the digits, and subtract 36 from 63; you have 27, a multiple of 9, and 2 + 7 = 9. Once more, the number 13 is the reverse of 31; the difference between these numbers is 18, or twice 9.

Again, the same property found in two numbers thus changed is dicovered in the same numbers raised to any power.

Take 21 and 12 again. The square of 21 is 441 and the square of 12 is 144; subtract 144 from 441 and the remainder is 297, a multiple of 9; besides, the digits expressing these powers added together give 9. The cube of 21 is 9261 and that of 12 is 1728; their difference is 7533, also a multiple of 9.